EARLY SCIENTIFIC ACTIVITIES OF JOHN PAUL COX

Charles A. Whitney Harvard-Smithsonian Center for Astrophysics

Cambridge, MA 02138

l, Introduction

I eagerly accepted the invitation to speak about John Cox's

scientific research, because I felt close, to John after our early

collaboration, and because I knew that the broader task of

assessing his work and putting it into context must wait for a

more objective historian. And the task of broadly summarizing

the fields that interested him has already been undertaken by

the organizing committee -- the program of this meeting is a list

of the numerous topics John's research enriched.

On the other hand, I doubt that I am in a position to

discuss this remarkable body of work in a way that might be

useful to this audience, which contains many who collaborated

with John and who know the work more intimately than I. So I

have chosen a more personal approach and I will confine myself

to John's early pioneering papers on the cause of pulsational

i n s t ab i l i t y .

2. A Brief Vita

John would have been 60 years old this autumn. He ~ras

born on November 4, 1926, and he died on August 19, 1984,

survived by his wife, Jane.

He did his undergraduate and graduate work at Indiana

U n i v e r s i t y , w h e r e he r e c e i v e d his Ph. D. in a s t r o n o m y in 1954

u n d e r t h e g u i d a n c e of Marsha l l Wrube l , The n e x t 8 y e a r s w e r e

spent t e a c h i n g a t Cornell U n i v e r s i t y - - w i t h t i m e out for s u m m e r

r e s e a r c h jobs. In t h e s u m m e r of 1957, he c a m e to Cambr idge .

We w r o t e a j o in t p a p e r a n d I b e c a m e his f irst c o - a u t h o r . In

1963, he m o v e d to Boulder , Colorado, w h e r e he b e c a m e a Fellow

of JILA a n d a professor a t t h e U n i v e r s i t y of Colorado. For n e a r l y

25 y e a r s he s e r v e d as a c o n s u l t a n t to t h e Los Alamos group t h a t

is hos t ing this c o n f e r e n c e .

In 1981, I w a s w o r k i n g on a h i s t o r y of t h e pu l sa t ion

t h e o r y a n d I w r o t e J o h n asking abou t his e a r l y i n t e r e s t in s te l lar

pulsa t ion . He sen t m e some l e t t e r s ( l a rge ly b e t w e e n t h e two of

us) t h a t he h a d p r e s e r v e d in his files. These l e t t e r s p rov ide a n

u n u s u a l gl impse a t J o h n ' s e a r l y c a r e e r , because he of ten w r o t e

l e t t e r s to c l ea r his m i n d a n d l a y ou t t h e possible d i rec t ions for his

work . Reading t h e m again has been an in t r igu ing lesson in

h i s t o r y for m e . In fac t , t h r e e lessons e m e r g e . First , we h a d

u n d e r e s t i m a t e d t h e d i f f i cu l ty of a p rope r n o n - a d i a b a t i c

t r e a t m e n t ' , second, we w e r e a t f i rs t mis led by a too l i tera l

a c c e p t a n c e of Eddington ' s idea t h a t t he pu lsa t iona l in s t ab i l i ty a n d

t h e s u r f a c e phase lag of t he flux w e r e i n t i m a t e l y c o n n e c t e d ;

th i rd , our e a r l y p e r i o d - l u m i n o s i t y re la t ion was r igh t for t h e

w r o n g reason .

5, J o h n ' s Doctoral Thesis

His Ph. D. thes is (1954) was a s t u d y of t he pu lsa t iona l

d r iv ing force p r o d u c e d b y n u c l e a r sources in g ian t s ta r s . The

resu l t w a s u n a m b i g u o u s a n d nega t ive . Using Epstein 's (1950)

adiabatic pulsation solutions for a new, highly condensed

red-giant star model, John was able to show that no nuclear

processes, either at the center or in a she11, could account for

the pulsation unless they had a temperature exponent of at least

one hundred million. This was clearly impossible, and at the end

of the paper, John pointed to the next region for the search --

the outer layers of the star. He said,

In order to have sustained pulsations ... it appears to be a necessary condition that the contribution to [dissipation] from the outlying "non-adiabatic" region rrn~st be sufficiently negative to balance exactly the positive contributions from the adiabatic region. This implies that whatever is the cause of the pulsation phenomenon, it must be in the regions occupying, roughly, the outer 15 per cent of the stellar radius.

It remains to be seen whether models with extensive hydrogen convection zones or different boundary conditions will remove these diff icul t ies .

The mention of hydrogen convection zones is an allusion to

Eddington's hypothesis that such a zone might behave as a heat

valve producing a phase lag in the emitted flux and causing the

star to act as a heat engine. John's thesis provided a proof

that such a valve mechanism was needed. Looking back in

1981, John wrote, "I became interested in the basic problem of

the cause of cepheid pulsations, I think, only when I realized

how inadequate nuclear sources were. " And in the abstract of

his 1955 paper, we find the statement that the "cause of the

pulsations must be sought ... where many of the usual

approximations are not va}id. " This turned out to be prophetic of

John's later work, which often involved careful formulations at

the analytical boundary of current pulsation theory.

~4. A o D r o x i m a t e T r e a t m e n t of L i n e a r n o n - a d i a b a t i c P u l s a t i o n

At Cornell during 1955 and 1956, John started looking for

the Eddington heat valve. He wrote in retrospect,

I remember during the early days (mostly while I was at Cornell) I was quite struck by the very small amount of work (in fact, essentially none) that had been done, or that was then being done, on the basic problem of the cause of the pulsations. It seems that Eddington, in his 1941-42 papers [pointing to the hydrogen ionization zone as the direct cause of heat-valve effect] was about the only person who had addressed that question. Yet i remember seeing quite a few papers on details of the shapes of light and velocity curves, etc. I found this quite an amazing fact.

John adopted an iterative approach to the linear

non-adiabatic equation for radial pulsation starting from the

quasi-adiabatic approximation. Progress was slow because of his

teaching load, but he began obtaining results in mid-1956.

That year, I returned from a Post-Doc with Ledoux and

%vrote John summarizing our work. We did not plan to publish

because it seemed so tentative and we knew that Evry

Schatzman was in the process of publishing independent work

that was quite similar.

None of us in the United States were aware of the seminal

work then being done by Zhevakin in the Soviet Union, and we

all still focussed on the ionization of hydrogen as the critical

process, and | told John that | felt that the best way to attack

the non-adiabatic pulsations would be "by setting up a

discrete-shell model for the star and using electronic computers. "

But John preferred an analytical and more general approach, and

he soon outstripped us all in his understanding of the essential

process. During the summer and fall of 1956, I received a

series of letters describing his steady progress with an iterative

approach to non-adiabaticity based on the formulations of Woltjer

and of Schwarzschild, and John analytically developed the

relationship between the run of £~amma in a stellar envelope and

the phase lag of the emitted flux during pulsation.

• . . J -

5. The Theoretical Penod-Lummosltv Relation

J o h n ' s f ocus a t th i s t i m e w a s on t h e p h a s e lag of t h e

o b s e r v e d f lux t h a t w e all t h o u g h t w a s d i r e c t l y r e s p o n s i b l e for

i n s t a b i l i t y . In a l e t t e r w r i t t e n to m e on A u g u s t 22, ! 9 5 6 , he

said:

My own work seems to be ~oin~ rather well now, I've succeeded in generalizin£~ the treatment somewhat, and it now appears possible to prove that a phase lag in the emitted flux is a necessary condition for instability, without making any assumptions regarding the non-adiabaticity in the region under consideration.

But it soon became clear that the relationship between the

envelope structure and the instability was more complex than he

had first judged. On November 2, 1956, he wrote that the

phase lag for maximum instability depended on the detailed run

of the wave function in the outer layers, so the purely schematic

models he had been using would not be adequate to answer the

question of cepheid instability. Then he added an exciting

development:

If it should turn out to be possible to say that sustained pulsation may in general exist only if the phase lag is near a quarter period, then a basis for a period luminosity relation seems to exist somewhere in the present theory, but I haven't yet been able to pin it down precisely.

Eight d a y s l a t e r , h e w r o t e w i t h his u s u a l c a u t i o n , "I s e e m

to h a v e f o u n d a p e r i o d - l u m i n o s i t y r e l a t i o n w h i c h a p p e a r s to

a g r e e r e a s o n a b l y we l l w i t h o b s e r v a t i o n . " In e f f e c t , h e s h o w e d

t h a t if, a long t h e c e p h e i d s e q u e n c e , t h e l a y e r of p a r t i a l i o n i z a t i o n

of hydrogen occurred at a depth corresponding to a constant

phase lag, then he cou ld derive a relation among period,

luminosity, and mass. He had estimated the phase lag as the

ratio of heat capacity to flux emitted in a full cycle -- essentially

the thermal time to the surface.

I suggested that he come to Cambridge the following

summer so we could work out the numerical details on a "larI{e

machine" that was being installed.

John presented a brief description of his work at the

Christmas, 1956, meeting of the American Astronomical Society.

Although limited to first-order non-adiabatic terms, it

constituted a mathematical statement of the hypothesis that had

been rather intuitively expressed by Eddington.

During the spring of 1957 John wrote a detailed discussion

of his iterative treatment of non-adiabaticity in the schematic

models. The paper (Cox 1958) was received by the Astrophysical

Journal on April 29. The gist of that paper was that no

particular phase lag was a necessary condition for instability,

although an abrupt drop in the radiative flux probably was. He

also broadened the search for the Eddington valve beyond the

hydrogen zone and stressed the weakness of the first order

theory.

During John's visit to Cambridge in the summer of 1957,

we performed some homology calculations based on his recent

paper, and we made his theory of the period-luminosity relation

quantitative, This was done without being able to calculate the

actual net dissipation. John had merely considered the condition

for minimum dissipation, and we were still unable to show the

net dissipation was negative, because we had not done the full

pulsation calculation. According to our formulation, the hydrogen

and neutral helium ionizations occurred too close to the surface to

explain the phase lag of classical cepheids, but the second

ionization of helium appeared to occur at the correct depth. At

the time, we still felt that Eddington's description was correct,

and the instability ought to be related to the phase lag of the

observed flux.

At the end of the summer, l received a translation of

Zhevakin's work on the non-adiabatic oscillation of discrete zone

stars and I sent a copy to John. It, too, pointed to the second

ionization of helium and it contained many important results on

pulsational instability. We emended our manuscript (Cox and

Whitney 1958) and added the references.

Zhevakin had insisted that the phase lag of the surface

flux was not related in a simple way to the pulsational

instability. What counted was the degree of non-adiabaticity at

the level of partial ionization. As it turned out, he was correct in

this. Ironically, the heat-capacity function that John and I used

to estimate the phase lag was actually a measure of the degree

of non-adiabaticity in the critical region, so we got the right

p e r i o d - l u m i n o s i t y re la t ion , bu t looking back, it s e e m s t h a t

Z h e v a k i n w a s m o r e n e a r l y co r r ec t t h a n w e w e r e a t t h e t ime ,

because he h a d a l r e a d y r e j e c t e d Eddington 's s imple r e l a t i on

b e t w e e n ins t ab i l i ty and s u r f a c e phase - l ag .

5_. Stems Toward The Exact Linear Non-adiabatic Treatment

This was my last substantial collaboration with John,

although we corresponded regularly for the next few years, and

he visited Cambridge several times to use our computational

facilities.

Dur ing t h e w i n t e r of 1957-58 , J o h n felt doubts abou t t h e

a d e q u a c y of his f i r s t - o r d e r t r e a t m e n t a n d he s t a r t e d w o r k i n g

w i t h t he Wol t j e r v - e q u a t i o n - - a m o r e complex bu t , he hoped ,

also m o r e a c c u r a t e p r o c e d u r e t h a n t h e S c h w a r z s c h i l d technique~

(In t h e course of this work , he also publ ished a pape r ex t end ing

his ana lys i s of t h e p e r i o d - l u m i n o s i t y r e l a t i on to r a d i a t i v e

envelopes.) Finally, on January 29, 1960, after a year and a

half, his paper on the approximate analysis was received at the

Astrophysical Journal. It was titled '~A Preliminary Analysis of

the Effectiveness of Second Helium Ionization in Inducing Cepheid

Instability in Stars" (Cox 1960).

Even the title of that paper reveals that John was not

Convinced by this approximate treatment either, and in a letter

dated Jan. 12, 1960 -- before he had submitted the paper -- he

~¢rote that he was already at work on the "exact linear

treatment but progress is temporarily slow because of a rush of

other things to do and because the algebra is setting to be a real

mess," The work went so well that he started an extensive

numerical study that spring, and when the approximate paper

came out, John had added a footnote in the proofs announcing

the successful numerical integration of the eighth-order system of

linearized pulsation equation.

The work on the full set of linearized equations was finally

submitted in July 1962. It was titled "On Second Helium

Ionization as a Cause of Pulsational Instability in Stars, " and it

ran 49 pages (Cox 1963). Here, at last, was a treatment that

could evaluate the net dissipation quantitatively. The paper has

a tone of authority that was lacking in the earlier "preliminary"

papers.

John concluded that helium second ionization "probably

accounts for the instability in classical cepheids and 111% Lyrae

variables and also (but less certainly) in W Virginis variables and

dwarfs cepheids of the 6 Scuti type. " But the phase lag came

out wrong. It was clear that his linear theory, which ignored

the ionizations of hydrogen and neutral helium, was not giving

the observed phase lag of the surface flux. This paper, with its

mathematical rigor, ~vas a key to our understanding that the

phase lag problem was quite distinct from the instability itself.

Writing with D. S. King (King and Cox 1968), John later said,

This study isolated the driving at small amplitudes, due to second helium ionization alone. It was possible, therefore, to obtain a clear picture of how this mechanism works and how it can lead to an instability strip which has the essential features of the observed strip.

But I~etting the proper phase lag w a s another matter, and

the clue came from independent, concurrent work of Baker and

Kippenhahn (1962), who had included hydrogen and neutral

10

h e l i u m a n d t r e a t e d t h e l i nea r . n o n ' a d i a b a t i c p u l s a t i o n s of a

c a r e f u l l y c o n s t r u c t e d c e p h e i d enve lope . T h e y f o u n d a la rge

pos i t ive p h a s e lag p r o d u c e d b y n e u t r a l h e l i u m a n d h y d r o g e n . I

t h i n k t h e s e p a p e r s w e r e t h e f i rs t c l ea r s igns c o n f i r m i n g

Z h e v a k i n ' s conc lu s ion t h a t t h e i n s t a b i l i t y a n d t h e p h a s e lag w e r e

two s e p a r a t e p r o b l e m s .

In his classic r e v i e w p a p e r of i 9 7 4 , J o h n c lar i f ied t h e

d i s t i nc t ion as follows:

It is a r e m a r k a b l e f ac t t h a t t h e cond i t i on for t h e a p p e a r a n c e of t h e p h a s e lag a n d t h e n e c e s s a r y cond i t i on for i n s t a b i l i t y a r e b o t h sa t is f ied . . . w h e r e m a n y c o m m o n t y p e s of p u l s a t i n g s t a r s a r e f o u n d . Because t h e s e t w o p h e n o m e n a ( i n s t ab i l i t y a n d p h a s e lag) a r e c a u s e d b y t h e ac t i on of t w o different ionization zones, it appears that the occurrence of the phase lag in pulsating stars is more or less an accident of nature; attributing both phenomena to a single physical mechanism, which was Eddington's view, is evidently not entirely correct.

In 1960, John began a series of fruitful collaborations with

t h e Los A l a m o s g roup , a t t h e i n s t i ga t ion of Ar t Cox, w h o h a d

b e g u n n o n - l i n e a r n o n - a d i a b a t i c c o m p u t a t i o n s . J o h n ' s e a r l y

a t t i t u d e t o w a r d t h e r e l a t i o n s h i p of l i nea r a n d n o n - l i n e a r

m o d e l i n g w a s desc r ibed in s e v e r a l l e t t e r s :

I feel t h a t th i s [ l inear] a p p r o a c h is still v a l u a b l e in v i e w of ou r p r e s e n t s t a t e of i g n o r a n c e r e g a r d i n g t h e cause of t h e pu l sa t i ons . H o w e v e r , it will u l t i m a t e l y be n e c e s s a r y , of cour se , to go in to a n o n - l i n e a r t h e o r y before a r e a s o n a b l y c o m p l e t e U n d e r s t a n d i n g is possible. [To A . N . C . , Nov. 2, 1959]

I ' m s o m e w h a t inc l ined to t h e v i e w p o i n t t h a t one shou ld a t t e m p t to e x h a u s t t h e possibili t ies of a l i n e a r n o n - a d i a b a t i c t h e o r y f i rs t . This shou ld , if n o t h i n g else, r e v e a l ~vhat f a c t o r s a r e l ikely to be i m p o r t a n t in a n o n - l i n e a r t r e a t m e n t . [To A . N . C . , Dec. 16, 19593

11

These remarks were made in the heat of his work on the

eighth-order linear system, and when that work had been

finished, John became a frequent collaborator of those who had

developed non-linear programs.

7. Concludin~ Remarks

I k n o w of n o t h e o r e t i c i a n w h o h a s h a d m o r e c o l l a b o r a t o r s

- - a n d m o r e r e p e a t e d c o l l a b o r a t i o n s - - t h a n J o h n . To l o o k a t

the record, you would think that he was a radio astronomer!

This remarkable record is a result of John's character. He knew

how to be a friend, and to commit himself to a scientific task.

He was generous in giving credit and gentle in criticism. (The

harshest phrase I have found in his work is that a particular

treatment was "not completely convincing. ") He did more than

his share of the writing, and he must have loved writing. How

else do we understand that one of his solo papers in the

Astrophysical Journal spanned 48 pages. He excelled in putting

a differential equation into %vords, and he could described physical

processes in the language of self-consistent mathematics. His

superbly balanced and thoughtful review papers were another

reflection of his humble and devoted spirit, as were his books.

John's concern was with the stars, not merely with the

properties of an admitted approximation to the stars. In his

search for the sources of instability in the stars, he was an

earnest realist. He was never satisfied with an approximation or

a model merely because it was tractable. He consistently sought

12

the relationship of the approximation to the tru~ situation.

And he seems never to have lost his sense of delight in

stel lar va r i ab i l i t y . One of his l a t e r pape r s (Cox 1982) w a s a

shor t no te in N a t u r e descr ib ing t h e p r e d i c t e d a n d n e w l y

d i scovered pu lsa t ions of DB w h i t e d w a r f s d r i v e n by h e l i u m second

ionizat ion. He pointed out t he r e m a r k a b l e fac t t h a t t he pulsa t ions

of the other type, the DA white dwarfs (ZZ Ceti stars), are

driven by hydrogen ionization, and this is the process that also

drives the Mira variables -- stars as different from white dwarfs

as one could imagine.

As n e w f o r m s of ins tab i l i ty a r e found, we will r e t u r n t i m e

and again to J o h n ' s r e m a r k a b l y r ich papers , w h e r e we will f ind

n e w insights and be r e m i n d e d of t h e m a n w h o so f r u i t f u l l y

c o m b i n e d his love of physics , a s t r o n o m y , and m a t h e m a t i c s w i t h

an a b u n d a n c e of h u m a n compass ion .

I wou ld like to t h a n k Carl J . Hansen a n d J a n e Blizard for

the i r g e n e r o u s help in collect ing m a t e r i a l for this pape r .

Yzelected References to John Cox's Papers

1955 "The Pulsat ional S tab i l i t y of Models of Red Giant Stars, " ~strot~hvs. J . . 122, 286.

1958a " N o n - a d i a b a t i c S te l la r Pulsa t ion , " Astro•hvs, J , , 127~ 194.

1958b "A Semitheoretical Period-Luminosity Relation for Classical Cepheids ," ,Astrophys. J . . 127, 561, ( w i t h C. A. W h i t n e y ) ,

195q "A S e m i t h e o r e t i c a l P e r i o d - L u m i n o s i t y Rela t ion for Cepheids With Rad ia t i ve Enve lopes , " Astroph~zs. d . . 150, 296.

13

i960 "A Preliminary Analysis of the Effectiveness of Second Helium Ionization in Inducing Cepheid Instability in Stars, " Astrophys. O.. i 3 2 . 594.

1963 "On Second Helium Ionization as a Cause of Pulsational Ins tab i l i t y in Stars, " AstroDhvs. J . . 138, 487.

1968 "Pulsat ing Stars , " Pub. Astron. Soc. Pacific. 80. 365, (w i th D. S. King).

1974 " P u l s a t i n g S t a r s , " ReD. Pro~, P h y s . 37. 563.

1982 "A New Type of P u l s a t i n g S ta r~" N a t u r e . 299. 402.

14

THE EVOLUTION OF VARIABLE STARS

Stephen A. Backer University of California

Los Alamos National Laboratory Los Alamos, NM 87545

ABSTRACT

Throughout the domain of the H-R diagram lle groupings of stars whose

luminosity varies wlth time. These variable stars can be classlfled

based on their observed properties into distinct types such as 8

cephel stars, 6 Cephel stars, and Miras, as well as many other cate-

gories. The underlying mechanism for the variability is generally felt

to be due to four different causes: geometric effects, rotation, erup-

tive processes, and pulsation. In thls review the focus will be on

pulsation variables and how the theory of stellar evolution can be

used to explain how the various regions of variability on the H-R dia-

gram are populated. To this end a generalized discussion of the evolu-

tionary behavior of a massive star, an intermedlate-mass star, and a

low-mass star wlll be presented.

PULSATIONAL VARIABLES AND THEIR LOCATION IN THE H-R DIAGRAM

There are many types of variable stars whose variability mechanism is

known or believed to be due at least In part to pulsation. Using

Glasby (1971), Hoffmelster, Richter, and Wenzel (1985), Kholopov

(1984) and information presented at thls conference as a guide, Figure

i was constructed to show the approximate locations of the various

types of pulsatlonal variable stars on a Mbo I vs, Log T e H-R dlagram.

Log T e was used instead of spectral type on the abscissa In order to

better represent the stellar evolution tracks. Approximate boundaries

of the spectral type classes (Lang 1980) are illustrated at the top of

Fig. I, and one can clearly see the nonlinear relationship between

temperature and spectral type. The appearance of Flg. 1 changes con-

siderably If spectral type or color index is used on the abscissa as

can be seen on p. 265 of Hoffmelster et. al° (1985) and Figure 1 of

Kholopov (1984).

16

. 1 0 ~ 0 . B A F K M , C , $ ~

- [ ~

.

RR Ly~ae RR L y , a e ' ~ - /

4.5

%o 0

4.0 3.5

tog Te

Figure i. The approximate position of various types of pulsatlonal variable stars on a Mbo I vs. log T e H-R diagram. Spectral class boundaries are given at the top of the diagram. The solid curved llne represents the Zero-Age Main sequence wlth numbers corresponding to the mass of the representative star in solar units. The dashed curves are the evolutionary tracks of i, 7, and 15 M O models

Detailed definitions of the various classes of variable stars are

given in Hoffmelster et. al. (1985), Kholopov (1984) and in these

Proceedings. For convenience a short summary is presented In Table I.

S Dot:

Cyg:

B Cep:

X Cen:

Be stars:

PULSATIONAL VARIABLE STARS

High luminosity eruptive variables whose mass loss may be due to a global pulsatlonal instability.

Quasl-periodlc superglants having amplitudes of 0.1 mag, possibly showing several radial and nonradlal modes.

Early B pulsating giants having periods of hours and amplitudes of around 0.I mag, some showing multiple modes and possibly nonradlal modes.

Possible class of B subglant variables having periods less than an hour and amplitudes of 0.02 mag.

Rapidly-rotatlng, mass-loslng B stars some of whlch show variability which may be due to pulsation. Example LQ And.

17

MAIA: Struve's hypothetical variable sequence between 8 Cep and sct. Probably doesn't exist, see McNamara this conference.

SRd: Semlregular yellow giants and superglants some of which show emission lines, exhibit periods of 30 to 1100 days and amplitudes up to 4 mag. Example: S Vul.

Cep: Radially pulsating (Pop I) variables having well-deflned periods of 1 to 135 days and amplitudes generally from 0.1 to 2 mag. Some show multiple modes.

6 Sct: Dwarf to giant A-F stars having periods of hours and generally amplitudes < 0.i mag. Some show multiple modes and possibly nonradlal modes.

PV Tel: Helium supergiants that appear to pulsate with periods on the order of days but with small amplitude ~ 0.1 mag

R Cot Bor: Hydrogen-deficlent eruptive variables which also may show quasl-periodlc pulsational behavior having periods of 30 -100 days and amplitudes > i mag.

RV Tau: superglant Pop II variables exhibiting a double wave light curve with periods generally from 30 to 150 days and amplitude up to 5 mag.

W vir : Radially pulsating stars somewhat similar to ~ Cep but arising from stars of much smaller mass. Periods generally 12 to 35 days.

BL Her: Radial pulsators related to W Vir class but show a bump on the descending part of the light curve and periods of 1 to 8 days.

Anomalous Cephelds:

RR Lyrae like variables of higher luminosity found almost exclusively In dwarf metal-poor spherlcal galaxies llke Draco.

RR Lyrae: Radially pulsating A-type giants of disk and Pop II composition having periods of about 1 day and amplitudes < 2 mag. Some show double mode behavior.

SX Phx: Subdwarf Pop II equivalent of the 6 Sct class having periods of hours and amplitudes < 0.7 mag. Some show multiple modes and possibly nonradlal modes.

Lc Slowly irregularly varying supergiants of type M showing amplitudes of 1 mag. Example TZ Cas.

SRcz Semlregular pulsating supergiants having periods of 30 to several thousand days and amplitudes of about 1 mag. Examples: u Orl, OH-IR stars.

Lb: Slowing varying irregular giants exhibltlng no lndlcatlon of periodicity. Example~ CO Cyg.

18

SRa,

SRb:

MIRAI

GW Virs

DB Variables:

ZZ Cstil

Semlregular giants showing MIRA-llke behavior but smaller amplitudes < 2.5 mag and periods of 35 to 1200 days. Examplez Z Aqr.

Semlregular giants showing periods of 20 to 2300 Rays that come and go. Example, AF Cyg.

Radially pulsating red giant and supergiant stars of disk and POp II composition having amplitudes ) 2.5 mag and periods of 80 to 1400 days.

Multiperlodlc, nonradlally pulsating white dwarfs of very high temperature.

Multlperlodlc, nonradlall¥ pulsating, helium white dwarfs.

Multiperlodlc, nonradlally pulsating, hydrogen white dwarfs showing periods on the order of minutes and amplitude from 0.O01 to 0.3 mag.

The reader should keep in mind that some overlap in the domains of the

various classes of variables may exist. In addition, some stars which

reside in a given region of variability may not be observed to be

variable (see e.g. Bidelman 1985). This behavior is due to other

factors besides luminosity and temperature (such as composition and

total mass) playing a role on whether a given star Is pulsatlonally

unstable. For example, a Pop I star of Intermedlate-mass would be

Pulsatlonally unstable inside the ~ Cepheld instability strip but not

Within the W vir instabillty strip due to the fact that the latter

requires stars to have much lower total mass. Finally, it should be

noted that our understanding of what excitation mechanism drlves the

observed pulsations of variable stars ranges from fairly well

Understood In the case of Cepheld varlables (see e.g. cox 1985) to

Still being investigated as in the case of the B Cep stars.

EV~OLUTION 0F MASSIVE STARS (M, > 10 M~)

The mass-losing 15 M e model of Brunlsh and Truran (1982) has been

used in Fig. i to represent the general features of a massive star

eVol~tionary track. Massive stars essentially begin their lives on

the Zero-Age Main Sequence (ZAMS) when the pressure generated by

nuclear burning of H by the CNO cycle has balanced the opposing force

of gravity and star ends its earlier phase of gravitational

19

contraction. The interior of a massive star on the main sequence

consists of a large convective H-burnlng core which may be surrounded

at various times by a semlconvectlve shell (depending on how

convective overshoot and rotation are treated) and a radiative outer

region. As the star evolves while burning H, the convective core

shrinks in size and becomes hotter while the star becomes brighter and

cooler. For whatever reason many massive stars become 8 Cep variables

after evolving off the ZAMS. If the star is a rapid rotator, It may

also manifest itself as a Be star. The most massive stars will evolve

into the S Dot region of variability.

The evolution of a massive star proceeds to the right in the H-R dla-

gram until the H abundance in the convective core becomes about 5% at

which point the whole star begins to contract again and the

evolutionary track reverses course and undergoes a short excursion to

the left. This phase of increasing luminosity and temperature ends

when the convective core disappears and H is exhausted at the center.

About 90% of a massive star's total lifetime Is spent during the core

H-burnlng phase, once H is exhausted in the center, a H-burnlng shell

forms around an inert He core and the star evolves redward in the H-R

diagram. Some massive stars will become u Cyg variables, SRd

varlables, and even long period Cephelds as they evolve toward the red

superglant region.

Massive stars are known to lose mass at significant rates and depend-

ing on the rate of mass-loss and the treatment of convective over-

shoot, core He-lgnltion may occur before or after a star becomes a

red superglant. If core He-lgnltlon occurs before becoming a red

supergiant, evolution will continue toward the red on a slower nu-

clear tlme scale and the lifetimes of the various variable phases wlll

be much longer than they would have been if only the H-burnlng shell

was active. The most massive stars lose mass at such a large rate

that they are unable to become red superglants and consequently, the

upper right portion of the H-R diagram is left unpopulated. For these

stars, once the H-rlch envelope Is evaporated the redward evolution

stops and reverses to the blue toward the domaln of the WR stars,

Perhaps some R Cot Bor and PV Tel variables are due to H-deflclent

20

massive stars evolving blueward. If instead the star ignites He as a

red superglant it may remain as a red superglant or it may in some

cases develop a blue loop during the core He-burnlng phase.

Once He is ignited, a convective He-burnlng core forms which slowly

grows in tlme unlike the previous convective H-burnlng core. He is

consumed by the triple alpha and various alpha capture reactions.

Surrounding the convective core Is a radiative He shell on top of

Which is the still active H-burnlng shell followed by a H-rlch

radiative layer, and depending on where the star is In the H-R

diagram, possibly a convective envelope. The core He-burnlng phase

OCcupies most of the remalnlng 10% of a massive star's lifetime. Upon

becoming a red superglant a massive star may become a Lc or SRC type

variable.

With core He exhaustion a He-burnlng shell forms around an inert C-O

core. This core contracts and heats up and soon c-burning begins

Which is quickly followed by respective phases of neon, oxygen, and

Silicon burning until an iron core is formed. Eventually the iron core

becomes unstable and collapses which may then cause the star to become

a supernova. These flnal evolutionary phases take place on a rapid

tlmeScale so that the probability of detecting a given star at this

Stage of evolution is small. The post core He-burnlng phases are

represented by the near vertical segment of the 15 M@ track at the

right portion of the H-R diagram. For more details on evolutionary

models of massive stars see the references listed in the review

article by Iben and Renzlni (1984).

~V~V_~OLOL~TION OF INTERMEDIATE-MASS STARS ~i0 M~>M,>2.25 M@)

The 7 Me, ( ¥ , g ) = ( 0 . 2 8 , 0 .02) model i n Becker (1981) has been used

i~ Fig. i to represent some of the general features of an inte~-

medlate-mass star evolutlonary track. The core H-burning phase of

intermediate-mass stars and massive stars are very similar except that

~aSS-loss and semlconvectlon do not play much of a role for the former

during this phase. The more massive intermedlate-mass stars may

become 8 Cep and X Can variables as well as Be stars as they evolve

21

off the ZAMS. The total llfetlme of the Intermediate-mass stars are

much larger than the massive stars and the main sequence phase for

this case comprises about 80% of the total lifetime.

Unlike the case for many massive stars, the H-burnlng shell phase for

Intermedlate-mass stars lasts from the end of the main sequence to the

red giant area. During this phase the first passage of the Cepheld

instability strip occurs on a thermal tlme scale wlth a lifetime of

around 10 3 to 105 yrs. Upon becoming a red giant the evolutionary

track changes from a horizontal to a more vertical slope known as the

red giant branch (RGB). Core He ignition occurs at the top of the RGB

and a convective He-burnlng core forms surrounded by an inert He

shell, a H-burnlng shell, a radlatlve H-rlch layer and a convective

envelope. After a period of adjustment to two central energy sources,

many Intermedlate-mass stars evolve off the glant branch on tracks

that have been given the name blue loops. Some Intermediate-mass stars

wlll evolve to a sufficiently hlgh surface temperature that they wlll

again intercept the Cepheid instability strip. The second crossing of

the Cepheld strip is generally the longest lived and can be greater

than i0 ~ yrs in duration. As a Cepheid, a star may undergo a

significant amount of mass-loss driven by pulsation (see Brunlsh and

Willson in these proceedings). The blueward evolution stops when the

convective core has nearly exhausted Its supply of He. In general the

blue loop extends to a greater temperature as the mass of the star

increases. Some lower mass Intermedlate-mass stars wlll not show blue

loops and they will remain on the RGB for all of their core He-burning

lifetime.

AS He is exhausted in the core, intermediate-mass stars evolve back

toward the RGB which completes the first blue loop track. A third

crossing of the Cepheld strip is possible and when it occurs it is

generally the second longest In duration. All told, the core

He-burning phase of an Intermedlate-mass star occupies about 15% of

the total lifetime.

u p o n becoming a red giant again a He-burning shell forms around an

inert C-O core and thls energy source gradually overtakes the

H-burning shell as the principle energy source. For certain

intermedlate-mass stars, depending on how much mass is lost and how

22

convective overshoot is treated, a second blue loop (not shown in Flg.

i) may take place lasting less than 1/20 as long as the first blue

loop. The second blue loop track may allow for two additional

crossings of the Cepheid strip.

Ultimately the star will evolve onto the asymptotic giant branch (AGB)

and the H-burning shell will reestablish itself as the main energy

SOUrce. The He-burning shell essentially becomes dormant except when

it Undergoes periodic shell flashes. Depending on the total mass of

the star, the AGB track w111 extend into the region of the MIRA, SRa,

SRb, Lb, SRc, and Lc variables. The big uncertainty in the evolution

at this point is the rate of mass-loss wblch controls the duration of

this phase and how far the AGB extends. It now appears that the

Stellar envelope is usually lost before the degenerate C-O core can

grow to the Chandrasekhar limit of 1.4 M~ (which otherwise would

lead to a c-deflegratlon supernovae), once the H-rlch envelope is

evaporated, the H and He-burning shellls quickly run out of fuel and

the star will evolve off the AGB essentially horizontally across the

H-R diagram. During this evolution the star might become an R cor

Bor, W Vir, and a FV Tel type variable. Finally, this horizontal

evolution ends when the white dwarf (WD) coollng sequence is reached

and evolution then proceeds along this path. AS the remenant star

COols, it may become a GW Vir, DB, or ZZ Cetl WD variable. For more

details on evolutionary models of Intermedlate-mass stars, see Becket

(1979), Iben (1967a, 1974), and Iben and Renzlnl (1983, 1984).

EVOLUTION OF LOW-MASS STARS IM, < 2.25 M~)

LOw-mass stars are ones that develop degenerate He cores prior to core

He ignition. As a result such stars evolve onto a common RGB which

extends to nearly 103 L~. Because of their very long lifetimes

lo~-mass stars can be either Pop I or Pop II composition. The models

of Iben (1967b), Swelgart and GROSS (1978), and Despaln (1981) were

USed to construct the general features of a i M 8 track shown in Fig.

I.

23

On the main sequence low-mass stars can be divided into two groups.

Stars > 1.2 M® will have H-burnlng convective cores driven by the

CNO cycle and the outer region of the star will be radiative. The

core H-burnlng behavior w~ll be very much llke that of the

intermedlate-mass stars. AS they evolve off the main sequence, some

stars in this grouping will evolve through the hypothetical MAIA

region and the domain of the ~ Sct variables. Some of the more

massive Pop II stars will also become SX Phx variables.

Low-mass stars ( 1.2 M@ behave differently as they evolve off the

main sequence. These stars burn H in radiative cores driven by the pp

cycle and they have convective envelopes. Unlike stars with

convective cores, radiative H-burning stars get hotter as well as

brighter as they evolve off the main sequence.

For either case, once H is exhausted in the core the star evolves to

lower temperatures in the H-R diagram toward the RGB powered by a

H-burning shell. Due to the effects of conduction and neutrino

losses, the inert He-core becomes degenerate and the core needs to

grow to about 0.5 M® before He ignition can occur. As the He core

grows in size the star evolves up the RGB. Stars undergoing this

phase have deep convective envelopes. Such evolution is illustrated

in globular cluster H-R diagrams. While climbing the RGB significant

mass-loss may occur.

Core He ignition is a dynamic event when it takes place in a

degenerate core and depending on initial conditions some envelope

ejection and mixing between the H and He layers may occur. During the

adjustment to core He igDitlon, the luminosity of the star drops and a

nondegenerate convective He-burnlng core forms. Eventually conditions

in the interior stabilize to the presence of both a He-burnlng core

and a H-burnlng shell and the luminosity decline stops. Depending on

how much mass remains in the outer envelope of the star and opacity

conditions, the continued evolution of low-mass stars can go in two

directions. Stars with thick outer envelopes will remain as red

giants for the rest of their core He-burnlng lifetime. Some of these

may become Lb, SRa, and SRb variables. If, however, the outer

envelope is thin and the opacity conditions are favorable the star can

leave the red giant region and evolve onto the horizontal branch. The

24

horizontal branch can extend much further to the blue than is

illustrated in Fig. 1 into the RR Lyrae region. Anomalous Cepheids

are RR Lyrae like variables that arise from extremely metal-poor

stars of about 1.3 M® evolving onto their equivalent horizontal

branch (Hirshfeld 1980). Stars burning He in the RR Lyrae region can

have lifetimes In excess of 107 yrs.

Once He is exhausted in the core, a He-burnlng shell forms and low-

mass stars will again evolve toward the red giant domain. Some stars

eVolving out of the RR Lyrae region will intersect the BL Her region

during this phase of evolution. Ultimately the H-burnlng shell re-

establishes itself as the primary energy source and evolution wlll

then proceed onto the AGB. At this evolutionary stage Such stars

might be observed as MIRA, semlregular and irregular variables, when

the mass of the stellar envelope becomes < 10 -3 M®, He-shell

flashes In some cases may cause looping evolution away from the AGB.

These loops may intersect the w vir and RV Tan domains. In any case

Once the outer envelope is nearly exhausted the star must evolve off

the AGB and begin a nearly horizontal track to the left in the H-R

diagram. If the atmosphere is still H-rlch the star might become a w

Vir or RV Tau variable during this phase. If the atmosphere is

H-poor, the star may appear as a R Cot Bor and then a PV Tel variable

during this evolution. Ultimately the stellar remnant will become a

WD and evolve down its cooling sequence. If the atmosphere has no H

the star can become a GW Vir and then a DB variable as it cools. If H

is Present, the star will evolve eventually into the ZZ cetl domain.

For more details on the evolution of low-mass stars see Renzini (1977)

and Iben and Renzlnl (1984).

TH-~OEORETICAL UNCERTAINTIES AND OBSERVATIONAL CONSTRAINTS

Although these topics were included in my original presentation, space

limitations prevented their inclusion here. The interested reader is

invited to see Becket (1985) for a similar discussion applied to

CSpheld evolution.

Thls work was performed under the auspices of the U.S. Department of

Energy Contract # W-7405-ENG.36.

25

REFERENCES

I. Becker, S. A. (1979) Ph.D. thesis, university of Illinois, Urbana-Champaign.

2. Becket, S. A. (1981) Ap. J. Suppl., 45, 475-505.

3. Becker, S. A. (1985) in "Cepheids: Theory and Observations," IAU colloquium 82, ed. B. Madore, pp. 104-125. Cambridge: cambridge University ~ess.

4. Bidelman, W. P. (1985) in "Cephelds: Theory and Observations," IAU colloquium 9~ ed. B. Madore, pp. 83-84. Cambridge: Cambridge University Press.

5. Brunlsh, W. M. and Truran, J. W. (1982) Ap. J., 256, 247-258.

6. COX, J. P. (1985) in "Cepheids: The6ry and Observations," IAU colloquium 8_22, ed. B. Madore, pp. 126-146. Cambridge: Cambridge University Press.

7. Despain, K. H. (1981) Ap. J., 251, 639-653.

8. Glasby, J. S. (1971) The variable Star Observer's Handbook, New York: W. W. Norton and Co. Inc.

9. Hoffmeister, C., Richter G., and Wenzel, w. (1985) variable stars. Berlin: Springer-Verlag.

10. Hirshfeld, A. W. (1980) Ap. J., 241, 111-124.

ii. Iben, I. Jr. (1967a) Ann. Rev. Astron. Ap., 5, 571-626.

12. Iben, I. Jr. (1967b) Ap. J., 147, pp. 624-649.

13. Iben, I. Jr. (1974) Ann. Rev. Astron. Ap., 12, 215-256.

14. Iben, I. Jr. and Renzlnl, A. (1983) Ann. Rev. Astron. Ap. ~!, 271 -342.

15. Iben, I. Jr. and Renzini, A. (1984) Physics Reports, 105, 329 -406.

16. Kholopov, P. N. (1984) Sov. scl. Rev. E. Astrophys., Space Phys., 3, 97 - 121.

17. Lang, K. R. (1980) Astrophysical Formulae, Berlin: springer-verlag.

18. Renzlni, A. (1977) in "Advanced Stages in Stellar Evolution," ed. P. Bouvier and A. Maeder, pp. 151 - 283, sauverny: Geneva Observatory.

19. Swelgart, A. V. and Gross, P. G. (1978) Ap. J. Suppl., !~, 405-437.

26

EVOLUTION OF CEPHEIDS WITH PULSATIONALLY DRIVEN MASS LOSS

W. M. Brunish ESS-5, MS F665

Los Alamos National Laboratory Los Alamos, NM 87545

and

L. A. Willson Iowa State University

Ames, Iowa 50011

ABSTRACT

We have run models of intermediate mass stars (5, 6, 7, and 8 M 8 with

Y=0.28, Z=O.O2) with pulsationally driven mass loss occurring in the

Cepheid instability strip. We used the new 12C(~,y!160 rates of

Caughlan et. al. (1985). The enhanced rate extends the tip of the

blue loop, allowing the 5 and 6 M@ models to re-enter the Cepheid

Strip, unlike the models calculated using the old rates (Becket,

1981). We attempted to see if mass loss during the Cepheid stage

Could redden the tip of the blue loop sufficiently to place it inside

the instability strip, thereby "trapping" the star, and allowing it

to lose mass for a period of time significantly longer than the nor-

mal crossing time. Our results show that this mechanism does in

fact work for a 7 M O star with mass loss rates as low as

~5xi0 -7 M O yr -I Observations of P-Cygni profiles in Cepheids indi-

Cate that this rate is not unreasonable. This behavior acts to

reduce the discrepancy between the evolutionary and pulsation-derived

masses for Cepheids. Another consequence is that the rates of period

change are decreased, bringing them into better agreement with

Observed values.

I. INTRODUCTION

It has been proposed by Lee Anne Willson and George Bowen (1984) that

variable stars may experience pulsationally driven mass loss. We

have attempted to study the effects on intermediate mass star evolu-

tion of mass loss in the Cepheid instability strip. We evolved 5, 6,

7 and 8 M O models of Population I composition (Y=O.28, Z=O.O2). We

Used the definition of the location of the Cepheid instability strip

Provided by Iben and Tuggle (~975). The mass loss parameterization

is given below:

27

: 5 x I0 -8 * (~/I000) * (R/35.O) 2

where L and R are the luminosity and radius in solar units.

Observational evidence for Cepheid mass loss (Welch and McAlary, 1986

and Deasy and Butler, 1986) have large uncertainties and depend

strongly on assumptions regarding the structure of the wind, but give

rates in the range of 10 -5 to 10 -9 M o yr -I .

II. REACTION RATES

We first studied the differences in Cepheid evolution without mass

loss caused by using the new 12C(~,~)160 rates of Caughlan et al.

(1985). (For more information on the effects of mass loss on Cepheid

models with the old reaction rate, see Brunish, Willson and Becket,

1986). The new rates cause the tip of the blue loop to be con-

siderably bluer for the 5, 6 and 7 M O models, compared to the models

of Becker (1981). Thus, models which only had one crossing of the

Cepheid strip now have at least three crossings and perhaps five.

The pulsational periods for the models are considerably changed also,

becoming quite a bit shorter for a given mass than those derived

using the old rates. With the old rate a 6 M O model has a period of

about eight days, while the same model with the new rate has a period

of only five days. This is because the new rate causes the models to

be more luminous for a given mass and effective temperature.

TABLE I

EFFECT OF 12C(~,~)160 RATE ON BLUE LOOPS

Blue Tip (old rates) Blue Tip(new rates)

Mass lo s T e lo s L/L e lo s T e

5 M e 3.665 3.013 3.763

6 M e 3.882

7 M O 3.875 3.670 3.992

8 M e 3.959

log L/L 9

3.078

3.453

3.752

3.941

28

III. MASS LOSS

Inclusion of mass loss while the models are in the Cepheid strip

caused the tips of the blue loops to be reddened, bringing them back

closer to the blue edge of the strip for the 7 and 8 M e models. For

all the models the time spent in the strip increased with mass loss,

With a concomitant decrease in the rate of period change (dP/P).

Crossing times were increased by factors of 3 and 5 for the 6 and

7 M e models, respectively. The luminosities were only slightly

decreased. The amount of mass lost while crossing the strip was

about 2 to 10%.

TABLE II

EFFECT OF MASS LOSS ON BLUE LOOPS

Blue Tip(no mass loss) Blue Tip(mass loss)

Mass io 6 T e lo~ L/L 0 log T e lo 6 L/L8

5 M 0 3.763 3.078 3.761 3.083

6 M e 3.882 3.453 3.808 3.392

7 M e 3.992 3.752 3.963 3.732

8 M O 3-959 3.941

IV. CONCLUSIONS

POStulated mass loss due to a pulsationally driven wind that occurs

While stars are in the Cepheid instability strip causes evolutionary

models to evolve more slowly with lower masses but only slightly

lower luminosities. This results in shorter periods and considerably

smaller rates of period change for Cepheids observed at a given

luminosity. Therefore these models are in better agreement with

Observed rates of period change and with pulsational masses deter-

mined for Cepheids than standard models.

This work was supported in part by the United States Department of

Energy. Lee Anne Willson would like to thank the Canadian Institute

of Theoretical Astrophysics and the Astronomy Department of the

University of Toronto for hospitality during the 1985-86 academic

year, and Iowa State University for granting a Faculty Improvement

Leave.

REFERENCES

Becket, S. A. 1981, Ap. J. Suppl., 45, 33.

Brunish, W.M., Willson, L.A. and Becket, S. A. 1986, B.A.A.S., 17, 894.

Caughlan, G. R. et. al. 1985, Atomic Data and Nuc. Data Tables, 32,197.

Deasy, H. and Butler, C.J. 1986, Nature, 320, 726.

Iben, I. Jr., and Tuggle, R. S. 1975, Ap. J., 197, 39.

Welch, D. L. and McAlary, C. W. 1986, Ap. J., in press.

Willson, L. A. and Bowen, G. H. 1984, Proceedings 3rd Trieste Workshop, ed. R. Stalio and J. Zirker.

30

MIXING CORE MATERIAL INTO THE ENVELOPES OF RED GIANTS

Robert G. Deupree ESS-5, MS F665

Los Alamos National Laboratory LOS Alamos, NM 87545

John Cox never worked directly on the core helium flash to my

knowledge, but he did influence some people who have worked on it.

The first example would be Edwards (1969), who utilized a convective

"phase lag" scheme similar to that proposed by Coxj Brownlee ~ and

Eilers (1966) for variable star applications. I do not believe that

John ever had much faith in this approach except as possibly an in-

dicator as to what effects convection might have, but the

disagreement between Edwards" results and observation prompted him to

think how this simplistic convective treatment might be refined. I

have also benefited from John's influence, because it was he who

first introduced me to finite difference techniques in numerical

fluid dynamics and to the core helium flash.

In this paper I extend the work by Cole and Deupree (1980, 1981) and

Deupree (1984a, b, 1986) to examine how much residue of the core

helium flash can be mixed into and above the hydrogen shell. The

Starting point for these calculations is a point source explosion on

the polar axis of a two-dimensional finite difference grid. Deupree

(1984b) showed that this well reproduces the results of a full

three-dimenslonal calculation initiated with a stellar evolution

struc ture.

The point source explosion produces a high temperature, low density

bubble of processed material whose peak temperature and hence com-

Position depends on the degeneracy at the position and time of the

~XPlosion. This bubble will be mostly helium and carbon, but may

have appreciable amounts of silicon and sulfur if the initial condi-

tions are sufficiently degenerate.

I have performed four core helium flashes with peak temperatures of

7-4, 8.3, 9.2, and 10.2x108 K (Cases i-4, respectively). In all

Gases carbon is the heavy element with the greatest amount of mass

mixed. This is followed by neon in Cases I and 2, sliicon in Case 3,

a~d sulfur in Case 4, a progression which reflects the temperature

31

dependence of multiple = captures on carbon during the explosion.

The amount of mass mixed for these elements in each case is given in

Table I.

TABLE I

-5 ABUNDANCES OF ELEMENTS MIXED INTO THE HYDROGEN SHELL (I0 MS)

Element Case 1 Case 2 Case 3 Case 4

C 1.10 37.1Z 76.18 57.8

O -- 0.i0 0.Z6 0.i

Ne 0. 15 0.67 1.54 0.53

Hg 0.011 0.50 1.12 0.52

Si 0.015 0.41 4.93 2.74

S -- 0.015 0.99 3.10

Under the assumption of complete mixing~ the enhancement of the en-

velope abundances can be computed from the mass mixed and the

envelope abundance of each element. This letter quantity requires

specification of t h e envelope mass and metal abundance if one assumes

the same relative distribution of heavy elements as the sun. With

these criteria, all enhancements in Case I are very small except for

very metal poor, low-mass envelopes. The carbon enhancement in

Case 2 is about an order of magnitude larger than the neon, mag-

nesium, and silicon enhancement. For Case 3 the carbon enhancement

is four times larger than that of silicon, and about eight times that

of neon, magnesium, and sulfur. The most complex situation is

Case 4, where the total amount mixed has decreased because of en-

velope expansion before the arrival of the bubble. Here the order is

carbon, sulfur~ and silicon, each Lower than the preceding element by

about a factor of two, with the other elements about an order of mag-

nitude lower.

There are a number of uncertainties in the quantitative results, not

least of which is the 12C(u,y) 160 nuclear reaction rate.

Calculations wlll be performed with the revised rate to determine its

e f f e c t s .

This work has been supported by the United States Department of

Energy. It is a pleasure to thank the Department of Astronomy at the

32

University of Toronto for their hospitality during a brief visit, and

to thank Drs. P. W. Cole and R. K. WaLlace for useful discussions.

REFERENCES

COle, p. W. and Deupree, R. G. 1980, Ap. J., 239, 284. , 1981, Ap. J., 247, 607. A N ~ Brownlee, R R. ~ and Eilers, D. D 1966~ Ap. J , 144,

1024 . Deupree, R. G. 1984a, Ap. J., 282, 274. -.-____, 1984b, ~ , 287 , 268 . -...___., 1986, Ap. J ,, 303, 649, EdWards, A. C. 1969, M.M.R.A.S., 146, 145.

33

PULSATIONS OF B STARS-A REVIEW OF OBSERVATIONS AND THEORIES

Arthur N. Cox Los Alamos National Laboratory

Los Alamos, NM 87545

ABSTRACT

I discuss the observational and theoretical status for several classes of variable B stars. The older classes now seem to be better understood in terjns of those stars that probably have at least one radial mode and those that have only nonradial modes. The former are the/3 Cephei variables, and the latter are the slowly rotating 53 Persei and the rapidly rotating ~ Ophiuchi variables. It seems that in this last class there are also some Be stars that show nonradial pulsations from the variations of the line shapes and their light. Among the nonradial pulsators, we must also include the supergiants which show pulsations with very short lifetimes. A review of the present observational and theoretical problems is given. The most persistent problem of the cause for the pulsations is briefly discussed, and many proposed mechanisms plus some new thoughts are presented.

I. INTRODUCTION

In the last ten years there have been many reviews of the variable B stars. One of my favorites was the one by John Cox (1976) who was able at that time to cover most of the ideas about possible mechanisms for the pulsation driving that are still being discussed today. While there had been many reviews of the observational da ta before that time, those with some quantitative theoretical interpretations are all more recent. In 1978, Stamford and Watson/1978) showed that the light and velocity variations suggest that in some 13 Cephei stars the pulsation mode was radial or at least not a quadrapole sectorial mode. In that same year, Aizenman and Lesh (1978) pointed out that the ~ Cephei variable mode is likely to be the first or even second overtone, if radial, or a low order p mode, if nonradial with a low I value. An extensive review of the theories for these ~ Cephei variables was given by Aizenman (1980), while at the same conference, Smith {1980) reviewed the newly established line profile variations in 53 Per variables that reveal nonradial pulsations.

While the problems of the D Cephei variables remain with us almost the same as presented by John Cox ten years ago, the attention of many observers has moved to the line profite observations for the slowly rotating (53 Per) variables and the rapidly rotating ones (~ Oph and Be stars). Sareyan, LeContel, Valtier, and Ducatel (1980) and Percy (1980) have both noted the great increase in the types of B stars that vary, from the supergiants, the line profile variables including the slow and rapid rotators and even the Be stars, to the short period B stars. All the aspects of the B star variability were discussed at a conference on pulsating B stars in Nice, and the report is given by LeContel, Sareyan, and VaItier (lg81).

Further reviews were given by Osaki (1982), Cox (1983), and Osaki (1985ab) with emphasis on possible mechanisms. Underhill {1982) has written a comprehensive review of all B stars. The most recent reviews of the observational data are by Baade (1985) and Smith (1986), who

36

detail the many problems in the line profile variable stars, and by Maeder (1985b) who gives

data and interpretations for the Wolf-Rayet stars and the early type supergiants. General parameters for the variable stars are that they have masses ranging from a little

less than 3 M O to over 20 M®. The radii range from about 3 to almost 15 solar radii, with the SUpergiants as much as 5 times larger. Luminosities then are from just less than 100 to over I0,000 solar luminosities for the main sequence B stars and up to 100,000 solar luminosities for the superg~ants. Spectral classes for these variables go into the O stars at about 50,000K, and

go cooler to just over 10,000K at spectral class B7. The internal composition for most of the variable B stars consists of normal solar-type

COmposition surface layers. Watson (1971) showed this for the ~ Cephei variables. Deeper there is a gradient with increasing helium down to either a convective core highly depleted in hydrogen or a hydrogen exhausted isothermal core. For the case of supergiant B stars, the central helium may have started to burn to carbon before evolution to later spectral classes. For a few B stars the atmosphere layers are helium poor, caused presumably by gravitational settling. For others, all of which are probably magnetic, the helium is enhanced (Osmer and Petersen, 1974) by the aCtion of a stellar wind that blows away more hydrogen than helium (Vauclair, 1975), The Wolf-Rayet and hot R CrB stars are different, because they are highly evolved with almost all the surface layers blown away to uncover helium, carbon, nitrogen, and even oxygen layers, the result of extensive thermonuclear burning of hydrogen and helium.

If. CLASSES OF VARIABLE B STARS

There have been many suggestions for different classes of the variable B stars, and here I will try to sort out these into those few that fit into the current ideas about these massive upper

main sequence stars. First of all, we must note that there are only two ways of making the observations: pho-

tometry and high resolution spectroscopy. Luminosity variations are generally not large. Even raost 15 Cephei variables do not show much amplitude in the light variations, but that is partly due to the fact that at maximum luminosity the stars are bluer with an even smaller fraction Of their light being able to pass through the atmosphere and telescope filter. Data from above the atmosphere in the ultraviolet show much larger variations in luminosity amplitude. The 53 Per variables and the supergiants also have light variations of less than about 10 percent (Smith a~Id Buta, 1979). Line profile data divide naturally into those from slowly and rapidly rotating Stars, where in the last category, we must include those known to be Be stars.

I suggest that there are really only three classes of B star variability- those that show at least one radial mode and two classes that display only nonradlal modes. The first class clearly deserves the name ~ Cephei or ~ Canis Majoris variables (Frost 1902). The other two classes seem to be the slow rotators (53 Per variables, Smith and Karp 1976) and the fast rotators (~" Ophiuchi variables, Walker, Yang, and Fahlman, 1979). Stars in these three classes can exhibit both light and line profile variations but the line profile variations are easier to detect.

With these classes, other kinds of variable B stars can be included as discussed here. The e~rly proposed Maia variables (Struve, 1955) are just 53 Per variables with both luminosity and line profile variations. The Abt (1957) supergiants would be either 53 Per or f Oph variables but not ~ Cephei variables. They are known from their radial velocity and light variations with Periods typical of nonradial modes. The ultrashort B star variables (Jakate 1979) and the 53 Psc variables (Sareyan et al. 1980) would be just 53 Per variables. The Be stars of which there

37

a r e examples ~ Oph (Vogt and Penrod, 1983}, ), Eri (Smith and Penrod, 1985) and many others would be in the ~" Oph class and they just happen to have emission at least occasionally in the Ha line. The slow variables found photometrically by Waelkens and Rufener (1985) are in the 53 Per class as suggested by them.

Stars such as the helium variables and the Wolf-Rayet variables also vary in light and spectrum, but they are not normal upper main sequence stars because of their highly evolved surface compositions. I do not discuss these stars much here, mostly because their pulsations are not yet well observed. It appears to me that the Wolf~Rayet variables can be radial pulsators (Maeder, 1985a), but nonradial g mode pulsations that have been reported (Vreux et al., 1985) seem unlikely in a star that has such an extensive convective core that cannot support these pulsation motions.

III. BETA CEPHEI VARIABLES

While there has been considerable new observational da ta on the line profile variable B stars, the key to understanding the variability seems to lie with the B Cephei variables. This is mostly because the radial mode oscillations are easier to interpret theoretically. For example, if the periods are between about 0.15 and 0.25 day, as they all are for the ~ Cephei variables, it seems at least that they cannot be low degree g modes. In eight ~ Cephei variables they observationally do not seem to be nonradial - re=l=2 p modes either because of their light versus wavelength variation (Stamford and Watson, 1978). Some tentative 1 and m values are compiled by Cox (19ss).

In addition to a well known list of 16 ~ Cephei variables that are slowly rotating and a list of 6 that are rapidly rotating, there are dozens of other candidates that have been proposed at one time or another. A few of these ~ Cephei variables are seen in galactic clusters. The recent discoveries by Balona and Englebrecht (1982}, Balona (1983), Balona and Shobbrook (1983}, and Balona and Englebrecht (1985), of 10 in NGC 3293 and 6 in NGC 6231 have been extremely valuable for settling the question of the evolutionary stage of these variable stars. Observed luminosities of 10,000 solar luminosities imply a mass of about 11 Mo, with a range seen from the field stars, of 8 to about 16 M®. Not all the stars :in this luminosity and mass range are observed to vary, however, just as suspected from field star data. Actually, for the more evolved NGC 3293 variables, all the stars in a effective temperature luminosity box in the Hertzsprung-Russell diagram vary, whereas~ at the younger age of NGC 6231, the variables are mixed among non-variables near the main sequence. Observable variability can occur over a range of luminosity, but it is not a sure occurrence among stars that otherwise look identical.

A few of the multiperiodic ~ Cephei variables are suspected to display nonradial modes mostly because the close periods cannot all be from radial modes. These nonradial modes must be low degree p modes however, to be observable. Unfortunately, no mode, radial or nonradiat, has been definitively identified.

There have been reports that periods of the ~3 Cephei variables are both increasing and decreasing. These data tabulated by Lesh and Aizenman (1976} and recently by Chapellier (1984) may not be wholly believable, but they may well indicate internal changes in the structure of the semiconvection zone.

A recent advance has been made by Englebrecht and Batona who have found that one of the ~ Cephei variables in NGC 3293 is an eclipsing variable with the primary eclipse of about 0.1 magnitude and the secondary of about 0.02 ma~uitude. The star 16 Lac is also an eclipsing

38

Cephei variable, discussed in some detail by Jerzykiewicz (1980), but it has only one very

shallow eclipse visible. More data for the cluster variable HD 92024 are needed, but it appears that the pair of stars have masses about 15 and 3 M O. The mode identifications are possibly

the first and second overtone for the massive star, but it is also possible that the two observed periods indicate that I--2 and rn=%2 and -2. These data reinforce the current ideas about the

mode identifications, but they are mostly based on theoretical pulsation constants. We need to mention also my favorite star Splca. It seems that the recent decay of its

pulsations (see Sterken, Jerzykiewicz and Manfroid, 1986, for recent data) may be clue to the

precession of the rotation axis. Then a presumed zonal mode with I--2 (m--0) would be hidden for perhaps 10 to 20 years starting in the early 1970's. If this suggestion by Balona (1986) is correct, one main llne ~ Cephei variable is not really a radial pulsator at all.

The eclipsing variable 16 Lac also seems to have decaying amplitudes for at least two of its three modes. Applying the same precession model as for Spica would seem appropriate because it is also a binary of short period. Balona (1986), however, notes that a rather high angle between the rotation and orbit axes is necessary to get this precession. With a high angle the m value would not necessarily be zero. If the two most rapidly amplitude varying periods differ by unity in rn, the Ledoux and Walraven (1958) C va|ue for the coriolis effect on the periods can

he calculated to be a very reasonable number, 0.318.

IV. SLOWLY ROTATING NONRADIAL PULSATORS

The nonradial pulsating B stars are usually easily identified as nonradial g mode pulsators because they have periods up to a few days, too long for either the radial modes or the p modes. The location of the nonradial pulsators on the Hertzsprung- Russell diagram surrounds the Cephei variables. The frequent occurrence all over this region is surprising, because for the

Yellow giants, pulsation occurs only in a well defined instability strip. Often the assumption is that the nonradial modes can be interpreted in terms of spherical

harmonics, with the quantum numbers I and m giving the surface structure of the modes. This

expectation is supported by the fact that these modes satisfy the hydrodynamic equations of motion in the gravitational field of the star. It is also possible, however, that the modes could

be toroidal with theoretically no radial motion, because these modes are also mathematically aCceptable.

For rotation speeds of less than 200 kin/s, all modes (except maybe 22 Ori) are prograde with -m-~l--2. Higher degrees are seen, but with the slow rotation, it is not easy to see their

many crests of the sectorial modes. Smith has studied many of the 53 Per variables and believes that he has been successful in identifying the few I and m values displayed. However, there

has been a problem in the apparent mode switching, because the changes seem very rapid (in a matter of days or months) compared to some theoretical expectations. Even within a mode, it

seems that the amplitude changes are too rapid also. An interesting observational fact is that the superperiod mP is apparently constant for all

modes with different values of-rn=l. This means that crests stay lined-up, and for low I the Period is much longer. It also appears that only even values of m participate in the superperiod

phenomenon. These observations must be of value in generating theoretical models, because

Perhaps not all the modes seen need be pulsationally unstable by thernselves. I need to point out that Balona (1985) has photometrically observed many of the 53 Per

Variables, and rarely is he able to detect the periods that are used to model the line profile

39

variations. He questions many of the identifications, proposing that they may be aliases instead. For e Per, however, a recent campaign by Smith, Fallerton, and Percy has shown that the strongest period at 3.85 hours, identified as -re=l=4, is seen in both light and line profile fitting. Balona suggests that the line profile fitting may require allowance for temperature as well as geometric effects and Balona and Englebrecht (1986) even wonder if some of the line profile variations are due to s tar spots.

V. RAPIDLY ROTATING NONRADIAL PULSATORS

For the B stars with rapid rotation, nonradial modes of very high degree t=m=16 have been suggested. They can be seen as doppler imaged crests on the stellar surface because of the large variations of radial velocity across the disc. Surprisingly, many of these modes seem to be retrograde, in spite of the rapid rotation, and theoretical interpretation of the mode excitation in terms of some aspect of a rapidly rotating core seen~s out of the question. Observationally, it seems that the difference between a Be and a Bn variable, both of which are rapidly rotating and display nonradial pulsations, is that the Be star has in addition to high 1 modes (actually only m is detected) an 1=2 mode that somehow is essential for the mass loss. Perhaps this low 1 and m puffs-up the star to allow the radiation and pulsation effects to promote mass loss.

The Be stars (recently reviewed by Percy 1985) are in this category, and they exhibit both line profile and luminosity variations. A current line of investigation is to see if the nonradial pulsations cause mass loss seen in the Be stars and others. There are a number of observational problems, including the apparent unequal spacing between crests of the sectorial modes, the occasional masking of a crest that results in either an amplitude change or even a disappearance often accompanying a Be s t a r outburst , and again rapid mode switchings.

VI. CURRENT THEORETICAL PROBLEMS

In spite of many recent papers on the evolution of massive main sequence stars, there are still questions concerning the mass loss by stellar winds, possible overshooting and mixing at the edge of convective core, and other mixing due to rotation. It seems certain that for the most luminous B stars a radiation driven wind causes extensive mass loss. However, at the lower luminosities of the ~ Cephei variables, there may be difficulty getting the wind started. Persistent suggestions that pulsations may cause mass loss should probably by taken seriously. The exact internal structure of the star modified by mixing and mass loss processes in the B star pulsation instability region may be essential to discover the cause of the pulsations.

The status is that some internal mixing is necessary to match evolution calculations with color-magnitude observations of clusters, but too much will prevent evolution into the giant and supergiant region of the Hertzsprung-Russell diagram. However, the ~ gradient that is left behind, exterior to the shrinking homogeneous composition convective core, is a strong barrier to the penetration of hydrogen downward and helium upward. Thus mixing should be minimal, and I believe the small amount now being included in the evolution calculations must be about right.

Mixing by rotat ion which gives slow interior currents must also be ineffective and probably small. Again the ~ barriers are difficult to penetrate, and the material that is mixed-in is essentially all the same composition even if from a different part of the star.

Additional problems of a theoretical nature involve the interpretation of observations of the line profile variations. The apparent lack of horizontal motions in the line profiles (detected at

40

the stellar limb) has led to the proposal that the observed motions are not of the spheroidal type but toroidat. This recent suggestion by Osaki (1985c) can produce line profile variations during the pulsation cycle similar to those observed, but detailed analyses by Smith (1986) seem to rule them out. A big problem is that there should be little light variations for these modes, but most often there is at least some detected.

VII. DEEP PULSATION MECHANISMS

The most severe problem in understanding B star pulsations is discovering their driving mechanism. I here review the six most discussed deep mechanisms and the three envelope ones. I close with three recent ideas that I have thought of, all of which need more development.

It has often been suggested that a star pulsates by the modulation of its thermonuclear energy source at the stellar center or in energy producing shells. We know that the amplitude of the motions is usually extremely small there, and only for the very massive main sequence st~rs, Where the central mass concentration is only slight, does this mechanism effectively operate. Nevertheless, many papers have proposed that this ~ mechanism can be strong enough to at |east help make the B stars pulsate. In my investigations, it seems that for low order g modes with amplitudes still significant in the evanescent convective core, there is almost enough driving to Overcome the radiative damping in the composition gradient zone and in the normal damping layer near the surface. For realistic models it seems however, that the e effect cannot produce enough driving for any radial or nonradial mode.

Another mechanism that was found to be destabilizing is the slow shrinking of the central convective core. The conversion of hydrogen to helium results in less pressure per gram, less support, and a slow core compression under the weight of the outer layers of the star. There are actually two aspects of this slow collapse: the strengthening of the apparent spring constant ~ d the release of potential energy each cycle to the form of heat. The first aspect is damping, but the second produces pulsation driving. Definitive studies by Aizenman and Cox (1975) show that this thermal imbalance does not give enough pulsation driving to destabilize B stars.

Osaki (1974) has suggested that the turbulent convection in the hydrogen depleted core is ~£tually more ordered due to the rapid rotation. This oscillatory convection has a time scale of the rotation, and it has been proposed that this periodicity can couple with a low order mode of the envelope to produce the observed B star pulsations. Further investigations on this general idea have been made by Lee and Salo (1886) who derive eigensolutions consisting of combinations of spherical harmonics. A paper in these proceedings describes this work.

Ando (1981) has championed the idea that the Kelvin-Helmholtz instability at the surface of the uniformly rotating convective core can excite envelope pulsations. It seems to me that this process will surely operate, but will not produce any periodic effect.

I (Cox, 1980) have made the suggestion that a sudden jolt, caused by overshooting at the Surface of this convective core, may mix hydrogen into the surface layers of the core. The increased pressure then would push out the core and envelope a bit, and on recollapse more ~ i n g would occur. This process would give self-excitation of a normal mode until additional ~ixing is no longer possible with the limited hydrogen available in the immediate layers of the COmposition gradient. This series of jolts would be driving, but is there enough to produce Observable amplitudes at the surface? My calculations show that the jolts do not have enough energy to cause the observed pulsations. Thus overshooting events, which actually must occur to ~ limited extent, do not seem to be the cause of B star pulsations.

41

Finally I must mention the Kato (1966) mechanism that operates in composition gradient layers that are superadiabatic. The dynamic instability of convective elements starts them moving, but the # gradient prevents a large excursion. During the time that a convective eddy is away from its equilibrium layer, it either loses or gains heat depending on the relative temperature between it and its surroundings. The/~ gradient forces the eddy to return to its equilibrium level, but when it does, it is either hotter or colder than its surroundings even there, and overshooting of its position occurs. This pulsation driving is only local, and most at tempts to have these motions couple to destabilize the entire star give, at best, instability in high degree unobservable modes.

VIII. ENVELOPE PULSATION MECHANISMS

There are three well discussed envelope mechanisms that seem to apply, and I suggest three other possibilities that might be significant in causing B star pulsations. The three well known ones are the ~, ~ and radius effects. All operate to restrain the flow of luminosity at minimum radius, and then release the energy flow during the expansion ph~tse. The energy flow timing then causes the pressure to peak a£ter maximum compression, and to reach a minimum after maximum expansion.

With solar type composition, the operation of the first two of these mechanisms at the appropriate mass depth where there is enough mass involved, but not too deep for the energy flow to be too slow, results in very little pulsation driving. This appropriate mass depth is at a level where the temperature is about 250,000K, and no ionization process for an abundant element occurs at that temperature for the density of the layers. There has been much hope that a small "bump" in the opacity versus temperature, due to the ionization of the last electron from helium at about 150,000K might give enough ~ effect. However this Stellingweff (1978) bump is just not large enough to make the observed B stars pulsate (see Lee and Osaki, 1982).

In spite of the results of Watson (1971) which show that the surface layers of the B stars have solar-like compositions, I have tried over the years to see if any special composition could be found to give the ~ mechanism periodic radiation blocking that causes at least the yellow giants to pulsate. The element that would give strong ionization at 250,000K would have an effective nuclear charge of the square root of 250,000/40,000 --6 times the Z=2 for helium which has its strong ionization at 40,000K. The only possible candidate is carbon that has both K and L shell electrons ionizing in layers of the star where pulsation driving can occur . Can we have enought carbon in most all B stars to make pulsations occur?

Figures 1, 2, and 3 give the work per zone plots for three composition structures in a 11.5 M® B star model at 24,700K and 6.074x1037 ergs/s. The first figure uses the composition X--0.70 and ¥--0.28, essentially the solar composition, clown through a mass fraction of 0.425 of the star to the top of the evolution-caused hydrogen depleted gradient. One sees that the outer 10 -4 of the mass for this first radial overtone mode is strongly damping for pulsations. Changing the composition to Z--0.044 to obtain some carbon and other heavier element enhancement in a layer between 10 - s and 10 -~ of the mass deep, produces less damping. Finally, making this layer 90 percent helium and 10 percent carbon by mass, one can see driving from helium (above 10 -6 of the mass deep) and from carbon (above 10 -5 of the mass deep). Since such outrageous compositions seem out of the question below the thin (10 -0 of the mass) convection zone, it appears that the ~ and ~ effects do not cause B star pulsations.

42

Another possible surface layer mechanism that I have considered when preparing this review is the continuous and possibly cyclical release of the overburden due to mass loss. Since all B stars display mass loss, this might be a good candidate. However, I have found that this weakening

of the effective spring constant with time and estimates of nonadiabatic effects are very small. This mass loss driving cannot be operating for the mid B class stars, because there the rate is extremely slow. Thus, as a universal mechanism for all B stars, this effect is not admissible.

Similar conclusions have been discussed by Castor (1986). I here suggest an entirely new mechanism that I have never seen discussed anywhere in

the B star pulsation literature. The temperature gradient in the layers just below the surface convection zone is subadiabatic by a small amount approximately 0.05 in dlogT/dlogP. It is possible during the pulsations for this gradient to become superadiabatic so that convection is Periodically started. The delay of the convection causes a luminosity lag similar to that for the

and "7 effects. Thus this is a driving mechanism, but it operates only if there is originally a large enough perturbation. I suggest that the B stars, all on which are not too subadiabatic in the pulsation driving part of the envelope, may pulsate by this hard self-excited mechanism.

Figure 4 shows the linear theory variation for the gradient dlogT/dlogP in the envelope at mass levels above 1.0xl0 - s of the stellar mass. This temperature is 260,000K and the density is lust over 1.0xl0 -6 g/cc. Driving or damping of pulsations are effective in zones 200 to 300, with deeper ones being too adiabatic for periods seen for the ~ Cephei variables and more shallow ones having too little mass to affect global pulsations. The normalization for this variation is ~r/r-~l.0 as is customarily done. If the linear theory is adequate for representing the internal variations of the variables, and scaling down by a factor of about ten or so matches the real B star pulsations, then indeed convection can be switched on and off each cycle with the equilibrium gradient being only 0.0S subadiabatic.

We need a mechanism to cause radial or nonradial pulsations in the period range of the tow order radial or nonradial p or g modes for B stars of spectral type earlier than about B7. Apparently we need the same mechanism also for the O stars. As has been emphasized by many, especially Osaki (1986 and others earlier), this mechanism is not specific to a limited effective temperature or luminosity range. He then says that this indicates a deep mechanism, but is that really true? Actually, at luminosities just about one magnitude above that for the ~ Cephei variables, the convection is already transporting energy, and its slow onset and decay during the pulsation cycle may give driving that is not hard self-excited.

The problem is to get this cyclical convection going without a finite amplitude pulsation. I can only suggest that either rotation or, bet ter yet, a binary companion can slightly influence the /~ Cephei star structure to produce an adiabatic temperature gradient in the pulsation driving region. Kato (1974, 1975) has commented on this and similar problems in binary systems. Previously it was thought that binaries could not play any role for the ~ Cephei stars because they were thought to exist only in a small part of the HR diagram, but with these variable stars now seen in a much wider region, the influence of unseen binaries may be admissible.

It is possible that this cyclical convection switching mechanism may operate properly in all early type stars to produce the observed prevalent pulsations. It is also possible that the unknown cause of pulsations in the hot helium stars (the hotter R CrB stars) is this convection switching in their envelopes that are quite similar to the normal B star envelopes. Whether this mechanism is significant in the Wolf-Rayet stars or the GW Vir variables needs further

investigation.

43

2.3

0.0

-2.5

-5 .(]

-?.~

-IO.(]

0 "L2,5

"15,0

-17.5

0.0 t2.0 I

IO.O

oLphovLr oprLLS6 1 1 . 5

, ,J , , ,.0 :0 ;.0 ;.0

-go~ ( l - q ]

-20.0

% ' t ,O

0,0

-B.6

~L -12.0

_7o - 1 6 . ( 1

o2LO i

o.o ~.6 ,Lot 6.0 ~,o !6.o 12,o -Loq ( 1 - q |

oLphovLr oprLLB6 1 1 , 5

Figure 1. Work/zone vs. external mass fraction for the first overtone radial mode with the composition Y=0.28 and Z=O.02

Figure 2. Work/zone vs. external mass fraction for the first overtone radial mode with the composition Y=0.354 and Z=0.044

oLphovLr aprLL86 I 1 . 5 ~.25

6.00

3,75

! .2S

0.0O -[ o -1.25

-1.50

°3,;'5

-5.00

'o

t r ~ I 0.0 .0 t .0 6.0 .0 10,0

-Loq( l -ql 12.0

O . B ;

C~ O.4

• - 0 . 4

- O . B

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FILPH~,VIR RPRILB6 ! 1 . 5 , ~ J t i i , ~ i i

- ! , 6

- 2 . 6 I ~ ~ I r I I , , I I

Zo~o

Figure 3. Work/zone vs. external mass fraction for the first overtone radial mode with the composition Y=0.90 and Z =carbon only=O.1

Figure 4. The variation of the gradient over the superadiabatic gradient during the nonradial P3 mode pulsation versus zone number. The peak at zone 810 is just below the helium driving at 150,000K

44

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Osaki, ¥. 1985a, in The Connection Between Nonradial Pulsations and Stellar Winds in Massive Stars (eds. D.C. Abbot, C.D. Garmany, C.J. Hansen, H.F. Henrichs, and W.D. Pesnell) P.A.S.P. 98, 30.

Osaki, Y. 1985b, in Highlights of Astronomy (ed. J.-P. Swings) p. 247. Osaki, Y. 1985c, in Seismology of the Sun and Distant Stars (ed. D.O. Gough)

NATO ASI 169, p. 453. Osakl, Y. 1986, in Highlights of Astronomy (ed. J.-P. Swings), 7, p. 247. Osmer, P.S. and Petersen, D.M. 1974, Ap. J. 187,117. Percy, J.R. 1980. in IAU Colloquium 58 Stellar Hydrodynamics (eds. A.N. Cox

and David S. King) p. 313. Percy. J.R. 1985 in Highlights of Astronomy (ed. J.-P. Swings) p. 265. Sareyan, J.-P., LeContel, J.-M., Valtier, J.-C. and Ducatel, D. 1980. in IAU

Colloquium 58 Stellar Hydrodynamics (eds. A.N. Cox and D.S. King) p. 353.

Smith M.A. 1980. hn Nonradlal and Nonlinear Stellar Pulsation, L#cture Notes in Physics 125 (eds. H.A. Hill and W. Dziembowski) p. 60.

Smith, M.A. 1986 in Proceedings of the Unno Retirement Conference. Smith, M.A. and Buta, R.J. 1979, Ap.J. Lett. 232, L193. Smith, M.A. and Karp, A.H. 1976, in Proceedings of the Solar and Stellar Pulsa-

tion Conference, Los Alamos LA-6544-C, p. 289. Smith, M.A. and Penrod, G.D. 1985, Proceedings Third Trieste Conference on

Relationship between Chromospheric/Coronal Heating and Mass Loss (eds. R. Stalio and J. Zirker) Trieste Observatory p. 394.

Stamford, P.A. and Watson, R. D. 1978, in IAU Colloquium 46 Changing Trends in Variable Star Research p. 504.

Stellingwerf, R.F. 1978, Astron. Journ. 83, 1184. Sterken, C., Jerzyklewlcz, M. and Manfroid, J. 1986, Astron. Astrophys. in press. Struve, O. 1955, Sky and Telescope, Sept 1955, p. 461. Underhill, A.B. 1982~ B Stars with and without Emission Lines. NASA SP. 456. Vauclair, S. 1975, Astron. Astrophys. 45, 233. Vogt, S.S. and Penrod, G.D. 1983, Ap.J. 275, 661. Vreux, J.-M., Anclrillat, Y., and Gosset, E. 1985, Astron. Astrophys. 149, 337. Waelkens, C. and Rufener, F. 1985, Astron. Astrophys. 152, 6. Walker, G.A.H., Yang, S. and Fahlman, G.G. 1979, Ap. J. 233, 199. Watson, R.D. 1971, Ap.J. 169, 343.

46

STABILITY OF RADIAL AND NON-RADIAL PULSATION MODES OF MASSIVE ZAMS MODELS

A. p. Odell*, A. Pausenwein, W. W. Weiss, and A. Hajek Institute for Astronomy--University of Vienna Tuerkenschanzstrasse 17 A-If80 Vienna, Austria

ABSTRACT: We have computed non-adiabatic eigenvalues for radial and non-radial pulsation modes of star models between 80 and 120 M O with composition of X=0.70 and Z=0.02. The radial fundamental mode is unstable in models with mass greater than 95 M~ , but the first over- tone mode is always stable. The non-radial modes are all stable for all models, but the ~ =2 f-mode is the closest to being driven. The non-radial modes are progressively more stable with higher ~ and with higher n (for both p- and g-modes). Thus, our results indicate that radial pulsation limits the upper mass of a star.

I. INTRODUCTION

It has long been assumed that the upper mass limit allowed during star formation is due to the occurrence of radial pulsation driven by the epsilon mechanism in the stellar core. Ziebarth (1970) used linear and non-linear radial codes to derive a limit of about i00 M G , a number somewhat sensitive to helium abundance. Recently Klapp, Langer, and Fricke (1986) used a modified version of the Castor (1971) code to Study this limit. They found that non-adiabatic damping just beneath the surface (in the region where radiation pressure becomes dominant) Produces stability up to 440 M O .

The purpose of this paper is to present results of non-radial as Well as radial linear stability analysis of massive stars. We want to determine whether non-radial modes are unstable at lower mass than radial, and therefore would limit the upper mass of a star during its formation.

II. METHOD OF COMPUTATION

The equilibrium models were produced separately from the pulsation analysis. The equilibrium models were homogeneous with X=0.70 and Z=O.02, they included nuclear burning by CNO Cycle (see Cox and Giuli, 1968), and they used analytic opacities from Stellingwerf (1975a, b). The effective temperature and luminosity were derived by converging a model from the core and surface simultaneously. However, the final equilibrium models were integrated from the surface only and left the inner 0.5% of the mass untreated in the core. The only effect of this was to slightly reduce the driving by nuclear reactions, and thus increase the lowest mass at which instability occurs.

The non-adiabatic pulsation code which we used for radial stability analysis was identical to that described by Castor (1971), but with nuclear burning taken into account.

The non-radial stability analysis was performed on the same equilibrium models with the Lagrangian code of Pesnell (1984, 1986).

* on leave of absence from Northern Arizona Univ., Flagstaff, AZ 86011.

47

III. RESULTS

Table I shows the results of the radial and non-radial pulsation stability analysis for the homogeneous (ZAMS) models with masses between 80 and 120 M O . The effective temperature and luminosity are given, as well as the pulsation period and fractional energy gain per period. The modes analyzed are the radial (~ =0) fundamental (F) and first overtone (IH), and the non-radial ~ =I, 2, and 3 gl-mode, f-mode, and pl-mode.

A mode is stable if AE/E is negative (energy decreases by that fraction each cycle), but the amplitude grows if &E/E is positive. Only the radial fundamental is unstable for M = I00 M~ , and all other modes were found to be stable. In the figures, the values of ~E/E are plotted as a function of mass for the models in the table. Fig. la shows the radial fundamental and first overtone, and fig. Ib shows .~=I gl and pl modes. In fig. 2a is shown the ~ =2 gl, f, and pl modes, while fig. 2b shows the same modes for ~ =3.

In the non-radial mode calculations, we find that the amplitude is always quite small in the core, where the epsilon mechanism is a source of driving. The convective core (which includes about 80% of the stellar mass) has a Brunt-V~is~l~ frequency of zero, which means that the low frequency g-modes are evanescent, and the amplitude decays exponentially in the core. In the case of the p-modes, the node(s) requires the amplitude to be small. The f-mode for low values always has a small amplitude in the core.

IV. CONCLUSIONS

We find that our calculations of stability for radial pulsations agree with earlier published results, i.e. that star models above about I00 M O are unstable in the radial fundamental mode, but that the overtones are all stable. This result conflicts with Klapp, Langer, and Fricke (1986), who find that strong damping in the sub-surface layers (where the gas pressure is dropping faster than radiation pressure) stabilizes models up to 440 MO • Weiss and Baker (1986) also do not find the strong damping, and we agree with them that the upper limit of stable main sequence models is about 95 M O .

Further, for non-radial modes in the same models, we find that all modes are stable, but the least stable mode (with the longest damping time) is the ~ =2 fundamental mode; all other non-radial modes are at least an order of magnitude more damped.

This project was done as part of a graduate course in non-radial pulsation at University of Vienna. We would like to thank Dr. W. Dean Pesnell for the use of his codes.

48

TABLE I, Periods and fractional energy generation per pulsation cycle calculated for radial and non-radial pulsation modes

MASS (Mo) gl F/f IH/pl Teff 2 log L/L e P(days) AE/E P(days) ~E/E P(days) AE/E

80 0 ...... .281 -2.26(-7) .139 -1.86(-4) 52550 1 . . . . . . . . . . 168 -3.12(-5) 5.96 2 .456 -1.19(-5) .217 -3.25(-6) .140 -1.63(-4)

3 ,347 -9.25(-6) .181 -1.60(-5) .125 -4.32(-4)

90 0 . . . . . . . 298 -1.14(-7) .145 -1.91(-4) 53520 i .754 -1,57(-5) . . . . 176 -3.13(-5) 6,05 2 .467 -1.16(-5) .225 -3.46(-6) .146 -1.68(-4)

3 .355 -9.04(-6) .188 -1.67(-5) .130 -4.50(-4)

I00 0 . . . . 317 +1.63(-7) .151 -i.94(-4) 54340 1 .832 -2.28(-5) . . . . . . . 184 -3.18(-5) 6,12 2 ,502 -1,58(-5) .234 -3.72(-6) .151 -1.76(-4)

3 .384 -1,25(-5) .195 -1.76(-5) .135 -4.73(-4)

Ii0 0 . . . . 326 +3,81(-7) ,151 -1,93(-4) 55290 1 1.960 -2.60(-4) ...... .188 -3.08(-5) 6o19 2 .447 -1.01(-5) .238 -3.74(-6) ,154 -1.73(-4)

3 ,883 -2,24(-4) .199 -1.76(-5) .137 -4.72(-4)

120 0 . . . . . . . 366 +7.56(-7) .170 -2.10(-4) 55150 I 2.177 -3.60(-4) . . . . . . . 202 -3.60(-5) 6.25 2 1.313 -2.98(-4) .254 -4.65(-6) .165 -2.07(-4)

3 ,976 -3.10(-4) .213 -2.10(-5) .147 -5.60(-4)

49

~_EE E

10

- 5

-10

t0

aE E

5

- 5

- I 0

~ = 0

IH

I I

i t

~ = 1

gl xIO5

- Pl x105

8'0 9'0 ,6o ,,o STELLAR MASS ( M e )

I

120

Fig. la and Ib: The energy gain per cycle for the radial and L = I non-radial modes

10

S

0

- 5

- lC

IOI-

5g

01-

i i !

.... Pl x 10'4

f x lO 6

2 .=3

- I 0

8'0

f x iO 5

Pl x 10 4 ~ -

\ gl x tO 5

I 0 l i i

9 100 t10 12_0

STELLAR MASS (M e )

Fig. 2a and 2b: The energy gain per cycle for the ~ =2 and 3 non- radial modes

REFERENCES

Castor, J. I. 1971, Ap. J. I~66, 109.

Cox, J. P. and R. T. Giuli 1968, ~rinciples of Stellar Structure, Gordon and Breach, New York.

Klapp, J., N. Langer, and K. J. Fricke 1986, poster paper presented at IAU Colloquium 123, Aahrus, Denmark, July 7-11, 1986.

Pesnell, W. D. 1984, B.A.A.S. 16, 409.

Pesnell, W. D. 1986, poster paper from this conference.

Stellingwerf, R. F. 1975a, Ap. J. 19~, 441.

Stellingwerf, R. F. 1975b, Ap. J. 199, 705.

Weiss, A. and >~. H. Baker 1986, preprint, submitted to Ap. J.

Ziebarth, K. 1970, Ap. J. I~62, 947.

50

LINEAR NONADIABATIC I~LSATIONS OF ~ S ZAMS STARS

J. H. Cahn Department of Astroncmy, University of Illinois

Urbana, IL 61801

A. N. cox Theoretical Division, Los Alamos National Laboratory

Ixgs Alaraos, NM 87545

D. A. Ostlie Department of Physics, weber State College

Odgen, UT 84408

A~TRACT

Current uncertainty about the most massive observed stars has led to a reexam-

ination of the most massive star that is stable against radial pulsation. The

nuclear energy generation equations in the INA linear, nonadiabatic code have been

considez-ably i~proved, so that it is now appropriate to redo the study to determine

the max~ mass of a ZAP~ star that will be stable against pulsation.

INTR0~JCTION

The question of the most massive star continues to motivate astronomers. The

existence of very luminous 03 supergiants, Huhble-Sandage variables, and Wolf-Rayet

stars have motivated evolutionary calculations in the mass range above 60 M O. The

inclusion of mass loss, both steady and in strong episodic ~ts in evolutionary

calculations, has made it possible to construct an evolutionary stellar sequence

from main sequenoe to H-S variables to Wolf-Rayet stars to supernovae.

~retical efforts have historically focussed on pulsational stability

(ledoux, 1941; Schwarzsc/aild and Harm, 1959; Aiz~, Hanson, and Ross, 1975) which

sUggested a lower limit of 60M o for stellar pulsation. The epsilon mechanism which

depends on a strong texture dependence of the CNO cycle of nuclear reactions,

Was shown to be capable of driving stable radial stellar pulsations for masses above

60 Mo. It was supposed that these pulsations would grow to sufficient amplitude

that the star would either disintegrate or by mass loss approach a more stable mass

rar~e. In 1970-1971 (AppeIzzeller 1970, 1970a; SJ_mon ard Stother 1970; Ziebarth

1970; Talbot 1971) proposed through nonlinear calculations that (I) the ar~plitudes

of surface zones might be small even though strong pulsational driving occurred in

the core or (2) that shock waves would develop damping further amplitude growth and

causing enhanced mass loss. Such limitations were thought to lift the domain of

stability to about 100M o. ziebarth, as a result of extensive INA modelling, stmmar-

ized the cc~position dependence of the critical mass for stable pulsatior~ by the

formula:

M c = i00(i.0 + 4.19Z - 0.83Y) _+ 2.0 M o.

Davidson CAm~hreys and Davidscn, 1983) has sear~ %E~IXX~fttlly for photc~etric

variations in prcmising 03 stars in Carina which may support this hypothesis.

51

As noted above, evolutionary calculations (Maeder, 1983) have gone far toward

explaining observations of H - S variables and Wolf-Rayet stars, in the mass range

above i00 Mo. Thus the question of pulsational stability has been raised again and

a recent preprint by Klapp, Langer and Fricke (1986) (herein after KLF) reexamines

this issue. A linear non-adiabatic (I/qA) analysis of homogeneous zero-age main se-

quence (ZAMS) stars with chemical cc~oosition (Y,Z) = (.277, .043) yields the

surprising result that stars below 400 M o were pulsationally stable against radial

pulsations.

Because of the challenge raised by the unusual results of KLF and the increased

sophistication of the nuclear energy generation programs, the authors decided to

redo the classical Ledoux- Schwarzschild-Harm calculations and those of the early

1970s to redeterm/ne the critical mass for pulsational instability.

In order to make a successful model of a star in which pulsational driving

ccmes from the epsilon mechanism, it is obviously very important to have the best

possible nuclear energy generation program, including details of the chemical and

~ t u r e dependence. The current program in the Los Alamos LNA code includes the

energy generation terms as well as the Fowler et al. (1975) reaction rates.

The calculations covered (i) the ~%ss range 70 to 160 M o at Z = 0.015 and (2) the

~ition range in 0.005 < Z < 0.043 for X = 0.689 and a mass of 130M O. The

dependence of the period, Po, and growth rate, Po/To, where T O is the e-folding

time, on mass for fixed cc~position are shown in Table I. It will be seen that the

periods form a generally increasing sequence with mass, while the growth rates are

~ch more erratic, as shown in Fig. i. Nonetheless, the growth rates become

positive between 80 and 90M o which is in agreement with the earlier results. In an

accumpanying article in this issue, O'Dell, Pausenwein, Weiss, and Hajek also obtain

similar results. The variation with Z is in Table II.

Table I.

Results of Fundamental Mode Radial Pulsation for Z = 0.015 and Y = 0.296. For ccmparison the average life of a 60 to 160 M o star is about 3 x 106 years

Mass log L/L o log Tef f Period Pc/To T O M O hours years

70 5.857 4.702 9.17 -2.944E-7 1,640 80 5.960 4.712 7.51 -2.356E-8 36,300 90 6.045 4.722 7.98 5.355E-7 1,700 I00 6.123 4.728 8.30 3.330E-7 28,400 ll0 6.189 4.734 8.89 9.807E-7 1,030 120 6.252 4.737 9.30 1.030E-6 1,030 130 6.304 4.746 9.17 6.394E-7 1,640 140 6.353 4.750 9.17 5.090E-7 2,050 160 6.443 4.752 10.61 2.125E-6 570

52

Table II.

Radial pulsation calculations for 130M o as a function of Z with X = 0.689 and 0.8Z=(~O

Z log L/L O log Tef f Period Pc/To T o hcp,~--s years

• 005 6.309 4.760 8.73 1. 312E-6 5,681 • O10 6. 252 4.737 9.30 1. 030E-6 6,858 .015 6.304 4.746 9.17 6. 396E-7 11,200 • 020 6. 303 4. 739 9 • 51 9. 037E-7 7,642 • 030 6.299 4.730 8.73 1. 312E-6 5,738 • 043 6.293 4.716 10.31 2. 397E-7 26,594

For cc~Darison, the results of KLF for cx:mioc~ition (X,Z) = (.687, .043) give Po =

9.43 hours, a growth rate of - 5.511E-7, and an e-folding time of 1,954 years. KLF

• aintain that not until about 440M O does a positive growth rate occur for pulsation

in the fundamental mode. The run of growth rates with Z is also shown in Fig. i.

The result of KLF is also shc~n. The variation of growth rate with Z is fairly

~%mbigucus. We did not go to a low enough Z to replace C~O entirely with p-p bur-

ning, so that our results do not apply to Population III stars.

The nature of the driving for the pulsation is shown in Fig. 2, which records

the PdV work per zone for each of the 359 zones. It will be seen that the driving

takes place in the first 55 zones at ~tures above 29 million degrees and

dam~ing thereafter. It is believed that the erratic driving at approximately one

million degrees is due to discontinuities in the opacity fit.

" ' l + , + " l + ~ , " l , , , l , , , , , , + i + + 201 +

~+ 15 +

I0

5

t~ 0

- 5 ~

7O

/ + .005

+~* ~ .010 / ~ .02o /\/ +--.-+/ -

/ + . 043 "

/

, + , I . . . . . I K I + ' F , , t + . Q 4 ) , , ,

9 0 I I0 130 150

S+ellor Moss (M O)

Fig. 1

Growth rates as a function of ~ass for Z = .015 and for var- ious Z at 130M

£ !

I,I,I, IS f

i I i l i i i

Z o . l

Fig. 2

Work per zone for the 120M o case

400

53

CONCLUSIONS

The location of the critical mass for pulsation for massive stars has been

redetermined to be very close to 80 M o. The location on the HR diagram of a number

of homogeneous ZAMS stars have been determined using the Los Alamos I_NA program in

good a ~ t with current evolutionary results. J. Cahn thanks S. Starrfield for

formatting and printing this paper.

R~ ~CES

Aiz~, M. L, Hansen, C. J. and Ross, R. R. 1975, Ap. J. 201, 387

Appenzeller, I. 1970, Astr. and Ap. 5, 355

....... . 1970a, Astr and Ap. 9, 216

Fowler, W. A., Caughlan, G. R. and Z ~ , B. A. 1975, Ann. Rev. Ast. and Ap. 13,

69

H~Ireys, R. M. and Davidson, K. 1984, Science 223, 243

Klapp, J., lar~er, N., and Fricke, K. J. 1986 IAU Colloq. 123, Aarhus,

Denmark, July 7-11

LedoILx, P. 1941, Ap. J. 94, 537

Maeder, A. 1983, Astr. and Ap. 120, 113

Schwarzschild, M. and Harm, R. 1959, Ap. J. 129, 637

Simon, N. R. and Stothers, R. 1970, Astr. and Ap. 6, 183

Talbot, R. J. 1971, Ap. J. 165, 121

Ziebarth, K. 1970, Ap. J. 162, 947

54

Thirteen-Color Photometry of BW-VULPECULAE: REVISITED

Manuel Alvarez._ Instituto de Astronomla, U.I#.J.~._ Apto._ Postal 877, 22830, Ensenada, Baja California, M~xico._

Ram6n Michel._ Escuela Superior de Ciencias, U.~.~.C._ Ensenada, Baja California, M~xico._

~lg~ivat_~_ol3__an_d_n~]_o_b_s~r~_a_t~_on~_- BW-Vulpeculae (HD 199140) a B2 III Well known ~-CMa variable is regularly observed on the UV and blue part of the spectra to study the most important features associated With its behaviour._ Trying to understand the physical picture that produces the observed periodicity, simultaneous spectroscopic and photometric observations have been carried out by several authors._ We report in this work some interesting results recently obtained and give some ideastrying to explain the observed phenomena._

On August 1984, we observed BW Vul with the 13-color photometric system, that has been described by Johnson and Mitchell (1975), as part of a coordinating campaign._ The observations covered from the extreme UV (912 ~) up to the IR (2 /u) and H~ was also monitored._ Preliminary results were reported by Barry 917_~i~- (1985). One interesting feature that came out from this work was the observation of a double-peaked light curve that is very clear in the IR.. This double peak is also present in our 13-color observations._ Our _Fi~uI~_l shows the light curves of filters '33', '45', '58', '72' and '80'. (Filter '33' is a medium-band filter: ZiTk = i00 ~, centered at 3371- ~).. Filter '33' shows only the well-known 'standstill' sistematically observed on this star with the U B V filters, while the double peak is observed with our 13-C filters at longer wavelengths._

5Z-

53-

~,m ::ram 54- :. O :'

,5 % / < "~

. , .<,_...._....o

6e • ~ o , ~ "i ] ]

\ -.. : .~ ~ -= ".. i

~. i~ ~ ~ I[ . 0 ira mr

oag"-.-.3' -.o".....2

8,1 I I ' ' "1 " o o o ~ io 15 2.o

P H A S £

Fig. i.- Phase diagram of BW Vulpeculae.. The UV variation is larger than the red._ The developig double maximum coincident with phase 0..75-0..80 is clearly seen from filters '45' and longer wavelegths._ Solid symbols are for August 7th, while open ones correspond to August 6th.. Dotted line is a spline fit to the data._ Magitude of the filters is given in the vertical

55

From our 13-C observations, we computed the energy flux density that is shown in ]~i~rg_~ for the moments of maximum (MX) and mini;.~um (ran) of the light curve._ There is an observed gradient in this energy flux density showing a variation of i.~004 x i0-14 watt cm-2 ~u-I at 3,300 and 0..0583 x i0-14 watt cm -2)/ -I at 8,600 ~ between the MX and mn of the light curve on the night of August 7, 1984.. The maximum absolute flux, integrated between these wavelengths is approximately 54% larger that the minimum._ This has to be considered in the theoretical models of this pulsating star._

I I I I I i i 1

- 13 .93

- 1 4 . 0 2

1 I I

1 3 - C o l o r P h o l o m e f r'

%

. - . . . - ,

,. 8 W V u l o, A u g 7 , 1 9 8 4 0 .J

~0 tOrO ,,, T T

99mp~ri~o~_w_i_tb_gp~_c_tr_o~Qpi~__obHeI~igD~ Young, Furenlid and Snowden (1981), have shown that the 'standstill' (in the 'y' filter of the 'ubvy') occurs at the time where the Rv curve reaches the maximum positive value, a few minutes before the 'discontinuity' that is observed in the radial velocity. As mentioned above, at the time of the 'standstill' in the UV, there is an observed maximum at longer wavelengths. The second (and larger) peak of the light curve occurs when the Rv changes abruptly from a large positive value to a medium negative value, where it remains aproximately constant for several minutes and the light curve start to decline._ After this, the Rv diminishes until it reaches the minimum and start to increase again._

_.General Di~uszi_oD=- Goldberg, Walker and 0dgers (1976) have considered a model of a pulsating star that ejects matter after the contraction phase of the star. According to them, this happens due to the strong outward acceleration observed when the Rv changes from >+i00 km/s to -36 km/s in less than 16 minutes as can be seen from the work of Young and collaborators._ The large value of the gravity and the slow rotational velocity of the star, inhibits this mechanism and the ejected material falls back-again into the atmosphere._ When the ejection of this small shell occurs, there is an opacity change that allows us to see the contribution of a higher temperature and probably deeper layers of the atmosphere of the star.. This is reflected in our observations in such a way that for short wavelengths, we see these high temperature regions and the light curve increases monotonically._ For longer wavelengths, however, there is a different contribution to the emitted continuum and hence, for the red and IR part of the spectrum, we see the observed double-peaked light curve._ Our 13-C observations show a small red excess that may be due in part to this small moving shell._

56

Kubiak (1972), considers that this effect can be done to a moving wave that heats the atmosphere without any change • in radius._ However, we believe that the Rv curve does not support this hypothesis, because after the sudden radial velocity discontinuity, the Rv monotonically diminishes until it reaches a value of -90 km/s between phases 0.Ii - 0.17._ This is the behaviour that we expect for the atmosphere to grow._ In fact, the atmosphere continues to grow until around phase 0.5, where the radial velocity reaches the 'systemic' velocity._ After this phase, the Rv shows a new contraction stage and the cycle starts again._

This work is part of the CONACYT-CNRS program 140106G202-160 on the Study of the variability of B stars._

Barry, D. C.w Holberg, J. B.~ Schneider, N.j Rautenkranz, D.~ Pol idan, R.~.~ Furenlid, I.jMargrave, T.~Alvarez,M.~Michel,R.j Joyce, R._ 1985.

Synoptic Observations of BW Vulpeculae. 1985th._ A.A.S.. Meeting._

Goldberg, B.~.j Walker,G.A._H.w and Odgers, G..J.~ 1976, A.J. 81, 433. Jonhson, H. and Mitchell,R._ 1975. Rev.jlex..Astr._Astrof.. i, 299._ Kubiak ,M.j 1972.. Act Astr._ 22, ii.. Young,A.w Furenlid, I. and Snowden, M.S.~ 1981. Ap.J._ 245, 998.

57

SPECTROSCOPY AND PHOTOMETRY OF THE OPTICAL PHOTOSPHERE OF BW VULPECULAE:

RADIATIVE TRANSFER, IONIZATION, AND OPACITY EFFECTS

ARTHUR YOUNG ~ CHESTER HAAG, AND GREG CRINKLAW

ASTRONOMY DEPT., SAN DIEGO STATE UNIVERSITY

SAN DIEGO, CALIF. 92182

AND

INGEMAR FURENLID

DEPT. OF PHYSICS AND ASTRONOMY, GEORGIA STATE UNIVERSITY

ATLANTA, GEORGIA 30303

Decades of observations and analyses of stellar spectra~ and of theoretical

studies of radiative transport have resulted in a general comprehension of the

emergent radiation from stars which are in stable equilibrium. Rapid~ global

pulsations, regardless of their origin, propagate through radiating stellar

atmospheres, altering the physical conditions even to the extreme of forming

shocks~ and leave imprints upon the emergent radiation. Our studies have been

directed to observing and analyzing the signatures of such pulsations~ attempting

to interpret them in terms of physical changes in the radiating zones using the

concepts which are based upon studies of static atmospheres. Our goal is to

identify which properties are most affected by the pulsation wave, and which are

most responsible for effecting changes in the spectrum of the emergent rad~atlon.

In the study we report here we have observed spectral absorption lines whose

equivalent widths range between 20 and 80 m~, and whose total ionization -

excitation energy ranges between ~ 30 and ~ 40 ev. Static model atmospheres for

stars with Teff'~ 22000°K and 10g g=4 indicate that these lines are formed deep

within the layers from which most of the optical continuum rad~atlon emerges. That

was our intended probe, and we therefore secured simultaneous b-band optical

photometry of that continuum radiation. The spectroscopic observations were

recorded on a CCD in the eoude spectrograph of the 2.1m telescope at the Kitt Peak

National Observatory. They span the wavelength region from ~ 5115 ~ to ~5165 o

with a two-plxel resolution of 200mA, a mean signal-to-noise ratio of i00~ and a

characteristic time resolution of 5 minutes. Thirty seven such observations were

secured over 75~ of the pulsation cycle. Synchronous photometric observations were

~de with the O.4m telescope at the Mt. Laguna Observatory~ with much higher (20s)

resolution time.

In another paper in these same proceedings, Furenlid, Young and Meylan give a

complete discussion of the kinematical studies resulting from those same data, and

in this paper we make use of those results without discussion. @

Figure i is the optical light curve of continuum radiation near~.4800 A, and

the fiduolals ¢i'- ¢2' and ¢~ ~rk the m~Jor e~ents in th~ light curve starting

58

With the cessation of increasing radiation ( ~I = 0"8340' the resumption of

increasing radiation ( ~ 2 = 0.935) and the occurence of maximum output (~3 =

0.00). The fourth fiducial (¢ h = 0.082) is derived from the velocities, and is

the time of onset of maximum outward acceleration (i.e. expansion) of the I l T I I l I I I I I I I I I I I I ' ~ I I ~

Photosphere.

,,o ; J2 % "x

5 04

MT.LAGUNA OBSERVATORY, IgB3 SEPTEMBER 27 U,T. ...... ~ ; ) ' '~ . . . . . . . . .

FIG. 1 OPTICAL LIGHT CURVE IN b-FILTER ( )% 4800 ~)

From the velocity curve we know that phase ~l occurs while the photosphere is

in rapid compression (V r = + 120 kms-l), phase (~) 2 marks the onset of nearly ~0

(V r = 0 kms -I in rest frame of the star)~ phase ¢ 3 is at the minutes of quiescence

midpoint of that quiescent period, which then ends at phase ~4"

Figure 2 shows the measured values of equivalent width of a AlgA-excitation

(32.6h ev) llne of Fe III (•5127 ~. $imi!ar behavior of the stron~ C II doublet

at ~6578-82 was reported by Young, Furenlid and Snowden (1981), but those lines are

formed well above the region from which most continuum radiation emerges.

@ •

e e ~

@ •

• • •

• o e

@ • @

@ o

o 0

,,1 GI i t s I ~ 1 J (, , , , , ~ , l , i t , 1 ,

- . 4 - . 3 - . 2 - .~ 0 A , 2 . 3 . 4

PHASE

g o

ao

70

(,4 60

4O

2 O

I0

0 ,5

FIG. 2 MEASURED

EQUIVALENT WIDTH OF

Fe III k5127 ~ LINF

59

Furthermore, the C II lines have a total ionization - excitation energy of

only 25.59 ev. All that is in common to the C II and the Fe III lines is the

continuous opacity in the Paschen continuum due to bound-free ionization of

hydrogen. We interpret the steep negative gradient in Fig. 2, between phases ¢1

and ~2' as being due to an enormous and rapid increase in the continuous

opacity,resulting from increased density, as the infalllng photosphere forms a

stationary shock . In the quiescent interval, -- ~ 2 to ~ 4 ' we observe no systematic

variation in the line strength, but it increases immediately after phase ¢ h when

the photospheric layers expand rapidly. The continuous opacity is then decreasing,

thus causing increasing strength of the line. The interval ~ 1 to ~2 is also the

time when the light curve (Fig. i) shows an abrupt and sustained cessation of the

erstwhile increasing photospheric radiation. The sam~ increasing continuous

opacity which weakens the absorption lines seems to be responsible for suppressing

the photospheric radiation at a rate comparable to its increasing thermal

generation at this wave3ength.

In Fig. 3 we plot the ratio of measured equivalent widths of the very high

excitation (~0.OOev) 0 II line, ~5106 ~, to those of the FeIII llne at ~ 5156 ~.

The ratio removes the sensitivity to continuous opacity, and the energetics of the

two species provides a temperature-sensitive indicator.

2,0

1 .8

, + , *

, 5 - , 4 - , 3 +$

I O

I I O • •

O • O •

• ~ o O

O @ • •

, i , t , ~ , i , i + i , I i

- . 2 - .A 0 .~ . 2 . 3 +4

PHASE

1.6

1.4

1,2

3

~J ~.o

.S

FIG. 3 RATIO OF EQUIVALENT WIDTHS IN THE SENSE 0 ll/Fe III

For iron, the ionization equilibrium is between Fe III and Fe IV, and for oxygen it

is between 0 II and 0 III. In both cases, thermal increases drive the equilibrium

toward the higher state. However, the 0 II llne arises from a state which is 26.~

ev above ground for that ion, and the Fe IIl line comes from a state only 8.6 ev

above its ground. The relative populations of those states are sensitlve to

but unequall~, in favor of 0 II. Between phases~ I_ and --~h we observe temperature ~

60

an increase and a decrease, with a peak at ~3' which recapitulates the

photospheric variation (particularly that which is observed in the far ultraviolet)

Which is itself attributed to changes in the temperature of the deep photosphere.

If our measurements are being interpreted correctly~ the implication is that

the exact shape of a light-curve of continuum radiation (at any wavelength) in a

pulsating star is vulnerable to significant modification by opacity effects and may

not be a faithful indicator of the actual pulsation wave.

REFERENCES

YOUNG, A. FURENLID, I., AND SNOWDEN, M. S. 1981, Ap.J:, 2h5, 998.

6~

BW Vulpeculae Pulsation Kinematics

Ingemar Furenlid and Thomas Meylan Department of Physics and Astronomy

Georgia State University Atlanta, Georgia 30303

Arthur Young Department of Astronomy

San Diego State University San Diego, California 92115

The work reported here is part of a long term study of BW Vulpeculae~ the

Cephei star with the largest known amplitude in brightness and velocity. The

primary motivation for studying these stars stems from the fact that the pulsation

mechanism is unknown. The observational work is done with the purpose to describe

in as detailed a way as possible the physical and kinematical behavior of a ~ Cephei

star in order to provide a sound basis for theoretical investigations and, ultimate-

ly, for an understanding of these stars. The large amplitude is important in

permitting the greatest possible detail to be revealed throughout each pulsational

cycle and in providing a pulsating star where the resonance condition is fully

developed.

This report has its roots in discussions in a paper by Young, Furenlid and

Snowden (1981), where the argument was made that the large observed velocity

amplitude might reflect the motions in high, tenous, line-forming regions and not be

representative of the deeper, continuum-forming strata, Published radial velocity

measurements invariahly refer to intermediate or strong lines formed at shallow at-

mospheric depths, leaving the kinematics of continuum forming layers unexplored.

The obvious way to study the motions of the deeper layers is to select very weak

lines of such properties of excitation and ionization that they are formed close to

the continuum.

Figure i shows the studied spectral region in BW Vul centered around 5140 A as

observed with the 2.1 m telescope and coude spectrograph at Kitt Peak; 37 spectral

frames were obtained. The spectral resolution is just under 0.i A per pixel on the

CCD and the time resolution around 5 minutes, the wavelength coverage is 50 A and

the signal to noise ratio is around 200. The lines are all very weak; note that the

zero point of the intensities is suppressed and that only the upper i0 percent is

shown. The weakest line has a depth of 2 percent and the strongest 8 percent of the

continuum intensity. In figure 2, which equals frame 14, the lines have changed

completely and here even the strongest line has a depth of only 3 percent. The

62

measurement of Doppler-shifts in spectra of this kind is a real challenge~ A

method developed by Furenlid and Furenlid (1986), using a cross-correlation (c-c

hereafter) technique, was applied and succeeded in disentangling the complex

motions quite well. Each line in figure I was replaced by a delta function, shown

at the top of the figure, and all frames cross-correlated with these delta

functions. The resulting c-c function is also the mean profile of all the included

lines and yields therefore more information than just the velocity shifts. The

rest of the discussion is devoted to the analysis of these mean profiles.

The peak of the c-c function of the frame in figure 1 defines an arbitrary

zero point of velocity shifts. Each shift of one pixel on the CCD corresponds to

a velocity change of 5.7 km/sec. Returning to frame 14 we can now see to which

extent the c-c function has succeeded in cleaning up the convoluted appearance of

the spectrum. Figure 3 shows the result of the c-c operation and it is immediately

clear that we now have a mean profile sufficiently well defined to make a meaning-

ful interpretation. We see a substantial spread in velocities, i.e. part of the

stellar disk is still in a state of contraction at the highest velocity observed,

While other parts have already reached what is essentially the stillstand velocity.

Other parts, in the middle of the profile, are clearly in a state of rapid decel-

eration and the overall picture is one of chaotic and turbulent motions in the

atmosphere of the star as it changes from a state of contraction into one of essen-

tially no radial motion. Similar turbulent behavior occurs in classical Cepheids

at the corresponding phase in the pulsations (Benz and Mayor, 1982).

I.OO

0 . 9 5

~ 0.90 51ZO 5130 514,0 5150 5160

WAVELENGTH (A}

Fig. i. The studied spectral region in BW Vul

~- I00

z o,95 w >

.J 0 . 9 0

i I I i i

1 f I f l 5120 5130 5140 5150 516,

WAVELENGTH (A}

Fig. 2. Same spectral region as fig. I., shown at the velocity discontinuity

Combining the c-c velocity determinations from all the frames gives us a very

Precisely defined velocity curve for the pulsation of BW Vul. The precision of the

CUrve comes from the high resolution of the spectra, the high signal to noise ratio

of the data and the use of the mean profile of all the lines in defining the Doppler

shift of each frame. The veloclty curve, shown in figure 4, should be thought of as

representing the motions of the layers forming the visual continuum in the star.

The variation in radius of BW Vul through its pulsatlonal cycle has been

measured by two independent methods; by integration of the velocity curve and by the

63

relation between radius, effective temperature, and luminosity.

Integration of the velocity curve determined above leads to an increase in

radius from smallest to largest size of somewhat more than 400,000 km, in good

agreement with the result of Goldberg, Walker, and Odgers (1976).

I ! I, I I I I I

- 15o - IOO-SO 0 50 I00 15o 200

& V (kin ~=J

Fig. 3. The cross-correlation function shown at the velocity discontinuity

150

L E I 0 0 .*¢

~ ~o

S o

_.1 - 5 0

I I- I I I

0.4 0.6 0.8 0.0 0.2 0.4

PHASE

Fig. 4. The radial velocity variations of BW Vul shown in the stellar rest frame

The luminosity of BW Vul has been obtained from integration of the flux dis-

tributions recorded by Voyager 2, the IUE satellite, and ground based observations

in the visible and the near infrared (Barry et al., 1984). Effective temperatures

have been derived by fitting stellar atmosphere models (Kurucz, 1979) to the flux

distributions, leading to photometrically determined radii. The change of radius

from minimum size to maximum is around 7% of the mean radius of 8.1 R O (Lesh and

Aizenman, 1978), or around 400,000 km. The star is hottest at smallest radius and

coolest at maximum distension, with a temperature difference of around 4000°K.

The velocity data can he used to find the systemic velocity of the star in

the following way. The central cap of the disk of the star will display the whole

range of radial velocities caused by the radial pulsation of the star, whereas a

thin ring around the limb will have no radial component at all. The high spectral

resolution in these data permits us to identify that part in each line profile

which is con~non to all phases of the pulsation; that part originates in the ring

around the llmb and equals the systemic velocity of the star. The velocities

plotted in figure 4 refer to a zero point of systemic velocity determined this way.

Using this systemic velocity we find the systemic, heliocentric radial velocity of

the star to be -I +/-3 km/sec.

The next point concerns the stillstand phenomenon in BW Vul, which has attract-

ed a lot of attention as can be easily gleaned from the literature on the star.

The somewhat paradoxical conclusion from the spectroscopic work reported here and

64

from Barry et al. (1984) is that the photometric stillstand is of no consequence,

but that velocity stillstand is highly significant; the two are not simultaneous.

The photometric stillstand appears to be only a transitory enhancement of

atmospheric opacity longward of the flux maximum and of little consequence for the

variation in total flux of the star.

A striking feature in the velocity curve is the apparent velocity discontinuity

Which follows the point of maximum positive velocity. We have found that this

discontinuity preceding the velocity stillstand is actually a rapid, turbulent,

but continuous deceleration. This interpretation emerges from the fact that we can

follow a continuous transition in the profiles from the high red shift of the first

dozen of frames to the group that represents the velocity stillstand. Figure 3

shows as mentioned above the mean profile of frame 14, which is located in the

middle of this transition from high positive velocity to stillstand. The rate of

deceleration is difficult to determine separately because of the turbulent behavior

of the atmosphere. We may, however, conjecture that the rate of deceleration equals

the rate of outward acceleration occuring after the velocity stillstand. The data

are fully compatible with such a hypothesis, which implies that we are witnessing

a transition from bulk motions in the atmosphere to a pressure wave propagating in-

Wards with the local speed of sound. If this picture is correct than it is log-

ical to pose the following question. If the deceleration is followed by a wave

traveling inwards and the acceleration phase is preceded by a wave going out, at

what depth in the star do the two waves coincide? Using an interiors model of BW

Vul kindly supplied by Art Cox we find the answer from integrating over the

Velocity of sound to be around 40,000 km, which equals the depth where we find the

He II ionization zone. It is clear that energetically the He II ionization is

insufficient as a driving mechanism in 8 Cephei stars, so another, additional,

mechanism is needed. The following scenario then suggests itself: the primary driv-

ing mechanism is located in layers sufficiently energetic to drive the pulsations

and the He II ionization zone only serves as a secondary, coupled drive,

The sharply defined locus in the H-R diagram of large amplitude B Cephei

Pulsators can then he explained by the fine tuning of the stellar structure needed

for the double resonance to occur. BW Vul must be close to the perfect configura-

tion as evidenced by its large amplitude of pulsation; the peak to peak amplitude in

radius variation is around 7% of the mean radius. Considering that pulsations are

relatively co,on in early B-type stars it might also be, that marked deviations

from resonance lead to small amplitudes, double or multiple periods, non-radial

Pulsations, or any combination of these. The possibility of a mechanism of such

broad implications makes it particularly important to search for and fin~ the process

that generates the 8 Cephel phenomenon.

65

References:

Barry, D.C., Holberg, J.B., Schneider, N., Rautenkranz, D., Polidan, R., Furenlld, I.,

Margrave, T., Alvarez, M., Michel, R., and Joyce, R. 1984, Bull Amer. Astr. Soc.,

16, 898.

Benz, W., and Mayor, M. 1982, Astr. Ap., iii, 224.

Furenlid, I., and Furenlld, L. 1986, in prep.

Goldberg, B.A., Walker, G.A.H., and Odgers, G.J. 1976, Ap. J., 81, 433.

Kurucz, R.L. 1979, Ap. Jo Suppl., 40, I.

Lesh, J.R., and Aizenman, M.L. 1978, Ann. Rev. Astr. Ap., l~, 215.

Young, A., Furenlid, i., and Snowden, M.S. 1981, Ap. J., 245, 998.

S6

HIGH RESOLUTION OBSERVATIONS OF IOTA HERCULIS

J.M. Le Contel, D. Ducatel, J.P. Sareyan, P.J. Morel, E. Chapellier, A. Endignoux

Observatoire de Nice, B.P. 139 06003 NICE CEDEX - France

IOta Her (B3 IV) has been known for a long time as a spectrum variable. Smith (1978,

1979, 1981) and Smith and Stern (1979) detected different periods in line profile

Variations and classified iota Her ~n their 53 Per group.

Recently the Nice group and S. Gonzalez-Bedolla in Mexico observed it in photometry

and spectrography at 12 A/mm. The main results are the detection of short period va-

riations (0.12 or 0.14 day period) in photometry, radial velocity and on the

He 1 4387/Mg II 4481 lines intensity ratio (Chapellier et al. 1986). These short

Periodic variations are superimposed on longer ones which were first detected by

Rogerson (1985).

We present here preliminary results of the 1985 campaign obtained with very high

Spectral resolution.

I.- OBSERVATIONS

We observed iota Her during 10 consecutive nights at the Haute-Provence Observatory

with the 1.93 m telescope and its T.G.R. Spectrograph (Baranne et al. 1967). A new

receptor, i.e. a photon counting camera, has been used. The field is limited to 1~rm,

so that only one line can be observed, due to the high dispersion of the spectrograph

(0.4 to 0.6 ~/mm at 4000 and 6000 A respectively; a pixel = 30 mA). A thorium lamp

allows wavelength calibration, and a tungsten one flat field corrections.

Real time control of the S/N ratio is achieved by CRT visualisation. This facility

Was used to improve time resolution, in order to study rapid variations (the S/N

ratio always being over 25, which is sufficient for position measurements). Several

Spectra were later added for line profile studies.

The instrumental stability was checked on the RV standard y Equ (V = -17 km/s) on the

Mg II 4481 doublet.

DUe to their sensitivity to non-LTE effects, we chose the He 1 5876 and the Si III

4552 lines, each one being observed for five nights, some spectra were also obtained

On the Mg II 4481 doublet.

After correction for flat field, the spectra have been calibrated in wavelengths.The

lines profiles were then smoothed using a polynomial filter. The resulting resolu-

tions are 50 and 70 mA at 4552 and 5876 A, respectively.

Although the reductions are not yet completed, some new results are already obtained.

67

II.- RESULTS

I.- Radial velocities

Fig. la shows the night by night RV of iota Her, obtained on the mean profiles resul-

ting from the addition of the individual spectra (only one spectrum for the Mg II

4481 line). The precision is about i 0.5 km -I. Fig. Ib shows at the same scale the

RV of y Equ, obtained from three lines in the 4481A domain.

One can note :

- The red shift of He I lines with respect to the other elements, already obsel~

ved in iota Her (Chapellier et al. 1986) and in other B sta~s, is confirmed on the

5876 ~ line. -I - The nightly RV of iota Her vary within a 8 km s range, with a time scale pro-

bably longer than one day. This confirms Rogerson's (1985) and Chapellier et al.'s

(1986) observations.

- Fig. 2 shows that both in the Si III 4552 and He 1 5876 A domains, we observe

short period variations on different nights. (These measurements were made at half

line intensity).

2.- Line profiles

a) Si IIl 4552

The average profiles in Fig. 3 were obtained by adding the 19 first spectra (about

3 hours, July 31-J.D.244 6278.) and 22 spectra about 3.7 hours, July 27-J.D. 244

6274.). A strong violet component appears at 0.25 A (14 km s -I) of the main peak on

July 27 th , while there is no such feature on the average of July 31 st. However

this component can be found in the individual spectra of July 31 st, after 24 hours

U.T. So this component, although it may appear in one hour, can last 3 to 4 hours

without any important change in RV. One can note that the apparition of the compo-

nent is associated to a jump in R.V. (Fig. 2).

b) He I 5876

Fig. 4 shows a series of profiles obtained on J.D. 244 6273 (July 26). Weak pertur-

bations of the peak mainly on its red part can be followed along the night. The first

three line profiles (U.T. 20h36 to 21h11) look like those recorded between U.T.

24h19 and 25h19.

III.- DISCUSSION

The existence of short period variations in radial velocity is obvious from Fig. 2.

Such variations also exist in photometry and line intensity ratio probably due to

temperature variations (Chapellier et al. 1986). We are not yet able to determine

68

a period in the line profile variations although there is strong suspicion that the

Si III line variations (and the night to night RV variations) are related to longer

period. These preliminary results confirm the complexity of the spectral variations

in iota Her : both short and long time scales are present. The improved spectral and

time resolutions show evidence for larger deformations and shorter time scales than

previously detected.

A 0.12 or 0.14 period has been proposed by Chapellier et al.. It means that the

COrresponding mode has a pulsational constant similar to that found in ~ CMa stars

(around 0.029). So iota Her is a very interesting star as it lies in a region of the

H.R. diagram, outside the classical region of the B CMa stars, where long time scale

Variations have larger amplitude(Waelkens and Rufener, 1985). It is also the third

53 Per star, after 22 Ori and u Ori (Balona and Engelbrecht, 1985), in which short

period variations are detected. We suggest that stars situated between B2(22 Ori)

and B3(iota Her) should be intensively observed to look for short period variations.

0nly simultaneous photometric and spectrographic observations performed at different

longitudes could lead to a better frequency spectrum determination and could let us

know whether the long time scale variations are due to pulsation or activity.

A_.~_nowledgements : One of us (J.P. Sareyan) gratefully acknowledges the financial

support of the Local Organizing Committee.

~ography.

Balona, L., Engelbreeht, C.A. : 1985, M.N.R.A.S. 214, 559

Baranne, A. et al. : 1967, Pub. Observatoire de Haute-Provence 9, 289

Chapellier, E. et al. : 1986, Astron. Astrophys. in press.

Rogerson, J.B. : 1984, Astron. J. 89, 1876

Smith, M.A. : 1978, Astrophys. J. 224, 927

Bmith, M.A. : 1979, Tucson workshop on Non Radial Pulsation, Ed. H. Hill

Smith, M.A., Stern, S.A. : 1979, Astron. J. 84, 1363

Smith, M.A. : 1981, Astrophys. J. 246, 905

Waelkens, C., Rufener, F. : 1985, Astron. Astrophys. 152, 6

69

-15

-18

I km/s -../ o o

o I b

J.D. 265

symbols :

a) iota Her b) y Equ

n He 1 5876 a Si III 4552 o Mg II 4481

J.D. 277

fig. I : night by night, mean radial velocity

I I I kmls ~

~+~+~ ++++++ + + L ~ ~ +#H + ++

O.I.J.D. 0. I J.D. J.D. 278, Si III 4552 J.D. 270, Hel 5876

fig. 2 : Radial velocity of iota Her

++ +++

++~++ + +

O.1 J.D. J.D. 273, Hel 5876

R B U.T. 20h 12

20h50

21H15

21h57

22h29

5~

Io.I

U.T. 23h12

23h43

25hi 5

25h33

26h14

average profiles

J.D. 278

J.D. 274

J.D, 278

fig. 3 : line profile of Silll 4552

R B U.T.

20h36

21h30

23h 11

25h52

o. 5~ H

I0.~

average profile

J.D, 273

fig.4 : line profile of He 1 5876

70

THE OBSERVATIONAL STATUS OF 8 CEPHEI STARS

L,A. Balona

South African Astronomical Observatory

P.O. Box 9, Observatory 7935, Cape, South Africa

I. Introduction

Our knowledge Qf pulsationsl instability among the early-type

Stars has changed considerably over the last few years. The consensus

of opinion seems to favour a view in which the S Cep variables are

Only a small group amongst a much wider sea of pulsational instability

which includes the 53 Per stars and the Be variables. In this picture

the distinction between B Cep variables and the other types of non-

radial pulsators must be sought in terms of modal differences. The

instability mechanism for these stars is presumed to be the same and

i s s t i l l one of the g r e a t e s t u n s o l v e d p rob lems o f s t e l l a r p u l s a t i o n .

While this picture may be true, photometric observations of 53

Per stars have failed to show the short-period variations found in the

early spectroscopic work (Balona & Engelbrecht 1985a). A reanalysis

of the light curve of 5) Per itself shows that it could be explained

in terms of a single period of 3.45d; a monoperiodic interpretation of

HR3562 and HR)600 is also possible (Balona & Laing 1986). The short-

Period light variations of Be stars are monoperiodic as well. It is

Possible that rotational modulation might be the source of variability

for these groups of stars. However, it is not clear whether this

hypothesis is adequate to explain the profile variations. ]he B Cep

stars are still the only group where NRP is certainly present.

2. E v o l u t i o n a r y S t a t u s

U n t i l r e c e n t l y , i t was t h o u g h t t h a t a l l 6 £ep v a r i a b l e s were

c o n f i n e d to the S-bend r e g i o n of s t e l l a r e v o l u t i o n . Ba lona &

E n g e l b r e c h t (1983) and Jaka te (1979) observed these v a r i a b l e s in the

YOung c l u s t e r s NGC3293 and NFIC4755 and showed beyond doubt t h a t they

71

were in a l a t e c o r e h y d r o g e n b u r n i n g s t a g e o f e v o l u t i o n and no t i n the

two o t h e r s t a g e s a s s o c i a t e d w i t h the S-bend r e g i o n . T h i s c o n c l u s i o n

was based not o n l y on the p o s i t i o n s o f the s t a r s in the HR d i a g r a m ,

bu t a l s o on the ] a r g e numbers o f B £ep v a r i a b l e s d e t e c t e d in t h e s e

c l u s t e r s .

The o b s e r v a t i o n s o f NGC3293 s u g g e s t e d t h a t a l l t e n 6 £ep

variables in this cluster defined an instability strip where no

constant stars are found. The subsequent discovery of seven of these

stars in the very young cluster NGC6231 (Balona & Shobbrook 1983,

Balona & Engeibrecht 1985b), did not confirm this picture. Again, Lhe

large number of ~ £ep stars found in NGC6231 strongly suggests that

this phenomenon is very common among the early-type stars. A recent

survey of another cluster NG£2362 (Balona & Engelbrecht, unpublished)

has confirmed this conclusion.

The B £ep v a r i a b l e e in NGC6231 e s t a b l i s h e d a v e r y i m p o r t a n t

r e s u l t : t h e s e v a r i a b l e s are not c o n f i n e d to the S-bend r e g i o n as

assumed u n t i l now. l h e v a r i a b l e s in t h i s ve r y young c l u s t e r a re

scarcely evolved, lhe conclusion is that B Cep variability is

probably found from the ZAMS until the end of core hydrogen burning.

Observations of two other unevolved field stars HR3058 and HR3088

(Jerzykiewicz & Sterken 1979) support this view.

lhe reason why this result had not been found earlier probably

lies in two selection effects. Firstly, the pulsation amplitudes are

generally larger near the end of core hydrogen burning. Secondly,

there are very few field stars which are relatively unevolved. For

evolutionary reasons, most ZAMS early-type stars are to be found in

young clusters. Surveys of these clusters to detect B Cep variables

have only been undertaken very recently.

3. Pulsation modes

Definitive mode identification in B £ep stars would permit the

study of asteroseismology in these stars, lhe most promising method

is based on the analysis of high-quality line profiles. Using this

method~ Smith (1981) has found that a11 ~ Cep stars seem to have at

]east one radial mode and suggests that these stars be distinguished

from other B-type pulsating variables by this fact. It would be very

important to re-observe some of the bright B £ep variables to confirm

this finding. Recent developments in analysing line profile obser-

vations on more objective grounds should offer the possibility of de-

finitive mode identification (Balona 1986).

72

The ten ~ Cep v a r i a b l e s in NGC3295 o f f e r an e x c e l l e n t o p p o r t u n i t y

of s t a t i s t i c a l mode i d e n t i f i c a t i o n . Because the r e l a t i v e masses,

r a d i i and tempera tures are b e t t e r determined fo r these c l u s t e r s t a r s ,

COmparison of observed pe r iods w i th those ob ta ined from models should

enable f a i r l y r e l i a b l e mode i d e n t i f i c a t i o n s . To [ h i s end, Enge lbrech[

(1987) has made i n t e n s i v e pho tome t r i c obse rva t i ons to determine t h e i r

Per iods. His a n a l y s i s shows a p re fe rence fo r the ~ = 2 (quadrupo le )

mode. At l eas t one (and p robab ly two) of the ~ Cep s ta rs in t h i s

C l U s t e r was found t o be an e c l i p s i n g b i n a r y ( E n g e l b r e c h t & B a l o n a

1986 ) . F i r s t and second o v e r t o n e r a d i a l p u l s a t i o n or a r o t a t i o n a ] l y

s p l i t q u a d r u p o l e mode a re p o s s i b l e i d e n t i f i c a t i o n s .

4. C o n c l u s i o n s

The n a t u r e o f t h e 6 Cep s t a r s and t h e i r r e ] a t i o n s h i p to t h e 53

Per and Be s t a r s i s as o b s c u r e as e v e r . Excep t f o r t h e mov ing bumps

seen i n t he l i n e p r o f i l e s o f some 8 and Be s t a r s i t may be p o s s i b l e [o

e x p l a i n t he p h o t o m e t r i c and l i n e p r o f i l e v a r i a t i o n s o f 53 Per and Be

S t a r s in t e rms o f r o t a t i o n a l m o d u l a t i o n . Only f u r t h e r o b s e r v a t i o n s

w i l l answer t h i s q u e s t i o n .

One o f t he most i n t e r e s t i n g f i n d i n g s o f t h e l a s t Few y e a r s i s

t h a t t h e 6 Cep phenomenon i s no t c o n f i n e d to t he S -bend r e g i o n o f

e v o l u t i o n as p r e v i o u s l y t h o u g h t , b u t o c c u r s f rom t h e ZAMS u n t i l f he

end o f c o r e h y d r o g e n b u r n i n g . T h i s a g a i n opens t h e q u e s t o f t he

i n s t a b i l i t y mechan ism f o r t h e s e s t a r s . F u r t h e r p r o g r e s s in t h e s t u d y

o f 6 Cep s t a r s i s l i k e l y to be made by h i g h q u a l i t y l i n e p r o f i l e

a b s e r v a f i o n s t o g e t h e r w i t h s i m u l t a n e o u s p h o t o m e t r y i n o r d e r to i d e n t -

i f y t h e modes o f o s c i l l a t i o n .

Re.•£erences B a l o n a , L . A . , 1986.

Ba]ona, L.A. & Engelbrecht, C.A., 1983.

202, 293.

B a l o n a , L . A . & E n g e l b r e c h t , C . A . , 1985a.

212~ 889.

B a l o n a , L . A . & E n g e l b r e c h t , C . A . , 1985b.

214, 559.

B a ] o n a , L . A . & S h o b b r o o k , R . R . , 1983.

205, 309.

Mon. N o t . R. a s t r . S o c . , 219, 111.

Mon. N o t . R. a s [ r . Soc.~

Non. Not. R. astr. Sot.,

Mon. N o t . R. a s [ r . 5 o c . ,

Mon. Not. R. astr. Soc.,

73

E n g e l b r e c h t , C.A. & Balona~ L.A.~ 1986. Men. Not . R. a s t r . Soc . ,

219, 449.

Engelbrecht, C,A., I~87. Non. Not. R. astr. Soc., submitted.

Jakate, S.M., 1979. Astr. J., 84, 552.

Jerzykiewicz, M. & Sterken, C., 1979. Changin 9 trends in Variable

Star Research (IAU colloq. 46), Waikato University Hamilton

New Zea land, 474.

Smi th , M.A. , 1981. Workshop on P u l s a t i n g . B..St...#rs , Nice O b s e r v a t o r y ,

317.

74

PERIODIC LINE PROFILE AND PHOTOMETRIC VARIATIONS IN MID-B STARS

C.L. Waelkens

Astronomlsch Instituut Katholieke Universitelt Leuven

CelestiJuenlaan 200B, B-3030 Heverlee (Belgium)

I. The variable mid-B stars

In a study in the Geneva photometric system of a large sample of B stars as free

from observational bias as possible, we found a larger-than-average scatter for the

mld-B stars (Waelkens and Rufener, 1985). This larger scatter is not a statistical

fluke but points to a genuine group of variable stars. The variations of the members

of the group reveal the existence of well defined periods. Light and color vary in

phase, with the amplitude of the (U-B) variations being always less than but still

of the order of the amplitude of the light variations. The amplitudes of the other

Colors are an order of magnitude smaller. The amplitudes are variable in time, and

the ratios of the amplitudes of color and light variations remain roughly constant.

Table I lists observational data for the best studied mid-B variables.

Table i: Data for mid-B variables: listed are the HD numbers, periods in days, epochs of observations, amplitudes, and projected rotational velocities (references: (I) Bright Star Catalog (2) Andersen (1986))

HD

74195

74560

123515 143309

160124

177863

181558

Period

2.78

1.55104

1.456 1.66760

(P2 = 1.67177)

1.92016

(P2 = 1.9175)

1.2378

1.1896

Epoch A my ~ (U-B)

1981 0.012 0.007 1983 0.019 0.012 1981 0.019 0.009 1983 0.015 0.007 1983 0.021 0.016 1979 0.036 0.026 1983 0.034 0.028 1984 0.052 0.038 1985 0.035 0.028 1981 0.043 0.031 1982 0.013 0.010 1983 0.027 0.019 1984 0.024 0.019 1985 0.043 0.034 1983 0.016 0.013 1984 0.016 0.016 1982 0.030 0.018 1983 0.029 0.021 1984 0.022 0.013

v sin i

4O (I)

22 (I) <39 (I)

6

i0

(2)

(2)

75

2. Identification of the variable mid-B stars with the 53-Persei stars

Despite the large amplitudes of some of the variables, the variable mid-B stars

were not defined earlier as a class of photometric variables. One reason for this is

observational bias. The spectra of these stars are not conspicuously anomalous, as

are those of Be stars and CP stars. Also, their periods are rather long, while the

intriguing problem of the ~ Cephel stars has stimulated observers to look mainly for

short-perlod variability. As a matter of fact, some known variables that I think

also belong to the mid-B-variable group -- HD 76566 (Burki, 1983), HD 37151 (North,

1984), HD 27563 (Mathys et al., 1986) and HD 54475 (Stlft, 1979) -- were found Incl-

dentally by observers looking for other phenomena.

Important for the interpretation is the similarity of the photometric variations

of my stars with those of 53 Per, the prototype llne profile variable (Smith, 1977).

53 Per itself and the other line profile variable Her are mid-B stars with a large

scatter in the Geneva photometry. I have therefore conjectured that the mid-B stars

I described are also llne profile variables. I was able to verify this conjecture

for HD 74195 and HD 74560 from measurements with the Coud4 Echelle Spectrograph at

the 1.4 m CAT-telescope at the European Southern Observatory. The lines of the Sill-

doublet near 4130 A of both stars show profile variations similar to those of 53 Per

with a time scale of the order of that of the photometry.

My observations thus provide a new insight into the problem of the 53-Per stars.

They offer the first proof that stable periods are present in these stars. I think

that photometric techniques are better suited than spectroscopic techniques in order

to unravel the complicated frequency spectra of these stars. The measurements are

less time consuming and can be done with smaller instruments, and the description of

the variability is less model dependent than with spectroscopy.

3. Rotation versus pulsation

Since his discovery of the line profile variations of 53 Per (Smith, 1977), M.

Smith has proposed that non-radlal pulsation is responsible for the variability of

this star. This interpretation has been challenged by L. Balona (this meeting), who

argues that a spotted-star model cannot yet be discarded. It is thus appropriate to

consider my data in view of both the rotation and pulsation hypotheses. My findings

lend some support on the latter hypothesis.

76

A first argument is the apparent existence of an instability strip~ the larger-

than-average scatter is restricted to the spectral range B3-B7, and I did not detect

Similar behavior in earlier or later B stars. It may then also be that the earlier

llne profile variables are not 53-Persei stars.

A second argument is the observation that all variable mid-B stars appear to be

slow rotators, although the photometric approach does not induce a selection effect

on v sin i. When we assume that the periods are rotation periods, we find that the

projected rotational velocities (Table i) and reasonable radii lead to inclination

angles which systematically are very small, of the order of some degrees only. This

conflicts with a random distribution of the rotation axes. Also, for near pole-on

Stars one does not expect important oblique-rotator variability to be seen.

A third and potentially conclusive argument would be multiperiodicity. The non

detection of multiple periods would in turn argue against the pulsation hypothesis,

Since for the periods we found here the mode density in frequency space is large, so

that many modes should be excited simultaneously. In order to find a second period,

it is essential to determine the mean amplitude of the primary oscillation with good

Precision, and this requires a large set of data covering all the phases of the beat

Perlod~ otherwhise prewhitening is a dubious procedure. Although dominant secondary

Peaks are present in the prewhitened power spectra of the two stars most frequently

observed, HD 143309 and HD 160124, their physical reality is doubtful at this stage.

Indeed, the beat periods are rather long, i.e. not much shorter than the time base

of the observations. Also, no shorter beat periods are found yet, despite evidence

for cycle-to-cycle variations. Nevertheless~ it is encouraging that precisely for

the most often measured stars indications for secondary periods were found. Since

the oCeurence of multiperiodicity is crucial In view of the interpretation, we will

continue our photometric observations of these stars.

4. An attempt to identify the pulsation modes

We have obtained 28 spectroscopic measurements with a resolution of I00,O00 of

the SiII-doublet at 4128 and 4130 A for HD 74195 (o Velorum~ m v = 3.6) during eight

COnsecutive nights in February 1985. An attempt was made to identify the pulsation

modes with the moment method outlined by Balona (1986). In this method the various

Parameters of the modes are determined from Fourier decomposition of the moments of

the llne profiles.

77

The results of this investigation (CrlJns, 1986) are inconclusive. The 2.8-day

period was recovered in the velocities, but other variations were also present. It

was not possible to distinguish periodic and erratic components in such a short data

string. Moreover, the non inclusion of the temperature effects on the llne profiles

complicated the interpretation.

5. Discussion

Lee and Salo (1986) have suggested that the mid-B star variability is induced by

overstable convection of a rapidly rotating core. However, the rotation velocities

of my stars are consistently low, despite there being no selection effect.

We have suggested (Waelkens and Rufener, 1985) trapping of high-order g-modes in

the Hell ionization zones of mid-B stars, so that these objects present similarities

with the DB white dwarfs. The hot edge of the instability strip would then be where

the ionization zone is located too near the surface. But then, what causes the red

edge, i.e. why is there not a continuous range of variables up to the limits of the

6 Scutl star strip? One reason could be that, the ionization zone being situated

deeper in the star, g-modes of lower order would have to be excited in stars cooler

than the mld-B variables. Such modes would have considerable amplitudes outside the

ionization zone and damping would overcome driving, since only a limited amount of

pulsation energy is available in the ionization zones of B-type stars. Thus, we may

expect that only the modes that are trapped in these zones can be excited.

References

Andersen, J., 1986, Private Communication to M. A. Smith. Balona, L.A., 1986, Mon. Not. R Astron. Soc. 219, iii-129. Burki, G., 1983, Astron. Astrophys. 121, 211-216. Crijns, S., 1986, Licence Dissertation University of Leuven. Lee, U., Salo, H., 1986, Mon. Not. R. Astron. Soc. 221, 365-376. Mathys, G., Manifold, J., Renson, P., 1986, Astron.Astrophys. Suppl. Ser. 63,403. North, P., 1984, Astron. Astrophys. Suppl. Ser. 55, 259-358. Smith, M.A., 1977, Astrophys. J. 215, 574-583. Stift, M., 1979, Inf. Bull. Variable Stars No. 1586. Waelkens, C., Rufener, F., 1985, Astron. Astrophys. 152, 6-14.

78

Nonlinear Behavior of Nonradial Oscillations in E Per

Myron A. Smith National Solar Observatory, Tucson, AZ. 85726, USA

Alex W. Fullerton and John R. Percy Department of Astronomy, University of Toronto, M5S IA7, CANADA

We have conducted a simultaneous spectroscopic/photometric campaign of ~ Per (BO.7 III) during five nights in November, 1984. The SPectroscopic data consist of 300 observations of the Si III XX4552- 74 triplet, while the photometric data were obtained at two different observatories. In both sets of data we find a dominant 3.85±.02 hr. period. The analysis of line profiles in the context of nonradial pulsation (NRP) indicates thls oscillation is caused by a -m=~ =4 mode. In this context the line profiles also indicate the Presence of a secondary -m=£ =6 mode with a period of 2.25±.03 hr, an oscillation below the detection threshold in the photometric data• These periodicities and mode identifications have been reported by Penrod (priv. comm.) on other occasions. They may be Considered to be stable except that their amplitudes vary from epoch to epoch.

The figure shows an example of strong blue-to-red traveling bumps arising from the £=~ mode on 4 November. Particularly strong bumps

P E R 3 NOV 1 9 8 4

1.0 "" " • " "1 *30

~6= 5.31 92 ~ '8B

10:15 U.T. .~4= ,.25 (~6 • 6 21

I'- ' ~ ~ I I I I I - - I

X . 4 5 5 2 OR X 4 5 6 T ( "B")

I t I 1 t t I 1 I I

"1 '/'." , .5o ~6" 6,69

i ~ "1548 J 12:12 U,T -I ¢4" 1.65

j ~ v , . O 8

I ' t l l l l : : { :

79

from the £=6 mode are visible in the censer and right (red) edge of Observation 48.

A modified Baade-Wesselink analysis has been performed (see Buta and Smith 1979, Ap. J., 232, 213). It shows that the average velocity amplitudes derived from the profiles predict a light amplitude for the £ =4 mode that is within a factor of 1.5 of the photometric observations. This success, considering the margin of errors, corroborates the assumption that vertical velocity patterns (and not horizontal velocities, or temperature modulations) are responsible for the line profile variations. The agreement also shows that the £ =4 mode is spheroidal and not toroidal in character. FinallY, the photometric data show no clear-cut color variations, which suggests that geometric effects dominate temperature effects in producing light variations.

E Per is remarkable in that it exhibits the consistently largest line profile variations of any suspected nonradial pulsator on the H-R Diagram (Smith, 1985, Ap. J., 288, 266). Both the profiles and light curves show significant changes in amplitude from cycle to cycle. We find that the spectroscopic amplitudes of both suspected nonradial modes vary from nearly zero to Mach 2 on a timescale comparable to the mode's superperiod on the corotating frame (18 hrs.). We find several correlations of the £ =4 amplitude wit~ departures from expected behavior. These include mismatches in the llne cores of our modeled profile fits, departures from the expected ephemerides within an ~=4 cycle (spectroscopic phases .25 and .75 dO not appear to be ~ cycle apart), and lengthenings or compressions of the period depending on whether the amplitude is large or small. The latter in turn are related to concomitant variations between the spacings of adjacent bumps on the line profile. The amplitudes vary in an apparently stochastic, rather than impulsive, manner, we suggest a natural picture in which strongly driven waves become supersonic near the stellar surface and develop nonlinearities in their wave form. This tendency causes them to lose their coherence with themselves and with their strictly periodic driving function in the stellar interior. As a consequence, the waves lose their global nature at the stellar surface and acquire a random fluctuation around their mean amplitudes and positions, resembling forced waves sloshing around in a closed container. This picture would also account for the general tendency of high degree nonradial modes in other B stars to depart from regular behavior with increasing amplitude (e.g.6 Sco; Smith, Ap. J., 304, 728, 1986). TheSe nonuniformities seem to increase until, as in ¢ Per, the average velocity amplitude reaches the atmospheric sound speed. The coincidence of this limit suggests that atmospheric dissipation restricts the growth of an apparently superficially confined mode in the stellar envelope.

The quantitative fits to over 60 observations representing 55 hours of data are very good except in two instances in which bump s suddenly disappear on the line profile. In one of these cases, two very pronounced bumps disappeared and remained absent on the following night, suggesting that (cf. Obsns. 48 and 53 in the figure)the waves may have quickly annihilated each other. We suggest that nonlinearities may play a part in the rapid, apparent disappearance of bumps on the line profiles, culminating in spectral transients and mass ejections in many Be stars when an additional, large-energy (£=2) mode is available. These data and model fits

80

Will be published in a forthcoming Ap. J. Suppl. The photometric data have already been presented by Percy and Fullerton (J.R.A.S. Canada, 79, 242,1985, poster paper).

APPENDIX M. Smith's Comments on L. Balona's "The Nature of 53 Persei"

Having nOt had an opportunity to answer criticisms by Balona on my llne profile work on B stars in the 1970's, I wish to do so now. I reply to his paper on 53 Persei in these proceedings as follows:

Balona has made several criticisms. The first concerns his inability in 1982-3 to detect in his photometry the periods that I had derived a few years earlier from line profiles in two other Stars, 22 Ori and Ori. This is not surprising. The profile variations in those stars are small and one should not expect the COrresponding light variations to be detectable at all. Balona's Periods for these two stars are based themselves on very small (I-3 millimags), unconfirmed semi-amplitudes. One period, 11 days, is extremely long, a few times longer than the longest other reported Periods for these stars. All of Balona & Engelbrecht's periods are inconsistent with the profile variations I exhibited in the cited references. Disagreements are now emerging over periods in other B Stars, e.g. ~ Eri, where Balona reports a period of 0.406 days Whereas Bolton (radial velocities), Percy (photometry), and Penrod (Profiles) and myself have found a period of .71 days. The period in this star has been stable over years, including the 1985-6 epoch of Balona's observations. These disagreements can mean anything one Chooses, particularly if the observations are made a few years apart, as is the case of the Smith et al. and Balona & Engelbrecht Observations that Balona compares. For example, they are consistent With recent work on B stars which shows frequent and large amplitude Changes for various modes leading to the impression of "mode switching,,.

There is no explicit criticism of the line profile data itself or of the manner in which periods were derived from the profiles, yet this is where the case should be examined. These data are presented in a form that lends itself to quantitative analysis by interested Parties. Smith and Stern (A.J., 84, 1363, 1979) have investigated the reliability of period extractions from simulated line profile Variations with realistic spectrophotometric and time-sampling Parameters. The basic result was that it would take a pretty Complicated multi-mode structure to fool one every time one

investigated a new data set.

As Balona suggests, it is possible to argue occasionally for rotational modulations, particularly for a minority class of the dOUble-wave Be variables. However, this argument is not easy to • ake for 53 Per itself. In particular, a 10 ° -200 inclination will not permit matches to the large amplitudes of llne profile Variations observed. For example, the line widths vary by nearly a factor of three in 53 Per. One cannot simulate these variations by arbitrarily increasing the amplitude of any variation parameter (Whether a wave or spot) to compensate the effects of small sinl. The narrow lines at certain phases set particularly strong eOnstralnts on the allowed inclination because one finds that above an amplitude limit the llne widths start to increase, and not to

81

continue to decrease, at narrow line phase. The conclusion is that the extreme line width changes requires strong velocity vector cancellations across the disk surface, which means an organized velocity wavefield. Additionally, temperature spots of any kind beget large, well-deflned variations in llne strength and R.V. with phase that are unobserved. There are several other stars, both normal (e.g. o Vel) and Be (e.g. o And, I Erl), which have long periods and moderate to high sini's. Some of these stars have profiles with traveling bumps indicative of high-degree NRP modeS. In these stars the condition for rotational~modulatlon, P(var) P(rot), is usually badly violated unless one forces slnl to low values -~ values that are forbidden by the line profile variations. These matters do not pertain to sparse observations. They seem to be serious objections to any spot modulation picture.

Concerning the periods of 53 Per, the values determined by Walt Fitch in Smith et al. (Ap. J., 282, 226, 1984) were obtained from photometry, not line profiles. Those values are not in error but refer to solutions derived from the entire 1977-81 data set rather than a portion of that set, which Balona analyzed. Also, I agree that f2 is weaker in the later data. I think that Balona and I would both agree that the llne profiles in 1977-8 and 1983 are compatible with two periods, each near ~wo days. One can see immediately the opposite-phase/llke-phase aspects in the profiles (asymmetry, width) observed on adjacent/alternate nights, respectively. We would also agree that these variations cannot arise from a single 3~ day slnusold. The double-lobed waveform arising from Balona's PDF analysis could arise from intermode beating or from an ~ = 2 mode with a 1.73-day period but with unequal lobes, similar to the kind of behavior occasionally found bY Penrod (prlv. comm.) in o And. In sum, I believe that the light and profile data for 53 Per can be fit well but not perfectly with two low-degree NR modes with periods near 2.3 and 1.7 days. Whether these modes exhibit short term variations in amplitude, or whether they are accompanied by secondary oscillations I do not knoW. Evidently they do suffer long-term changes in amplitude which preclude a fit with a fixed set of parameters for these two modes.

Finally, an oblique rotator model requires a constancy in period, and therefore it should fit all the photometric data. Balona has arbitrarily excluded two nights from the 1983 data set, as well as the entire ~977 data set. By this exclusion, he has in effect retreated from his own starspot hypothesis which demands strictly stable periods that fit all the data. In criticizing the work on 53 Per stars, Balona has first questioned the interpretation of unstable NR periods (better said: variable amplftudes) and them introduced a new model for which in effect the requirement of stable periods is dropped. Overall, my current feeling, based in ¢ per work, is that any spectral/photometrlc comparison had better refer to simultaneous data. I fear this is true even of the long period modes with slowly changing characteristics. Both our works suffer this criticism.

82

THE NATURE OF 53 PERSEI

L.A. Belone

South Afr ican Astronomical Observatory

P.O. Box 9, Observatory 7935, Cape, South Af r ica

1. In t roduc t ion

The 53 Per stars are s group of shar ~ l i n e d ea r l y - t ype s tars

(09.5 _ BS) showing va r i a t i ons in spec t ra l l i ne p r o f i l e s . The ear ly

Work on these stars (Smith 1980) showed timescales of 5 to 22 hre, but

the per iods are unstable, of ten doubl ing or halv ing or swi tching to an

e n t i r e l y d i f f e r e n t per iod. The v a r i a b i l i t y is explained in terms of

non-radial g-mode pulsations.

Photometric observations of 53 Per itself (Buts & Smith 1979;

Smith et el. 19B4) show the presence of two more or less stable

frequencies at 0.464 and 0.595 cycles per day. Although these

frequencies dominate the line variability of 53 Per during 1977-1981,

the overall fits are still fairly unsatisfactory and point to the

P~esence of additional frequencies.

Recently, extensive photometry of two other 53 Per stars 22 Ori

and u Ori (Balona and Engelbrecht 1985) have failed to show the

frequencies observed in the line profiles. If the wariations are

indeed periodic and coherent, they suggest e period or periods longer

than 3 days for 22 Ori, while u Ori couid be e very low amplitude B

Cap star (defined such that the longest period may be a p-mode).

Balona and Laing (1987) obtained intensive photometric

Observations of two more suspected 53 Per stars HR3562 and HR3600.

lhey find that in both stars only one dominant frequency is

Present. Although one could interpret the residual variations in

terms of further eigenfrequeneies, these are below the level of

detection. There are, however, changes in the shape of the light

CUrve from cycle to cycle, a fact which could explain why they were

thought to be multiperiodic.

83

Finally, recent observations have shown that many Be stars are

periodic on s timescale comparable to rotation. The light curves of

these stars can be described by one period, but in many cases there

are also cycle to cycle variations. Some also display a double-wave

light curve. Examples are the four Be stars observed in NGC3766

(Belona and Engelbrecht 1986).

These observations have prompted us to re-examine the photometric

data on 53 Per published by Smith et sl. (1984) with s view to

explaining the variation in much the same way as for HR3562, HR3600

and the Be stars. The Buts & Smith (1979) data are unpublished and

could not be analysed, but in any case they are less numerous and

poorly spaced for a periodogram analysis. If it~can be shown that a

single dominant period with small cycle to cycle variations adequately

describes these observations, then it may not be necessary to invoke

NRP as an explanation. It is then possible that the 53 Per and Be

stars may owe their photometric variability to some kind of rotational

modulation (as in the Bp, Ap RS CVn and BY Dra stars). Interpreting

the line profile variations in this way could prove fruitful.

f I I I I " ; . i

@ • • O 8 •

5 10 15 20 25

JD24.~.;9~*0+

Fi 9. i : The l i g h t curve of 53 Per f i t t e d wi th a two-frequency Four ie r curve. The t i c k marks on the magnitude ax is are spaced by 0.02 msg.

2. Results

That the l i g h t curve of 53 Per i s m u l t i p e r i o d i c seems q u i t e

ev iden t at f i r s t e i g h t . Fig. I shows the best f i t t i n g curve using the

84

(::3

' , r ,<¢

t !

t t

O.Z 0 I

O-4 0"6

FREQUENCY 0.@ 1.0

Fi~: Top - the Fourier periodogram of the light curve of 55 Per. 6ottom - the same after prewhitening by the principal frequency fl.

two most probable frequencies fl = 0.4302 and f2 = 0.5940 cycles per

day. The value of the dominant frequency, fl, is actually different

from the value given by Smith et al. (1984) owing to an error in their

Period finding technique. Although the general trend is correct, the

fit is not very satisfactory as they have pointed out.

Fig. 2 shows the periodogram of the raw data and after

P~ewhitening by fl = 0.4302. The aliasing problem is severe and a

definitive choice of f2 is not possible. On the other hand, if the

light curve is indeed a superposition of Fourier components, more than

one frequency is required to describe the data adequately.

One problem with the period finding technique based on Fourier

decomposition is the implicit assumption that the data are well

~ep~esented by s superposition of Fourier components. While this

85

I ' '" ] "'' '" "' " " '""'" "" i '"" " l

J 0 0-2 0.1. 0.6

FREQUENCY O-@ 1-0

Fig. 3: The phase dispersion minimization periodogrsm of 53 Per.

ought to be t rue for a pu l sa t i ng s ta r , i t need not be the case in

genera l . For example, the l i g h t curve of a detached e c l i p s i n g b inary

is far from sinusoidal and its period is not easily found using

Fourier techniques. The double-wave light curve of some Be stars is

another example.

The phase dispersion minimization (PDM) technique (Stellingwerf

1978) is one way in which this difficulty can be avoided. Fig. 3

shows the PDM periodogram of the 53 Per photometry. The last two

nights have been omitted as they are distantly separated from the rest

of the data, but their inclusion does not alter the periodogram very

significantly. The strongest peak in Fig. 3 corresponds to a

frequency of 0.29 cycles per day. The resulting light curve, together

with s least-squares Fourier fit, is shown in Fig. 4. The Fourier

curve fits the data with a r.m.s, error of 5 millimags, close to the

expected observational error.

3. Conclusions

These results for 53 Per are very similar to those found in

HR3562, HR3600 and the Be s ta rs . From Fig. 4 we see tha t the l i g h t

v a r i a t i o n s of 53 Per can be i n t e r p r e t e d aa a ra the r complex

double-wave l i g h t curve wi th a per iod of 3.45 days. From the observed

v sin i = 17 km s -1 and adopting a radius of 4 solar radii, the

period implies an inclination of about 20 ° . Whether such e low

inclination is sufficient to explain the light amplitude and profile

B8

.... ' ' . . ' ' .... ' . .... ........ '!

• I [ ' ' - ]

0.0 0"2 0.t+ 0-6 0-8 1.0 1"2 PHASE

~iD~ M" 4: The light curve of 53 Per phased with the period given by the DM technique, P = 3.~5d. A least-squares Fourier fit is shown. Tick

marks are spaced by 0.01 meg.

variations needs to be investigated.

Of course this does not disprove NRP, but suggests that an

alternative explanation is possible. There are difficulties in

modelling the profile variations in terms of NRP (notably the 'K

Problem'). The rotational modulation hypothesis might answer this

Problem as well as providing a link between the Be and 53 Per stars.

ReFerences Ba lona , L .A . & E n g e l b r e c h t , C . A . , 1985. Hon. Not . R. a s t r . S o t . ,

214, 559.

Balona, L.A. & Engelbrecht, C.A., 1986. Hon. Not. R. astr. Soc.,

219, 131.

Balona, L.A. & LBing, D.L., 1967. Men. Not. R. sstr. Soc., in press.

Buts, R.J. & Smith, M.A., 1979. Astrophys. J., 232, 213.

Smith, M.A., 1980. Current Problems in Stellar Pulsation Instabilitx,

NASA, 391.

Smith, H.A., Fitch, W.S., Africano, J.L., Goodrich, B.D., Halbedel, W,

Palmer, L.H. & Henry, G.W., 1984. Astrophys. J., 282~ 226.

Stellingwerf, R.F., 1978. A strophye. J., 224, 953.

87

Preliminary Results of a Survey for Line Profile Variations Among the 0 Stars

A. W. Fullerton I, David Dunlap Observatory, University of Toronto

D. R. Gies, Mcdonald Observatory, University of Texas at Austin

and

C. T. Bolton I, David Dunlap Observatory, University of Toronto

Over the past 18 months we have collected high dispersion, high

slgnal-to-nolse ratio spectra of a large sample of O-type stars using

the coude spectrographs and the Reticon detectors of the Canada-France-

Hawaii Telescope and the McDonald Observatory. The aim of this survey

is to search for the absorption line profile variations which are usually

attributed to nonradial pulsations (NRP) and which appear to be endemic

among the early B stars (Smith 1986). There is a growing body of evidence

that NRP play an important role in the atmospheres of some B stars,

especially Be stars (see reviews by Baade 1986 and Percy 1986). SimilarlY~

NRP may have important consequences for the atmospheres, winds, and

episodic mass loss events which have been observed among the O stars

(Conti 1985). A further goal of our survey is to determine whether there

is a "blue edge" to the region in the HR diagram where line profile

variability occurs, and whether such variability is correlated with vsi~ ~i'

luminosity, etc. We hope that the systematic acquisition of this infor-

mation will provide additional constraints on the excitation mechanism

or mechanisms which cause pulsational activity in early type stars.

These issues are currently of great interest, but clearly the first step

must be to determine the incidence of line profile variability among the

0 stars. This paper presents our preliminary findings.

Our sample consists of 46 0 stars spanning all luminosity classes

and spectral types from 09.7 through 04. The sample is apparent magnitude

limited to permit high S/N (about 300) observations of the faintest stars

(V37.1) to be obtained without sacrificing time resolution (AtS30 mln.).

In spite of this constraint, each spectral type - luminosity class bin

contains at least two stars. However, the sample is biased towards

brighter or nearer stars. The sample was not deliberately biased towards

iVisiting Astronomer, Canada-France-Hawaii Telescope.

either large or small vsini values. Known double-lined spectroscopic

binaries were not observed, or were observed only when the lines were

clearly separated.

We have collected approximately 20 spectra for each of the stars on

our program. We usually observed the C IV doublet at ~5801, 5812 ~ and

the He I triplet line at X5876 ~. These features are in absorption

through the full temperature range spanned by the O stars, although

emission is frequently present in the He I llne. The simultaneous obser-

vation of three lines provides invaluable redundancy which aids in as-

sessing the reality of subtle features in the llne profiles. This region

is reasonably free of telluric contamination (especially in the CFHT

Observations) and is also near the sensitivity peak of Reticon detectors.

In some cases, however, the C IV lines are blended either with each other, o

owing to rapid rotation, or wi~h the interstellar feature at ~5797 A.

For several of the brighter stars observed in the McDonald portion of

our program we have acquired Octicon observations of many lines simul-

taneously.

Our preliminary results are based on inspection of the time series

spectra of a subset of 35 of our program stars. We have not modelled the

Variations in any way to date, and have been conservative in deciding

Which stars are variable. We distinguish three classes of behavior:

those stars which show line profile variations consistent with NRP,

those which show line profile variations which may or may not be due to

NRP, and those stars which appear to be constant. While we have adopted

NRP as a working hypothesis to account for some of the line profile

variations we observe, we are aware that other interpretations are pos-

Sible. However, since there is no direct evidence that O stars individ-

~ally or as a class possess large scale magnetic fields, chemical patches

or any other kind of "starspot", we feel fully justified in interpreting

the data within the framework of our working hypothesis.

In order to fall in the first class ("NRP"), at least two absorption

lines must show consistent asymmetries and the sense of the asymmetry

must progress in the manner expected for NRP. About 26% (9 stars) of the

Subset of our program stars fall into this class. The stars in the second

category ("Var") consist of objects for which we have insufficient ob-

Servations to determine whether the variations are NRP-like, objects

Which appear to be newly detected double-line spectroscopic binaries,

or objects which show variability in the He I llne which may be related

to wind activity. Approximately 23% (8 stars) of the subset exhibit this

~ort of variability. The remaining 51% (18 stars) of our subset did not

~how significant variability on any timescale sampled by our data,

and fall into the third class ("Constant"). As a caveat we add that the

fraction of stars in the "Constant" category is probably an upper limit

to the number of truly nonvariable 0 stars since we do not possess suf-

ficient observations over a long enough baseline to establish rigorously

that a star is strictly constant.

The distribution of our program stars in the HR diagram and the

variability class into which they fall are illustrated in Figure I.

The absolute magnitudes and effective temperatures used to construct

this figure are taken from the calibrations of Walborn (1972) and Conti

(1973), respectively, and are intended to be illustrative rather than

definitive. The B stars which Smith (1986) considers to be NRP are also

indicated in Figure i. This diagram illustrates~three points: i) line

profile variability is a continuous phenomenon, spreading from about B5

to at least 06 along the main sequence, 2) there is apparently no "blue

edge" to the region occupied by line profile variables~ and 3) variable

and constant stars coexist in the same regions of the HR diagram. The

enormous range of physical conditions which exist in the domain spanned

by the line profile variables suggests that a single opacity source is

unlikely to be the excitation mechanism of these pulsations. Further-

more, the coexistence of variable and non-variable stars in the same

region of the HR diagram poses considerable problems for any type of

excitation mechanism~ including rotatlon.

The analysis of our spectra is just beginning. We plan to charac-

terize the line profile variability we have observed according to the

number of bumps in the profile, their amplitudes, the timescales involved

in their passage across the visible hemisphere of the star. For a few

of our program stars we have acquired enough observations to attempt

to model the variability through disk integration simulations, and we

plan to do this to investigate compatibility with the NRP hypothesis

and to undertake a preliminary mode-typing of the pulsations. We encour ~

age photometrists to search for light variations which might accompany

these line profile variations, and urge theoreticians to consider the

p~isational properties of these massive objects in more detail.

Acknowledgements

The stimulating comments and valued encouragement of Drs. D.C.

Abbot~, D° Baade, H.F. Henrichs, J.R. Percy, M.A. Smith, and "proto-Dr°

G.D. Penrod are gratefully acknowledged.

90

References

Baade, D. 1986, Hi~hiights of Astronomy, 7, 255.

Conti, P.S. 1973, Ap. J~, 179, 181.

Conti, P.S. 1985, in 'Reports on Astronomy', ed. R.M. West (Dordrecht:

Reidel), p. 353.

Smith, M.A. 1986, Unno Retirement Conference, ed. Y. Osaki (Tokyo:Univ.

of Tokyo), in press.

Percy, J.R. 1986, Highlights of Astronomy, 7, 265.

Walborn, N . R . 1972, A. J . . , 77, 312.

r o I . . . . . . . . . I I " 1 "

>

I

N !

4 . 8

0

0

0

+ 8 *p

, 00

* 0 0 +

0 #

00

,04 o~, 08 po 4,7 4.6 4.5

z~

&

&

B 2

4.4

Iog(Teff)

...... I I I

I I 9 5 I a B 8

4.3 4.2 4.1

~Igure i: The domain of line profile variables in the HR diagram.

Asterisks represent "NRP" stars, crosses indicate "Var" stars, and

diamonds signify "Constant" stars. The B stars which Smith (1986)

Considers to be nonradia! pulsators are denoted by triangles

91

THE MAIA STARS - A REAL CLASS OF VARIABLE STARS

Dr. Bernard J. McNamara Department of Astronomy

New Mexico State University Las Cruces, NM 88003/USA

ABSTRACT

In 1955, O. Struve suggested that a sequence of variable stars having

periods between about 0.1-0.3d and spectral types in the range B71II-

A2 V-II might exist. These stars were called "Maia" stars after the

presumed prototype Pleiadid, Maia. Struve later abandoned this pro-

posal when he concluded that Maia was neither a light nor velocity

variable. Despite Struve's disclaimer, the search for short period

variability in this spectral region has continued. This paper sum-

marizes the observational data relating to the "Maia" variables and

suggests that little if any reason exists for maintaining the "Maia

star" designation.

PROS CONS

Struve 1955 Sky and Tele. 14,461

Maia B81II Suspected short period

velocity variable.

yU Mi A3 II-III Possible short

period light and velocity

variable.

Fernie 1969 J.R.A.S. Canada 63,133.

Percy 1970 Pub A.S.P. 82,126.

~Cor Bor AO IV Suspected short

period light variable.

Breger 1972 ApJ. 176, 367.

Percy 1978 Pub A.S.P. 90,703.

McNamara 1985 Ap.J. 289, 213.

Struve, Sahada, Lynds, Huang

1957, Ap.J. 125, 115.

Maia light and velocity studies

designed to detect vari-

ability are all negative.

Percy 1978 Pub A.S.P. 90,703.

yU Mi No light variations

greater than ~V=0.01 mag.

detected.

Tippetts and Wilcken 1970 Pub

A.S.P. 82,1156.

Percy 1978 Pub A.S.P. 90,703

yCor Bor No light variations

greater than ~V=O.01 mag

detected.

92

Antonello, Arienti, Fracassini,

Pasinetti 1978 Astr. and

Ap. 66,37.

~And A2 IV Large short period

0Peg A2 IV velocity and equiva-

2Lyn A2 iv lent width changes

observed.

-~Breger, Light, sch01tes 1979,

Astr. and Ap. 78,11.

oAnd No equivalent width

8Peg changes detected, pre-

vious results ascribed

to photographic errors.

~-Jard ....................... sley, Zizka 1980 Current Pro- Neubauer L.O.B. 17,109,193.

blems in Stellar Pulsation Insta-

bilities, NASA Technical Memorandum

80625, 421.

~Lyr AO IV Periodic velocity vari- ~Lyr Strongly disputes radial

ations claimed, P-0.19d velocity variability.

~0Vir A1 IV Periodic velocity vari- --

ations claimed, P~0.15d

¥Gem AO IV Periodic velocity vari- --

ations claimed, P-O.13d

B~ardsley, Worekl King 1980 current McNamara 1"985 Ap.J. 289, 213.

~Problems in Stellar Pulsation Insta-

bilities, NASA Technical Memorandum

80625, 409.

Alcyone B7 III Radial velocity --

period of 0.27d "might" exist.

~aia B8 III Radial velocity period Maia light constant to Ay=0.002

of 0.10d claimed to exist, mag.

Taygeta B6 IV Radial velocity period --

of 0.27d claimed to exist.

Cox A.N. (observations by Broiley)

Saas Fee Lecture Series (in press).

Alcyone B7 III Photometric obser-

Atlas B8 III vations suggest

Electra B6 III that Electra, Mer-

Maia B8 III ope and Taygeta

Merope B6 IV might be small,

~aygeta B6 IV short period

l~ght variables.

McNamara 1986 Ap.J. (in press).

Atlas Light variable, P=2-4d,

amp(y)=0.00B

Electra Light constant to

Ay=O. 002 mag.

MaJa Light constant to Ay:0.O02

mag.

Merope Light variable, P=O.5d,

amp(y)=O.OO6

Taygeta Light constant to

Ay=O.O02 mag.

Above periods are well outside

the "Maia" range.

93

CONCLUSION

When first discussing the location of this class of variable star in

the HR diagram, Struve stated: "The group labeled Hypothetical Maia

Sequence is at present quite uncertain." Since that time, not a sin-

gle confirmed member has been round. The designation "Maia Star" thus

has some historical interest but its continued usage is not supported

by observational results. The term re£ers to a class of stars con-

raining to members.

In a possible ironic twist, Struve et al (1957) reported that Maia

showed a "non-periodic" change in its helium line strength. Such a

change might be related to the 53 Per phenomenon discussed by Smith

(1977). If so, the Struve observations represented a lost opportunity

ol discovering tLe 53 Per stars nearly two decades before their actual

discovery.

ASSUMED LOCATION OF MAIA STARS

W F

J o 6t

- - 4 . 0

- - 2 . 0

0 0

+ 2 . C

+ 4 0

.... l ' I L I i i i I

B E T A C A N I 5

'::' '; MAJORIS ~6 ̂~ ~ . . . . Classical .,~11 ',.; ~ J ueta fcams Cepheids III II.' /l,Z':;;ii: .... ~//Majoris Sequence ,,/~l//

, MA,A ( . . , ) . i

/ " ~ .,.., .... ~ H y p o t h e t i c a l

~"~i '~i~i i ' : : . , : "-" Maia Sequence W Virginis -- G A M M A P E G A S I ";::',':~:i;" " , ~ " ' " (a sJ') ,.~k4~ ~ Cepnelos

=~ , , D E L T A S C U T I

OA,, ,A U. ,AE M , . O . , . - " ' zs ̂ ~ . / ,l~t/tllW~:,l ./ ( , ,/

RR Lyrae Stars 1t1//II1!1 ~ ' ' ' : ; i ~ ti I, vz c ~ r ,

D arf Cepheid ~ ~'T"PHOeNIc,S:: Sequenca (,,.,,,̂ )

t , I I I f I I , _ _ I o B A F G

S P E C T R A l , . C L A S S

94

The 0.103 day Radial Velocity Period of Maia

+ 4 0 M A I A l , ' ',

s ~

+ 3 0 ,"

* 2 0

ssJ i

T + I 0 '~

0: - , o (

- Z 0

- 3 0

t I l I I ! I , i i I ,, " O 0 0 . 2 0 , 4 0 . 6 O I L 0

PHASE

The evidence for this period consists of a phase plot of the radial

Velocity data obtained by Henroteau (1921 Pub. bom 0.5,45) during the

nights of Dec. 16, 1919 (filled circles) and Aug. 16, 1920 (open cir-

cles). Taken from Beardsley, Worek, and King (1980). ~ z i i ......

40

"~10

E x 0 >

-10

3 0

2 0

C')

- 2 0

I . . . . . . . I ! ........... 0 . 1 8 0 . 0 0 .2 0 .4 0.G t , O

P h a s e Same plot as above but the additional velocities obtained by ilenroteau

for the nights Aug 17,18,20,22,1920 are inclut~ed. Periodicity not seen.

95

EMPIRICAL-THEORETICAL MODELING OF Be VARIABLE MASS-LOSS VIA

VARIABLE: PHOTOSPHERIC MASS-OUTFLOW; CORONAL OPACITY; RADIA-

TION-AMPLIFIED WIND-PISTON, DRIVING PULSATING COOL-ENVELOPE

Vera Doazan (Obs.ParJs) and Richard N.Thomas(Radiophysics,Inc)

I.INTRODUCTION: No decade of s te l lar observations has been so thermodynam-

ically-educational on s te l lar structure as the present: on normal stars, and on

peculiar stars l ike the Be. Stars arise via local concentrations of the in ters te l lar

medium,then evolve i r revers ib ly via energy and mass return to it. IF such fluxes

• were only radiative-energy, s te l la r structure might be ~nly quasi-static thermal.

Such was indeed yesterday's speculative-theoretical model: rest r ic t ion of" stars

to being closed thermodynamic systems: no mass-flux. Today: two decades of far-

UV observations suffice to identify al l well-observed stars as o ~ n thermodynam-

ic systems; which return to their parent medium additional fluxes of mass, non-

radiative energy, and momentum --- aerodynamically, not thermal ly; and contin-

uously, not just at highly-exceptional epochs. Thermodynamically-consistent mod-

eling of such phenomena requires for a l l stars some nonthermal structure -- sub-

atmospheric as wel l as atmospheric; wi th a wide var iety of possible nonthermal

modes (Thomas,1973a,b). Speculative-theoretical attempts to res t r ic t the struct-

ural changes under open-system models to only an outer-atmosphere are of two

variet ies: (l) those with only thermal mass-loss --- requir ing a coronal or igin;

(i i) those in i t iat ing and dr iv ing a nondisstpative mass-loss by l ine-f luxes in that

radiation f ield produced only by the photosphere --- which permits only an extend-

ed photosphere. We have documented the inabi l i ty of ei ther (i) or (ii) to represent

the empir ical-thermodynamic structure of any well-observed s te l lar atmosphere;

and reiterate that the open-system character must persist into the subatmosphere

and in ter ior (Thomas, 1983). So the stel lar-depth to which "open-hess" effects are

felt is controversial , for a l l those stars exhibit ing, observationally, mass-loss

open-system (Thomas,1973a). We sought to c la r i f y the problem by studying stars

showing, observationally, strongly-variable mass-outflow (Doazan et al, 1980).

The current decade of farUV observations couples wi th a century of visual ob-

servations of Be stars:(a) to conf irm strongly-variable photospheric mass-outflow

as dist inguishing them from normal-B stars; whi le (b) giving f i rm empir ical sup- port to Be models producing such variable mass-outflow by some variety of sub- atmospheric nonthermal structure (Doazan, 1982~ 1986). We doubt that the basic

nonthermal character of mass-outflow or ig in can di f fer between normal-B and Be

96

stars; in proof, we compare the empirical atmospheric structures and underlying

observations. In brief, the atmospheric structural ~ , up through the corona,

agrees for Be and normal-B. But the variable photospheric mass-outflow for Be

introduces: (a) var iabi l i ty in values of some thermodynamic parameters in each

region; (8) variable coronal opacity, corona] radiation-field, post-escape outflow-

acceleration; (7) variable pulsation, geometry, and mass-content of the transient-

ly observed cool envelope producing the Be and shell phases. Consequently, this

Be local environment acts as a variable-valve on mass-loss. Photospheric mass-

outflow of normal-B stars always measures the mass-loss; of Be, usually not.

Ih EMPIRICAL DISTINCTION BETWEEN Be AND NORMAL-B STARS: The hlst-

oric visual-spectral characteristics of" Be-stars l ie in a set of phenomena long-

associated with a photospheric mass-outflow producing an extended atmosphere.

That the mass-outflow is also variable is implied by these data, but variability is

histor ical ly treated as of secondary importance to extstance of mass-outflow. The

phenomenological set was f i rs t defined by the Be-spectrum: Ha emission, variab-

le in existance and in strength for a given star. Then i t was extended to include

variably-existing narow absorption-cores in low-ionized atoms: the "shell-spect-

rum". So long as observations were "snap-shots", each of these types of spectrum

was taken to define a different type of B-star. Long visual-spectral observational

sequences of a given star established that actually Be stars vary between 3 "phas-

es": B-normal, Be, B-shell: in time-scales incomparably-shorter than evolution-

ary. Such data delineated phase-change as the major Be-varJabtlity seen in the

visual spectrum. It was recognized that such observational changes reflect ther-

modynamic changes in the envelope, which imply var iabi l i ty in the mass-outflow

feeding it. But no observations were able to study,nor even to delineate, the t ran -

sit ion regions between photosphere and envelope. This missing information was

ersatzed by thermodynamically-incomplete speculation(cf Doazan summary, 1982)

FarUV observations: (i) delineate the transition-region between hot Be photo-

sphere and coot envelope as a chromosphere+corona (CO), each much hotter than

the photosphere, each part of an outward-accelerated mass-flow reaching outer-

coronal velocities that much-exceed those nbserved in the overlying Ha envelope;

(ii) exhibit such hot CC, and rapidly-expanding corona, as common to both normal

B and Be stars; (iii) distinguish Be From normal-B stars by strong variability of

these farUV features. That the immediate post-photosphere is heated, at least as

a consequence of the observed mass-outflow (Thomas, 1973b), is one part of that

thermodynamics ignored in classical Be-models. So farUV observations establish

that mass-outflow and post-photospheric heating do not distinguish Be From norm-

al-B; but the strong var iabi l i ty of these features, plus the existance and variabi l-

i ty of the visually-delineated Ha-envelope, do distinguish Be from normal-B.

97

Long sequences of only farUV observations of a given Be-star delineate the

var iab i l i t y character ist ics of these farUV-discovered regions. Long sequences of

simultaneous farUV + visual observations delineate the association+interaction

between photosphere, mass-outflow, CC, and cool envelope. None of these regions

nor mass-outflow can be modeled in isolation. All. these data are necessary to re-

place speculative ' modelin~ by a_thermodynamically:sound empir ic ism. Af ter we

determine what is necessary to represent the observations, we can ask why.

Likewise, i t is only from such decade-long, simultaneous, observational se-

quences that we have discovered a dist inct ion between normal-B and Be B-normal

phase stars. As a B-normal phase of (9 CrB(B6Ve) evolved: in a number of reson-

ance lines, sequentially according to ionization-level, the ~ l ine-width monotonical-

ly increased, accompanied by a secular r ise in the displacement of the l ine 's ab-

sorption minimum (Doazan, et ai, 1986b,c). These are the symptoms of a photo-

spheric mass-outflow increasing with t ime. Figs. t ,2 i l lus t rate such l ine-prof i le

var iabi l i ty , which impl ies mass-outflow var iab i l i t y in regions below the cool en-

velope: for 0 Orb during a B-normal phase; for the photosphertcally-hotter 59Cyg

(BI.blVe) as a B-normal, or weak B-shell, phase evolves into a Be phase.

A s im i l a r long-duration sequence of simultaneous farUV + visual observations

shows, for the f i rs t t ime, an apparent association between thermal state of, and

mass-outflow from, the photosphere (88 Her B6Ve;Doazan,et a l , t986d,e) . Visual

observations with poor time-resolution have long-shown a luminosity drop associ-

ated • with the growth of the cool enveIope producing a shell-phase. Speculatively,

some associated this with the envelope's increased absorption of a time-constant

photospheric luminosity. On the contrary, these simultaneous farUV+visual data,

when self-consistently analysed~ show the luminosi ty drop to preceed the s h e l f

spectrum. Apparently, this effect ref lects a decrease in photospheric T e as the

mass-outflow begins to increase. Indeed, as the shell-spectrum progressively

strengthens, we observe a luminosi ty-r ise. We can dist inguish envelope From pho-

tospheric effects only because, for the f i r s t t ime, we have simultaneous farUV +

visual data, which permits replacing speculation by empir ical guidance.

I:__[. Be-MODELING: Because a century of visual observations identif ied mass-

outflow as the pr imary Be-distinguishing characteristic, pre-farUV "theory" foc-

used on identifying any mechanism to produce such. Struve (193 t) suggested Be-

stars rotate at the c r i t i ca l velocity; hence produce an equatorial mass-loss and

discoidal-envelope. The idea was wholly-qual i tat ive; no algori thm for computing

mass-outflow size, nor latitude to which i t was confined, nor direction of outflow

has ever appeared. A!I its modelin[ ha s been ad hoc~ simply assumin[ values of.

these quantit ies , without demanflin~ thermodynamic consistency. To the h is tor ic

lack of any evidence for c r i t i ca l rotation; the lack of any idea how the mechanism

98

can produce var iabi l i ty; the presence of some normal-B s t a r s rotat ing as rapidly

as any Be --- one now adds the fa rUV evidence of some mass - lo s s in all s t a r s .

All these data negate s imple mass-outf low, and reac t iva te s t rong-var iabi l i ty of it

as the distinguishing Be-characteristic to be modeled: delineated as above, but

noting that "strong" is as yet unclear, Cf Ooazan's (1982) "gradualness".

From the results of a decade's simultaneous observations in farUV+visual,

we can now extend our p re l imina ry empir ica l models of normal-B and Be a tmo-

spheric structural patterns (Doazan, i 982;Thomas, i 983) .Pr imar i ly the extension clar i f ies the cool envelope found in Be but not normal-B. We focus on that prob- lem: given a variable photospheric mass-outflow for Be, but not for normal B, do

we obtain a cool-envelope uniquely for Be, and what is its cha rac te r? We ask not

cause of var iable outflow: only effect; our model can live with a lmos t any cause.

We model sequentially outward f rom a quasi- thermal photosphere -- defined as

a gas in thermal-hydros ta t ic and radia t ive equil ibria (HE,RE),with outflow-veloc-

ity U ~ q/3, q = [kTe/P] 1 /2 __ which is structurally the same for normal-B and

B e. Such l im i t on U ensures no departure from HE exceeds 10 %. Photosphere is the deepest region seen at any X; demanding i t be quasi-thermal at r < 1 for hot stars and log g ~4 gives an upper-l imit on mass-outflow M' [ = 4Trr2U#] < 10 -4, the observed value for Wolf-Rayet stars, the h is tor ic example of nonthermal ob-

jects. So for normal-B and Be alike, a quasi-thermal p~otosphere guarantees: (i)

that standard thermal models (HE,RE,U=0) give i ts p-distribution; (ii) U(r) lot- '

lows from this p(r) and a given M'; (i i i) any change in M' that retains quasi- thermal does not change p(r), only U(r); (iv) the f low is subthermic, effectively incompressible, so any changes M'( t) , U(t) propagate upward at q(r).

The chromosphere is defined as: nonRE; HE at the Te(r) fixed by' nonRE; and U ~ q/3. Under M'( t) , U is U(r, t) , propagating upward as q(r); p(r) is unchanged

The chromosphere and HE end where U > q/3: at the thermal-po!nt. Our tables

(Ooazan,1982) locate this point as f(M',g,Te); the effect of" M' predominates. The lower-corona is the "trans-sonic" region of the mass-outflow: U~q+e; e is

small, fluctuating about 0. The f low exhibits the instabi l i t ies long-observed in i ts aerodynamics/ball istics studies; both U,p fluctuate. When U accelerates > q, a

shock arises, energy dissipates, U decelerates < q, the cycle repeats. Thomas (t973a,1983) discussed the configuration's atmospheric role. Mean values are:

U(r) ~ q(Te),SO fixed by nonradiative-heating; with p(r,t) fixed by M'(t) , q(Te). I f

M'(t) occurs without Te(t), a U(t) in the photosphere+chromosphere is replaced by a p(t) in the lower-corona; the effect of M'(t) propagates upward as q(r). p(t)

produces a var iable CC optical thickness,rce,Which increases as M'(t) propagates

up. The change in CC radiative-acceleration propagates as £; so U(r,upper-corona)

increases before M'(t) reaches that region. The t ime- lag can be a few days.

99

The lower-corona ends in tile upper at the escape-point: U 2 = q2= GM/2resc" Our tables (Doazan, 1982) show, for Te(max) ~ 107K, a small chromosphere ( Ar/R <~ 0.01 ) and larger lower-corona ( Ar/R ~ 2-I0 ). In normal-B, the out-

Flow is continuously accelerated above the escape-point. We suggested a CC rad-

iative acceleration for both norma]-B and Be because of t-variations in Be --

ranging From 200 to 2000km/s in some Be. The above variable Tee can provide variable acceleration, independently of Te, accompanying M'(t). Nothing else sat- isfies the complete set of observations.

For Be-stars, the upper-corona ends in the Ha-envelope, within which U(r) ~< S0-200km/s; which begins at r / R ~ 3-5, extending to 10-20, all values varying.

This implies that collision with a pre-existing slower~ outflow decelerates the

post-escape, CC-accelerated outflow; which implies continuous existance of" some

mass in a slowly-moving cool envelope. It also implies epochs of negligible CC

radiative acceleration: hence lower Te(max) and/or rcc. Thus the Be-photospher-

ic variable mass-outflow produces an upper-coronal mass-outflow varying in p,U,

rcc; a variable nonradiative heating produces variable T e and Uesc; and the total-

ity forms a piston of variable mass, momentum, and energy which runs into an amorphous "local environment" of the Be-star. This ]ocal environment both puls-

ates and "valves" the mass-outf[ow originally produced in the photosphere and ac- celerated by the CC radiation field it causes to vary.A proper .aerody99m~c treat- ment of this "pulsation" and "mass-loss-valve" is today's .Be modeling problem.

Litt le has yet been done on it; we try: empirically-theoretically.

References: Doazan,V. 1982, B and Be Stars, NASA, SP-456; A.B. Underhiil ,V.Doazan

t986a, Be Modeling; IAU Coltoq.No.92;Physics of Be Stars, in pub]. Doazan,V.,Kuhi,L.V.,Thomas,R.N. t 980, Astrophys.J.Letters 235 L.20 Doazan,V.,Mar]borough,J.M.~Morossi,C.,Peters,G.-].,Rusconi,L.,Sedmak,G.,Stab

io,R.,Thomas,R.N.,Will is,A. 1986b, Astron.Astrophys. 158,1 Doazan,V.,Morossi,C.,Stalio,R.,Thomas,R.N. t 96c, Astron.Astrophys.in press

Doazan,V. ,Barytak,M., 1986d, Astron.Astrophys. t 59,65 Doazan,V.,Thomas,R.N.,Barylak,M., 1986e, Astron.Astrophys. 159,75

Struve,O., t 93 t , Astrophys.J. 73, 94 Thomas,R.N., i973a, IAU Syrup.49,3; WR Stars; ed.M.K.V.Bappu, J.Sahade

t 973b,Astron.Astrophys.,29, 297 t 983 ,Stellar Atmospheric Structure, NASA SP-47i

100

1 6 4 0 1644 1648 1 6 0 Z W A V E L E N O T H

~ 0

4,s

$.6 ̧

~ 2.7

~ 1.8

o,g

4,S

X

t~ >

_1 I1:

0,~

59 Cy~

WAV~ILENGTH(A)

. . . . . . . . ! I , ,

WAVELENGTH(A)

Fig. 2

Fig. i

101

Low Frequency Oscillations of Uniformly Rotating Stars and

a Possible Excitation Mechanism for Variable B Stars

Umin Lee and Hideyuki Saio

Department of Astronomy, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

I. Introduction and Summary

In order to search for the excitation mechanism of variable B star pulsations, we have conducted a study of low frequency nonradial oscillations in a uniformly ro- tating massive main sequence star with a convective core. The first intention of our study (Lee & Saio 1986a) was to examine Osaki's mechanism (1974), which was originally proposed for the excitation mechanism of B Cephei pulsations. Subse- quently, we have tried to make clear the general nature of low frequency oscilla- tions of rotating stars (Lee & Saio 1986b). In this paper, we summarize our present theoretical understanding on the properties of low frequency oscillations of rotat- ing stars.

Under the adiabatic assumption, we obtained purely real frequencies of convec- tive (inertial) modes, gravity modes, and rotational modes, and also complex fre- quencies of overstable convective modes. We found that these overstable convective modes in the core could penetrate into the envelope as a result of their resonance coupling with high order g modes in the envelope. In such a case, the amplitude of oscillations in the envelope is comparable with that in the convective core. The overstability of the mixed convective-g mode oscillations persists even when the

nonadiabatic effects are included. The frequency of such oscillations observed in an inertial frame would be a few times the rotation frequency, which is consistent with the observed periods of the variable B stars.

2. Equations

We assume the perturbed quantities are proportional to exp(iot+im%), where m is an integer and o is the frequency of oscillations observed in an inertial frame. For example, the linearized equations of momentum and continuity are given by

-(o+m~) 2 p~ + 2i(o+m~)pQx~ = -grad P' + 0' grad P - p grad @' (I) P

p' + d i v ( p ~ ) = 0 , ( 2 )

where primes (') and 6 denote the Eulerian and the Lagrangian perturbations, respec- tively, ~ is the Lagrangian displacement vector, and the other symbols have their usual meanings.

For a given index m, the displacement vector ~ and the perturbed scalar quan- tities 5' are assumed to be represented in the form (e.g., Zahn 1966; Berthomieu et al. 1978);

~Visiting Fellow, 1986-87, Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado, Boulder, Colorado 80309-0440.

102

= r I {erS£(r) + r) ~ ~ ] + ~ qml ~ ~e[H~ ( ~+Tz(r) sin 0~ ~

3 T£(r) ~ Y~(O exp(iot) , + ~[H (r) sin 0a~ ~]} ,~) (3) and

f' = ~ fi(r) Y~(8,~) exp(iot) , (4)

Where ~r' ~8' and e# are the unit vectors in the r-, 8-, and ~-directlons, respec- tlvely.

Substituting equations (3) and (4) into linearized equations such as (I) and (2), we have an infinite system of coupled linear differential equations. The sys- tem of the differential equations describes the two decoupled groups of modes, i.e., even modes and odd modes. Even modes are represented by a summation of terms as- soclated wi h the sphorlca harmonics of de ees =Iml ml+2 ImI+ ..... w lle odd modes are represented by a summation of terms assoc{ated with that of £=~ml+l , ImI*3 Ima+ ..... .umeri=al calculations are carried out by solving the finlte system of differential equations which are obtained by truncating the infinite SyStem leaving only the first two components associated with degrees £=ImI,Iml+2.

~:...Numerieal Results

The model star is a I0 M@ main sequence star with convective core ~M c = 3.25 M@; ~ V-?ad) = 1.0 × 10-3]. The physical parameters are (X,Z) = (0.70,0.02), log(L/L@) = • 731, log(Teff) = 4.412. The azimuthal index m of the spherical harmonics is set

equal to -2. The deformation of the equilibrium configuration due to rotation is neglected. The effect of rotation is taken into account only in the linearized equations, in which angular frequency of rotation R is treated as a free parameter. in this paper, we present the results only for even modes.

3-~!i~ Overstable convective modes

NUmerical results for even modes are summarized in Fig. i, in which eigenfrequencies ~(~o+m~) in the co-rotating frame are plotted as functions of the rotational fre- quency, R. These angular frequencies are normalized by using (GM/R3)I/2 (=2.78 x I0"~ s-l). Modes with m R > 0 are prograde modes, while modes with ~R < 0 are retro- grade modes. In the adiabatic analysis, if a complex frequency is an elgenvalue, its complex conjugate is also an eigenvalue. Therefore, if an eigenfrequency has a nonzero Im(~), there exists an unstable (overstable) mode. In Fig. I only positive Values of Im(~) are shown.

Modes which tend to a convective mode gn associated with £ = ml as ~ + 0 are labeled as Bn, while modes which tend to g; modes with £ = Iml~2'as ~ + 0 are labeled as An, where the subscript n denotes the number of nodes of an eigenfunction Which appear in the convective core when ~ = 0. Beyond a critical value of R for a given n, B n modes come to have a mixed mode character, i.e., elgenfunctions have large amplitudes both in the convective core and in the envelope. This phenomenon • ay be regarded as a penetration of overstable convective modes into the envelope as a result of resonant coupling between the overstable convective mode and a high Order g+ mode in the envelope (cf. Osaki 1974). (Such a phenomenon never appears in A n modes.) In fact, the eigenfunctions have a large number of nodes in the en- Velope which amount to about 80 in our calculations. An example for such eigen- ~unctions is illustrated in Fig. 2, in which the radial displacement IrSI of the 81 mode at ~ = 0.162 is shown as a function of log P, For this mode, =R = 3.31 x

By performing a full nonadiabatic analysis, we confirmed that the overstability of these modes persists even when the nonadiabatic effects are included.

103

O OI 0.2 ~ O~ O~

Fig. I. Eigenfrequencies in the frame rotating with ~ for even modes with m = -2 are plotted as functions of the ro- tation frequency. The abscissa is the rotation frequency and the ordinate of the upper (lower) frame is the real (imaginary) part of the eigenfrequency

S /

16.7 i4 , ( ] H . 3 (] .5 5-8 3.1

LOS P

Fig. 2. The radial displacement !r S I of the even B I mode with m = -2 f r ~ = 0.162 is shown as a function of log P. The solid curves and the broken curves

the components with £ = Iml and indicate £ = Iml+2, respectively. The short' 'ver- tical line attached to the upper hori- zontal axis shows the location of the convective core

3.2. Gravity modes and rotational modes

Numerical results are summarized in Fig. 3, in which purely real eigenfrequencies are plotted as functions of ~. The way of mode labeling is as follows: modes which tend to those of £ = Iml, and £ = Iml+2 as ~ + 0 are, respectively, designated as g_ and gn modes (r and ~ modes), where the subscript n denotes the number of node~ of an elgenfunc~lion which appear in the envelope.

Some interesting phenomena are: (a) avoided crossings between gravity modes gn and gk (Fig. 4); (b) resonance couplings between gravity modes and convective (iner- tial) modes; (c) resonance couplings between rotational modes, convective (inertial) modes; and (d) the frequencies of gn modes tend £o const, x ~ as n + ~.

Detailed discussions on these low frequency oscillations are given by Lee & Salo (198bb) who used an asymptotic method.

References

Berthomieu, G., et al. 1978, Astr. Astrophys., 70, 597. Lee, U. & Saio, B. 1986a, Mon. Not. R. astr. Soc., 221, 365. Lee~ U. & Saio, H. 1986b, Mon. Not. R. astr. Soc., in press. Osaki, ¥. 1974, ~trophys. J., 189, 469. Zahn, J. P. 1966, Ann. Astrophys., 29, 313.

104

I ..... /

/

~ /" g4o

0.05 /

-0.00

-01

-O J!

- /

g,o

\L , , \g~o

42~ t ' , , \ ~,, , o.I o~ 0.3

Flg. 3. Eigenfrequencies in the co-rotating frame for even modes with m = -2 are plotted as functions of the rotation frequency. The abscissa is the rotation frequency and the ordinate is the elgenfrequency. Eigenfre- quencies of gravity modes (gn,gn) and rota- tional modes (r_,~.) are presented. The dashed parts of"th~ curves represent modes which are difficult to calculate because of their coupling with other modes. The thin dashed lines labeled as L 2 and L 4 are asymp- totic lines for the gn modes and the gn modes, respectively

01354

0 ~352

0029 0030 0.031 0086 0087

(b) ' I / '

/

7 < - . 2 /---<

gso

0085

),1278

31276

3A274

Fig. 4. Large-scale views of selected regions of Flg. 1 to show the avoided crossings (a) between g20 and g~0 and (b) between g20 and g60

105

OSCILLATIONS IN Y~SSIVE SPINNING STARS

D. Narasimha and S.R. Sreenivasan Department of Physics

The University of Calgary

Many early-type supergiants are known to exhibit variability and the variation

in luminosity is observed to be generally small. Also, ~arge variable macroturbulence

has been observed in many of them, as well, (cf. de Jager et al., ]984) although no

systematic periodicity in radial velocity can be definitely identified in most cases.

However, evidence for the existence of both prograde modes (e.g., E Persei, by

Smith, 1985) and retrograde modes (e.g., H Con and y Ara, by Baade, 1984) has been

cited. Smith and Ebbets (1981) have, in addition, argued that multiperiod non-radial

oscillations are excited in the B-type supergiant 0-Leo. In the present work, we

examine the stability characteristics of oscillatory modes in a differentially

rotating stellar model of ZAMS mass 24 M@ during its main sequence life-time as well

as its immediate post-main sequence phase.

i. The Equilibrium Model

A stellar model of ZAMS mass 24 M~, with composition X = 0.7, Z = 0.03 and a

surface rotation velocity of 112 km/sec was computed. The following rotational

velocity profile was adopted for the zero age model. Rotational frequency f~ =

constant in the convective core, specific angular momentum ~r 2 = constant in the

radiative envelope and a parabolic fit for ~ over one pressure scale height distance

adjacent to the convective core. The effects of mass loss and semiconvection were

included during the evolution although semlconvection has little relevance in the

presence of mass loss. Angular momentum is assumed to be redistributed due to

viscosity. The main features of the evolutionary models have been given in Narasimha

and Sreenivasan (1986).

Linear non-adiatic inviscid oscillatory modes were studied for two models;

during the core hydrogen burning phase and at the commencement of core helium burning.

The model characteristics are given below:

MODEL I MODEL II

Age 5.6 x 106 yrs 6,7 x 106 yrs Mass 21.3 M@ 20.8 M@ Luminosity 8.2 x 104 L 0 1.3 x 105 L@

Tef f 30140 K 9240 K Central hydrogen abundance X c = .2 0

106

2. Linear Stability Analysis

The governing equations are the usual hydrodynamic equations for the conservation

of mass, momentum and energy applicable to inviscid, thermally conducting fluids. We

treat the radiative flux in the diffusion approximation. The equation of state is

that for a perfect gas including the effects of ionization and radiation pressure.

Since the centrifugal force is small compared to gravity,~we assume a spherical

geometry for the equilibrium model. The Lagrangian perturbation in the hydrodynamic

equations and the Eulerean perturbations in the radiative flux and the gravitational

potential were considered to analyze the acoustic and gravity modes, in terms of the

Spherical harmonics. The effect of convention was not considered in the perturbed

equations and the ~ cos 8 term in the Coriolis force was neglected so that the normal

mode analysis is straightforward. Such an approach is reasonable in regions where

lq/m] << 1 (~ is the angular velocity and m is the frequency of the mode) since our

main aim is to study the overstability of the modes.

These equations, with the appropriate boundary conditions (see Unno et al.,

1979), were solved by a finite-difference method as an algebraic eigenvalue problem

for the complex frequency, m. The first order corrections were computed to test the

reliability of the eigenvalues following Antia (1979). The numerical computations

Were carried out on CYBER 175 and CDC 205 at The University of Calgary.

3. Numerical Results and Discussion

We have investigated the oscillatory modes trapped in the stellar model, prograde,

retrograde and standing waves for a range of horizontal harmonic numbers, ~ < 16 for

model I and ~ < 8 for model II.

(a) Main Sequence Model: All radial modes studied turned out to be stable. This is

not surprising because for stars in this mass range and evolutionary status, neither

the K-mechanism nor the E-mechanism is strong. Likewise, among the axisymmetric non-

radial oscillations, all the p-modes turned out to be stable. However, the g-modes

for ~ in the interval of 4 to i0 and having periods in the range of 9 to 14 hrs (g3 -

g6) were found to be marginally overstable although their stability coefficients

(N = mReal/~) are not larger than 10 -6 . These modes are trapped in the steep ~-

gradient zone adjacent to the convective core and they have substantial amplitude

Only in those deep layers where the gradient of the rotational frequency is also the

largest. Shibahashi (1980) has argued that these overstable g-modes could be

indirectly responsible for redistribution of angular momentum in the star through

mixing.

Among the prograde modes (m < 0), the overstable and the stable modes coexist

throughout the spectrum though most of the modes are stable. The overstability of

the few prograde modes, however, is caused entirely by differential rotation.

Although our numerical scheme is not very reliable in analyzing the stability

107

characteristics of prograde modes, we can demonstrate the overstability of adiabatic

prograde modes by appealing to formulation given by Lynden-Bell and Ostriker (1967).

As the strength of differential rotation increases, the modes trapped near the region

where (Vad - V) ~ 0 produce a stable and an overstable component. The non-adiabatic

effects do not decrease the stability coefficient of these modes appreciably because

the modes do not propagate very far outwards. However, when we impose the condition

that IV X Vlmax < 2 ~ for the overstable prograde modes, their velocity amplitude at

the surface becomes very small. We believe, therefore, that observationally, they

cannot easily be detected. Nevertheless, as pointed out by Ando (1981) earlier, these

modes could be important in transporting angular momenum outwards from the core. All

the retrograde modes studied were found to be stable.

Our results indicate that no single oscillatory mode is likely to be dominant in

massive stars during the core hydrogen burning phase. Further, in view of the very

small growth rate, the overstable g-modes individually would have very little

observational significance; but non-linear interaction between the modes, described

by Perdang (1983) could be important for these waves, because they have nearly equal

periods and are excited in the same region of the star. The probable outcome of the

interactions between these marginally overstable g-modes could be a quasiperiodic

variability, consisting of a high frequency component having a period of a few hours

(~ ~ <~>) and a superimposed long-period motion (m ~ A~) over a time scale of a few

days to months. It should be noted that modes energized by stellar winds cannot be

neglected in these stars (Narasimha and Chitre, 1986). These waves would behave

like local disturbances with a comparatively small vertical wavelength.

(b) Post-Main Sequence Model: The radial modes were found to be highly stable due to

radiative damping in the core as well as in the outer envelope. There are two classes

of non-radial modes - the modes trapped in the ~-gradient zone in the interior and

those trapped in the helium ionization zone in the envelope. All of the axisymmetric

and retrograde modes were found to be stable. But, some of the prograde modes are

still overstable because the strength of differential rotation increases as the core

contracts and the envelope expands. However, we find that it is unlikely that any of

the observed oscillations in this phase of evolution are energized in the interior

of the model.

4. Summary and Conclusions

In the core-hydrogen burning phase of the supergiant models, gravity modes having

a narrow range of period and low horizontal harmonic number are marginally overstable.

Their cumulative effect might be observable since their narrow frequency range and

common trapping zone indicate the possibility of non-linear interaction. An

observational test for the existence of such oscillatory waves will be the isolation

of a short period component with a slowly varying part in the quasiperiodic velocity

108

profile of early-type supergiants. A detailed version of the present investigation

Will be published elsewhere.

The work was supported by an NSERC research grant (to SRS) and The University

of Calgary.

References

Ando, H., 1981. Mon. Not. Roy. Astr. Soc. 19__7, 1139. Antia, H.M., 1979. J. Comp. Phys, 30, 283. Baade, D., 1984. Astron. Astrophys. 135, I01. de Jager, C., Mulder, P.S. and Kondo, Y., 1984. Astron. Astrophys. __141, 304. Lynden-Bell, D. and Ostriker, J.P., 1967. Mon. Not. Roy. Astr. Soc. 136, 293. Narasimha, D. and Chitre, S.M., 1986. (Submitted to Astrophys. J.). Narasimha, D. and Sreenivasan, S.R., 1986. Ball. Am, Astr. Soc. 1~7, No. 4, (Houston

Mtg. AAS), 895. ~erdang, J., 1983. Sol. Phys. 82, 297. Shibahashi, H,, 1980. Pub. Astron. Soc. Japan 32, 341. Smith, M.A., 1985. Astrophys. J. 288, 266. Smith, M.A. and Ebbets, D., 1981, Astrophys. J. 247, 158. Unno, W., Osaki, Y., Ando, H. and Shibahashi, H., 1979. Non-Radial Oscillations

of Stars, University of Tokyo Press, p. 198.

109

Rapidly Oscillating Ap Stars and Delta Scuti Variables

H]romoto Shibahashi Department of Astronomy, University of Tokyo,

Bunkyo-ku, Tokyo 113, Japan

Abstract

The discovery of the rapidly oscillating Ap stars has raised many questions con- cerning the relationship among ~ Sct variables, Ap stars, and other apparent non-varying stars in the 6 Set instability strip. In this paper, the present status of our under- standing of the rapidly oscillating Ap stars and the % Set stars is reviewed, but the former class is discussed more in detail.

i. Introduction

The pulsating variables in the lower portion of the Cepheld instability strip near the main-sequence are known as the Delta Scuti variables. Since this class of variables was first distinguished from the RR Lyrae stars by Eggen (1956), many stars have been counted as the members. Breger (1979) listed 129 stars in his review and the number is still increasing as a result of the succeeding discovery of new members. Their periods are in the range of 30 min - 4.7 h (typically ~2h), and the amplitudes a ~e mostly smaller than 0.3 mag. The interesting thing is that not all the stars in this region of the instability strip actually are observed varying and only about one third of the stars in this region are 6 Sct variables. I1owever, most of these variables pul" sate with small amplitudes arld the histogram of amplitudes observed in 6 Sct stars shoWS that the number of 6 Sct variables increases nearly exponentially with decreasing amplitude. Therefores we cannot exclude the possibility that many of apparent non- varying stars in the 6 Sct-instability strip vary but with undetectable amplitudes.

All the Ap stars have been included in the apparent non-pulsating stars. The variability in Ap stars has been searched but until quite recently explicit evidence fo~ the pulsation in Ap stars has not been discovered. Therefore, the chemical peculiarity and the pulsation have been empirically regarded as exclusive Bach other. The most promising, conventional, theoretical explanation for their exclusiveness was as folloWS: In the case of a slowly rotating A-type star, the material diffusion is induced in its atmosphere so that a peculiarity in the spectrum appears and the star is recognized aS an Ap star. The helium abundance in its envelope is too little as a result of depleti0o to make the star pulsate by means of the x-mechanism of helium. On the other hand, is the case of a fast rotating A-type star, the meridional circulation induced by the rot~" tion inhibits the material diffusion and there is enough helium in the envelope to ma ke the star pulsaLe. This hypothesis has seemed to explain qualitatively well the ex- clusiveness of Ap stars and the 6 Sct stars.

However, the basic conception of the exclusiveness of the chemical peculiarity and the pulsation was recently broken by the discovery of the rapid pulsations in some of ~ stars (Kurtz ]982). These stars were named the rapidly oscillating Ap stars and so far eleven stars have been counted as the members. They are cool, magnetic Ap stars of St" CrEu-type pulsating with short periods in the range of 4 - 15 min and small amplitudes

Am~10 mag. Detailed description on each of these members is available in Kurtz'S review (1986a, b) (see also Weiss 1986). Their periods are much shorter than those o f6 Sct variables, although these two groups overlap each other on the fiR-diagram. The ver~ high frequencies of the oscillations in these Ap stars are most likely to be explained by very high overtone p-mode oscillations, while the 6 Sct variables pulsate in low o r~ der modes. One of the most noticeable characteristics of the rapid pulsations in Ap stars is that the amplitudes of the oscillations are modulated with the same period and phase as the magnetic strength variation. This fact suggests that the excitation mechanism of the oscillations in Ap stars is somehow related to the strong magnetic fields. As above, the characteristics of the rapid oscillations in Ap stars are quite different from those of the 6 Sct variables. However, all the rapidly oscillating Ap

stars so far discovered are in the @ Sct instability strip, and no rapid oscillation has been hitherto detected in Ap stars outside of it. Therefore, there may be some physical connection between pulsations of ~ Sct stars and those of Ap stars despite

112

their different characteristics. The discovery of the rapidly oscillating Ap stars has raised many questions: What

is the relationship between these stars and the @Sct variables? What is the relation- ship between the pulsation and the anomalous chemical abundances? Why do these stars

Pulsate in such high overtones? Why are the pulsations aligned with the magnetic axis? What is the excitation mechanism of the rapid oscillations? Are their oscillations use- ful to the asteroseismological approach to Ap stars? Most of these questions have not Yet been answered.

In this paper, the present status of our understanding of the rapidly oscillating Ap Stars and the 6 Scuti stars is reviewed, but the former class is discussed more in detail. There are many nice reviews on the 6 Sct stars or A type stars in general Written before the discovery of the rapidly oscillating Ap stars and the readers should consult them (e.g., Breger 1979, Cox 1983, Wolff 1983). The mode identification of the rapid oscillations in Ap stars is first reviewed in section 2. Nonradial pulsations in a rotating, magnetic star are discussed in section 3 and the asteroseismological approach

to Ap stars is discussed in section 4. The excitation mechanisms of the rapid oscilla- tions are discussed in section 5. Theoretical investigations on the pulsations of the Set variables are reviewed in section 6.

2.' Mode Identification of Rapid Oscillations in Ap Stars Pulsations in many of ~ Sct variables consist of multiple modes. Although the

Periods are of the same order of the radial fundamental mode, the modes of oscillations are not always identified and sometimes even whether the modes are radial or nonradial and whether they are p-modes or g-modes is not yet certain. On the other hand~ the rapid oscillations in Ap stars are quite naturally interpreted as p-modes, because the frequencies are much shorter than that of the radial fundamental mode.

As is well known, a normal mode of oscillation Js specified by three integers (n, £, m), where Z is the degree, m is the azimuthal order, and n is the radial order. As for the rapid oscillations in Ap stars, among these quantum numbers, the degree £ and the azimuthal order m are determined as follows: The amp]itudes of rapid oscillations in Some Ap stars are modulated with the rotation period of the star in the sense that the amplitudes are correlated with the phase of the magnetic strength. In order to explain this character, Kurtz (1982) has proposed a model called the oblique pulsator model, in Which the observed oscillations are interpreted as zonal (m = 0) nonradial oscillations of low degree ( £ = I and/or 2) whose symmetry axes are aligned to the (dipole) magnetic axis but oblique to the rotation axis of the star. As the star rotates, the amplitudes of oscillations are modulated in parallel with the magnetic strength. The conception of the oblique magnetic axis is consistent with the oblique rotator model for magnetic Ap Stars, which is now widely accepted to explain the variation in the magnetic field Strength with the stellar rotation. The modulation is dependent on the geometric con- figuration as well as the degree Z of the mode, that is, the angle between the magnetic ~Xis and the rotation axis, ~, and the angle between the rotation axis and line of Sight, i. Therefore, from the analysis of pulsations, we can infer these angles. A fine example is HR 3831 (HD 83368), in which six frequencies are obtained as two sets of triplet (Kurtz 1982, 1986a; Kurtz and Shibahashi 1986). Kurtz (1982) suggested that the low frequency triplet is due to £ = 1 oblique pulsation and that the high frequency triplet is due to Z = 2 oblique pulsation. That led to the prediction of i = 86 n and $ = 36°, and as a consequence, the polarity of the observed magnetic field has been ex- Dected to reverse. Also at the quadrature, the phase of the pulsation has been expected to he shifted by ~. The magnetic polarity reversing was later confirmed by Thompson {1983), and the phase jump was confirmed by Kurtz (1986a) (see also Kurtz and Shibahashi 1986). Good accordance of the observations with the predictions for this star supports the oblique pulsator model and thus the pulsation modes are now regarded as £ = l and/or 2 D-modes which are axisymmetric (m = 0) with respect to the magnetic axis.

An alternative explanation is Mathys' (1985) spotted pulsator model. In this model, a star is supposed to be pulsating in a single eigenmode whose symmetry axis coincides with the rotation axis of the star, but an inhomogeneity in the stellar sur- face related to the oblique magnetic field makes the observed amplitude of the oscilla- tion vary with the stellar rotation. Though this model can phenomenologically explain the observed properties, the theoretical treatment of eigenoseillations of a rotating, ~agnetic star supports the oblique pulsator model rather than the spotted pulsator model (Sziembowski and Goode 1985, 1986; see also next section).

As for the determination of the radial order n, since the oscillation modes seem to

113

be very high overtones with n >> I, the asymptotic expression for eigenfrequencies is useful. According to the asymptotic theory of oscillations (Tassoul 1980), the angular eigenfrequency m of the mode with the radial order n and the low degree~ (n >>~i) n,~ ~s , to flrst order, given by

where

~n,~ = 2~0 (n + ~/2 + s) , (1)

s R

~0 ~ [ 2 c -I dr ]-i , (2)

0 E is a constant of the equilibrium model and c(r) denotes the sound velocity. Equation (i) means that frequencies , v ~ m/2~, of p-modes with even and odd ~ alternate with a separation of v0/2. Detailed power spectrum analyses have revealed that some of rapidly oscillating Ap stars are pulsating in several modes with uniformly spaced frequencies (Kurtz and Seeman 1983). The observed frequency spacing (e.g., ~33 ~Hz for HD 24712) is consistent with the theoretical values of ~0/2 for stellar models with M s 2M o in the main-sequence stage (Shibahashi 1984, Shibahashi and Saio 1985, Gabriel et al. 1985), which indicates that the observed oscillations are an alternation of even and odd degree p-modes. The odd modes are considered to belong to ~ = 1 since they have triplet fine structure. The combination of the asymptotic formula equation (i), and ob" served frequencies yields n = 40 for HD 24712 if ~ = I. A~ for the even degree modes appearing in the middle of ~=i modes in the periodgram, the degree is likely to be £ = 0 or 2, otherwise the stellar surface is divided into small, many regions oscillating in different phases and the contribution of each region is canceled by others so that the

total amplitude of the variability of the star is too small to be detected. According to the theoretical calculation of eigenfrequencies of the nonmagnetic A-type stars, the mean separation of ~ ~ - m - is slightly smaller than that of m -~ ~ - ~ . while the

n~± 0~ n~l u nt± mean of ~ . - ~ ~ ~ is sllghtly larger than that of ~n 2 - ~n i" ~omparison of this

n ]. n--ltL " " ~7 • result wit~ the observed frequency distrlbutlon for HD 24 12 then suggests that the even modes are likely to i: 0 rather than ~= 2. However, the amplitudes of these modes are modulated with the stellar rotation and this fact indicates that these modes are non- radial ones (Kurtz, Schneider, and Weiss 1985), i.e., ~ = 2. Probably, the perturbation of eigenfrequency due to the magnetic field must be taken into account in the theoreti- cal calculation of eigenfrequencies in order to solve this contradiction.

7n most of rapidly oscillating Ap stars, the number of detected modes has been several. Matthews et al. (1986), however, recently reported the detection of some ten modes in HD 60435.

3. Nonradial Pulsations of a Rotating Magnetic Star

Kurtz's (1982) oblique pulsator model was generalized by Dziembowski and Goode (1985, 1986), who took account of both the oblique magnetic field and advection to for- mulate the pulsations of a rotating magnetic star as an eigenvalue problem. According to their formula, if a single mode is excited, (2~+l)-frequency components are observed

as a fine structure in the power spectrum and their relative amplitudes are not equal each other but dependent on the rotation and the magnetic field of the star. This result leads to a possibility of using the fine structures of oscillation frequencies a~ diagnosis of the internal magnetic field of Ap stars. In this section, we discuss non- radial pulsations of a rotating, magnetic star following Dziembowski and Goode (1985, 1986) and Kurtz and Shibahashi (1986).

In the case of a non-rotating, nonimagnetic star, its eigenfunctions are repre- sented in terms of spherical harmonics Y~[8, }), in which (8, %) are associated with any axis since the star is spherically symmetric and hence it has no specific preferential direction. The introduction of the (dipole-like) magnetic axis settles the axis of pulsations, and the eigenfunctions are now approximately given by

Y~ (eB, *B) exp [ i (~(0) + ~ ) t ] , (3)

where O~, = 0 is the magnetic axis, (0) denotes the frequency in the absence of magnetic

~ d i ~ ~i~i~ ~n ~ ~ ~ r ~ t ~ n a ~ ~nt~ magnetic field. Let us now also o ] t the magnetic field. We assume here

for simplicity that the rotation is uniform and that the effect of the magnetic field oo the oscillations dominates that of the Coriolis force. Then, a form of eigenmodes

114

labeled by m (m = -Z .... , ~) is given by (Kurtz and Shibahashi 19867

[ Y~ (SB, CB) + ~jm mag 7 ~l)mag J=-Z ~Jl ( ) (1)mag

× exp [ i (~.0. + w + mC~ cos6) t ] , (4) m I where

(~) ~(.~)(~) = [ k d (~) (6) d (~) (5) 3m jk mk '

k=-~ Z" means the summation over j except for j = fm, ~ denotes the rotational frequency, and C is the rotational splitting constant which,~ depends on the equilibrium model and the mode under consideration. The matrix {d~X~)(8)} is composed of the following elements

d (B) = Y£ (O R, ~R ) (6)

where {OR, ~R) are the polar angles measured with respect to the rotation axis and B denotes the angle between the ro[ational axis and the magnetic axis. It should be remarked here that the matrix{~. )(~7} is a t~diagonal matrix This fact means that a ]m " normal mode Y~(8 B, cB ) in the aSsence of rotation is now modified by the weak Coriolis for~e only to'have two extra components of ym-i (-£< m+ 1 < £) The explicit forms of <~ ~ - _ . ~jm (8) for Z = I, 2, and 3 are seen in Shibahashi (1986). As the star rotates, the aspect of the eigenoscillation varies and the observable luminosity variation due to a single mode represented by equation (4) is given by (see Kurtz and Shibahashi 1986)

hL/L = m, ~ ~(-i) {d (8) + C~I ~ ~(£)(8) m ~o =- k=_£ km (I) g (°i) ag d(-)(i)

Iml -elkl

where i is the angle between the rotation axis and the line of sight. Equation (77 means that (2i+l)-frequency components (m' = -i, ..., Z) with spacing equal to ~ are ob- served as a result of the rotational modulation for a single eigenmode and that the amplitudes of components are not equal each other (Dziembowski and Goode 1985).

Since, as described in the previous section, the oscillations in Ap stars are well exnlained by m = 0 modes, we restrict ourselves to consider the case of m = 0. Let A(] )

= " " . ~ ( 0 ) + m~l)mag + m'~ denote the amplitude corresponding to a frequency component ~,. = Equation (7) leads to simple relations among the amplitudes A~) :

.(£), A~£) (£) d ( l ) , . , 3(£) ~(Z) (A (~)m, + ~-m ') / = 2 d0m,(6) m,0 tz) / ~00 (6) u00 (i) (8)

and

( A ( ~ ~ - A(~ !) / (A(£') + - m A(£~)= m ' C ~ - m / (~ l )mag- m~l)mag) (9)

Equation (8) means that the ratio of the summation of the amplitudes of ±m'-components to the amplitude of the central component provides information about the geometrical configuration of the star. Equation (9) means that the relative difference of the amplitudes of im'-components gives a ratio between the effects of the rotation and of the magnetism on the oscillation. By using these formula, if we obtain fine structures in the periodgram, we can compare the relative importance of the effects of the magnetic field and of the rotation on the oscillation and determine the geometric configuration

of the star. It may be instructive to consider the other extreme case, that is, a rapidly rotat-

ing star with a weak magnetic field, although in the rapidly oscillating Ap stars so far discovered the magnetic effect on oscillations seems to dominate over that of the Coriolis force. In this ease, the eigenfunctions are approximately given by

[ Y~(8R, ~R)+ ~ ' (k~.)(6)dkm I~I J=-£ kT_ {d (£>(8>" mag}/(m-j>C~×Y~(@R, }R> ] x

exp [ i{{ ~(0)+mC~+ [£[ { (~) (nmag x =_ d km ( 6 ) }2 ~lk l }} t ] , ( ~ 0 )

w h e r e g ' means t h e s u m m a t i o n o v e r j e x c e p t f o r j = m. The o b s e r v a b l e l u m i n o s i t y v a r i a - t i o n d u e t o a s i n g l e mode l a b e l e d b y m i s g i v e n b y

115

m' £ d(£)to~d(~)~, (1)magl = m E dm,0(i ) x AL/L [ C-l) {6=,+ ~ ,D, km,,mJWlk I ,(m-m')C~} (£)

m'=-~ k -£

}2 m(1)mag] × cos [{{ ( 0 ) +mC~+ ~ [ { d ( ~ ) ( ~ ) - m ' ~ } } t ] . (1]) k = _ ~ km [ k l

Since IC£I >> ~(1)mag in this case, the amplitudes of the frequency components other than ~ = m (0) + m(l-C)~ + E[{d(£)(~)}2~tk}l)mag] are small and hence the rotational

l | modulation is not conspicuous.

4. Asteroseismology of Ap Stars The discovery of several eigenmodes (differing in n and £) in individual Ap stars

prompts us to develop a field of research called asteroseismology, in which we may probe the stellar internal structure using oscillations. Such seismological studies have been quite successful as for the Sun, of which as many as 107 eigenmodes are identified. For example, the sound velocity distribution in the solar interior can be well inferred from the oscillation data. The number of modes so far observed in an individual Ap star or 6 Sct star is much smaller than the solar case, and hence the seismological approach to these stars is much more difficult than to the Sun. Nevertheless, if the developments in observations in future enables us to identify many eiged~nodes even with only 0 ~ Z 4 in an individual star, we will be able to infer the internal structure of these stars by applying some useful techniques used in the helioseismology. The unique aspect of the asteroseismology based on the rapid oscillations in Ap stars is the possibility of inferring the internal magnetism in these stars by applying equation (9), which tells us the relative importance of the effects of the magnetic field on the oscillations and of the rotation. Table 1 gives some results of the application of equations (8) and (9) to HD 6532, HD 60435, and HD 83368° The oscillations in these stars are identified as the dipole (~ = i) modes, and in the case of £ = i, the right hand side of equation (8) is reduced to tanStani. The second column of table gives tan~tani thus obtained and the third column gives C~/[{0} i)mag - ~l)mag] derived from equation (9). The fourth and fifth columns give the rotational periods and the surface magnetic field strengths ob- tained from other independent observations. The magnetic field strengths of HD 6532 and 60435 have not yet been measured.

So far we have not specified a form of ~mag Accordingt1~ to Dziembowski and Goode

(1985, 1986), in the case of a dipole magneti~ field, wI~[mag is given by

(1)mag , (12) elm I = [£(£ + I) - 3m 2 ] / [4£(£ + i) - 3 ] K mag

frequency component A(l~'-m Table I shows that this is the case for HD 653~, 60435, and 83368. Uslng equat!on (12) and the observed values of C[2/[~I l)mag - ~0 (ljmag] and Prot listed in table i, we obtain the values of K mag by setting C = 10 -2 (Shibahashi and saio 1985). The values of K mag thus obtained are listed in the sixth column of table i, and

they provide us a measure of the internal magnetic fields in the Ap stars. Though the directly observable magnetic field strength, He, gives us the magnetic field strength a~ the photosphere, the information provided from the oscillations gives us a field strength somehow averaged with eigenfunctions in the stellar interior. A lot of data

(1)mag (1)mag sets e I - ~0 would provide us an integral equation with an unknown function concerning the internal magnetic field, and therefore, in principle, the internal mag" netic field could be inferred by solving the integral equation.

Table I. Application of equations (8) and (9)

C£ He Reference HD tanBtani ~l)mag_ w~l)mag Prot Kmag

6532 60435 83368

2.14 - 0.295 1.78d 2x10 -6 s -I Kurtz and Kreidl 1.3 - 0.25 7.66 6x10 -7 Matthews et al. 9.65 - 0.i0 2.85 -700~+700 G 4x10 -6 Kurtz and ShibahaS 5

116

Since a magnetic field is regarded as a very important factor to the physics of Ap Stars, understanding of magnetic fields through the asteroseismological studies is highly desirable.

5. Excitation Mechanism for the Rapid Oscillations The pulsations in the Delta Scuti variables are excited primarily by the <-mech-

anism working in the helium ionization zone, as is true in the Cepheids, which occupy the upper portion of the instability strip. In fact, linear stability analyses of models of 6 Sct stars show that models with normal helium abundances are unstable against the radial, fundamental mode, the following several radial overtones, and many nonradial modes. The problems to be solved are, rather, why most of the unstable modes are not observed and how only a few observed modes overcome the others.

On the other hand, as for the rapid oscillations in Ap stars, the situation is quite different. We have to answer both why these stars do not pulsate in the modes of~ Sot stars and why they pulsate as they do, and these problems still remain to be solved. Let us first consider the possibility of the K-mechanism. A model in the ~ Sot instability strip with a normal helium abundance is pulsationally unstable against some OVertones of p-modes. The highest overtone among the radial unstable modes depends on the treatment of convection. It is 7H-mode in the case of calculations by Stellingwerf (1979) and Lee (1985b) whose period is still too long to be responsible for the observed rapid oscillations, but it is the 15-th overtone in the case of Dolez and Gough (1982) and its period is 15 min. Until the rapidly oscillating Ap stars were discovered, it has been thought the diffusion process occurring in Ap stars lets helium sink down and hence the K-mechanism does not work to excite any modes (Cox et al. 1979). In fact, Dolez and Gough (1982) demonstrated that even the most unstable mode mentioned above be- Comes stabilized in the hellum-depleted atmosphere. However, the diffusion process of chemical elements is supposed to occur along the magnetic axis, resulting in chemical Stratification which is not spherically symmetric but axisymmetric with respect to the magnetic axis. Therefore overstability due to K-mechanism, if it works, is likely to induce oblique pulsation, because the <-mechanism is sensitive to the chemical stratification. Recently, Dolez et al. (1986) supposed that helium accumulates in some appropriate depths in the magnetic polar region as a result of the balance between the diffusion process and the stellar wind, and demonstrated some short period pulsations are excited by the <-mechanism. There may be two problems to be studied further~ one of which is to construct a reasonable equilibrium model with such an inhomogeneous en- Velope and the other is the boundary conditions for the calculation of pulsations.

Dziembowski (1984) and Dziembowski and Goode (1985) considered that the magnetic field plays an important role in the nonadiabatic effect through the distortion of the Star. They derived the work integral W by taking account of the distortion in a form of

W= I [wo + w2 P2 (cos@)] IY~I 2 sin@ de , (13)

where the second term in the square bracket represents the effect of the distortion while the first term gives the usual work integral. Even if the radial modes are Stable, the dipole, axisymmetric modes can be selectively excited through the distortion of the configuration. Their suggestion seems to explain naturally the coincidence of the pulsation axis with the magnetic axis, the degree Z and the azimuthal order m of the Observed modes. Constructing an appropriate equilibrium model and the numerical cal- culation of pulsations with proper boundary conditions are desirable.

Another effect of the magnetic field has been considered as the excitation ~echanism of the rapid oscillations by Shibahashi (1983) and Cox(1984). In the presence of strong magnetic fields, the ordinary convective instability is suppressed even in a SUperadiabatic layer, but it manifests itself as overstable convection, that is, oscil- latory motion with growing amplitudes. In the case of a magnetic A-type star, such OVerstable convection manifests mainly in the magnetic polar region. Based on the local analysis and the Boussinesq approximation, Shibahashi (1983) estimated the frequency of Oscillatory motion in the superadiabatic layer in the envelope of the magnetic A-type Star and showed that the frequency is in the range of the observed rapid oscillations. He Suggested global p-mode oscillations resonate with the oscillatory convection and they are responsible for the observed oscillations. This magnetic overstability model Seems to explain the coincidence of the pulsation axis with the magnetic axis, the axial Symmetry of modes and the short periods of excited oscillations. However, there are also problems. One of them is that Shibahashi's (1983) treatment is based on the local

117

analysis while overstability should be determined by the global analysis. Furthermore, whether global p-mode oscillations are induced or not should) be investigated by the global analysis without Boussinesq approximation.

Although all the proposed mechanisms described here seem promising, much work is necessary before a definitive answer is given as for the excitation mechanism for the rapid oscillations of Ap star.

6. Multiple Oscillations in a Delta Scuti-Type Star Let us now turn to the problems of the Delta Scuti stars. One of the observational

characteristics of pulsations in these stars is that light curves of some of these stars significantly vary. This fact suggests that the observed pulsations in 6 Scuti stars consist of multiple eigenmodes so that the beat among these modes causes the apparent changes of the light curves. In some stars, nonradial pulsations have been considered to be involved. These situations are quite different from the pulsating stars in the upper portion of the instability strip; Most of them are stably pulsating in a single mode (the radial fundamental mode or the first overtone) and even some exceptional variables, the double-mode Cepheids, are pulsating only in two different modes, and they do not seem to pulsate in nonradial modes. These differences come from that the 6 Sct stars are less evolved than the pulsating stars in the upper portion of the instability strip as discussed below.

In an evolved star, in general, the Brunt-V~is~l~ frequency, N, which is the characteristic frequency of gravity waves, increases greatly with depth toward the stel- lar center due to the high central condensation of the mass and it is higher than the order of the frequency of the radial fundamental mode (GM/R3) I/2. In this situation, any nonradial eigenmode with a moderate frequency ~ behaves like a gravity wave in the deep interior of the star while it behaves like an acoustic wave in the envelope. The radial wavenumber k r of gravity wave in the core is estimated as

k = N / w x Z / r . (14) r

The travel time that it takes for a wave packet to traverse the core and to return to the envelope is then given by

= ; ,-i dr ~ 2£ / ~2 I N / r dr . (15) ~tr 2 IVgroup,r core core

Here v stands for the group velocity in the radial direction In the case of the group, r pulsating stars in the upper portion of the instability strip such as the Cepheids and the RR Lyrae stars, the central condensation of the mass is very high and N 2~ 10 7 x (GM/R 3) ~10 7 m 2. The damping time ~damp of nonradial oscillations in the core of the star is very short due to strong radiatlve dissipation as a result of the extremely short wavelength nature of the gravity waves and is the order of the period, e-i (GM/R3) -I/2. It is much shorter than the travel time and hence the wave may be damped to a negligible intensity long before being reflected at the center. In such a situation, standing nonradial oscillations extending from the center to the surface are impossible, and the survivals are only the radial modes and the nonradial p-modes which are well trapped within the envelope but leaky through the evanescent zone into the core

(Dziembowski 1977, 0saki 1977). As for the surviving nonradial modes, the damping rate due to the leakage is estimated as (Shibahashi 1979, Unno et al. 1979, Lee 1985a)

~I / ~R ~ [ 4 (n + 1/2) ~ ]-I (r I / r2)-~ , (16) P

where n means the number of nodes in the envelope (n D%I), that is, the radial order of P

the mode, and r I and r 2 are the outer and the inner r~dius of the evanescent zone be- tween the envelope p-mode cavity and the core. Therefore, the low degree nonradial modes are likely to be stabilized by the effect of leakage even if the <-mechanism works to destabilize. Since the effect of leakage becomes small as the degree ~ increases, the nonradial p-modes with high degree £ may be excited by the <-mechanism. However, os- cillations of such a high degree divide the surface into small, many regions oscillati~g in different phases and the contribution of each region is canceled by others so that the total amplitude of the variability of the star is too small to be detected. As a result, we can conclude even from the linear analysis that the observable, self-excited modes in the pulsating stars in the upper portion of the instability strip are restricted only to a few radial modes.

118

On the other hand the ~ Scuti stars are less evolved than the pulsating stars of other class in the instability strip. The Brunt-V~is~iM frequency in the interior of a 6 Sct-type star is much lower than that in the Cepheids and it is N2 ~ 104x(GM/R 3) ~ 104 2

even in the case of a ~ Sct-type star having sub-giant structure in the shell hydrogen- burning phase. Then the wave traveling time in the core is not long enough and hence the radiative dissipation in the core is not enough to inhibit the existence of standing Waves. However, since the 6 Sct stars but for those on the main sequence are higly COndensed, (we should be reminded of N 2 ~ 104 ~2) any eigenmode with moderate frequency

% (GM/R3i I/2 behaves like a gravity wave with short wavelengths in the deep interior while it behaves like an acoustic wave in the outer envelope. The eigenmodes consist of modes which mainly oscillate in the core rather than in the envelope and modes which mainly oscillate in the envelope rather than in the core. The number of nodes ng in the

Core is given by

f ng k r dr / ~ ~ £ / (~w) N / r dr (17)

and it is estimated as ng £ 30, if ~(GM/R3) I/2. Therefore the frequency spectrum of

eigenmodes (in particular, gravity-wave like modes) is quite dense and the frequency difference in adjacent overtones of nonradial modes differing in ng but with the same

degree £ is

A~ / ~ = n - 1 << 1 . (18) g

Since the frequency difference between adjacent radial mode overtones is approximately given by [see equation (i)] 2~v 0 ~ (GM/R3) 1/2, there are as many as [~/(2~2v0)] ×/N/rdr

nonradial modes in the frequency range of two adjacent overtones of the radial mode. Whether or not these nonradial modes are self-excited depends on the competition between the destabilizing effect due to the K-mechanism in the helium and/or hydrogen ionization Zones and the radiative dissipation in the core. With increasing the degree £, the acoustic wave zone in the envelope and the gravity wave zone in the core become decoupled each other and the nonradial modes are clearly depicted into two classes: the pseudo-gravity modes and the pseudo-acoustic modes. As for the former modess the radia- tive dissipation overcomes the <-mechanism, and they are pulsationally stable. The lat- ter modes are excited by the K-mechanism. As for low degree modes (Z = I~ 4), the inner and the outer wave zones are not well separated and then the eigenmodes have more or less the dual characters. As a consequence, many nonradial modes with low degrees can be pulsationally unstable as well as radial modes. Among them, the low degree (£ = l) modes having frequencies quite near those of radial modes have p-modes characters rather than g-mode characters and then they are excited as easily as the radial modes. Figure 1 shows the results of numerical calculation and we can see the characteristics men- tioned here in this figure. As seen in this figure, several low order radial modes are unstable and many nonradial modes are simultaneously excited. The most unstable mode is not the radial fundamental mode but the higher overtone as first noticed by Stellingwerf (1979). Therefore, from the point of view of the linear analysis, many modes are likely to be excited in 6 Sct stars at the shell hydrogen-burning stage; which occupy the upper, right (luminous My, low Teff) portion of the ~ Sct region on the H-R diagram.

- - 3

- - 5

.~ _ ,

x x x

x x x

- - l =3

4 5 6 7

Fig.l. Growth rates of unstable radial and

low degree nonradial modes of a Sct-type star. The frequency• ~R is normalized by (GM/R3) I12. As

for the radial modes, the most

left cross means the F-mode. As for nonradial modes, each dot corresponds to an overtone. (from Lee 1985b). Copyright by Astronomical society Japan

119

In performing numerical calculations of nonadiabatic, nonradial oscillations of the Cepheids and RR Lyrae stars in the upper portion of the instability strip, we have only to treat the envelope as an isolated pulsator and to adopt the leaky wave condition as the inner boundary condition (Dziembowski 1977, Osaki 1977). AS for the 6 Sct stars, however, we should use an evolutionary model including not only an envelope but also a core part. The leaky boundary condition for the Cepheids and RR Lyrae stars was adopted by Fitch (1981) in order to treat the 6 Sct stars, but it is not justified. We should also be reminded that the eiqenmodes even with ~ (GM/R3) I/2 have quite many nodes in

the core. Therefore a fine mesh may be necessary but numerical calculations based on a fine mesh are sometimes difficult in practice (e.g. Cox and Clancy 1982). Dziembowski (1975, 1977, 1979) and Lee (1985a, b) resolved the difficulty by first solv" ing analytically the wave equations in the core with the use of WKB method and then using them as the boundary conditions for the numerical calculation in the envelope. Figure 1 shows the result thus obtained and the growth rate ~i/~w is plotted against the

frequency ~R" One of the remaining problems is what modes are really sustained after nonlinear interactions among those linearly unstable modes. This problem has been at- tacked by Stellingwerf (1980) and Dziembowski (1982, 1983, 1985), but the treatment of fully nonlinear nonadiabatic nonradial oscillations has not been established and remains to be solved.

If multiple modes are sustained and observable, they provide much information and the asteroseismological study may be useful to probe the~internal structure of the @ Sct stars. To resolve observationally many frequencies in each 6 Sct-type star is highly desirable for this purpose. In addition to the photometric observation, the spectroscopic observation of variability in line profiles is also useful (Campos and Smith 1980, Smith 1982, Odell and Kreidl 1982).

7. Concluding Remarks The discovery of rapid pulsations with very small amplitude in Ap stars has raised

a question concerning the distinguishment of apparent pulsating and non-pulsating stars in the 6 Sct instability strip. We will now have to examine very carefully whether or not the apparent non-variable stars in this region are really non-pulsating stars. The detection of small amplitude pulsations in Ap stars is possible because their periods are quite short as compared with the characteristic time scale of terrestrial atmos- pheric variation. If their periods would be as long as periods of the £ Sct stars it would be quite difficult to detect such small amplitude oscillations from the ground- based photometric observation. We should be reminded that the amplitude distribution of the 6 Sct stars increases exponentially with decreasing amplitude. Hence we may expect that the apparent non-varying stars, which are two third of stars in the 6 Sct regions on HR diagram, may be pulsating but with undetectable amplitudes. Furthermore, as for Ap stars, since the pulsation amplitude is modulated, we cannot conclude the con" stancy of a star even if the star does not show luminosity variation in an observational time. Long, uninterrupted observations are needed and international co-observations and observations from the space are favorable for this purpose. Even for the known variable stars, such observations are preferable, since many eigenmodes are expected to be in- volved and the resolution into the eigenmodes is useful to the asteroseismological ap- proach to these stars. On the theoretical side, effects of nonlinear coupling among various nonradial and radial modes should be investigated. The excitation mechanism of pulsations for the rapid oscillations in Ap stars remains to be solved. The global os- cillations of a magnetic, rotating star should be formulated as an eigenvalue problem. In performing such a calculation, the realistic equilibrium structure of Ap star must be first constructed. The asteroseismology will be useful to justify the model. In this respect, Campbell and Papaloizou (1986) presented a method to calculate adiabatic non- radial oscillations of a magnetic star.

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(D. Reidel, Dordrecht), p.417. Kurtz, D.W. 1986b, in Highlights of Astronomy, ed. J.-P. Swings (D. Reidel, Dordrecht),

p.237. Kurtz, D.W. and Kreidl, T.J. 1985, Monthly Notices Roy. Astron. Sot., 216, 987. Kurtz, D.W. and Seeman, J. 1983, Monthly Notices Roy. Astron. Soc., 205, ii. Kurtz, D.W. and Shibahashi, H. 1986, Monthly Notices Roy. Astron. Soc., in press. Kurtz, D.W., Schneider, H., and Weiss, W.W. 1985, Monthly Notices Roy. Astron0 Soc.,

215, 77. Lee, U. 1985a, Publ. Astron. Soc. Japan, 37, 261. Lee, U. 1985b, Publ. Astron. Soc. Japan, 37, 279. Matthews, J.M., Kurtz, D.W., and Wehlau, W.H. 1986, paper presented in IAU Symp. 123:

Advances in Hello- and Asteroseismology held in Aarhus. Mathys, G. 1985, Astron. Astrophys., 151, 315. Odell, A.P. and Kreidl, T.J. 1982, in Theoretical Problems in Stellar Stability and

Oscillations, eds. A. Noels and M. Gabriel (Universit~ de Liege, LiEge), p.148.

Osaki, Y. 1977, Publ. Astron. Soc. Japan, 29, 235. Shibahashi, H. 1979, Publ. Astron. Soc. Japan, 31, 87. Shibahashi, H. 1983, Astrophys. J. Letters, 275, L5. Shibahashi, H. 1984, Mem. Soc. Astron. Ital., 55, 181. Shibahashi, H° 1986, in Proc. Workshop : Hydrodynamic and Magnetohydrodynamic Problems

in the Sun and Stars, ed. Y. Osaki (University of Tokyo, Tokyo), in press, Shibahashi, H. and Saio, H. 1985, Publ. Astron. Soc. Japan, 37, 245. Smith, M.A. 1982, Astrophys. J., 254, 242. Stellingwerf, R.F. 1979, Astrophys. J., 227, 935. Stellingwerf, R.F. 1980, in Nonradial and Nonlinear Stellar Pulsation, eds. H.A. Hill

and W.A. Dziembowski (Springer-Verlag, Berlin), p.50. Tassoul, M. 1980, Astrophys. J. Suppl., 43, 469. Thompson, I.B. 1983, Monthly Notices Roy. Astron. Soc°, 205, 43p. Unno, W., Osaki, Y., Ando, H., and Shibahashi, H. 1979, Nonradial Oscillations of Stars

(University of Tokyo Press, Tokyo). Weiss, W.W. 1986, in Proc. IAU Colloq. 90: Upper Main Sequence Stars with Anomalous

Abundances, eds. C.R. Cowley, M.M. Dworetsky, and C. Megessier (D. Reidel,

Dordrecht), p.219. Wolff, S.C. 1983, The A-Stars: Problems and Perspectives (NASA SP-463, Washington).

121

THE UNUSUAL DELTA SCUTI STAR @z TAU

Michel Breger

Astronomy Department, University of Texas, Austin, TX 78712, USA, and

Institut f~r Astronomie, Universitat Wien, T~rkenschanzstrasse 17, A-If80, Wien, AUS

E. Antonello and L. Mantegazza

Osservatorio Astronomico di Brera, Via E. Bianchi No. 46, 1-22055 Merate, Italy

Huang Lin, Jiang Shi-yan~ and Guo Zi-he

Beijing Observatory, Academy of Sciences, Beijing, China

The variability of the bright Delta Scuti star @ 2 Tau ~HR 1412, HD 28319, V =

3~4, A7III) in the Hyades cluster was :first announced by Horan (1977), and confirmed

by Duerbeck (1978). A multifrequency study by Antonello and Mantegazza (1983) did

not lead to a satisfactory frequency solution for this star, but indicated one or

two dominant periods at 0.0754 and 0.0756 days, respectively, as well as an addi-

tional complex frequency spectrum. A comparison with other Delta Scuti stars

(Breger 1979) suggests that the length of the dominant pulsation period is unusually

short for such a luminous Delta Scuti star.

Multiple photoelectric observations of @2 Tau were obtained during 31 nights in

the time period December, 1982, to February, 1984, with the 0.6m telescope at Xing-

Long Station, China, the 0.9m telescope at McDonald Observatory, Texas, USA, and the

1.02m telescope at Merate Observatory, Italy. V and ~ filters were used, which have

almost identical effective wavelengths. A discussion of the reduction procedures

will be given elsewhere. Two comparison stars were used: HR 1428 (Am) and HR 1422

(FOV). No evidence for variability of these two stars was found. From the scatter

of the comparison star observations an internal uncertainty of about ±0~003 is esti-

mated for the average brightnesses.

We investigated the possibility that @z Tau or one of the comparison stars might

be a long-period variable (e.g. the possibility of nonradial g modes). The fact that

some observing runs at the different observatories were only hours apart allowed an

intercomparison of possible long-period trends. No convincing variations could be

found.

The two components of the 140~728 binary system (Ebbighausen 1959) have recently

been resolved during lunar occultations. Peterson et al. (1981) find AV = i~i0±0~05

and a secondary marginally bluer than the primary. They suggest a system with an

A7IV primary and an A5V secondary. The photometric indices of @2 Tau and their cali-

brations in the uvbyB system can be combined with the known distance of the Hyades

cluster to calculate the properties of possible components of the @2 Tau system.

Computer modeling of the photometric properties in the uvb~ system of possible bi-

nary components was undertaken. The main requirement was that the difference in

122

apparent magnitude between the components should equal the difference in the absolute

magnitudes computed from photometry. We also stipulated that each star should have

photometric indices typical for stars in the Hyades (Crawford and Perry 4966, Johnson

and Knuckles 1955). An important result for the computer modeling was the realization

that both stars should have similar temperatures. A range of magnitude differences

between the components from AV=O up to gV~1~5 is permitted by the photometry. The

solution proposed by Peterson et al. on the basis of occultation measurements is,

therefore, supported by photometric arguments:

Primary: V = 3.75, B-V = 0.47, A71V

Secondary: V = 4.85, B-V = 0.]6, A5V

The mass function of 0.126 together with the mass ratio T~2/~ = 0.8 indicates an

orbital inclination of i ~ 47 ° , which rules out the possibility of eclipses.

The multiple-frequency analysis of the 02 Tau photometry was performed with a

package of computer programs employing single-frequency (Fourier) and multiple-

frequency techniques of frequency ana]ysis. Our multiple-frequency least-squares

analysis avoids prewhitening by assuming a given number of simultaneous frequencies

and finding the best fit (lowest residuals in brightness) of these frequencies.

The top part of Figure i shows the spectral window pattern based on the times

of the available observations. The alias patterns are quite low in power, which is

a consequence of observing at multiple sites.

For the available data, the optimum number of frequencies appeared to be four.

We cannot, however, rule out the possibility that the star might have, for example,

six or eight frequencies with small amplitudes and frequency values slightly differ-

ent from those in our solution. Table i shows the four frequencies and their ampli-

tudes found in the present analysis. The improvement in the power spectrum up to the

noise level can be seen in Figure i. The average residual of the four-frequency fit

has a low value o~ ±0~003 and gives reasonable (but not perfect) fits to the obser-

Vations.

TABLE 1

Multifrequency Solution for ~2 Tau

Frequency fi fi - fi-I Semi-amplitude

(cd -I) (cd -l) (mag)

13.2297 0.007

13.4809 0.2512 0.003

13.6936 0.2127 0.004

14.3176 0,6240 0.003

The phase shifts caused by orbital motion allow us to estimate which component

pulsates. While in principle this analysis could be performed for each individual

frequency, the amount of available data only permits a check with the whole four-

frequency solution. The conclusions may also be dependent on the completenes s of

123

-5 0.8I ' 0.4

0 - 1

0.4

0.2

0

r ~

tuO. I 3: 0 0 O .

0.1

0

0.1

0

0.I

0

0 , I , I ' ' ' '

L SPECTRAL , , i I I I I I I '

IO 15 ! | | " ' " I ! I ' I l I ! I

~ ~ ~ i ~ i DATA

5 I ' ' /

WINDOW

I

20 I

J~_ DATA- IP

DATA-?_P

DATA- 3P L ~ _ . d l l ,L . . . . . .

DATA-4P

I I t i i I ,,, t I I I l

I0 15 20 FREQUENCY (CYCLES PER DAY)

Figure ]: - Power spectra of @2 Tau before and after removal of one or more periods.

We note that a four-period solution leaves mainly noise in the power spectrum.

Also notice the low aliasing in the spectral window due to multisite observations

1 2 4

this solution. The following procedure was adopted: The optimized four-frequency

Solution for the data without orbital llght-time corrections was subdivided into six

groups spaced in time with different orbital phases. The following average absolute

(O-C) values were found: i.i minutes for pulsation of the primary, 1.7 minutes for

no llght-time corrections, and 3.1 minutes for pulsation of the secondary. While the

Uncertainties of these (O-C) values are not known, pulsation of the primary, rather

than the secondary, appears the most probable.

In order to attempt pulsation mode identifications of the four frequencies

found in the previous section, we must first determine the pulsational Q values. If

we consider the dominant frequency of 13.2297 cd -I and the standard equation

log Q = -6.454 + log P + 0.5 log g + 0.I M + log v Teff

together with the log ~ Tef f calibration for uvbyB photometry by Breger (1977), we

find the following values for the primary: M =0.5, log g = 3.8, Teff=8200K and Q = d d Vd

0.019±0.002. The radial fundamental mode (Q0=0.033) and the first radial overtone

(QI=0~025) are excluded. For the other three frequencies similar conclusions apply.

For pulsation of the secondary we find: M v = 1.6, log g = 4.0, Tef f = 8300K and Q =

0~032±0~005. If one or more of the observed frequencies originate in the secondary

COmponent, the rather simple pulsation in the radial fundamental or first overtone

mode would not be excluded. We note, however, that orbital light-time arguments

presented in the previous section favor pulsation of the primary for the dominant

mode.

The closeness of the four frequencies indicate that at most, one radial frequen-

cy can be excited in @z Tau. This can he demonstrated by considering the extreme

frequencies fi=13.22970 cd -I and f4 = 14.31756 cd -I. The observed period ratio

becomes 0.924, which is too large for radial overtones (e.g. see Fitch 1981). It

can therefore be regarded as certain that the pulsation of @2 Tau is nonradial.

The actual nonradial modes excited in 82 Tau cannot yet be determined. The rea-

Son lies with the large number of possible identification with modes of different

values of (k,l,m). Especially the large range of permitted values of rotation fre-

quency (yu~sin i = 80 km s -I) allows for considerable uncertainty. Simultaneous

light, color and spectroscopic measurements are planned for the future.

REFERENCES

Antonello, E., Mantegazza, L.: 1983, Hvar Ob. Bull. 7, 335

Breger, M.: 1977, Publ. Astr. Soc. Pac. 89, 55.

Breger, M.: 1979, Publ. Astr. Soc. Pac. 91, 5.

Crawford, D. L., Perry, C. L.: 1966, Astron. J. 71, 206.

Duerbeck, H. W.: 1978, Inf. Bull. Var. Stars 1412.

Ebbighausen, E. G.: 1959, Publ. Dom. Astrophys. Obs. ii, 235.

Fitch, W. S.: 1981, Astrophys. J- 249, 218.

Horan, S.: 1977, Inf. Bull. Vat. Stars 1232.

Johnson, H. L., Knuckles, C. F,: 1955, Astrophys. J. 122, 209.

Peterson, D. M., Baron, R. L., Dunham, E., Mink, D.: 1981, Astron. J., 86, 1090.

125

PERIOD VARIATIONS IN SX PHE STARS: CY AQR, DY PEG AND HD 94033

J. H. Pefia I , R. Peniche I, S. F. GonzAlez ~ , M. A. Hobart 2

ABSTRACT. An analysis of the times of maximum light of three SX Phe star was carried out. The results support a monotonous decrement of the period which is consistent with the theoretical models of pre-white-dwarfs of 0.2

M O •

Key words: Stars - SX Phe, Delta Scuti stars - variable stars pulsation.

The period variations in pulsating stars can be compared directly with those calculated from evolutionary sequences and the evolutionary stage of the stars can be determined.

In the case of SX Phe stars, it is believed that they belong to an older population than the normal Delta Scuti type and, therefore, an exact determination of the period variation is specially interesting since this fact can be related with evolutionary stages and these stars can be described by either of the following two models: "Young" Pop II stars leaving the main sequence or they could be found in a more advanced evolutionary stage which would correspond to pre-white-dwarfs that still generate their energy through the burning of hydrogen in the shell.

Up to now we have found detectable variation in the period of some stars which have been most extensively observed in the last forty years: CY Aqr and DY Peg and an indication of the period variation of HD 94033.

A comprehensive study of the times of maxima for CY Aqr with a time span of fifty three years has been carried out. The results support a monotonous decrement of the period at the ephemeris found is given by 2440892.637+0.061038318 E - 4.58xlO-l~E 2 . Models of low mass (~0.2 solar masses) explain the parameters of this star.

An analysis of a complete list of the times of maximum light of DY Peg that exists in the literature has been carried out. With 681 times of maximum light covering a time span of 46 years, the following ephemeris was deduced: T~i x = 2437178.3729+0.07292633 E - 2.20xlO-~aE 2 which implies a period variation of dlnP/dt -3.O2OxI0 -~ yr -I . A direct comparison with the models suggests that this star is in the stage of a pre-white-dwarf of 0.2 M O as suggested by the theoretical models of Dziembowski and Kozlowski (1974). New observations of the maximum light of HD 94033 allow a determination of an ephemeris Tmax= 2442516.1585+05951012 E and the O-c residuals suggest a period variation. Hence, the three stars which were considered here share the typical SX Phe characteristics (low metal content, high space velocity or high galactic latitude) and also show a detectable period variation.

REFERENCES

Dziembowski, W. and Kozlowski, M. 1974, Acta Astron6mica 24, 245.

Mexico, i Instituto de Astronomla, UNAM. " Facultad de F~sica, Universidad Veracruzana. M6x]co.

126

FAR-ULTRAVIOLET OBqERVATIONS OF THE

DELTI-SCKFrl VANIABLE BETA CAK~OPEIAE

Thomas R. Ayres 1'2 and Jeffrey O. Bennett 1

1Center for Astrophysics and Space Astronomy, University of Colorado.

2Guest Observer, International Ultraviolet Explorer.

Introduction

Beta Gassiopeiae (HD 432 :F2 m-IV) is the nearest and brightest of the 6-Scuti variables (McNamara and Augason 1962), and is an intense far-ultraviolet emi~ion-line source having surface fluxes in excess of 20 times the average Sun (Linsky and Marstad 1981). Like the F-type secondary star of the Capella system (Ayres and Linsky 1980), fl Cas probably only recently has evolved into the Hertzsprung gap from the upper main sequence, and has not yet shed the rapid rotation of its ZAMS progenitor. At the same time, the surface temperature of fl Cas has cooled and a vigorous, though shallow, convective envelope has developed. On the one hand, it is plausible that the "fossil" fast rotation and newly acquired convective activity have conspired to energize a strong magnetic dynamo: the enhanced chromospheric emi~ions of fl Cas might then be analogous to those of magnetic "active regions" on the Sun. On the other hand, it is possible that the intense ultraviolet emi~ions are a byproduct of the pulsation phenomenon, itself, as has been proposed by Schmidt and Parsons (1982) for the Cepheid variables.

The n e w O~tJOn~

In September 1984, we acquired a number of far-ultraviolet (1150-2000 /~ ) spectra of fl Cas wi th the International Ultraviolet Explorer. During a 3.5 hour period on 28 September, we took a series of seven 5 A resolution spectrograms (5 minute exposures) to monitor the chromospheric H I Lya emission of the star over the course of its 8 Sct pulsation period (2.4 h). Prior to the low-dispersion series, we took a sequence of 0.15 • resolution echellograms ranging in exposure time from

127

2.5 to 37.5 minutes. Subsequently, we exposed two deep echellograms (150 and 185 minutes) to optimally record the faint spectral region near C IV 1548.2 ,~ , typically the most intense em!~ion of a solar-type "transition region" (I0 s K).

Method of Analysis

We reduced and calibrat~l the low-dispersion and high-dispersion images at the Colorado ]-UE Regton~ Data Anc, Zys~s Fac/Uty in Boulder using standard techniques. We measured the fluxes of promirtent emission features in the low-dispersion spectra using an automatic numerical algorithm: an example of the application of the fitting procedure is illustrated in Fig. I. The fluxes of Lya in the ~ven observations are plotted as a function of time in Fig. 2. The half-hour gaps between the individual 5-minute observations are mused by the unavoidable overhead of the READ/PREP cycle of the SWP vidicon ~mera.

Fig. 3 compares the high-dispersion profiles of C IV 1548.2 in /~ Cas and Procyon; the latter is a sharp-lined F-type star similar in temperature to /Y Cas, but closer to the main sequence and less active in terms of its chromospheric emissions. In both cases, the spectra are composites of several independent exposures that were co-added, and smoothed somewhat, to improve the signal-to-noise.

Analysis and discussion

It is clear from Fig. 2 that a rather dramatic increase in the f lux of the hydrogen emi~qion occurred in the last three observations compared wi th the first four. The horizontal bar in Fig. 2 depicts the duration of the 8 Set pulsation of /~ Cas: the factor-of-two increase in the Lya flux apparently occurred over a comparable time~cale.

Unfortunately, scheduling constraints prevented us from monitoring the emission of ~ Cas for a sufficient time to establish whether the enhancement of the Lya emi~ion is related to the pulsation, itself. Indeed, it might simply represent a transient "flare"-like event, or the rotation onto the visible hemisphere of an unusually bright "active region" (Prot --" 2 d). Clearly, additional observations wi l l be required to properly address the question of pulsation-induced heating in the chromosphere of ~ Cas; nevertheless, the present results are quite intriguing.

Concerning the high-dispersion r~'~ordings of Lya in Procyon and /~ Cas (not i~ustrated here): in spite of the geocoronal corruption of the inner core of the /~ Cas profile, it nevertheless is clear that the outer edges of the emi~ion are significantly wider than those of Procyon. In

128

o .'2 !

0 v - - -

I I I ............................ I ' ' I I ' I I 10 S24065L: HD 432 (BETA CAS) DATE: 84,272 EXPT: 300s . _ ,

6i d 4

2F ! ,, - 2 U I I . . . . . . . . . . . . . . .

- 4 ~ , , , ~ ~ ........ ~ . . . .

1150 1250 1350 1450 1550 1650 1750 1850 1950

W A V ~ . ~ q G T H (A)

Figure 1. One of the 5-minute low-dispersion exposures (SWP 24065) of fl Cas fitted by means of an automatic numerical procedure. The emission lines are modelled by least-squares Gaussians: tic marks indicate significant detections; arrows indicate upper limits. The spectrum longward of 1700 ~ is heavily overexposed.

Figure 2. Fluxes of H I Lya measured from the 7 low-dispersion spectra of fl Cas taken over a 3.5-tl period. The error bars (1¢) were determined from the empirical deviations of the observed profiles from the Gaussian fits. The horizontal bar indicates the duration of the 6 Set pulsation cycle.

4 . 0

3.0

2.0

1 . 0

0 . 0 i

0,32

|

i i i J o'aG 0:40 0,44 0:48 1 9 8 4 DAY 272+ (LFI~

4.0

2.0

0.0

i 4,0 2.0

0 .0

~ ' rA CAS ~F~ m-r0~

P~t0C'YON IF5 ~-Vl

1 5 4 0 1 5 4 5 1 5 5 0 1 5 5 5

W A ~ G T H (A} 1 5 6 0

Figure 3. High- dispersion profiles of C IV 1548.2 and 1550.7 in Cas and the comparison sharp-Line F-tyi~ star Procyon. Although the B Cas profiles are noisy, they clearly are broader than those of Procyon. The ordinate is the relative monochromatic surface flux.

129

both cases, the F W H M s are in excess of 300 km s -1, and cannot represent thermal and rotational Doppler broadening, alone (vsini[[3cas]ffi 72 km s-l: Baglin et at. 1973). In the case of C IV 1548.2 (Fig. 3.), the enhanced broadening of the ~ Cas profile compared wi th that of Procyon is even more dramatic: the F W H M is comparable to that of Lya , and is among the largest recorded for any "normal" star of late spectral type; even the fast-rotating secondary star of the CapeIla system (vstni= 36 km s -1) - one of the most "active" F-type stars in the solar neighborhood - has a C IV F W I t M of less than 200 km s -1 (although the C IV surface f lux is about twice that of ~ Cas). The exceedingly large velocity wid th of the C IV emission feature of ~ Cas presents another significant puzzle for future work: one possiblity, which has been suggested in the case of CapeUa (Ayres and Linsky 1980), is that the atmospheric structures which are bright at temi~cratures of 105 K -- presumably some type of f i lamentary magnetic "loop" -- might extend an appreciable fraction of a stellar radius above the surface, and thereby produce enhanced rotational broadening if they are forced to co-rotate wi th the photospheric layers (which is the case w i th the so-called large-scale structures of the solar corona). If true, the stellar transition zone must consist of a large number of such entities in order to produce a comparatively symmetric net emimion profile.

Acknowledgemezlts

This work was supported by NASA grant NAG5-199.

ReXerences

Ayres, T~R, and [.in~ky, J .L 1980, Ap. J , 241, 279. Baglin, A., et at. 1973, Astr. Ap., 23, 221. McNamara, D.H., and Augason, G. 1962 Ap. J., 135, 64. Linsky, J.L., and Marstad, N.C. 1981, in The Universe at Ultraviolet

Wavelengths: The F~rst Two Years o f 1-(/E, ed. R. Chapman, N A S A Conf. Pub. 2171, p. 287.

Schmidt, E.G, and Parsons, S.B. 1982, in Advances in Ultraviolet Astronomy: Four Years o f I-tiE Research, eds. Y. Kondo, J. Mead, and R. Chapman, N A S A Conf. Pub. 2238, p. 439.

130

RECENT OBSERVATIONS OF SOME RAPIDLY OSCILLATING Ap STARS

D. W. Hurtz Department of Astronomy University of Cape Town Rondebosch 7700, South Africa

Over 90 hours of new hlgh-speed photometric observations of the

Rapidly Oscillating Ap star HD 6532 were obtained from the South

African Astronomical Observatory and the Mount Stromlo and Siding

Spring Observatory in 1985 by Kurtz & Cropper (1986). A frequency

analysis of these data shows the presence of three principal frequen-

cies at fz=2.39620±0.00003 mHz (P=6.95546±0.00009 min), f2=2.40216 ±0.00003 mHz (P=6.93820±0.00009 mln) and f3=2.40810 ±0.00003 mHz

(P=6.92109±0.00009 mln) which are equally spaced by 5.95±0.04 ~Hz.

These three frequencies are a complete description of all of the rapid

light variations in HD 6532 above an amplitude of 0.15 mmag. The

first harmonlc of f2 at f~=4.80430±0.00003 mHz (P=3.46911±0.00002 mln)

appears in the data at a high confidence level. Two further frequen-

cies at fs=2.37868 mHz and f&=2.42567 mHz appear Just above the noise

level and are equally spaced from f2 by 23.5 ~Hz. The reality of

those two frequencies is not firmly established. There is no further

evidence of the 1.17-mHz peak seen in the amplitude spectra of two of

Kurtz & Kreldl's (1985) nights of observations in 1984.

From the mean light variations in HD 6532 a rotational period of

either Pmt=0.8943±0.0005 day or Pr0t=l.7886±0.0005 day is derived inde- pendently from both the 1984 data of Kurtz & Kreidl and the 1985 data

of Kurtz & Cropper. Following the arguments of Kurtz & Kreidl (1985),

we point out that the rotation period derived for HD 6532, Pmt=0.8943 -i

day, implies a rotational velocity of vr0t)100 km s , assuming that

R~I.8R®, a typical value for an A5 main sequence star. This rota-

tional velocity is rather large, but not impossible, for a magnetic Ap

star. We therefore presume that it is more likely that HD 6532 shows

a double wave mean light curve with the period P t=1.7886 day which

implies the more reasonable value of v t~50 km s Many Ap stars

show double wave mean light curves; this longer period of Pmt=l.7886

day for HD 6532 implies that the magnetic field variations in this

star will be found to be polarity reversing when they are eventually

observed.

Neither of the possible rotational periods is compatible with an

oblique pulsator model interpretation of the frequency triplet, f123,

since that triplet is split by &f=5.95±0.04 uHz which corresponds to

P~at=l.945±0.013 day ~ P~t=l.7886±0.0005 day. We suggest, therefore, that the frequency triplet in HD 6532 may be due to pulsation in rota-

tlonally perturbed m-modes (as long as second order terms which would

make the frequency splitting unequal are negligible). If this is

true, then we can observationally determine the rotational splitting

constant Cn, i=0.08 for the first time in any star other than the sun.

The problem with this determination of Cm, i is that theoretical calcu-

131

lations by Shlbahashi & Salo (1985) for A star models and by others

for the sun indicate that C .~0.01 for high overtone p-modes. Because M,I of the equal frequency spacing of f123' these three frequencies cannot

be due to pulsation in three modes of (n,E), (n-i,£+2) and (n-2,~+4).

The above derivation of C | rests critically on the rotation

period determined for HD 6532 from the mean light variations. While

it is not likely that the comparison star, HD 6491, is also variable

(which could confuse the period determination for HD 6532), that

possibility has not yet been completely ruled out observationally.

Observations to do this are now in progress.

The Rapidly Oscillating Ap star that has been best observed and

that presents the best case for the oblique pulsator model is HR 3831.

Kurtz & Shibahashl (1986) analysed 135 hr of 1981 observations along

with 43 hr of 1985 observations of HR 3831. From the relationship

between the phase of the principal 1.42801-m~z oscillation and the

rotational phase of HR 3831 (determined from the magnetic field and

mean light variations) they demonstrated that this principal oscilla-

tion is essentially due to a long-llved oblique dipole pulsation mode.

They also showed that there is a slight, but significant, difference

in the oscillation phase diagram between the 1981 and 1985 data sets.

They compared the Rapidly Oscillating Ap star models of Dolez & Gough

(1982) and Dzlembowskl & Goode (1985, 1986) and found that the model

of Dziembowski & Goode, in which the magnetic field rather than the

rotation dominates the oscillations, provides the best framework in

which to explain both the asymmetry in the perlodograms of the light

variations and the slight change in those periodograms.

I have obtained an additional 60 hours of new high-speed photo-

metric observations of HR 3831 in 1986. This 60 hours of observations

combined with the 43 hours of 1985 observations and the 135 hours of

1981 observations allow a frequency solution to the light variations

of HR 3831 without any alias ambiguities. This frequency solution is

given in the following table.

Table I. Frequency solution to the 1981-1986 HR 3831 data

frequency Amp Phase

mHz mmag

±0.0000003 ±0,02

ft 1.4239546 2.09 -1.959±0.010

f2 1.4320710 1.70

f3 1.4280126 0.39

f~ 2.8560254 0.42

fs 2.8641416 0.19

f& 2.8479085 0.18

f7 4.2880967 0.II

-2.037±0.012

2.372±0.053

1.106±0.049

1.399±0.107

1.543±0.112

-1.566±0.193

fs-fl=4.0580 ±0.0004 wHz

f2-f~=4.0584 ±0.0004 uHz

f~-f6=8.1169 ±0.0004 uHz=2(4.0585 uHz)

132

fs-fw=8.1162 ±0.0004 uHz=2(4.0581 uHz)

2fs-f~=-0.0000002 mHz

fT=3fs+4.0589 ~Hz

The zero point for the phases in Table 1 is T0=JD2444000.00000. An

interpretation of these frequencies is currently in progress.

I have recently discovered that HD 166473 is a Rapidly Oscillat-

ing Ap star. This m =7.5 star is classified Ap SrEuCr by Houk

(1982); It has Str~mgren photometric indices of b-~=0.223, ml=0.311,

and ci=0.517. This gives derived indices of 6mi=-0.133 and 6ci=-0.092

both of which a~e typical of the Rapidly Oscillating Ap stars (Kurtz

1986).

A preliminary frequency analysis of 46 hours of observations of

HD 166473 indicates that at least three frequencies are present in the

oscillations of this star with perlods near 8.9 minutes and wlth

amplitudes in the range 0.3 to 0.5 mmag. The separations between

these frequencies is on the order of 50 ~Hz which is about the separa-

tion expected for consecutive overtones of the same degree. Further

details of the oscillations in thls star will have to await a more

complete analysis.

References

Dolez, N., & Gough, D. 0., 1982. in Pulsatlons In Classlcal and Cataclysmic Variables, eds. J. P. cox and C. J. Nansen, JILA, Boulder, p. 248.

Dzlembowski, W., & Goode, P. R., 1985. Asgrophys. J., 296, L27.

Dzlembowski, w., & Goode, P. R., 1986. in Selsmologg of the Sun and Distant Stars, ed. D. O. Gough, D. Reldel Publ. Co., Dordrecht, Holland, p. 441.

Houk, N., 1982. Michigan Spectral Catalogue, vol. 3, Department of Astronomy, University of Michigan, Ann Arbor.

Kurtz, D. W., 1986. in Seismolog~ of the Sun and Distant Stars, ed. D. O. Gough, D. Reldel Publ. Co., Dordrecht, Holland, p. 417.

Kurtz, D. W., & Cropper, M. S., 1986. Non. Not. R. astr. Soc., in press.

Kurtz, D. W., & Kreldl, T. J., 1985. Non. Not. R. astr. Soc., 216, 987.

Kurtz, D. W., & shibahashi, H., 1986. Non. Not. R. astr. Soc., in press.

Shlbahashi, N., & Salo, H., 1985. Publ. astr. Soc. Japan, 37, 245.

133

A R E T H E R E A N Y T R U E • Sc t A p S T A R S ?

Tobias J. Kreidl Lowell Observatory 1400 West Mars Hill Rd. Flagstaff, AZ 86001 U.S.A.

ABSTRACT. Searches over the last 30 years for 8 Set variability in ~ p stars have yielded about 20 candidates. The validity of such detections, as well as the ~ Set classification of these stars, has been a subject of much controversy. An interpretation of the accumulated data, including photelectric photometry undertaken by the author of a number of candidates, is given. One problem involves the prccision of the spectroscopic classification. Other difficulties include noisy data, the lack of viable confirmations, and instrumental problems. While all candidates have not been examined thoroughly yet, there remain very few possible Ap stars tha t might show consistent 6 Sct variability.

1. INTRODUCTION

Many reports on the finding of short-period (15 min < P < 4 hr) light variability in Ap (also known as CP2) stars have appeared in the literature over the last 30 years. Two main problems are evident, however. First, for the majority, it has not been possible to verify consistent variability with additional observations. Second, the spectral classifications of some stars are uncertain. In fact, some of the legding candidates may not be actual Ap stars, but normal ~ Set stars. A review of the topic of low-harmonic pulsation in Ap stars has appeared recently (Weiss 1983). Clearly, follow-up observations are needed to try to confirm short-period variability. Such an effort has been underway at Lowell Observatory since 1983. Differential photoelectric photometry, much of which is obtained with a duM-channel photometer, is being utilized to search for possible periodicities.

2. OBSERVATIONS

Differential photoelectric photometry has been conducted at Lowell Observatory utilizing 0.6-, 0.8- and 1.1-m telescopes, and at CTIO using the 0.9-m telescope. Generally, a two-channel photometer has been used. Great care has been taken to perform the differential photometry by constantly switching between the variable and comparison star(s). This helps to suppress low-frequency drifts in the sky transparency which can induce false periodicities. Only nights of sufficiently good quality are used for these observations, since variability of generally less than 1% is being sought.

Table 1 lists the most important stars that have been investigated. It should be pointed out tha t this list does not represent an exhaustive literature searchj but at least gives the most prominent candidates and most of the relevant publications associated with them. Much controversy exists for a number of

]34

the well-observed stars. Potential problems with observations of these stars include overinterpretation, poor quality nights~ equipment instabilities, telescope tracking errors, and inadequate da ta sampling. As pointed out by Kurtz (1983), the use of Fourier and other period searching techniques must be conducted properly and interpreted correctly. I have obtained data for ten of these candidates, most of which have been observed on several nights for several hours on each night, simultaneously in two colors. Only for the stars HD 3326 and HD 4849 was short-period variability continuously present. The typical minimum amplitude tha t would have been detected in any of the data is about 0.005 mag. In the next section, some comments on some of the more interesting stars are given.

3. REMARKS ABOUT SOME OF THE CANDIDATES

\

Comments on a few of the candidates are given here. Table 1 contains references tha t should be consulted for more detailed information. HD 3326: This star varies regularly (cf. Kreidl 1985). A recent spectral classification (W. Bidelman 1985, private comm.) states tha t it is probably 6 Sct, but good classification spectroscopy is needed. It is probably not an Ap star.

HD 4849: W. Bidelman (1985, private comm.) states tha t this star, like HD 3326, is probably just a 8 Sct star and not an Ap star. It, like HD 3326, varies on a regular basis (cfi Kreidl 1985, op. cit.).

HD 62140: More observations are needed. Several hours of photometry obtained by the author in 1986 show no short-period variations.

HD 65339 (53 Cam): This star is one of the more controversial objects. Convincing data have been published, but variability is clearly not Mwavs present. Coordinated observations would be very useful.

HD 92664: Only one night with a marginal result exists. Follow-up observations should be made.

HD 108945 (21 Corn): Of all candidates, this is the most controversial. For reviews on this star, see Weiss, Breger, and Rakosch (1980) and Weiss (1983). Oddly, no magnetic field has been detected (Borra and Landstreet 1980), although there is little doubt tha t 21 Com is an Ap star. It is only certain tha t short-period variability is rarely present. Not even the rotat ion period of this s tar is known.

HD 119288: This one-time detection clearly needs confirmation, especially since this is such a cool (F3Vp) star. Possibly it is misclassified as Ap (cf. Matthews and Wehlau 1985).

HD 219749: Hildebrandt etal . (1985) seem to find from 1977 to 1981 two periods of nearly the same frequncy. SchSneich (1982) mentions tha t this s tar is a spectroscopic binary and t ha t short-period vari- ability only occurs in the half-period around periastron. Thus, tidal interactions may be responsible for the variability.

HD 224801: Many data have been collected, but only some published. Many observations exist showing no short-period variability.

4. COMMENTS AND CONCLUSIONS

It should be mentioned tha t searches for short-period variability have also been conducted spectroscopically, notably by a group at the Crimean Astrophysical Observatory (cf. Polosukhina etal . (1981) for work on 58 Cam). The coordination of photometric and spectroscopic observations would be useful for verifying periodicities in an independent way.

Considering the massive amounts of observing time already spent on this group of stars and how little we really know about their odd behavior, it would appear tha t the trend of future observational campaigns

135

should be towards coordinating multi-site observing runs and pooling efforts. This approach has already proven to be very successful for interpreting the frequencies of ~ number of rapidly oscillating Ap stars (cf. Kurtz 1985; Weiss 1985). It should also be pointed out tha t relatively few negative results have been published, which of course skews the number of publications towards those where positive detections are claimed. For this reason, I also list in Table 1 a number of searches tha t were negative to at tempt to produce a fairer picture of what has actually been found by observations.

We really do not know for certain if any Ap stars undergo low-harmonic pulsation consistently. As men- tioned earlier, this assertion depends strongly on what the correct spectral classifications of HD 3326 and HD 4849 are. There is, however, convincing evidence that at least some do show occasional short-period variability. Many are quite hot, and one needs to ask whether they may lie outside the S Set instability strip~ which would require a different excitation mechanism from that of the ~ Set stars. The upper tem- perature at which pulsations no longer are present in Ap stars cannot be accurately stated due to the fact tha t the upper boundary of the instability strip is not really known. Nominally, one would not expect Sct pulsation for an Ap star with a spectral type earlier than about A2 since T~fl,~ 8800 K for the hottest known ~ Set stars (Breger and Bregman 1975). The added complexities of the overabundance of certain elements plus often strong magnetic fields may produce conditions in which pulsation can occur only occa- sionally. Since the degree of peculiarity varies enormously, some of the ~nomalous physical characteristics of Ap stars may play an important role in how the ability to pulsate is affected. This contrasts strongly with the rapidly oscillating Ap stars (Kurtz 1985, op. cir.). The rapidly oscillating Ap stars have much shorter periods (4 to 15 rain.), generally lower amplitudes (<0.016 mag in B), and vary on a constant basis. It is certainly a mystery tha t only a small number of the cooler Ap stars have been found to be members of the group of rapidly oscillating Ap stars, i.e., are evidently high-harmonic pulsators.

Coordinated efforts are clearly going to be the best way to sort out many of the observational problems, although large amounts of small telescope time will be needed. This seems a more economic route to go, compared with the limited progress made in the last three decades.

Acknowledgements are due to Drs. W.W. Weiss and D.W, Kurtz for useful discussions and encouragement, and to Dr. W. Bidelman for his comments on the spectral classifications of HD 3326 and HD 4849.

REFERENCES

Borra, E.F. and Landstreet, J.D. (1980) Ap. J. Suppl., 42,421. Breger, M. and Bregman, J.N. (1975) Ap. J., 200, 343. Hildebrandt, G., SchSneich, W., Lange, D., Zelwanowa, E., and Hempelmann, E.

(1985) Pub. Astrophys. Obs. Potsdam, Vol. 32, No. 3. Kreid], T.J. (1985) Mort. Not. R. astr, Soc., 216~ 1013. Kurtz, D.W. (1983) LB. V.S., No, 2285. Kurtz, D.W. (1985) in Proceedings, NATO Workshop, Seismology of the Sun

and Distant Stars, D. Gough, ed., Cambridge. Matthews, J.A. and Wehlau, W.H. (1985) LB. V.S., No. 2225. Polosukhina, N.S., Churner, K.K., and Malanushenko, V.P. (1981)

Bull. Crimean Astrophys. Obs., 64, 37. Sch6neich, W. (1982) Pub. Astrophys. Obs. Potsdam, 32, No. 3, 109. Weiss, W.W. (1983) Hvar Obs. Bull., 7~ 263. Weiss, W.W. (1985) in Proceedings, LA. U. Coil No. 90, in press. Weiss, W.W., Breger, M., and Rakosch, K.D. (1980) Astron. Aatrophys.,90, 18.

136

Table 1. List of candidate Ap stars showing short-perlod variations.

WD No, mag spectral Perlod(S) (name) (V) type (mln) .. ................ ,. .........

33~6 6.1 A5-Tp St? 43,0

4849 6.5 Fp "80 Ag-FOIII 70.9

?g,3

I0088 7o7 AOp -86 none none

11503 ~.8

32633 7.0

621~o (49 Cam)

65339 (53 Cam)

Bgv+ 150 Alp Si none

88 SiCr 106 A2p SICr

6.6 A5 SrCrEu FOp

6.0 £2p SrCrEu

71297 5.6 A5 I I I - IV (marg. Am)

92664 5.5 AOp Si

108945 5.5 A2p CrSr

106 none none

61.2

none

none none irregular 21,0

20.I,27.6, 79.2 none none

5~.71

196

32 32 32.2 ,39.6 ~8,36 22 none 5.4,5.9 5.9 [247] none

119288 6.2 F3Vp

125248 5,9 AOp CrEu (CS Vlr)

173650 6.5 Bgp SlCr:

17,24

~60 none

242,2h7

184905 6.5 AOp $iSrEu

185139 6.3 A5-FZp SrCr:Si:

204411 5-3 A6p CrEu:

25-30

none

94,116

irregular

irregular

2197~9 6.5 Bgp Si 35-135 130,152

129,155

224801 6,4 Bgp SIEu t23 127 none

I ~0

references

Kurba (1982) MNeAS,200 Kreldl (}985) MNRAS.216

Weiss (1979) A&A Suppl.,35 Kurtz (1982) MNRAS,200 Kreldl (1985) MNRAS,216

Weiss (1983) A&A,~28 Kreidl (1984) IBVS,NO.2460 Kreidl (I984) IBVS,NO.2602

Rakoaeh, Pied le r (1978) A&A Suppl.,31 Weiss (1979) A~A Suppl.,35

Bakosch (1963) Lowell Obs. Bull,,6 Preston, Steplen (1968) Ap.J.,151 Percy (1973) A&A,22 Kreidl (1984) IBVS,NO.2472

Matthews, Wehlau (1985) PASP,97

Kreldl (1986) unpublished

Percy (1975) AJ,80 ~werko (1982) Bull. £Str, Inst. Czsch.j 33 Panov (1982) Comm. Konkoly Obs.,83 Polosukhina et el. (1981)

Bull. Crimean A. Obe.,6~ Burnash~v et el. (1983) SOY.

Astron, Lett..9 Kreidl (1985) MNRAS,2]6 Mabthewe, Wehlau (1985) PASP,9?

Kurtz (]984)MNRAS,206

M~gessier et el. ( t985) A&A Suppl.,S9

Bahner, Navrldls (1957) Z. Atrophys.,ql Percy (1973) A&A,22 Percy (1975) AJ,80 Weiss etal. [1980) ~&A,gD Toboehava, Zh l lJaev (1981) As t r , Nachr. ,302 Jarzebowski (1982) Comm. Konkoly Obs.,83 Muzie lok, Kozar (1982) IBVS,No,2237 Garrldo, Sanohez (1983) IBVS,No.2368 Weiss, Kreidl, Elllott,Felerman,

Huan8 (1983-85)unpubllshed

Matthews, Wehlau (1985) IBVS,No.2725

Maltzen, Moffat (1972) A&A,16 Kreidl (~985) MNRA5,212

Hlldebrandt et el. (1985) Pub. Aetropbys. Obs, Potsdam,32

Panov (1981) Pub. Spec, kstrophy. Obs.,32

Kreidl (1984) IBVS,NO.2607

Kurtz (1982) MNRAS,200

Rakoseh (1963) Lowell Obs, Bull.,6

Hildebrandt et el. (1985) Pub. Astrophys. Obs, Potsdam,32

Panov (1982) Comm, Konkoly Obs,,83 SchSneieh (1982) Pub. Astropbys.

Obs. Potsdam,32 Hildebrandt et el. (1985)

Pub. Astrophys. Obs. Potsdam,32

Rakoseh (1963) Lowell Obs. Bull,,6 Nittmann, Rakosch (1981) A&A,gT Weiss, Kreldl (1975,1976,

198~)unpubltshed Hildebrandt et ,el. (1985)

Pub, Astrophys, Obs. Potsdam,32

comments

probably not Ap (delta Set)

probably not Ap (delta Set)

probably Am

not really Ap

need more observations

need more observations

need more observa t ions

probably not Ap (delta Set)

need more I n tens i ve observat ions

specbroscopie binary; tidal Interactions?

137

SEARCHES FOR RAPID LINE PROFILE VARIATIONS OF TWO PULSATING CP2 STARS: HD 128898 AND HD 201601

H. Schneider Universit~tssternwarte G~ttingen Geismarlandstr. Ii D-3400 GSttingen, F. R. Germany

W. W. Weiss Institut fur Astronomie TUrkenschanzstr. 17 A-1180 Vienna, Austria

T. J. Kreidl Lowell Observatory 1400 W. Mars Hill Rd. Flagstaff, AZ 86001, U.S.A.

A. P. 0dell Dept. of Physics and Astronomy Northern Arizona University Flagstaff, AZ 86011, U.S.A.

I. INTRODUCTION

The group of rapidly oscillating CP2 stars gained considerable interest in recent years due to the possible application of astroseismology and the resulting insights in stellar structures. A more detailed description of this group of stars can be found in recent reviews, for example in Kurtz (1985) and Weiss (1986). One major problem, however, is related to the question of the mode identification. Classical techniques, based on the determination of phase lags between color and flux variations, proved to be unsuccesful (Weiss, 1986, op.clt.) although first applications gave reasonable results (Kurtz, 1982; Weiss and Schneider, 1984).

Presently, mode identifications seem to be possible on the grounds of either a very carefully determined pulsation frequency spectrum or of observed spectrum line profile variations. An example for the first method can be found in Kurtz, Schneider and Weiss (1985). For the second method theoretical line profile calculations due to non-radial pulsation, adapted to the oblique pulsator model, are given in Odell and Kreidl (1984) and Baade and Weiss (1986).

Based in part on observations collected at the European Southern Observatory, La Silla, Chile, with the financial support of the Austrian "Fonds zur F~rderung der wissenschaftlichen Forschung", project No. 4170. TJK and AP0 were visiting astronomers, Kitt Peak National Observatory, which is operated by AURA, under contract with the National Science Foundation.

138

The following limiting parameters can be estimated for a SUCcessful mode identification by currently available spectroscopic means:

Spectral resolution R better than 50 000. Signal-to-Noise ratio of several hundred. Pulsation velocity amplitude of more than a few km/s. Projected radial velocity of more than about 15 km/s. Very few pulsation modes simultaneously present. Suitable geometry of the oblique pulsator.

2. OBSERVATIONS

Observations of ~Equ were obtained by TJK and APO on the night of !984-July-08~09 with the Coud$ Feed at KPNO. A thinned TI 800x800 CCD was used as a detector. Combined with grating B and camera 6, a resolution of about 0.07 ~ was obtained over the wavelength range of approximately 5852 to 5899 ~. Due to poor weather conditions only about two hours of data out of four nights could be obtained.

The spectra were registered in the following way: A relatively long slit was utilized and the star kept fixed on one portion of the slit for about a tenth of the 12.44-minute period, then moved down so that the phase information could be retained. After nine individual Spectra were obtained, the chip was read out. The typical continuum level for an individual pixel was about 13000e- and the system's readout noise was about 16e-. In Figure 1 we present a section of the Spectrum which shows ten phases and the result of coadding all phases.

HS and WV~ obtained their observations of ~Equ and ~Cir at the European Southern Observatory, La Sills, C~ile, with the Coude AUXilliary Telescope (CAT) feeding an Echelle Coud~ Spectrograph (CES). The detector in use was a single line Reticon array. Since this detector does not provide for any resolution orthogonal to the direction of dispersion, a procedure similar to the CCD observations was not possible. We therefore decided to take many individual spectra with an exposure time equivalent to i/i0 of the pulsation Period and coadded these low S/N spectra later after proper phasing. To determine the pulsation phase, we observed simultaneously our PrOgram stars with the Danish 50cm telescope equipped with a four-channel Str~mgren photometer. During the nights from 1984-0ctober-27/28 to 30/31 we accumulated about 2500 individual Spectra of ~Equ and ~ Cir which were later reduced with the image Processing system IHAP at E80 in Munich, FRG.

3. RESULTS

~Equ (HD 201801): Both data sets from KPNO and ESO do not give evidence for llne profile variations larger than 1% and thus exceeding the 3~ limit. Cross correlation of individual phases with the mean of all spectra as well as line centers of several blend free spectral lines do not indicate radial velocity variations which exceed 300 m/s. Light variations were extremely small, if at all present, and did not eXCeed the noise of about 0.6 mmag (Figure 2). This result is COnsistent with the photometry obtained by David Kilkenny (1984) at the South African Astronomical Observatory in the nights of 1984-July-6/7 and 7/8. He determined an uDDer limit of 0.5 mmag in Johnson-B.

139

Cir (HD 128898): This program star was the more promising one, since its photometric amplitude was known to be fairly large (Kurtz, Allen and Cropper, 1981; Schneider and Weiss 1983; Weiss and Schneider, 1984, op.cit.). For the ES0 observing run an amplitude in StrSmgren-v of typically 6 mmag was observed (Figure 3) which was clearly above the noise of about 0.5 mmag. Line profile variations did not exceed 0.5% which basically is the noise limit (Figure 4). We are still working to find appropriate criteria for a better quantitative determination of minute profile variations which are nearly buried in the noise. For six lines we have determined radial velocities and could not find variations exceeding 120 m/s, which again corresponds to the noise.

4. CONCLUSION

Neither for ~Equ nor for ~Cir was cl~ar positive evidence present for spectral variations which could be attributed to non-radial pulsation. However, our instrumentation was just at the limit of what would be required for a detection of the subtle effects which can be expected for the pulsating CP2 stars. A spectral resolution at least twice as high as that presently used should be attempted for future investigations, as well as a considerable improvement of the S/N ratio of the individual spectra. Obviously, telescopes of the 3 m class and larger are required for such observations, even if the program stars are very bright.

REFERENCES

Baade, D., Weiss, W. W.: 1986, Astron.Astrophys.Suppl.Ser., in press

Kilkenny, D.: 1984, priv. comm.

Kurtz, D. W.: 1982, Mon.Not.Roy.Astr. Soc. 200, 807

Kurtz, D. W.: 1985, in Proceedings NAT0-Workshop "Seismology of the Sun and Distant Stars", D. Gough ed., Cambridge

Kurtz, D. W., Allen, S., Cropper, M. S.: Inf.Bull.Var. Stars No.2033

Kurtz, D. W., Schneider, H., Weiss, W. W.: 1985, Mon.Not.Roy.Astr. Soc. 215, 77

0dell, A. P., Kreldl, T. J.: 1984, in Proceedings 25th Liege Internatl. Astrophys. Coll. "Theoretical Problems in Stellar Stability and Oscillations", Li4ge, p.148

Schneider, H., Weiss, W. W.: 1983, Inf.Bull.Var.Stars No.2306

Weiss, W. W.: 1986, in Proceedings IAU-Colloquium No. 90 "Upper Main Sequence Stars with Anomalous Abundances", C. Cowley et al. eds., Reidel

Weiss, W. W., Schneider, H.: 1984, Astron.Astrophys. 135, 148

140

}

~@4g

Figure 1: CCD spectra of ~Equ. Individual phases and mean spectrum (198~/07/08)

HD20. 601 30./31,05:B4 S|romgren v

Frequency (I/ho~r)

F i g u r e 2 : P o w e r s p e c t r u m o f S t r b m - gren-v photometry of ~Equ.

o ~ ; :;. . : / ;~ ; : : " I " I I " , . I o I I I : - - -

.... ::" :: '..Z""'_ "";" - -_/-../" . : - -

~ ooo~ '".., ...'".:/ :.;: :

E %%; ,--"."

% • • * : - " " . / . " . "J t

0 0 ~ = - "1" *1 ' " l l ' 51romgren v 30, /3105 t9~4 23 - 5+4 L~T+

HOI2BBgl] AJpho Ci~ -~ 0os- 50¢m Danish letem¢ope [50 Lo 5ii1¢~

h I 0 S I I i ~ 0 I too ~o, Io~ 1o~ ;o , o= o~ Iou o~ o~,r

Figure 3: Light curve of o~Clr

o . f ~1/ [ v

i e l

O, Cir spechum end residuGIs 5 0113 I I 0 ~ 1984 @zO T.I

~o oee

wavelengfh [Angstram l

Figure 4: Reticon spectrum of~Clr Mean spectrum and residuals for individual phases.

t41

NONRADIAL PULSATIONS OF 6 SCUTI STARS

D.S. King The University of New Mexico

Albuquerque, NM 87131

A.N. Cox Los Alamos National Laboratory

Los Alamos, NM 87545

I. INTRODUCTION

Delta Scuti variables are known to pulsate in nonradial as welt as radial modes of oscillation. Theoretical models using the linear, nonadiabatic, nonradial approximations have still not been able to convincingly match the periods and mode excitation of real stars. Calculations typically come up with a number of unstable modes with a variety of periods. For the most part they

do not match the observed nonradial periods. What is the problem? What we would like to emphasize in this paper is that in order to analyze a stellar model for nonradial stability it is necessary to have a good evolution model to start with. Calculations by Fitch (1981) and Clancy

and Cox (1982) suffered from this problem. They tried to match (f Scuti itself and noted that the

behavior of the eigenveetors in the deep interior made interpretation of the theoretical results difficult and that at that time it did not seem possible to match the periods of & Scuti.

We had hoped that it would be possible to start with a complete stellar model obtained

through evolution calculations and to vary the stellar parameters such as luminosity, mass and radius by small amounts and still have a satisfactory model for our envelope code. Even though we can construct a model which has its interior boundary very near the center it is still an inward integration for which the solutions tend to diverge as the center of the star is approached. We

found that this does not work and that we must use a complete model.

II. MODEL

The preliminary model discussed here should be referred to as & Scuti "like" since the only

detailed evolution model available at the time was slightly too luminous and too cool to be in

the observed instability strip (cf. Fitch, 1981). The model we used was kindly provided by S. Becker (1986). It is useful to present the results for this model since it should tell us whether or

not we are able to obtain meaningful eigenvectors for the nonradial modes in the deep interior

of the star. Table 1 gives some of the details about the model. In table 1 q is the mass fraction interior to a given point in the star. The Iben fit to opacities

and equation of state data is used in the deep interior so that our model will track that of Becker.

Near the surface we used the Stellingwerf fit since it provides better detail in the hydrogen and helium ionization zones. The evolution models typically do not go to temperatures lower than about 100,000K. The mass contained in the inner, inert ball is approximately equal to that of

the shell next to it.

142

III. S T A B I L I T Y ANALYSIS

Even though our model is slightly cooler than observed variable stars in this region of the H-R diagram we still expect to obta in modes that are unstable since, as is well known unless a

proper t r ea tment of convection is included the red edge is not found with our pulsat ion models. In Table 2 we present an analysis of our model for the first three radial modes and I = 2, g modes. We see in table 2 that as in previous work the radial modes tend to be more unstable as we go to higher order modes. The periods here are about a factor of two t imes those for 6 Scuti. This is to be expected with our larger mass and radius. The g modes indicated are all unstable. Higher

order modes, as well as those of lower order than gs are found to be stable. For modes near the s table-unstable boundary region growth rates tend to be somewhat unreliable. The reason

for the band of unstable modes is that at low order the eigenvector samples the hydrogen and first hel ium ionization driving but not the second helium driving. Higher order modes are more unstable since the second hel ium ionization driving region is sampled. At the very highest order modes, above about g16 you sample the most radiation damping at T > 70,000K and damping

in the Iz gradient region near the center. Figure 1 is a plot of the radial eigenvector for the g8 mode. It does appear tha t we have made a marked improvement over previous calculations.

The nodes near the center are well defined and do not exhibit the noise evident in the work of Clancy and Cox. The work plot shown in Figure 2 is of interest. We see that the driving is coming from the outer ionization of hydrogen and helium. The double outer peak is primari ly an equat ion of s t a t e ( r ) effect due to the ionization of hydrogen and first ionization of helium. The details of the driving and damping are of course a complicated interact ion of the kappa and

gamma effects.

I .[:X~3

| _~ 0.875

!

g "'~ 0.375 ...) o

o c o 0 .12S c

0 . 0 0 0

f . q,o

-0 . i 2 5 I l i I

50 t O0 t SO 2O0

u ' " i •

I I I I I

2S0 300 350 400 450 500 550

Zone

Figure 1. 6r/r vs. zone number for the g8 nonradial mode.

143

C3 -- )0.0

• --" 2o.0 |

is.o

~:L I0 .0

o~ 5 . 0 O~ (_

w. o.o

r 0 - 5 . 0

t.. - 10 .0 0

- 15 .0 0 50 I00 150 2.00 250 300

Zone 350 400 450 500 550

Figure 2. W o r k / z o n e vs. zone n u m b e r for the gs nonrad ia l mode.

IV. S U M M A R Y AND C O N C L U S I O N S

We have used a n improved stel lar evolut ion model , w i th its deta i led compos i t ion s t ruc tu re to look a t the nonrad ia l behav ior of a s t a r nea r the region in the H-R d i ag ram where the ~ Scuti

variables are found. We are able to o b t a i n sa t i s fac tory solut ions to the l inear nonad iaba t i c

equat ions and in tend to use newly ca lcula ted evolut ion models to fu r the r s t udy the pulsat iona] s tabi l i ty of th is class of var iable s tars . Lee (1985) has recent ly ca lcu la ted & Scuti models for

/=0,1,2 and 3 modes . He finds a large n u m b e r of uns tab le modes. W i t h all the work done to

date , there still appea r s to be a conflict be tween the n u m b e r of ca lcu la ted uns t ab l e modes and

those observed.

T A B L E 1

& Scuti "like" Model

M = 3 M o , L = 6 4 . 6 L o , T e n = 6 1 6 6 K

Compos i t ion S t ruc tu re

Surface to q = 0.57 q = 0.57 to 0.15

q = 0.15 to 0.09

q = 0.09 to 0.07

q = 0.07 to 0.00

X = 0.69, Z = 0.03

X = 0.69 to 0.67

X = 0.67 to 0.27

X = 0.27 to 0.00

X = 0.00

Mate r ia l P roper t i e s

Stel l ingwerf fit f rom surface to T = 300,000K

Iben fit f rom T = 300,000K to center

144

Table 1 continued

Convection

Mixing length with I/Hp = 1.0

Central Ball Surface

M = 2.85 x l O - Z M , L = 8.17 x 1 0 - 3 L ,

T = 4.5 x 107K

R = 1.2 x l O - S R ,

TABLE 2

Radial Modes

Mode Period(days) Period(hrs) ~/ Q(days)

F 0.3765 9.036 4.3 x 10 - s 0.0347 1H 0.2862 6.869 4.1 x 10 -4 0.0264 2H 0.2271 5.450 1.7 x 10 -3 0.0209

Nonradial Modes (l = 2)

Mode Period(days) Period(hrs)

gls 0.397 9.53 3.8 X 10 - 6

g14 0.349 8.38 1.7 x 10 -5

g13 0.309 7.42 1.1 x 10 -5

gl2 0.279 6.70 1.2 X 10 - 4

g l l 0.242 5.81 6.6 X 10 -4

glo 0.221 5.30 5.3 X 10 - 4

99 0.195 4.68 2.2 x 10 - z

g8 0.173 4.15 1.6 x 10 -3

V. R E F E R E N C E S

Becker, S.A. 1986, (private communicat ion) . Clancy, S.P. and Cox, A.N. 1982, in Pulsations in Classical and Cataclysmic Vari-

able Stars, Joint Ins t i tu te for Labora tory Astrophysics, p. 264.

Fitch, W.S. 1981, Ap. J. 249, 218. Lee, U. 1985, Pub. Astron. Soc. Japan 3T, 279.

145

CEPHEIDS: PROBLEMS AND POSSIBILITIES

Norman R. Simon Department of Physics and Astronomy

University of Nebraska Lincoln, NE 68588-0111

Abstract: We assess the current state of several areas in

Cepheld research. The problem of mass determination is dis-

cussed and a number of methods evaluated. ~e examine a recent

interpretation of Cepheid masses which adopts a "short" distance

scale and relies heavily on the Baade-Wesselink technique. This

interpretation requires an evolutionary dichotomy, and is critl-

eized on the basis of the great regularity observed among

Cepheid light curves. The homogeneity of the Cepbeld sample is

further invoked to argue against the introduction of physical

mechanisms which increase the complexity of stellar evolution.

We touch briefly on the question of opacity and mention some

benefits which would be derived from an increase in absorption

by heavy elements. Finally, we discuss the implications ("long"

distance scale and "cool" temperature calibration) arising from

the recent determination of the mass of SU Cyg from its orbit.

The importance of verifying this mass is emphasized.

I wish to begin this review and to spend most of it discussing the

difficulties. The problems that have been around for 10 or 20 years

are still with us. These include possible conflicts between evolu-

tion and pulsation theory, and the inadequacy of our hydrodynamic

codes in modeling the phenomenon of double mode pulsation and in

reproducing the details of observed light curves. A major symptom of

these difficulties is that we still cannot deduce with confidence the

masses of the classical Cephelds.

I. Methods of Mass Determination

Various methods of mass determination have been reviewed in detail by

Cox (1980). Table I lists five of these. We have divided their

requirements into the categories theory-lnterlor and observation-

exterior. The former refers to the level of theoretical modeling of

the stellar interior necessary to determine the mass; the latter

includes observational requirements, and, perhaps, some modeling of

148

the atmosphere. In all cases, the pulsation period (let's say, the

fundamental mode period, Pc) must be measured - a relatively

Stralght-forward task. TABLE I

Methods of Mass Determination

M e t h o d Theory-Interlor Observatlon-Exterlor

EVolutionary M-L relation from tracks Pc, Te (or L)

Wessellnk Period-p relation Pc, R

Pulsational Perlod-p relation Pc, L, T e

Beat Accurate linear model Pc, PI

Bump Accurate nonlinear model Accurate light curve

The Evolutionary mass determination (note that Cox 1980 called this

the "theoretical mass") employs a mass-lumlnoslty relationship

Obtained from evolutionary tracks which pass through the instability

strip. In addition to this, an observational quantity is required,

USUally the temperature, T e. It is ironic that MEV, which is usually

taken as the standard of comparison for the various mass

determinations, is perhaps the most difficult of all to obtain and

the least well-known. It depends not only on a measured temperature

which may be off by a few hundred degrees, but is also subject to all

the physical and numerical uncertainties which may lead to errors

anywhere along the evolutionary track.

The Wessellnk and Pulsatlonal mass determinations may be called

"exterlor-llnked", since their interlor-theoretlcal underpinning

Conslsts merely of the perlod-mean density relation (Cox 1980). The

Wessellnk method attempts to deduce the Cepheid radius from the ob-

Served variations. It requires reasonably accurate, and preferably

Simultaneous, light and veloclty curves (certainly attainable with

mOdern techniques), as well as some theoretical construct linking

them. Various versions of this technique (e.g., Caecln, et.al. 1985;

COulson, Caldwell and Gieren 1985) have sometimes yielded eonflletlng

results, and the accuracy of these determinations remains unknown.

The Pulsatlonal mass determination requires, on the observational

Side, a measurement of the luminosity and the temperature. The prob-

lems involved with this were discussed i n detail at the Toronto

meeting, particularly in the review by Pel (1985). Once more, dif-

ferent investigators have obtained different values for these quanti-

ties; the Cepheid luminosities, especially, remain controversial. We

Shall return to this point later.

149

The "interior-linked" masses are MBEAT and MBUMP. The former makes

use of the dozen or so known classical Cepheids which pulsate stably

and simultaneously in the fundamental and first overtone modes.

Observatlonally, only the two pulsation periods (Pc and PI ) are re-

quired, and these may be obtained with high accuracy. These two

observed quantities may then be made to yield both M and L via the

Petersen (1973) diagram which displays various M-L loci on a plot of

PI/P o vs. Pc" However, the M-L loci are very sensitive to the physi-

cal properties of the models which produce them, e.g., opacity, mag-

netic fields, composition anomalies, etc. Thus, a physically

accurate and trustworthy linear pulsation calculation is indlspen-

sible for the beat-mass determination.

The Bump mass is calculated by modeling of the Hertzsprung progres-

sion, i.e., the dependence on period of a secondary maximum appearing

on the light curves of classical Cepheids with 6 ~ Pc ~ 18d. It was

this mass determination by Christy (1968) and Stoble (1969) that gave

the earliest hint that, despite the larger Hyades distance, all might

not be well in our understanding of the Population I Cepheids.

Observatlonally, the determination of MBUMP requires accurate light

curves with good phase coverage. These are readily available now in

the literature thanks to the recent work of Pel (1976), Moffett and

Barnes (1980; 1984) and others. The crudest application of this

method must include some definition of the bump phase, as well as a

nonlinear pulsation model which also (hopefully) produces a bump.

However, one can refine this mass determination using the idea that

the bump phase is governed by an accidental period resonance and thus

depends upon the period ratio P2/Po, where P2 is the period of the

second overtone (Simon and Schmldt 1976). For classical Cephelds,

the resonance center, P2/Po = 0.5, occurs near Pc = 10 days. A fur-

ther refinement may be obtained by employing Fourier decomposition,

rather than bump phase, to quantify the shape of the light curve. In

this technique, one fits observed and/or theoretleal light curves

with a Fourier series

V(mag) = A o + A I cos(~t ÷ ¢I) ÷ A2 cos (2~t ÷ ~2)

÷ A 3 cos (3~t + ¢3 ) + ....

and describes the structure of the light curve using combinations of

the low-order coefficients (Simon and Lee 1981):

R j l ~ Aj/AI , ¢ji ~ ~j-J¢1.

Figure I shows the plot vs. period of the Fourier quantities R21, ¢21

and ¢31 (Simon and Moffett 1985) for the V-magnitude observations of

Moffett and Barnes (1980;1984). The shape of these plots is governed

150

05 ........... l I l I I l

0.4

02

Oo

i , o•

• • t ,

, m i i

oo • og % o e

t

- - . ~ ~ i • I L . ~3 2 0 3 0

P (days)

I I

> 60

50

40

L ~ 3o~ 4O 5O

I I I I ]

o o •

6"

DT o

P { doys )

50

=,

II0

9~

70

5.0

3.0

I.O

i i [ ~ i I

• o •

• t • ~oo

i r

~) I , L~ 1 3o I L I 0 1

P (doys) 50

Fig. I . Fourier quantities R21, @21 and @31 vs. period.

151

by the resonance whose presence appears strikingly near 10 days. If

we had h~_~[£dynamle models whose ~h~sics and numerics we could trust,

then M, L and Te could be found to high precision by matching Fourier

quantities from observed and theoretical li~_~urves. The observa-

tions necessary for this are already in place.

2. Critt£ue of a Recent Mass Determination

Recent evidence has seemed to point to the conclusion that the luml-

nosity of Population I Cepheids is perhaps considerably lower than

has been believed since the change of the Hyades distance scale 20

years ago. Independent lines of evidence supporting this idea come

from recent studies of star clusters in the galaxy (Schmldt 1984;

Walker 1985a, b) and in the LMC (Schommer, Olszewski and Aaronson

1984) and from an investigation of Cephelds with giant companions

(Bohm-Vitense and Proffitt 1985).

Using the Schmldt (1984) distance sea/e, along with Wesselink masses

averaged in bins of size &log P = 0.1, Bohm-Vitense (1986) has found

the following relationships among the various mass determinations:

For P > 6d: MEV > MWE S m Mpu L > MBUMP

For P < 6d: MEV ~ MWE S > Mpu L > MBEAT.

Thus for P > 6d, the two "exterlor-llnked" mass estimates agree. The

masses determined by Bohm-Vitense and Proffitt (1985) from Cepheids

with giant companions also agree with the Wessellnk and Pulsatlonal

masses. However, the evolutionary masses (calculated from standard

tracks, e.g., Becker 1985) are considerably higher than MWE S and

MpUL, while the Bump masses are somewhat lower.

Turning to the shorter-perlod Cepheids (P < 6d), Bohm-Vitense finds

higher Wessellnk masses, approaching the Evolutionary values. The

Pulsatlonal masses are now considerably smaller, and the Beat masses

smaller still.

Note that the two "interior-llnked" masses are lowest in their re-

spective period ranges. This is somewhat misleading, however, be-

cause what is really meant is that standard interior models yield

period ratios larger than those actually observed. The observed

values are:

For the Beat Cephelds: PI/Po m 0.70 at Pc = 3-4 d.

For the Bump Cephelds: P2/Po m 0.50 at Pc m 10 d.

While the discrepancy can be corrected by lowering the mass of a

model at given luminosity, the period ratios, as mentioned above, are

also very sensitive to the interior physics. Thus MBEAT and MBUMP

152

Can be raised to the Evolutionary values if one is willing to accept

and include changes in the opacity law, composition anomalies, mag-

netic field effects, etc.

If, following Bohm-Vitense (1986), one accepts the average Wesselink

masses as correct, then one finds that the Cephelds with P < 6d and

those with P > IOd have similar masses, say between ~ and 6 M@.

While adopting low masses for the long-perlod stars solves the prob-

lem of an overabundance of these objects relative to standard pre-

dictions (Becker, Iben and Tuggle 1977), it requires an evolutionary

dichotomy (Bohm-Vltense 1986) whereby the longer per iod Cepheids

(say, P > 8d) cross the strip at a considerably higher luminosity

than do the shorter period stars (say, P < 6d).

In the opinion of this reviewer such a scheme does not seem tenable.

To argue this let us consider once more the Hertzsprung sequence and

its dependence upon the period ratio P2/Po . We remark, first, that

this dependence is well-established. Not only do bumps appear on

hydrodynamic light and velocity curves when P2/Po ~ 0.50, but the

lOW-mass, short-period Type II Cephelds, which lle in another regime

of the H-R diagram in which P2/Po ~ 0.50, also display light curve

bumps. Finally, the relationship between the resonance P2/P o = 0.5

and the bumps has been shown directly in nonlinear expansion theo-

rles, e.g., Klapp, Goupll and Buchler (1985).

If indeed an evolutionary dichotomy appears near, say, 7 days, then

the shorter period stars will have a larger value of P2/Po (since

their M/L ratio is larger) and the longer period stars a smaller

Value. But since the shapes of the light curves and thus the values

of the Fourier quantities R21 , ¢21 and ¢31 depend upon P2/Po, we

OUght to see a discontinuity in the plots of Figure I near the evolu-

tionary transition. No such discontinuity is apparent. In fact, the

regularity in the Fourier quantities is so great that it has even

Stronger consequences.

3. ~ £ ~ £ ~ Z of the Ty£~ I Cephelds

To begin t h i s sec t i on , l e t us s e l e c t a per iod at which Cephelds are

numerous, say P = 5d, and, using standard assumptions, es t imate the

range of masses c o n t r i b u t i n g to the sample of these s t a r s . Table 2

d i sp lays the r e s u l t s of some l i n e a r nonad laba t i c (LNA) p u l s a t i o n

Ca leu la t l ons . The M-L r e l a t i o n fo r g iven composi t ion comes from the

models of Becker, Iben and Tuggle (1977) and Becker (1985). The

masses which produce 5-d@y Cepheids are seen to have the range

153

4.7 ~ M/M@ ~ 7.3. The upper cutoff comes because models more massive

than about 7.3 MS can only give P = 5d beyond the blue edge. The

lower cutoff is due to the failure of low-mass models to loop blue-

ward far enough to penetrate the instability strip. We note in pass-

ing that the large mass range indicated here hints that the scatter

among Weaselink masses in Figure I of Bohm-Vitense (1986) could be

largely real. In that case, any attempt to average the masses in

period bins would be ill-advlsed.

TABLE 2

LNA Models with Standard Assumptions, P - 5d

Model M L T e X Z Po ~2 Number M@ ~@ Po

1 7 . 3 2570 5800 0 . 6 9 0 . 0 3 5 . 0 5 0 . 5 9 6

2 6 . 0 2089 5700 0 . 7 0 0 . 0 2 5 . 0 6 0 . 5 8 6

3 5 . 0 1995 5850 0 .71 0 .01 4 . 9 9 0 . 5 7 6

4 4.7 1585 5600 0.71 0.01 4.97 0.572

The last column in Table 2 gives the LNA period ratio, P2/Po. As

mentioned previously, the period ratios associated with standard

models are, in general, much too high to agree with the Hertzsprung

sequence. However, for the present dlseusston we are interested only

in the range of period ratio which, according to Table 2, is

&(P2/P o) = 0.024 (I)

We shall now use the above range to estimate the expected scatter in

the Fourier quantity ~21 at a period of 5 days. To do this, we first

use the slope of the {21 - period diagram (Figure I) at 5-days to

find the derivative

~ 2 1 ~ 0.2. (2 ) 6P o

Next, we have calculated a series of LNA models at each mass and

luminosity and obtained from them the derivative

_ ~ n _ _ ~ 100 . (3) 6(P2/P o )

Finally, combining relations (I) - (3), we calculate the expected

range of values of ~21 at a given period, P = 5d:

A~21 = 6__~l _~__ &(P2/Po) ~ 0.5 (4) 6P o 6(P2/P o )

However, returning to Figure I, we see that the observed scatter in

#21 at 5 days Is much smaller, of the order 0.2. The actual range is

somewhat narrower still, since some of the scatter (though probably

154

not much) is d u e to observational error. This very tight relation

between ¢21 and period seems to imply that either there is some defi-

Ciency in the standard evolutionary tracks or that the range of Z

among 5-day Cepheids is considerably smaller than the one we have

Chosen. Note, also, that we did not vary Y in the above calculation.

Had we done so, it is probable that the exercise would have yielded

an even larger range than that given by Eq. (4)

Another important point that should be made here is that the great

regularity in the Fourier diagrams argues strongly against the intro-

duction of any mechanism (e.g., core overshoot, magnetic fields,

He-poor winds, mass loss) which increases the physical complexity of

stellar evolution. Such mechanisms, which depend on a variety of

Parameters and on initial states, will likely spread the M-L relation

at a given period even further and thus increase the expected spread

in ¢21 and the other Fourier quantities.

4. Op_aciA~_~nges

We have already emphasized that standard models produce period ratios

far higher than those observed. One way to reduce the theoretical

Values is by an increase in the heavy element opacity. It was shown

by Simon (1982) that an augmentation of this opacity by a factor of 2

Or 3 would suffice to bring the period ratios down near observed

ranges. We now wish to demonstrate that such an increase in opacity

Will also greatly narrow the expected range in ¢21-

Becket (1985) has shown for a model with typical parameters that

Changes of the above order in the metal opacity alter the evolution-

ary tracks very little. If this result holds generally, then the

range of masses contributing to 5-day Cepheids will be the same as

With standard models. Table 3 displays parameters from a number of

LNA models, calculated with augmented metal opacities (AMO), exactly

as described by Simon (1982). The last column shows that not only

are the period ratios substantially lower than those in Table 2, but

the range is also greatly restricted, having the value

A(P2/P o) ~0.006. The narrowing of this range is easy to understand

81nee the high Z models which show the largest P2/Po (because their

M/L ratio is largest) are also the ones which are most affected by

the opacity changes and thus have P2/Po reduced the most. The low Z

models, which have smaller period ratios to start with, are affected

less. Calculating as above, the new range of P2/Po translates into

a~ expected range of ¢21: A¢21 mO.1.

155

TABLE 3

Augmented-Metal-Opacity Models, P ~ 5d

Model M L T e X Z Pc [2 Number W e ~@ Pc

la* 7.3 2570 5900 0.69 0.03 5.10 0.552

2a 6.0 2089 5800 0.70 0.02 5.06 0.553

3a 5.0 1995 5950 0.71 0.01 4.91 0.558

4a 4.7 1585 5650 0.71 0.01 5.02 0.553

* Model is pulsatlonally stable

We need to point out here that not only arc the AMO models completely

ad hoc, but strong objections have been raised against opacity In-

creases of the needed magnitude on the basis of atomic physics and

other arguments (Magee, Merts and Huebner 1984). Nonetheless, the

high opacity models seem to have so many desirable properties that

perhaps the subject should be pursued further.

5. Possibilities

Suppose we were to adopt higher metal opacities. This would have the

interesting result of bringing the two interior-linked masses (Beat

and Bump) into llne with the Evolutionary mass. What about the

exterlor-linked masses? While the Wesselink mass determinations are

generally viewed as very uncertain, the acceptance of a new, "short"

distance scale would now leave the Pulsation mass anomalously low.

An important result that bears on this question has recently been

reported by Evans and Bolton (1986a,b). Based upon extensive ground-

based and IUE data, these authors have determined the mass of the

classical Cepheld, SU Cyg, from orbital velocities in the triple

system of which this star is a member. The value they obtained is

6.3 M 8 (with quoted errors of +0.7 and -0.4 M@, respectively). If,

following Evans and Bolton (1986b), we use the "long" distance scale

of Caldwell (1983), then the luminosity of SU Cyg is log L/L@ ~ 3.21,

and a model with M m 6 M@, T e ~ 6000 K, Z ~ 0.02, Y = 0.25 will sat-

isfy the constraints imposed by both the orbital mass determination

and the standard evolutionary tracks.

However, more needs to be said on this subject. Moffett and Barnes

(1985) give mean colors for SU Cyg, <B>-<V> = 0.57; according to

Evans (1986, private communication), this value needs to be increased

by 0.04 in order to subtract the effect of SU Cyg's companions.

156

Folding in the reddening E(B-V) = 0.12 (Dean, Warren and Cousins

1978) we obtain an unreddened color (B-V) o = 0.49; which, in turn,

Yields a temperature for SU Cyg, T e m 6200 K, according to the recent

calibration of Teays and Schmldt (1986). We may now calculate a

Pulsational mass for SU Cyg once a luminosity has been assigned.

Employing, as above, the PLC relation of Caldwell (1983), an LNA

PUlsation integration yields Mpu L ~ 4.2 M e.

This mass is only about 70% of the quoted ~we~ limit for the Evans-

Bolton orbital mass. One way of increasing Mpu L is by the adoption

of a lower reddening for SU Cyg. The value given above is an average

of the results of a number of investigators. The smallest reddening

of those tabulated by Dean, Warren and Cousins (1978) is E(B-V) =

0.08. If we use this value, the Teays-Sehmidt temperature falls to

Te ~ 6000 K and Mpu L rises to 5.9 M@, provided we again adopt a luml-

nosity from Caldwell (1983). In this case, the orbital, Pulsatlonal

and Evolutionary masses are all brought into accord.

The above agreement is important because it could be achieved only by

adopting quoted limits for all relevant parameters, i.e., the minimum

orbital mass from Evans and Bolton (1986a,b), the minimum reddening

from Dean, Warren and Cousins (1978) and, most notably, the Teays-

Schmldt temperature calibration which is essentially the "coolest" in

the literature, and the Caldwell distance scale which is essentially

the "longest". For example, choosing the Pal (1976) temperature

calibration, with everything else as above, reduces Mpu L to about 4.2

M@; similarly, dimming SU Cyg by 0.4 mag (to represent a very "short"

distance scale) forces Mpu L down to the extremely small value, 3.2

M@. Thus, the orbital mass of SU Cyg appears to set meaningful llm-

its on the observational determination of both luminosity and tem-

Perature. It is clear that verification or refutation of the

Evans-Bolton result should have a high priority.

The author wishes to thank N.R. Evans, E.G. Schmldt and T.J. Teays

for helpful discussions, and to acknowledge support from the National

Science Foundation under grant number AST-8316875.

~rerences

Backer, S.A. 1985, in C~phelds: Theor~ and Observations, ed. B.F. Madore (Cambridge: Cambridge University Press), p. I04.

Beaker, S.A., Iben, I. and Tuggle, R.S. 1977, A~. J. 218, 633. BOhm-Vitense, E. 1986p A~. J. 303, 262. BOhm-Vitense, E and Pr~ffi~t, Ch. 1985, Ap. ~. 296, 175. Caccin, B., et.al. 1985, in ~pheids: Theor~ and Observations, ed.

B.F. Madore (Cambridge: Cambridge University Press), p. 43. Caldwellp J.A.R. 1983, The Observatory 103, 244. Christy, R.F. 1968, Quart. J.R.A.S. 9, ~7

157

Coulson, I.M., Caldwell, J.A.R. and Oieren, W. I~85, in Cepheids: Theorz and Observations, ed. B.F. Madore (Cambridge: Cambridge University Press), p. 48.

Cox, A.N. 1980, Ann. Rev. Astr. Ap. 18, 15. Dean, J.F., Warren, P.R. and Cousins, A.W.J. 1978, MNRAS 183, 569. Evans, N.R. and Bolton, T.C. 1986a, in Stellar Pulsation: A Memorial

to John P. Cox (Berlin: Springer-Verlag), this volume. 1986b, in New Insights in Astrophysics: Eight Years of UV

A ~ [ ~ £ m ~ , with IUE, ESA (SP-263) . Klapp, J., Goupil, M.J. and Buchler, J.R. 1985, Ap. J 296, 514. Magee, N.H. Merts, A.T. and Huebner, W.T. 1984, ~, J. 283, 264. Moffett, T.J. and Barnes, T.G. 1980, Ap, J. Suppl, 44, ~27.

1984, ~. ~. Suppl. 55, 389. 1985, Ap. J. Suppl. 58, 843.

Pel, J.W. 1976, Astr. Ap. Suppl. 24, 413. 1985, in £~pheids: Theory and Observations, ed. B.F.

Madore iCambrldge: Cambridge University Press), p.1. Petersen, J.O. 1973, Astr. Ap. 27, 89. Sehmidt, E.G. 1984, ~. J. 285, 501. Schommer, R.A., Olszewskl, E.W. and Aaronson, M. 1984, Ap. J. Lett.

285, L53. Simon, N.R. 1982, ~. J. Lett. 260, L87. Simon, N.R. and Lee, A.S. 1981, Ap. J. 248, 291. Simon, NR. and Moffett, T.J. 1985, PASP 97, 1078. Simon, N.R. and Schmidt, E.G. 1976, Ap. ~. 2~5, 162. Stobie, R.S. 1969, MNRAS 144, 485. Teays, T.J. and Schmidt, E.G. 1986, in Stellar Pulsation: A Memorial

to John P. Cox (Berlin: Springer-Verlag), this volume. Walker, A.R. 1985a, MNRAS 213, 889.

• 1985b, MNRAS ~!~, 45.

158

A Possible So lu t ion to the C e p h e i d Mass P r o b l e m ? Erika BShm-Vitense

University of Washington, Seattle, Washington

A b s t r a c t With new, smaller distances of the Cepheids, as determined recently by Schmidt (1984) and by

BShm-Vitense (1985), smaller pulsationa] masses are obtained than previously. Giant companions of Cepheids show that the luminosities of the Cepheids are too large in comparison with those of the giants. If increased mixing, for instance by convective overshoot at the boundary of the convective core during the main sequence stage, is responsible for this, then we expect an increase in the luminosity of the Cepheids of a given mass by approximately a factor of 4 as compared to conventional evolution calculations. Taking into account both of these effects we find good agreement between the corrected evolutionary masses, the pulsational masses, the dynamical raasses, the giant companion masses and the Wesselink masses. The bump masses are only slightly smaller than the other masses.

I. I n t r o d u c t i o n In the early 70's the previous differences between evolutionary and pulsational masses of the

Cepheids disappeared due to an increase in the adopted distances and a decrease in Tell (iben and Tuggle 1972, Cox 1980). The beat and bump masses still c a m e out much lower and so did the Wesselink masses for most of the long period Cepheids (P > 9 days).

The new distance determinations by Schmidt (1984) based on trigonometric parallaxes and on StrSmgren photometry yields now smaller distances, so did the distance determination from raain sequence companions of Cepheids by BShm-Vitense (1985). With these smaller distances the pulsational masses come out much smaller and approach the bump masses (Schmidt 1984). This opens up again the gap between evolutionary and pulsational masses.

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, ~ ,_ , ,:::~ . , ,= . . . . ~ ' , , ! . , , : ' , , ,

4.Z 4 .0 3 .8 3.6 4 .4 4 .Z 4 . 0 3-8 ~.6 4 .4 4 .Z 4 . 0 ~.8 ~. l l 4 ,4 4 . t 4 . 0 3 .8 3 A

Figure 1. Evolutionary tracks of intermediate mass stars with different chemical abundances (given at the bottom of the figures) as given by Becker, Iben and Tuggle 1977, are shown in the luminosity, Teff diagrams. Also shown are the observed positions of the Cepheids and their giant companions. No matter which chemical composition we are considering the Cepheids are always too luminous in comparison with their giant companions. (From BShm-Vitense and ProflFitt 1985).

159

Ralher uncer*am determinat ions of dynamical masses for S Mu~ and V 636 Sco neverthe- les~ ~upport the lower mass determinat ions for Cepheids [BShm-Vitense 1986), as do the mass det, ernLinations from giant companions of Cepheids (BShm-Vitense and Proffitt 1985).

When comparing the relative positions of Cepheids and their giant companions in the HR diagram we see that the luminosities of the Cepheids are too bright in comparison with those of their giant companions, see figure 1. Since the evolution t imes along the giant sequence are so short the Cepheids and their giant companions must have nearly the same mass, they should therefore fall essentially on the same evolutionary track, but the Cepheid luminosities are too high by about a factor of 2.5 in comparison with the giant companions. This discrepancy can not t~e explained by mass loss which would reduce the luminosity.

So far the general a t t i tude with respect to these problems has been: the theory must be correct , so let us see what might be wrong with the reductions and interpretat ions of the obser- vations. Yet there are large uncertaint ies in the theory, especially with respect to mass loss and mixing in early stages of stellar evolution. Semiconvection as well as convective overshoot mixing are not yet well understood. I would therefore here like to t ake ' the opposite approach and say: if we take the observations at face value what would they imply for the theory? Especially what would they tell us about stellar evolution and about the Cepheid mass problem?

II. Mix ing in Early S t a g e s o f S t e l l a r E v o l u t i o n An increase in relative luminosity of the Cepheid with respect to the giant stage is obtained

in stellar evolution calculations if convective overshoot mixing at the boundaries of the convective cores of massive stars is taken into account for the early stages of stellar evolution (Backer and Cox 1982) as seen in figure 2. We also see that due to this mixing the luminosity at the giant stage is also increased by about half the amount in Alog L, the increase in log L between Cepheid and giant. The observations yield Alog L -~ 0.4. For a given mass the luminosity of the Cepheid is therefore es t imated to be larger by Alog L (mass) = 0.6 4- 0.2 as compared to conventional evolution calculations.

• .............. . . %

~~ 9u~¥,2~lC.2e, O.C3~ . . . . ~] Figure 2. The theoretical evolutionary 42 .... ~0RVAL / track for a 9 M~, (Y,Z) = 0.28, 0.03) 4. - - %0"°'05 ~---~___._ / model is shown in the W~s, Luminosity di-

agram for the s tandard evolution (dashed 4( " , ~ - - - . , l i ne )and f o r a model with convective over-

. / "-.< ..... ] shoot (solid line}. Note how convective

/ ". ' . . ~ overshoot lengthens the main sequence .~3 ° - ..... ~...- . ..__ ;~ J ". ~ 2 ' i : : tracks and results in brighter post main _ sequence evolution. F rom Backer and Cox g ~ / , ......... :::(J (1982). 3 . 7 / ".

3 6 • .

3 . 5 "

,og 1~

We tentat ively use Alog L (mass) -- 0.6. Using hecker, Iben and Tuggle 's s tandard relation this correction leads to the tenta t ive mass luminosity relation

(1} log M = 0.219 log L - 0.099

This change reduces the evolut ionary masses. The pulsational masses depend strongly on the adopted distances. We use both P.C.L. rela-

tions, the one given by Schmidt

(2) Mv = - 3 . 8 log P + 2.70(< B >o - < V >o) - 2.21

a n d o u r s

(3) Mv = -3 .425 log P + 2.52 < B - V >o - 2 . 1 6

160

For the actual determination of the pulsational mass we use the relation given by Becker, Ihen and Tuggle (1977)

(4) log PF = 0.713 + (-3.34 - 1.10A log L - 0.16(A log L)2)A log Te~

+( -0 ,62 - 0.26A tog L + 1.9A log Tee)A log M

+(0.85 + 0.07A log L)A log L

where A log Tefr = log Te~ - 3.75. We use the Teft(B-V) calibration given by us (BShm-Vitense 1981).

III . N e w Cephe id Masses In figures 3 and 4 we compare the evolutionary and pulsational masses obtained with the above

relations with Cepheid masses obtained in other ways. There is essentially no mass discrepancy a n y more except for the low beat masses.

SU Cyg appears to be a triple system, with the companion being a close binary system itself. The mass ratio of the close binary system to the Cepheid is approximately one. The dynamical taass of SU Cyg depends on whether the close binary consists of a B9V and a much less massive Star or whether it is a B7.SV star with an A1V companion, (see Bolton and Evans this volume). The lower mass limit given in Figures 3 and 4 assumes the first case with M(B9V) : 2.9 Ivl e and M(?) = 0.6 M e. The upper mass limit given assumes the second possibility with M(A1V) = 2.3 Mo and M(B7,5V) = 3.2 M e (see BShm-Vitense 1986, Figure 5).

I would like to suggest that convective overshoot mixing is very important for massive stars and that the masses of the Cepheids for periods around 10 days are close to the bump masses. It also appears that the masses of most of the long period Cepheids are not much larger than those of the shorter period Cepheids and are indeed close to the Wesselink masses.

The good agreement of all the mass determinations also argues in favor of the smaller dis- tances.

The smaller masses for most of the long period Cepheids would atso explain the discrepancy in the observed and theoretical ratios of the numbers of long period and short period Cepheids (Becket, ]ben and Tuggle 1977).

6

- T , , , i I , , ' 1 1 ' i w P~ p D~r ~ Gem

distances f r o m e q u a t i o n { I )

, , - - i ¸ ,

o a • 6

o *

- - D . . . . . . . b u m p

o - o

. b e a t

t '~ ci, i i c ,~ vi , .~ s,,,, s u . sv ~ , , w ~

..i..L, , I , i l , , , l i I , , il , ,

. 8 .8 1 to i Per iod I.Z

I l ' . . F - ~

i C*r -

o ] 0

• I l v a L

o pul / : 6

• dy'a.

* l l teult camp,

l . l 1.6

' ' ' q ' ' ' 1 1 ' AI e~ ~ ~ r t r,,,*m i ~lr

$ c h r a l d t d l i t a n c e l

o o o

u - _ _ _ u . . . . . . . . barap - evoL

o p u l l . a W,Im~

be.at * dye.

t ~ , ° " , 1 , i l , , , l l 1 , , l i , , , I , , ,

.6 .8 I Io i P t r to t t 1,1 l , t

I

1 1 .6

Figures 3 and 4. Evolutionary and Pul~ational masses obtained with our corrected distances and mass luminosity relations are compared with Cepheid masses obtained in other ways as indicated in the figures. In figure 3 we show the results obtained with our P.C.L. relation, and in figure 4 we show the results obtained for Schmidt's P.C.L. relation, (notice the difference in scale). The agreement is equally good for both P.C.L. relations except for the longest period Cepheids for which our P.C.L. relation gives better agreement between the difference masses.

1 6 1

References Becker, S. A., and Cox, A. N. 1982, Ap.J., 260, 707. Becker, S. A., Iben, I., and Tuggle, R. S. 1977, Ap.J., 218,633. BShm-Vitense, E. 1985, Ap.J., 206, 169. BShm-Vitense, E. 1981, Ann. Rev. Astr. Astrophys., IO, 295. BShm-Vitense, E. 1986, Ap.d., 303, 262. B6hm-Vitease, E. and Proffitt, C. 1985, Ap.J., 296, 175. Cox, A. N., 1980, Ann. Rev. Astr. Astrophys., 18, 15. Iben, I. and Tuggle, R. S. 1972, Ap.J., 173, 135. Schmidt, E. 1984, Ap..I., 285, 501.

Acknowledgements This research was supported by NASA grant No. NSG-5398,~which is gratefully acknowledged.

162

The Mass of the Classical Cepheld SU Cygni

Nancy Remage Evans 1,2,3 and C. Thomas Bolton 2,3

Abstract

~adial velocity differences between the classical Cepheld SU Cyg and a hot Companion have been measured from IUE high dispersion spectra. The blue star is itself a member of a close binary system, The triple system solution provides the ratio of the Cepheid mass to the stnn of the masses of the companions. IUE low dispersion spectra are used to infer the masses of the companions. A mass of 6.3 ~{® is derived for the Cepheid, with a range from 7.0 M® to 5.9 ~'f®, which is in agreement with the evolutionary mass.

Introduction

Disagreement between evolution and pulsation masses of classical Cepheids is a persistent problem which represents an uncertainly in the basic parameters of these stars. Accurate radial velocity differences between a binary Cepheid

and its blue main sequence companion can be measured on IUE high dispersion long Wavelength spectra. The IUE data can be combined with ground-based data to provide the orbital velocity amplitude ratio, and hence the mass ratio between the two Stars. Radial velocity studies of two other Cepheids have recently been discussed by Bohm-Vitense (1986).

Results: High Dispersion

SU Cyg is a particularly important case, since it has an orbit with a velocity a~plitude more than twice the amplitude of any other known Cepheid orbit. Kadial Velocities were measured on IUE high dispersion long wavelength spectra taken by

Us and by Bohm-Vitense. The relative velocities between the Cepheid and the hot ¢°~panion were corrected to an absolute system using the orbital and pulsational Velocities of the Cepheid from the ground-based orbit (Evans, 1987),

The IUE spectra revealed that the blue companion is itself a member of a short Period binary. The velocities of the blue star from 7 IUE spectra taken at care- fully selected phases in the long and short orbital periods give an excellent fit

to a circular orbit with a period of 4,7 days with a standard deviation of 2.3 ~/sec. The mass ratio from the solution is:

_M[(co~pa~{ons) = 0,919 ~ 0.022 M(Cepheid)

The data and the solution for the short period orbit are shown in Figure I.

Kesults: Low Dispersion

In order to derive the mass of the Cepheid from the mass ratio, the masses of the two companions must be determined. An IUE SWP low dispersion spectrum has been

fitted with spectra of standard stars representing the Cepheid and both companions to investigate the preperties of the companions. The method used is the same as

described in Evans and Arellano (1986), Table I lists the images used.

i Computer Sciences Corporation, IUE Observatory

2 David Dunlap Observatory, University of Toronto

IUE Guest Observer

163

Table I. Low Dispersion Images

Star Phase Spectral SWP LWR Type

SU Cyg 0.34 15304 11815 49 Erl BTV 15788 12159

18 Tau B8V 8148 7079 Gam UMa AOV 8198 7124

Since the hottest star dominates at the shortest wavelengths, the analysis was started by matching the spectrum from 1150 to 1500 A to a spectrum midway between B7V and B8V, corresponding to a temperature of 13000 K. Because the flux at 1180 A is about 25~ lower at B8V (compared with the flux at 1300 A) than is is at B7V, the

spectral type and temperature determinations are unaffected by small uncertainties in the reddening. Figure 2 shows the comparison between the SU Cyg spectrum and the composite BZV + B8V spectrum.

Adopting the composite B7.5V spectrum an an accurate match to the hottest

star in the SU Cyg system, its flux was ratioed with the SU Cyg spectrum to search for excess flux in the 1800 A region from the third star in the system. As can

be seen in Figure 2, 20~ excess flux is found at wavelengths longer than 1600 A.

This would be produced by a star with the colors and absolute magnitude of an AOV star.

Discussion

If we adopt 3.5 M e for the B7.5 star and 2.3 M e for the AOV star (Popper, 1980), then the orbital solution leads to a mass of 6.3 ~V[® for the Cepheid. The

uncertainties listed in Table 2 are estimated by considering what companions are ruled out in the spectral fitting.

Table 2. The Mass of SU Cyg

I. Uncertainties

Companions M/•,.'[ e Upper Limit B7V ÷ B9V 7,0

Best Value B7.5 + AOV 6.3 Lower Limit B8V ÷ AIV 5 . 9

M~,~/M(~ t~p.t/Mo

II. Comparisons

Mv My Caldwell Schmidt -3.36 -3.11

5.8 5.5 4 . 7 3.5

evolutionary mass

pulsation mass

164

0

-20

-40

- 80

, , , , ,

I ! '

0 PHASE

Figure I. Radial velocity orbit of the short period system. The radial ve-

locities of the blue star have been corrected for the orbital motion of the long period orbit, and are shown as asterisks. The solid line shows the circular orbit for the short period motion.

><

1

0 1200 1~00 1400 1500 1(100 1"/00 11~00 1@00 20U~

WAVELENGTH A

Figure 2. Comparison of the spectrum of SU Cyg (solid line) and the B7.SV Spectr~Lm (dotted line). The spectrum of the AOV star Gam UMa is shown at the bot-

tom. Flux is in units of 10 -12 ergs cm -2 sec -I A -I. All spectra have been scaled

for comparison with the SU Cyg spectrum. A ten point boxcar filter has been used

~o smooth the data. Geocoronal emission has been removed from the center of the

LF ~ in the SU Cyg spectrum.

165

The Cepheid mass can be compared with the evolutionary mass from Becket, Ibsn, and Tuggle (1977, Z = 0~02 and Y = 0.28). Evolutionary masses for two luminosity calibrations (Caldwell. 1988 and Schmidt, 1984) are shown in Table 2. The temper-

ature used in Table 2, 6313 K, was calculated from E(B-V) = 0.12 (Dean, Warren, and Cousins, 1978), a Kraft temperature scale as discussed by Cox, (1979), and

a small correction to B-V due to the effect of the companions. Pulsation masses from the pulsation constant as parameterized by Cox (lg79) are also listed in Ta- ble 2. The evolutionary mass for the brighter luminosity scale is in agreement

with the orbital mass found here. Since small uncertainties in the luminosity and

temperature affect the pulsation mass much more than the evolutionary mass, the pulsation mass is the least well determined of the three, but is is smaller than the observed mass for reasonable values of luminosity and temperature,

Acknowledgements

We are happy to thank the Director of David Dunlap Observatory for generous and repeated allocation of telescope time and Ron Lyons for the timely determina-

tion of the triple system solution using the D. D. O. version of the program SBC~

Financial support was provided by a NASA IUE grant (NASA contract NAS 5-28749 to CSC) and a Helm Travel grant (University of Toronto) to Dr. N. R~ Evans, and NSERC

grandt to Drs. J, R. Percy (for NRE) and C. T. Bolton.

References

Becket, S. A., Iben, I., and Tuggle, R, S. 1977, Ap. J., 218, 633, B6hm-Vitense, E. 1986, Ap. J., S03, 262. Caldwell, J. A. R. 1983, The Observatory, 103, 244. Cox A. N. 1979, Ap. J., 229, 212. Dean, J. F., Warren, P. R., and Cousins, A. W. J. 1978, M. N. R. A. S., 183, 569. Evans, N. R. 1987. in preparation.

Evans, N. R. and Arellano-Ferro, A. 1986, this conference. Popper, D. M. 1980, Ann. Reu. Astr. Ap,, 18~ 115. Schmidt, E. G. 1984, Ap. J., 285~ 501

166

BM CAS: ROSETTA STONE MANQUE

J.D. Fernie David Dunlap Observatory University of Toronto

For nearly two decades now we have lived with the cepheid mass

discrepancy. The mass of a classical cepheld as determined from stellar

evolution theory does not agree with that obtained from the theory of

PUlsation, while alternative methods such as an application of the Baade-

Wesselink method offer yet other values of cepheid masses. The size of

the discrepancy has changed over the years as distance scales, reddening

Scales, and other parameters have changed, but as yet we have not resolved

this problem in any generally satisfying way.

What is clearly needed is a more direct way of determining a cephe-

Id's mass, and that of course raises the hope of finding a cepheid in a

suitable binary system. It is thus startling to find that just such a

system was supposedly found and thoroughly analyzed some thirty years

ago (Thiessen 1956). That system is the 197-day eclipsing binary, BM

Cas.

BM Cas has an A7 lab primary star (Bidelman 1982) and a secondary

of unknown type. It is also a slngle-line spectroscopic binary (Popper

1977). Thiessen suggested this secondary is a 27-day classical cepheid

mainly on the basis of apparent out-of-eclipse variations in the light-

CUrve, and proceeded to derive i~s properties from the binary solution.

Unhappily his results lie well outside the range of the present contro-

Versy: a mass of 14.3~, as against modern values of 5 to 100 for a star

of this period (e.g. T Mon), and a radius of 225.50 , compared with today's

estimates of 150-1600 .

Thiessen's data were obtained with what by today's standards was

~ather unsatisfactory equipment, in a north European climate~ and in

Some cases at large airmasses. I am therefore presently engaged in re-

Observing the lightcurve (in the StrSmgren and RI systems) to provide

improved and extended data.

Meanwhile other considerations make it seem unlikely that the sec-

Ondary component of BM Cas is really a cepheid. The combination of a

less-evolved, more-lumlnous A-supergiant with a more-evolved, less-lum-

167

inous cepheid is improbable. Moreover, the cepheid pulsations should

be most prominent during the six-week primary eclipse (of the hotter

A-star), yet my observations to date show no non-geometric effects near

minimum exceeding the observational errors of about 0.008 mag. There

was also no discernible change in b-y colour during primary eclipse,

suggesting that the secondary star is quite similar to its mate in tem-

perature or that it is of too low a luminosity to be a cepheid.

Nevertheless, supergiant eclipsing systems are rare enough to make

BM Cas worth pursuing, but it is unlikely to be the long sought-after

Rosetta Stone of cepheid research.

REFERENCES

Bidelman, W.P. 1982, Inf. Bull. Var. Stars, No. 2112.

Popper, D.M. 1977, Publ. Ast. Soc. Pacific, 89, 315.

Thiessen, G. 1956, Zeitschr. f~r Astrophysik, 39, 65.

168

CEPHEIO PERIOD-RADIUS RELATIONS

t~stract

Thomas J. Moffett Department of Physics

Purdue Universitg West Lafayette, Indiana 47907

and Thomas O. Barnes III McDonald Observator g

The University of Texas at Austin Austin, Texas 78712

Using the vlsuat surface brightness technique, we have determined radii for 63 Cepheids. The resulting

P-R relation is in better agreement with the P-R relations determined from theory and the cluster Cepheids than

the older Beade-Wess~link solutions. We find no evidence that the long and short period Cepheids have different

ternpereture scales.

introduction

An accurate knowledge of Cepheid radii is essential to the understanding of their structure, mosses, pulsation

properties and luminosities. Four approaches have been used to studg Cepheid radii: Beade-Wesselink methods

(hereafter BW), theoretical models, cluster and ~ociation Cepheids, and beat/bump Cepheids. Fernie(1984)

reviewed the the P-R relations resulting from these four distinct approaches, and concluded that their

agreement could be best described as, "a sorry situation". We want to emphasize that the BW techniques are the

most direct means of radius determination since the other three require the ¢xbption of a particular theory,

temperature scale or luminosity scale. For this reason, improvements in BW solutions should be vigorously

Pursued,

B==le- Wes~.~ltnk Solutions

Following Fernie's (1 984) suggestion, the term "Baede-Wesseiink" wi l l be used in a generic sense to

describe methods which employ photometric and radial velocity data to determine a Cepheid's radius. The older 8W

SOlutions used (B-V) as a predictor of effective temperature and yield the smallest r~di i , for the long period

Cepheids, of the four methods. Evans (1980), and Benz and Mayor (1 982), pointed out that the effects of

microturbulence, which verg with Cepheid phase, influence the (B-V) color index. Bell and R(x~rs ( l 969)

showed that the changing electron pressure in the dgnamic ~tmospheres of Cepheids a~so makes (B-V) an imperfec~

Predictor of effective temperature.

169

Modern BW solutions have tried to overcome these difficulties by adopting a color index other than (B-V)

as a temperature indicator. Coulson eta[ (I 986) have clearly demonstrated how the choice of color index affects

the determined radius in the BW method. Using the same data set, they showed that the radii of Cepheids increased

by 20-35~ using (V-l) k rather than (B-V). Barnes e/el. (t 978) also demonstrated that (V-R) was a better

predictor of a star's visual surface brightness than (B-V). The BW technique appears to be valid but one must

exercise caution in selecting an appropriate color index for its application.

Visual Surface Br ightness

The visual surface brightness technique is one of several "modern" BW methods. The visual surface

brightness parameter, FV, can be expressed in the following forms;

F V = 4.2207 - 0.1V o - 0,5 Log~

F v = b + m(V-R) o

FV= LOgTeff + 0.1 B,C.

(1)

(z)

(3)

where V o and (V- R) o are the apparent visual magnitude and col6r, corrected for interstellar reddening, and ~ is

the stellar angular diameter in milliseconds of arc.

If the values of the zero-point, b, and the slope, m, in Eq. 2 are known, then the observed color, (V- R)o,

yields F V which then allows Eq. I to be solved for $, the angular diameter of the star. In the case of a pulsating

star, integration of the radial velocity curves yields the linear displacements, AD, during its pulsation cycle. The

linear diameter of the the star, D, in Astronomical Units is related to its distance, r, in parsecs and its angular

diameter, ~, in milliseconds of arc by:

D= tO'3r ~ (4)

For a radial pulsator of mean diameter Om, the instantaneous displacement from the mean, AD, can be

expressed ~:

AD + D m = t 0-3r (~ (5)

By performing a regression analysis of ~ against AD, one can solve Eq. 5 for both the distance and the mean

diameter of the star. The distance depends on both the values of the slope, m, and the zero- point, b, in Eq. 2 but the

mean diameter only depends on the slope and is independent of the adopted zero- point.

The Slope - m

Thompson (I 975) devised a method for determining the slope of a surface brightness relation but his

method provides no information concerning the zero-point. Briefly, in the Thompson method one performs a BW

170

~olution under the assumption that a relation like Eq. 2 exist, then the computed changes in surface brightness can

be compared with the color changes and thus, the slope, m, is determined.

We used the Thompson method to determine the slopes of the 63 Cepheids in our sample yielding the

following least-squares solution as a function of period:

m = -0.372 + 0.009 Log P (6) (±.oo6) (±.oo6)

showing that the slope in not a function of Cepheid period.

Gieren (1986) conducted a similar study of Southern hemisphere Cepheids and found a very weak

dependence on period which he was not ready to accept as real due to the lack of long period Cepheids in his sample

of 28 stars. It appears that the slope of the visual surface brightness relation is constant with respect to Cepheid

Deriod.

Thompson (I 975) investigated a suggestion by Schmidl ( 1971, 1973) that two temperature scales,

~pendent on period, might exist for Cepheids, but h~s analysis indicated a single Cepheld temperature scale.

Inspection of Eq. 3 shows that the VlSUal surface brightness parameter, FV, is a strong function of effective

temperature slnce the bolometric correction for Cepheids is always small. If two temperature scales exist, this

Would manifest itself as a period dependence on the slope, m, in Eq. 2. The constant slope indicated by Eq. 6 argues

in favor of a single Cepheid temperature scale

The S u r f ~ Brightness Period-I~lius Relation

The radii determined from the visual surface brightness method depend on the photometry, radial

velocities, the slope, m, and the value adopted for p, the conversion factor from observed to pulsational velocity.

We considered two cases; p = 1.31, as suggested by Parsons (1972), and the new values determined by Hindsley

and Belt (1986). They determined projection factors for velocities obtained with photoelectric radial velocity

spectrometers, which is appropriate for our velocities, and found a value of 1,36 for the long period Cepheids and

1.34 for Cepheids with periods less than 20 days.

Using the Hmdsley and Bell ( 1 986) p-values, we get the following P- R relation:

LogR= 1.131 + 0.734LogP (7) (±.033) (±.034)

For the case of constant p = 1.31 {Parsons ( 1972)}, we find:

LogR=1.110+O.740LogP (8) (±,033) (±.035)

These results are shown in Figure 1, along with Fernie's(1 984) mean relations for the "theory" and

"cluster" P-R relations.

171

25

22

~19

~6

13 0 4 0=7 ItO

p : variable case Clumters- "

' ' ' I13 ' I'6 ' LOG P

2 5 , , , ....... ; ............. , . . . . ' 7 2 p = ~31 c a ~ ' " / / / / /- j -

" Clutters

19

I g I ~ T h e ° ¢ Y * -

1.9 04 07 I.O I 3 16 19 LOG P

(a) (b)

Fig. 1. The Period- Radius relation determined, by the surface brightness method, in this paper. Case (a) uses the variable values of p suggested by Hindsleg and Bell and case (b) uses constant p = 1.3t. The mean "theory" and "cluster" relations of Fernie ( 1 984) are shown for comparison.

Our results are in much better agreement with the "theory" and "cluster" relations than the older BW

relation given by Fernie. Agreement of the four methods is stii] poor, but our modern BW solution reduces the

scatter among the different approaches.

This research was supported by NSF grants AST-8417744 (TJM) and AST-8418748 (T6B).

References

Barnes, T.0., Evans, D.$., and Moffett, T.J. 1 978, M. N/Z if. 8.., 183;, 285.

Bell, R.A., and Rodgers, A.W. 1969, M N./Z A. 5, 1 42, 161.

Benz, W., and Mayor, M. 1982, mtron, a~d/~lrophy~, ! I ! , 224.

Coulson, I.M., Caldwell, J.A.R., and Oieren, W.P. 1986,/(a. J., 303 ,273 .

Evans, N.R. t 980, NASA Yech. MemoEOd25, p. 237.

Fernie, J.O. 1984, ~o. J., Z82,641.

6ieren, W.P. 1986, Ap. J., 306 , 25.

Hindsleg, R., and Bell, R.A. 1 986, Ap. d., (in press).

Parsons, S.B. 1972, ,4,0. d, 1 74, 57.

Schmidt, E.6. 1 971, .4#'. d., 165 ,335.

Schmidt, E.6. 1 973, M N ,~. A. 8., 163, 67.

Thompson, R.J. 1 975, M. N. ,~..4. 5, 172,455.

172

THE CEPHEID TEMPERATURE SCALE 1 2

Terry J. Teays ' and Edward G. Schmidt I

Behlen Observatory

Department of Physics & Astronomy

University of Nebraska

Lincoln, NE 68588-0111 U.S.A.

The question of the temperatures of classical Cepheids has

been studied extensively in the past (for reviews, see Pel 1985

and Teays 1986), and the present study was undertaken to attempt

to resolve the disagreement between the earlier results. Our

approach was to obtain energy distributions, using spectrum

scanners, and compare them to the emergent flux predicted from

model atmospheres.

The Northern Hemisphere data were obtained with the

Intensified Reticon Scanner ( IRS ) of Kitt Peak National

Observatory. The IRS uses a two-channel Reticon array, which

allows simultaneous measurement of all of the wavelengths, and

produces a photometric quality, low-resolution spectrum. The

instrumental parameters chosen for this project yielded a scan of

1024 data points, spaced approximately 3 Angstroms apart, in the

general region of 4000 - 8000 Angstroms.

The Southern Hemisphere data were obtained with the two-

channel scanner of Cerro Tololo Inter-American Observatory.

Fluxes were measured between 3448 and 7530 Angstroms, at

wavelengths which avoided strong lines.

The sample of observed stars was restricted to well-observed

Cepheids in open clusters, viz., CF Cas (NGC 7790), DL Cas (NGC

129), CV Mon ("CV Mon cluster"), S Nor (NGC 6087), U Sgr (M25),

and EV Sct (NGC 6664). Reduction of the scans was made using KPNO

IRS standards, which are based on the Hayes and Latham (1975)

calibration of Vega. The scans were corrected for the effects of

interstellar reddening by using the reddening curves of Nandy

(1966, 1968), which were scaled using the color excesses of

1 2 Guest observer, Kitt Peak National Observatory; guest

Observer, Cerro Tololo Inter-American Observatory, divisions of

the National Optical Astronomy Observatories, which are operated

by AURA Inc. for the National Science Foundation.

173

Schmidt (1980a, b, 1981, 1982a, b, 1983). Schmidt's color

excesses were derived from Stromgren photometry of the early-type

stars in the respective clusters to which the Cepheids belonged,

and represent a distinct improvement over those previously

available for these stars.

Thirty nine energy distributions of acceptable quality were

obtained for these six stars, at a variety of pulsation phases.

The energy distributions are then compared to the emergent flux,

at each wavelength, calculated from the model atmospheres of

Kurucz (1979). The models were interpolated in temperature every

50 K, and the best fit was deemed to be ~he star's effective

temperature, at that phase. Twenty-four of the energy

distributions could be reliably matched to a model atmosphere.

The details of the observations, reduction procedures, and

the specific temperature results will be published elsewhere.

The temperatures obtained were then compared to the unreddened B-

V color index of the Cepheid, at the same phase. The color

curves used were those of Moffett and Barnes (1984) or Dean et

al. (1977), for the case of S Nor. (The color indices were

corrected for reddening by converting Schmidt's color excesses to

equivalent B-V color excesses.)

The temperature scale, i.e. the temperature-color relation,

is shown in Figure I, where the closed circles represent the

energy distributions that were judged to be reliably matched by a

model atmosphere, while the open circles represent cases for

which a reliable temperature could not be determined and only a

rough estimate was made. The open square represents the non-

variable F supergiant NGC 129 A. A least-squares fit to the

filled circles, gives the line shown in Figure i, which

corresponds to:

LOG T = 3.904 - 0.237 (B-V) eff 0

This temperature scale has a steeper slope than most of the

previously published scales, and is generally cooler through most

of the instability strip. If this scale is adopted, along with

Schmidt's color excesses and luminosity scale (Schmidt 1984), the

mean photometric parameters of Moffett and Barnes (1985), normal

solar abundances, and Faulkner's (1977) formula for Q, then the

174

3.8~

O

3.75 ~- O O

\ i I I t I .

.4 .,5 6 .7 .8 .9 (B-VI 0

Pulsation masses are still slightly lower than the evolutionary

masses, though the discrepancy is reduced compared to previous

Work.

FIG. 1 - Log of effective temperature vs. un- reddened B-V color index for all of the Cepheid energy distributions. Filled circles represent those scans which could be reliably fit to a model atmosphere, while the open circles represent scans for which the tem- peratures were uncertain. A linear least- squares fit is shown by the solid line. The open square represents the non-variable F supergiant, NGC 129 A.

REFERENCES

Dean, j. F., Cousins, A. W. J., Bywater, R. A., and Warren, P. R.

1977, Mem. R.A.S., 83, 69.

Faulkner, D. J. 1977, Ap. J., 218, 209.

Hayes, D. S. and Latham, D. W. 1975, Ap. J., 197, 593.

Eurucz, R. L. 1979, Ap. J. Suppl., 40, i.

Moffett, T. J. and Barnes, T. G. 1984, Ap. J. Suppl., 55, 389.

~- 1985, Ap. J. Suppl., 58, 843.

Nandy, K. 1966, Pub. Roy. Obs. Edinburgh, 5, 233.

.......... 1968, Pub. Roy. Obs. Edinburgh, 6, 169.

Pel, j. W. 1985, in Cepheids: Theory and Observations, ed. B. F.

Madore, (Cambridge: Cambridge University Press), pp. 1-16.

Schmidt, E. G. 1980a, A. J., 85, 158.

-~ 1980b, A. J., 85, 695.

.............. 1981, A. J., 86, 242.

"" 1982a, A. J., 87, 650.

"- 1982b, A. J., 87, 1197.

.............. 1983, A. J., 88, 104.

~-- 1984, Ap. J., 285, 501.

Teays, T. J. 1986, Ph. D. Thesis, University of Nebraska.

175

PROPERTIES OF THE LIGHT CURVES OF s-CEPHEIDS

E. Antonello and E. Poretti

Osservatorio Astronomico di Brera

M~lano-Merate, Italy

i. Introduction

According to the definition reported in the General Catalog of Variable stars, s-

CepheJds (DCEPS) are Delta Cephei type variables with l~ght amplitude below 0.5 mag (V)

and almost symmetrical light curves; as a rule, their periods do not exceed 7 days;

possibly these stars are first overtone pulsators and/or are in the first transition

across the instability strip after leaving the main sequence.

In order to study the structural properties of their light curves, we have applied the

Fourier decomposition to the s-Cepheids with good photometric observations, and we have

tried to detect possible effects which could be ascribed to the pulsation in a mode

different from the fundamental one.

2. Data Analysis and Results

We have considered a group of about thirty s-Cepheids. For nine of them we have taken

the Fourier coefficients from Simon and Lee (1981, SL) and Simon and Moffett (]986, SM)

papers, while for the other stars we have collected the published data and Fourier

decomposed the V-light curves. Moreover, in order to increase the number of short per-

iod classical Cepheids with Fourier decomposed light curves, we have considered twenty-

three classical Cepheids which are not in SL and SM samples. Here we report the main

results of the analysis. A detailed discussion of the results will be published else-

where (Antonello and Poretti, 1986). The formula used in the analysis was

V = A 0 + Z A i cos [ i ~ (t-To) + ¢i] •

Following SL, the amplitude ratios and phase d~fferences were defined Rjl = Aj/A 1 and

¢jI = Cj- J¢l" A second order fit was sufficiently good for s-Cepheid light curves;

however the significance of the third order term increases progressively with the per-

~od. Figures I and 2 show R2] and ¢21 vs. period (P) for classical and s-Cepheids; only

the s-Cepheids with the best Fourier parameters are reported in the Figures. The mean

formal errors in R21 and ¢21 for these stars are 0,028 and 0.21, respectively. The

dispersion of the points near P = 3 d in Figure 2 and the regular trends shown by the

s-Cepheids in Figure 1 and 2 are real and cannot be explained by the uncertainties in

the Fourier parameters. Also SM noted that it is possible that, in their diagrams, some

of the d~screpant points represented by stars with very short periods (s-Cepheids)

could be real.

The Figures remind us in part of the respective diagrams for classical Cepheids with

P near i0 d in SL paper, and this suggests that there should be a mechanism affecting

the pulsation of s-Cepheids which is similar in part to the mechanism responsible for

the Hertzsprung progression of classical Cephe~d light curves. This conclusion is re-

176

R 2 1

O.e

i 7 T T • ] T r • T l T T ~ i i , X ~ ' ] . . . . i % , , , i

+ ,+

+ • • ¢*

÷ • • '* + +

+ • • • •

o

2 , 4 , O, I . P

Fig. 3. The amplitude ratio R2] vs.

P (days). Crosses: classical Cepheids;

dots: s-Cepheids.

+

t l

o

l , , i i l l b l , 1 J , 1 , i t L , i 1 1 ~ 4 , , [ 1 L l l l

2, 4 . 6 , 6 . P

Fig. 2. The phase d~fference ¢2] vs. P (days). Same symbols as in

Figure I.

4,31

9.C

8.O

7.O

6.C

5.0

4.0

3.0

2.C

o~o~ ~o

/

o/ "; io j

l . / . I x / I ,~

o/o - . j . ~ . ..~ ~.~.. x

t

I l I I I i I • !

2 4 6 8 10 p(d)

I " '""' ! " ' ! " " ~

• . ~ ' e " " " e

I

./{"

Y. ./

12 14 16 18

Fig. 3. The phase difference ~31 vs. P. Dots and crosses:

classical Cepheids (see SL); open circles: s-Cepheids.

infOrced by the inspection of the ¢31 vs. P diagram for the s-Cepheids with suffi- Ciently accurate Fourier parameters (Figure 3). The main difference between s- and

Classical Cepheids ~s that the effects on s-Cephe~d light curves are not very evident,

~nd only the Fourier decomposition is able to detect them. Figure 4 shows the possible

D~ogression of the light curve shape of s-Cephe~ds with the period. At first s~ght the

177

0.0 0.5 i , 0

Phase

S~419

GH Cir

~d. 445

0.0 0.5 1.0

Phl ie

Fig. 4 . The possible progression of the light curve shape of s-Cepheids.

Differently from the Hertzsprung progression, here we consider some

samples of light curves rather than average light curves (see text).

light curves displayed in Figure 4 are not very different one from the other; however

it is possible to detect some small differences. Starting from the shortest period

(SU Cas), the light curves tend to become symmetric for increasing periods, and for

P near 3.3 d some stars have a slightly steeper descending branch than the ascending

one. After that, the light curves tend to be again asymmetric in the usual way and not

very different from those of classical Cepheids w~th similar periods. One may note ths~

the characteristics of the light curves of s-Cepheids are determined non-uniquely by

the period; in particular, the ~2] values for stars with P ~ 3 d are not related in a

single-value manner to the period. This reminds us of the case of bump Cepheids (Efre-

mov, 1975). According to Efremov, only the average l~ght curves of classical CepheidS

with close periods will abey the Hertzsprung relation, and this should also be valid

for s-Cepheids.

The results of s-Cepheid light curve analysis may be interpreted in terms of a reson -

anee between pulsation modes as in the case of classical Cepheids. We were attracted

by the possible resonance between P3 and PO (third overtone and fundamental mode, re-

spectively), P3/Po = 0.5, for Cepheid models with P near 3 d; however, there are some

indications that s-Cepheids are first overtone pulsators, and in this ease one should

search for a possible resonance between P4 and Pl (fourth overtone and first overtone,

respectively), P4/~ = 0.5. Let us remark again that, whichever the explanation of the

phenomenon is, it would have been very difficult to reveal the progression without the

Fourier decomposition, and this Js another proof of the Dower of the method reintroduce8

by SL.

178

3. S-Cepheids in External Galaxies

The study of s-Cepheids in other galaxies is important in order to verify if the stars

are really first overtone pulsators. A numerous group of s-Cepheids were found by

Payne-Gaposchkin and Gaposchkin (1966) in the Small Magellanic Cloud. These authors

report that, for a given period, these stars are more luminous than the other Cepheids

and the difference is of about 0.5 mag. This difference is easily explained by the

~Ulsation of s-Cepheids in the first overtone. The period distribution, however, shows

that there are no s-Cepheids with P~ 3.2 d in the Small Magellan~c Cloud; th~s galaxy

has also a peculiar distribution of periods of classical Cepheids. Therefore we pre-

ferred to turn our attention to the Andromeda Nebula, because the light curves of

Cepheids in this galaxy indicated the presence of some possible s-Cepheids, and the

~eriod distribution is similar to that of our Galaxy (Baade and Swope, 1965). However,

Cepheids with very short period are lacking because probably they were not detected

OWing to their low luminosity. We have attempted to construct the R21 vs. P and @21

Vs. p diagrams for the Cepheids in M31. Of course, the photographic photometry does

net allow to get accurate Fourier parameters. The preliminary results of the analysis

of the photographic light curves are not very encouraging, in the sense that it is not

POSsible to verify adequately the effects of the presence of the s-Cepheids. Differ-

ently from our Galaxy, it is possible that some s-Cepheids in M3], with P between 5

and 6 d, have higher @21 values than normal Cepheids with s~milar period. Moreover

it seems that the possible s-Cepheids are brighter than the other Cepheids with similar

De,ind. It would be important to make accurate CCD observations of the variable stars

in M31 in order to confirm these indications.

As a final remark, we note that some tests made by us show that the Fourier parameters

of Cepheid light curves, i.e. amplitude ratios and phase differences, are not very

affected by the presence of a bright companion star; the main effect of this presence

is the strong reduction of the amplitude.

We believe that the study of s-Cepheids can give some new insights into the pulsation

Phenomenon and its relation with the evolutionary theory, and moreover it can improve

the reliability of the distance of nearby galaxies by separating possible spurious

Objects, such as first overtone pulsators, from normal classical Cepheids.

References

Antonello, E., Poretti, E.: 1986, Astron. Astrophys., in press.

Baade, W., Swope, H.: 1965, Astron. J. 70, 212.

Ef~emov, Yu.N.: 1975, in 'Pulsating stars' ed, B.V. Kukarkin, New York: J. Wiley, p. 42.

Payne-Gaposchkin, C., Gaposehkin, S.: 1966, Smithsonian Contr. 2"

~Imon, N.R., Lee, A.S.: 1981, Astrophys. J. 248, 291 (SL).

~i~on, N.R., Moffett, T.J.: 1985, preprint (SM).

179

LIGHT CURVES FOR CEPHEIDS IN NGC 6822

Edward G~ Schmidt and Norman R. Simon Department of Physics and Astronomy

University of Nebraska~Lincoln Lincoln, NE 68588

I. Introduction

In recent years, the use of Fourier components has become recognized as

a powerful way to characterize the form of variable star light curves.

It has been used to study the dependence of light curves on period, to

compare different types of variable stars, to identify overtone pulsa-

tors, to delineate subgroups of variables and to make comparisons be-

tween observation and theory. Unfortunately an attempt to apply the

method to Cepheids in Local Group galaxies (Teays and Simon 1982) was

unsuccessful due to the low accuracy of existing photographic photome-

try~

Panoramic detectors now available have both higher quantum efficiency

and larger dynamic range than photographic plates. This will allow

greatly improved photometry of faint stars seen against the background

of their parent galaxy. We have thus undertaken a project to obtain

accurate light curves of Cepheids in Local Group galaxies. Although

the phase coverage obtained so far is inadequate for Fourier decomposi-

tion and further efforts are needed to obtain optimal extraction of

magnitudes from the images, preliminary light curves for several stars

in one galaxy, NGC 6822, show some interesting features.

NGC 6822 Is a dwarf irregular galaxy at a distance modulus of about

23.75. The Cephelds were studded by Hubble (1925) and more recently by

Kayser (1967). A total of 13 Cepheids are known from these studies but

none have periods shorter than I0 days. This cut off is consistent with

the sensitivity of the surveys but we will be able to identify shorter

period stars from our new observations.

II. The Observations and Reductions

This report is based on 13 images taken through an R filter of a field

in NGC 6822. The 0.8-m telescope at McDonald Observatory was used wit~

an unthinned RCA charge coupled device. To achieve accuracies of a few

percent for six day Cepheids, exposures of 45 mlnutes were used.

180

While the peak intensity in the images of the Cepheids is between 6700

and 7600 photons per pixel, the background light constitutes well over

half of this. About two-thirds of the background is night sky bright-

ness and the remaining third Is from NGC 6822. The galaxy Background

is irregular due to the presence of individual stars and constitutes

the most serious source of error in the extraction of stellar !ntensl-

ties.

The reductions were done using the DAOPHOT image reduction package at

the MidAmerica Image Processing Laboratory at the University of Kansas

in Lawrence. This program fits a point spread function simultaneously

to all the stars in a preselected group. In our case, the groups were

chosen to contain all the stars within several image diameters of each

variable. The result of this fitting is then used to obtain the magni-

tudes. After all the frames of each field have been analyzed, we will

COmbine all the fits to obtain the best estimate of the background.

This will then be used in obtaining the final extracted magnitudes. For

the present, however, we are using the magnitudes extracted in the first

Pass.

III. Results

Of the five known Cepheids in our field, three were sampled at a large

enough range of phases to glve some insight into the form of the R

light curves. They are identified by Kayser's numbers.

a. V5

This star has a period slightly greater than thirteen days. Our obser-

vations cover minimum light reasonably well but there are none near

• aximum light. A bump with an amplitude of about 0.3 magnitudes ap-

Dears to be superimposed on minimum light. This contrasts with classi-

cal Cepheids of similar period such as TT Aql which has a much smaller

bump about 0.I cycle later in phase.

b. V 21

The maximum of this seventeen day Cepheid is also missing from our ob-

Servations but most of the rest of the light curve is reasonably well

delineated. Like V 5, there is a strong bump but it occurs about mid-

way down descending light. Again a comparison wlth a classical Cephe-

Id of similar period, such as CD Cyg, shows that the bump in V 21 occurs

earlier and is larger.

181

c. V 6

The phases of our observations seem to define both minimu~m and maximum

light reasonably well for V 6. While we can not see any bumps, we can

not exclude their existence until better phase coverage is obtained.

Classical Cepheids of similar period, about twenty days, show very

asymmetric light curves with rapid rise and slow decline. V 6, in con-

trast, seems to have nearly equal durations of rising and declining

light. Additionally, the R amplitude of at least 1.2 magnitudes is

larger than for classical Cepheids.

The presence of a bump in the light curves of the two shorter period

Cepheids at phases earlier than in classical Cephelds may suggest that

both the radii and the masses are less than their galactic counterparts.

If so, the radii might be between 50% and 80Z a~nd the masses less than

half those of classical Cepheids. This indicates that the use of these

stars in the calibration of the cosmic distance scale will require the

use of a different period-luminoslty law than for classical Cepheids.

ACKNOWLEDGEMENTS

The use of the facilities and the hospitality of the staff of McDonald

Observatory are greatly appreciated. This work is supported by the

National Science Foundation through grant number AST-8312649.

REFERENCES

Hubble, E. 1925, Ap. J. 62, 409 Kayser, S.E. 1967, A, J. 72, 134.

182

The Luminosities of the Binary Cepheids SU Cyg, SU Cas. and W Sgr

Nancy Remage Evans I

Computer Sciences Corporation, IUE Observatory and

David Dunlap Observatory, University of Toronto and

Armando Arellano-Ferro Instituto de Astronomia, UNA Mexico

Abstract

Absolute magnitudes for binary classical Cepheids have been derived by ra- tioing IUE low dispersion spectra in regions where the blue companion dominates With spectra of main sequence spectral type standards. IUE spectra are also used to determine the magnitude difference between the Cepheid and the blue companion. Preliminary absolute magnitudes determined in this way for SU Cyg and W Sgr are in agreement with the absolute magnitudes of Sandage and Tammann, and Caldwell and also with those of Schmidt within the om2 estimated uncertainty. The abso- lute magnitude of SU Cas is in better agreement with the PLC relations if it is Pulsating in the first overtone mode.

Introduction

Direct measurement of t he flux of a Cepheid and a blue main sequence binary Companion is possible on IUE spectra. The Cepheid spectrum dominates a typical

pair at 3000 A. but in the SWP region (1150 to 2000 A) provides no measurable con- tribution to the spectrum. This means that absolute magnitudes inferred from the temperatures of the main sequence companions can be used to provide distances to the Cepheids.

Method

IUE low dispersion spectra can be used to determine the absolute magnitudes of the Cepheid in two ways. First the SWP spectra can be matched to spectra of main Sequence standard stars. The companion/standard star flux ratio can be combined With an absolute magnitude--spectral type calibration (Schmidt-Kaler, 1982) to provide a distance modulus to the system. A similar calibration has been done

by BShm-Vitenee (1988) using model atmospheres instead standard star spectra. In the fitting procdure used here. a grid of standards was set up by interpolating

between the spectra of representative standard stars from the IUE Spectral Atlas (Wu, et at, 1983). Reddenings were taken from the same source. Differences of a quarter of a spectral subclass produce differences in the standard deviation of

the ratio of the companion to the comparison. Figure i shows the match between the SWP spectrum of W Sgr and an AOV star.

The absolute magnitudes of the companions have been estimated as shown in Table i. For each companion, absolute magnitudes are presented for mean values of MK spectral classes and also for the zero age main sequence, both as given by SclZmidt-Kaler (1982). The adopted absolute magnitudes in the table, which are a revision of the previous version of this calibration (Evans and Arellano. 1986) 0 include a small amount of evolution off the main sequence. The absolute magnitudes

here are preliminary because further work is planned, both to make more detailed

I IUE Guest Observer

183

use of models to estimate the amount of evolution, and also to obtain IUE spectra of cluster stars of similar age to use as standards.

Table I. Absolute Magnitudes of Companions

Cepheid Companion MK I Adopted tAMS 2

Spectral Type

SU Cyg BT.5V -om4 om2 om4

SU Cas Bg.5v 0.4 1.0 1.1

W Sgr AO.OV 0.65 1.2 1.3

I Mean absolute magnitudes for MK spectral classes from Schmldt-Kaler, 1982 2 Zero age main sequence absolute magnitudes from Schmidt-Kaler, 1982

The absolute magnitudes derived in this way for SU Cyg, SU Cas, and W Sgr are listed in Table 2 as row A. The absolute magnitudes have been corrected for the effect of the companion on the measured V magnitude where necessary and corrected to mean light. Note that SU Cyg is actually a triple system (Evans and Bolton, 1986) but this has been taken into account in the fitting.

Table 2. Absolute Magnitudes of Cepheids

SU Cyg SU Cas SU Cas I W Sgr

A: SWP spectra -3ml -3m3 -4mo

B: SWP and LWR spectra -3.2 -3.1 -3.9

Sandage and Tammann -3.23 -2,24 -2.75 -3.89

Caldwell -3.36 -2,27 -2.83 -4.11

Schmidt -3.11 -2.02 -2.58 -3.86

I First overtone pulsation

The second method is to fit the flux in the 2000 to 3000 A region to compar- ison stars for both the Cepheid and the hot companion. A final correction for the

LWR sensitivity degradation has not yet been adopted, but because the observations were taken within a small time interval, the corrections amount to only about 2~. The procedure is to adopt the normalized comparison spectrum from the 1150 to 2000

A region and subtract it from the composite Cepheid spectrum. The normalization between the remaining Cepheid spectrum and a nonvariable supergiant spectrum pro-

vides the magnitude difference between the Cepheid and the main sequence companion. Figure 2 shows the composite Cepheid spectrum in the long wavelength region for W

Sgr and the flux contribution from the AOV star, The magnitude difference between

the Cepheld and the companion must be combined with the absolute magnitude of the

blue companion to derive the absolute magnitude of the Cepheid, which is listed

in row B of Table 2.

Discussion

The absolute magnitudes from 3 period--lumlnosity--color relations (Sandage

and Tammann, 1969; Caldwell, 1983; and Schmidt, 1984) are listed in Table 2 for

184

x

10

6

4

2

o 'J200 . . . . 1 I I I 1300 1400 1500 1000 1700 1800 1900 2000

WAVELENGTH A

Figure 1. The short wavelength spectrum of W Sgr. Flux is in units of 10 -13

ergs cm -2 se¢ - I A - I " The s o l i d l i n e i s the Cepheid spectrum. The dot ted l i n e i s

the spectrum of Gam UMa (AOV) scaled to the W Sgr spectrum. All spectra have been SmOothed with a i0 point boxcar filter.

x

1o

8

8

4

I 2i

o

--w ........... , i ..... w ,

W S@R

2000 2400 2600 2800 3200 WAVELENGTH A

Figure 2. The long wavelength spectrum of W Sgr. The solid spectrum on the top is the spectrum of W Sgr. Flux is is units of ]O -12 ergs cm -2 sec -I A -I .

OVerlaid using a dotted line is the summed spectrum of the two standard stars AOV

÷ G2Ib, scaled for comparison with W Sgr. Also shown is the AOV spectrum alone.

A ten point boxcar filter has been used to smooth the data.

185

comparison with the present results. We estimate that the uncertainty for our luminosity determinations is at least om2. and stress that these results are pre-

liminary for the reasons discussed above. However. this technique, which provides a new list of Cepheid calibrators, gives results in agreement with previous studies for SU Cyg and W Sgr.

SU Cas has previously been suspected of being an overtone pulsator, and the absolute magnitudes corresponding to first overtone pulsation as well as funda-

mental pulsation are listed in Table 2. The absolute magnitude derived from the companion differs from the predicted absolute magnitude for fundamental pulsa- tion by five times the om2 uncertainty in the determination. It is in much better agreement with overtone pulsation, as shown in Table 2.

Work is in progress applying this technique to the entire sample of binary Cepheids.

Acknowledgements

Financial support was provided by a NASA IUE grant (NASA contract to CSC NAS 5-28749) and a Helm Travel Grant (University of Toronto) to NRE° and a NSERC grant to Dr. J. R. Percy.

References

BThm-Vitense, E. 1986, Ap. J., 298, 16g. Caldwell, J. A. R. 1983, The Observatory, 103, 244. Evans. N, R. and Bolton, C. T. 1086, this conference.

Evans, N. R. and Arellano Ferro, A. 1985, Eight Years of UV Astronomy with [UE, ESA Pub. ESA SP-263.

Sandage, A. and Tammann, G. 1969, Ap. J., 157~ 683. Schmidt. E. G. 1984, Ap. J., 285, 501. Schmidt-Kaler. T, 1982, La.dolt-BOrnstein Vl2b, eds. Schaifers, K, and Voigt, H. H.

(New York: Springer Verlag). p. 18.

Wu, C. C.. Ake. T. B., Boggess. A., Bohlin, R. C.. Imhoff. C. L., Holm, A, V., Levay, Z. G.. Panek, R. J., Schiller, F. H., and Turnrose, B, E, 1983, IUE Newsletter, 22, 1.

186

FREQUENCY ANALYSIS OF THE UNUSUAL SHORT-PERIOD CEPHEID EU TAURI

Jaymie M. Matthews University of Western Ontario (Canada)

Wolfgang P. Gieren Observatorio Astronomico Nacional (Columbia)

Bac..__kground

EU Tauri has one of the shortest periods among the known classical

Cepheids. From the first report of variability in 1949 through the

early 1970's, there had been conflicting claims about the variations

and period of this star. However, later photometry by Guinan (1972),

Sanwal and Parthasarathy (1973), Waehmann (1975), and Szabados (1977)

established the star as a low-amplitude Cepheid with a period of 2~I025.

In a Fourier decomposition analysis of the light curves of 57 known

Cepheids, Simon and Lee (1981) singled out EU Tau (and two other stars

With periods near 2 d, including SU Cas) as not falling within the locus

for short-period stars in their plot of the phase difference between

Fourier components vs. period. They speculated that these three stars

may in fact be overtone pulsators. This prompted Gieren (1985a) to re-

Observe EU Tau.

Ob~servations a~d frequency analysis

Gieren has obtained new UBVRI photometry of EU Tau on four nights,

8 - ii Jan 1984, using the #2 0.9-m telescope at the Kitt Peak National

Observatory. He also collected 43 CCD spectra (with moderate dispersion

and S/N > i00) of the star with the KPNO coud~ feed telescope during

4 - 27 Jan 1984. These spectra were measured for radial velocities

Using a cross-correlation with a velocity standard, as described by

Gieren (1985b).

When the V and velocity data are plotted according to the accepted

Period of 2~i025 (Figures l(a) and 2(a), respectively), systematic devi-

ations from smooth curves can be seen. The discrepency between the V

Observations on 8 and Ii Jan, which overlap in phase, is particularly

obvious. Such deviations suggested ghat the star might be multiply

Periodic. Frequency analysis of the Gieren data, and that of previous

observers, was undertaken to search for any additional periods present.

Three techniques were employed: a modified Fourier periodogram for

unequally-spaced time series (Matthews and Wehlau 1985), Stellin~verf's

(1978) "phase dispersion minimum" technique, and an approach similar

187

to that of Lafler and Kinman (1965). An example of a Fourier amplitude

spectrum of the EU Tau velocity data is shown in Figure 3.

7.9

8.0

8.1

> 8,2

~ zg~

8+0

8.t

8,2

(a) 1 + +

P" 2~.1025 ~ S J i n

it ' J ~ ' ~ 9 jan

I ~ I ..... ( l I

(b) P,. 2qoam5 f

, ! ' Z

\ / (c) P+- 1+04475 - R / 2

o'.o o'++ oi, oI~ o'.8 Lo PHASE

0

E ~I0

10

r~

0

-10

5 o 0

i ......... + . . . . . . ,

P o 2+1025 (a)

+- ; I ~ I I

R - 2 d 0 8 9 5 (b) \

( I I I I I

P= - ld04475 • P,/2 (c)

i l +

PHASE

FIGURE I, (a) Gieren's V photo- metry of EU Tau, plotted at P = 2.1025 d. Solid lines join obser- vations from the same night. (b) The data in (a) at Pl = 2.0895 d. The dashed line is a sinusoid of the same period. (c) Residuals resulting from the subtraction of the sinusoid in (b). The dashed line here is a sinusoid of P2 = ½PI"

FIGURE 2. The same as Figure l(a)(b)(c), except now for Gieren's radial velocity data, The tri- angles indicate values from the first night of the observing run which appear discordant in all of the plots.

A possible period change

The frequency analysis indicates first that a slightly shorter period

should provide a better fit to the Gieren data. The amplitude spectrum

of the RV observations (Figure 3) has its largest peak at a frequency

fl = 0,479 ± 0.001 d -I', i.e. P1 = 2.088 _+ 0,004 d. (The frequency fa in

that spectrum is the expected (l-fl) d -I alias; the other adjacent peaks

are the contributions of the respective spectral windows of fl and fa.)

The Lafler and Kinman technique gives its deepest minimum at P1 = 2.089

± 0.001 d. The best fit to the photometric and RV data is achieved with

the value P1 = 2~0895. This is demonstrated by the phase diagrams of

Figures l(b) and 2(b). The earlier discrepencies are markedly reduced.

188

However, previous photometry of EU Tau is not well represented by

this revised value, so there does not appear to have been a large error

in the original period estimates. An (O-C) analysis demonstrates that

a gradual period change is also unable to account for the difference.

The remaining explanation is an abrupt period change sometime through

the lO-year interval prior to the Gieren observations, during which EU

Tau was monitored only once (in 1982) by Burki (1985), who has not pub-

lished a table of his photometry. There is precedent among the short-

Period Cepheids for such sudden - though smaller - shifts in period;

e.g. IR Cep and V465 Mort (Szabados 1977).

I?

© 9 E

La.J 6 EZ3

i---

-J 3 CL

| a

0 t '" ' ' '

0.00 0.50 ]-00 1.50 2.00

FREQUENCY (c /d)

FIGURE 3. A Fourier amplitude spectrum of the RV data shown in Fig. 2. Peaks at frequencies fl and fo are discussed in the text. Peaks at f , l+f_, l+f , and l-f_ a~e known aliases. The remaining structure is t~e res61t ofathe spectral window for this data sample.

A~._~s@cond period.?

The peak labelled f2 in Figure 3 occurs at a frequency of 0.964 ±

0.002 d-l; the frequency ratio f2/fl is 2.01. Both sets of Gieren data

Were examined to determine if a second frequency 2f I (or a nearby value)

is also present in the RV and light curves.

If pure sinusoids of period Pl and appropriate amplitudes (A V = 0~15,

ARV = 9.75 km/s) - the dashed lines in Figures l(b) and 2(b) - are sub-

tracted from the respective data sets, the residuals show a reasonably

tight fit to sinusoids with periods PI/2. These are plotted in Figures

l(c) and 2(c).

Of course, one expects a Fourier transform to attempt to describe a

n0n-sinusoidal periodic curve in terms of pure sine waves at a funda-

mental f~'equency m and its harmonics, 2~, 3m, 4m, and so on. The compo-

189

nent f2 likely reflects merely the first harmonic term in an asymmetric

light(velocity) curve, but the tightness of the fit to only two sinusoids

is somewhat surprising. It is noteworthy that in the Simon and Lee

(1981) Fourier decomposition analysis, EU Tau was one of two stars whose

light curves could be adequately represented by only two cosine terms,

with frequencies ~ (= 0.4756 d -I for EU Tau) and 2~.

Discussion

It is clear that the Gieren observations are not best described by

the single period of 2~i025. The phase discrepencies in the light and

RV curves can be at least partially resolved by a new period, P = 2~0895.

Our analysis also indicates that either i) these curves have shapes which

are remarkably simple to describe in terms of ~ourier harmonics, or ii)

there may be a 2:1 resonance present in the pulsations of the variable.

Either interpretation may hold some physical significance for the nature

of EU Tau.

The only way to confirm whether the (primary) period of EU Tau has

indeed changed is through further observations of the star. Gieren's

photometric data in particular cover too few cycles to be an extremely

precise determinant of the period. If Burki's 1982 observations do not

fit the proposed new period, and the current validity of that period can

be proven, then we could narrow the time of the period shift to between

1982 and 1984.

The period ratio implied by the simple Fourier composition of EU Tau'S

light and RV curves would be unique among the known double-mode Cepheids.

Burki's (1985) radius determinations for nine short-period Cepheids iden-

tified EU Tau as a probable second-overtone pulsator. If two periods are

present in the star, this implies a weak 2:1 resonance between the second

and a higher overtone, analogous to the fundamental-to-second-overtone

resonance proposed to explain the bump Cepheids. We recommend a search

for an appropriate resonance in Pop I models compatible with EU Tau.

References

Burki, G. 1985. in "Cepheids: Theory and Observations", ed. B.F. Madore, (Cambridge University Press: Cambridge), p. 34.

Gieren, W.P. 1985a. ibid, p. 98. ............ 1985b. Astron. Astrophys. 148, 138. Guinan, E.F. 1972. P.A.S.P. 84, 56. Lafler, J0 and Kinman, T.D. T965. Ap. J, Suppl. ii, 216. Matthews, J.M. and Wehlau, W.H. 1985. P.A.S.P. 97, 841. Sanwal, N.B. and Parthasarathy, M. 1973. Astron. Astrophys. 13, 91. Simon, N.R. and Lee, A.S. 1981. Ap. J. 248 291. Stellingwerf, R.F. 1978. Ap. J. 224, 953. Szabados, L. 1977. Mitt. der Sternwarte #70 (Budapest). Wachmann, A.A. 1975. Astron. Astrophys. Suppl. 2_~3, 249.

190

DOUBLE MODE PULSATORS IN THE INSTABILITY STRIP

1 ],2 1 E. Antonello , L. Mantegazza and E. Poretti

10sservatorio Astronomico di Brera

Merate, Italy

2 Dipartimento di Fisica NNcleare e Teorica

Universit~ di Pavia, Italy

Introduction. The more carefully we study the variable stars inside the instability

strip, the more we find multimode pulsators, probably multimode pulsators are far more

numerous than it was believed only a few years ago. Many of these stars were unrecog-

nized as such either because of the small amplitude of their secondar M variation or

because the limited number of observations induced to consider them as semiregular vari-

able stars. In the following we describe our results on four interesting objects that

have proved to he double-mode pulsators: the two newly discovered best Cepheids CO Aur

and EW Set and the two high amplitude Delta $cuti stars HD 200925 and HD 37819. All

these objects have been observed with the photometer attached to the 50 cm reflector

of the Merate Observatory. The data analysis technique adopted by us is described in

the paper by Antonello, Mantegazza and Poretti (1986).

CO Aurigae. At: first this star was classified as a semiregular variable star (Smak,1964;

Du Puy and Brooks, 1974). A reanalysis of Smak's observations (Mantegazza, ]983) showed d

that the star is a double mode Cepheid with P =].783 and P2=1.430° The ratio between 1

the periods (0.800), which is unique among beat Cepheids, produced some debate about

the reality of the second period which has only a semi-amplitude of 0.04 mag. in the V

c~lor. An analysis of the photographic observations at the Postdam Observatory {Fuhrmann

et al., 1984) has given a first confirmation of the two periods. New photoelectric ob-

Servations were performed at Merate Observatory in ]983 and 1985 (Antonello and Msnte- gazza, 1984; Antonello, Mantegazza and Poretti, 1986). The new data confirmed the two

d d Periods and permitted to improve their values which resulted P$=1.78304 and P2=l.4277B.

The ratio between the periods is 0.8007~0.0001. It is interestlng to observe that this

ratio is coincident with that obtained for the high ampl~tude Delta Scuti star VZ Cnc

(P2/Pl=0.80063, Cox et al., 1984).

EW Scuti. This star has always been classified as a Cepheid, however there were some

Uncertainties about the regularity and the period of the light variations (Bakos, 1950;

Eggen, 1973). Following a suggestion of Figer (1984) Cuypers (1985) reanalyzed both

Bakos' and Eggen's observations and found that this star is a double-mode Cepheid with

191

d d PI=5.8195 and P2=4.0646. Since Bakos' data are visual estimates on photographic plates

and Eggen's measurements have times rounded to an integer of a day, no precise shape

of the light curve has been obtained. We observed EW Sct during 5 nights in 1984 and

34 nights in 1985 in the V color. The analysis of the data has confirmed the periods

found by Cuypers. A simultaneous least-squares fitting of our data with the terms fl'

2fl,3fl,f2,2f2,fl+f2 and f2-fl gives a mean r.m.s, residual of 0.013 mag., which is

only slightly larger than the mean r.m.s, data error (0.008 mag.). Therefore we think

that the light variations of EW Sct are satisfactorily explained by this model. Fig.l

shows the light curves of the two periodicities. Each curve has been obtained by sub-

tracting the other mode and the non-linear coupling terms from the data and phasing

the residuals with the respective period.

8o0

8.2

@.4

~ , T ~ I r ; ~ [ ~ T , ~ , i , i , i , i , i • i , i , q • I , r , I ~ I • i , J ,

÷+ ~, • ÷ ÷

%* % 1 *

°,O 0,8 %2 1.6 2 ~

0,|

GC

eLI

** ÷ ÷* .

* % ** % ~.

*o*"~** ÷ , b ' t *

Fig. I - Light curves of the two periodicities of EW Sct.

Dr. Sterken's group has performed uvby B.observations of this star at ESO. Some of these

data have been privately communicated us . The results obtained from their analysis are

in excellent agreement with those of our data.

Finally we were puzzled by the spectrl type assigned to this star :KO (e,g. Eggen,19?3).

If this was true the star would lie well outside of the instability strip. We took a

Reticon spectrogram at Asiago Observatory on June 27, 1986. The energy distribution in

the spectrum of EW Sct shows the presence of a strong ~nterstellar reddening, however

the comparison of the line intensities with those of some standard stars indicates a

spectral ripe very near to FaII. this value agrees nicely with that derived from the

perlod-mean spectral type relationship for Cepheids.

HD 200925 (V1719 Cyg). HD 200925 was discovered as a variable star by Bedolla and Pena

(1979). subsequent observations by various authors (for a complete list see Johnson and

1 9 2

Joner, 1986) permitted to establish a primary period of 0~267299 and suggested the pos-

Sible presence of a secondary one. Mantegazza and Poretti (1986) have shown that it

Was possible to obtain an excellent fitting of the 260 UBV observations by Poretti (

1984) by introducing a secondary period of 0~2138 . Very recently Johnson and Joner

(1986) have published over 700 uvby8 observations of this star. Since most of these

data have been obtained in the same season of Poretti's data, the two sets have been

Put together. In fig.2 the spectrum obtained by introducing Pl and Pl/2 as known con-

stituents is shown.

o

N

5 I ~11~11 ~ldll,,L ill, l o ;;_2__,~ ........... o I 2. 3. 4. 5. 6. 2. 8. 9, i0.

FR£0u£Nc~ It/d)

Fig. 2 - Power spectrum derived from Poretti's and Joner and

Johnson's data which shows the second period in HD 200925.

In this spectrum the aliases are strongly reduced in intensity because of the different

geographical longitude of the two observatories. The'peak in the spectrum shows without

any doubt the presence of a second period whose frequency coincides with that found by

Mantegazza and Poretti (1986). The ratio between the two periods is quite unusual being

0.7998. This value is very similar to those found in VZ Cnc and CO Aur. Also the shape

of the light curve of the primary period is quite unusual and bears no resemblance with

those of the other high amplitude Delta Scuti stars, in fact it has the descending

branch steeper than the ascending one.-

HD 37819 (V356 Aur). HD 37819 was discovered as a variable star by Burki and Mayor (

198~, who performed photometric and radial velocity observations. They found a period

of 0.18916, however their attempt to identify the pulsation mode by means of the Bslona

and Stobie's technique (1979) was unsuccessful, one of the possible reasons could be

the presence of an undetected secondary periodicity. Padalis a~d Gupta (1984) performed

further photoelectric observations and refined the period to 0.189266. However their

data are too scattered in time and their precision is too much low for allowing the

detection of the secondary period~ This star has been observed at the Merate Observ-

atory during 6 nights between Jan. 4 and Jan. 16, 1986 in thedB,V standard colors. The

analysis of these observations gives s secondary period of 0.15642 (fig.3). If we exam-

ine Burki and Mayor's data we find that these too can be fitted by the two periods.

193

°~ I !

Z. 4. 6. 8. 10, 12. , , . , . -o, [~]

Fig. 3 - Power spectrum derived from our B data which shows the 2ndperiod in HD37819

The ratio between the two periods is 0.826, a value which does not agree with the usual

theoretical ratios between radial modes. We have tried to see ~f it was possible to

identify the pulsation modes by means of Balona and Stobie's technique (1979). However

the uncertainties on the phase lags between light and color curves are too much high,

and it is not possible to decide whether the modes are radial or non-radial.

References

Antonello,E.,ManteKazza, L.,1984:Astron. Astrophys.133,52

Antonello,E.,Mantegazza,L.,Poretti,E.,1986:Astron. Astrophys.159,269

Bskos,G.A.,1950:Ann.Sterrenwaeth Leiden 20,177

Balona,L.A.,Stobie,R.S.,]979:Monthly Not.Royal Astron.Soc.189,649

Bedolla,S.F.G.,Pena,J.H.,1979:Inf.Bull.Var. Stars No.1615

Burki,G.,Mayor,M.,1981:Astron.Astrophys.97,4

Cox,A.N.,Mac Namara,B.J.,Ryan,W.,1984:Astrophys.J.284,250

Cuypers,J.,1985:Astron.Astrophys.14__55,283

DuPuy,D.L.,Brooks,R.C.,1974:The Observatory 94,71

Eggen,O.J.,1973:Puhl.Astron. Soc. Pacifie 85,41

Figer,A.,1984:GEOS N.C.403,1

Fuhrmann,B.,Luthsrdt,R.,Schult,R.H.,1984:Mitt.Ver. Sterne,Bd.lO,79

Johnson,J.B.,Joner,M.D.,1986:Publ.Astron.Soc.Pacific 98,581

Mantegazza,L.,1983:Astron.Astrophys.l18,321

Mantegazza,L.,Poretti,E.,1986:Astron.Astrophys.158,389

Padalia,T.D.,Gupta,S.K.,1984:Acta Astron.344,303

Poretti,E.,1984:Astron.Astrophys.Suppl.57,435

Smak,J.,1964:Publ.Astron.Soe.Pacifio 76,40

194

FOURIER DECOMPOSITION OF LMC CEPHEID LIGHT CURVES

G.K. Andreasen and J.O. Petersen

Copenhagen Unive r s i ty Obse rva to ry

Oster Voldgade 3

13K-1350 Copenhagen K, Denmark

in t roduct ion

For the g a l a c t i c Cepheids Simon and Lee (1981) gave a successful , q u a n t i t a t i v e descript ion

of the Her t z sp rung sequence by means of Fourier decomposi t ion pa rame te r s . They found a

sys temat ic va r i a t i on wi th per iod of bo th phase d i f f e r e n c e s and ampl i tude ra t ios probably

due to the n2 / r l o = 0.5 r e sonance a t a per iod n o ~- 9". In the p r e sen t s tudy we analyse

165 l ight cu rves of Cepheids in the Large Magel lanic Cloud t aken from Wayman e t al.

(1984). The main purpose of our s tudy is to compare the Fourier descr ip t ion of the LMC

sample wi th the a c c u r a t e da t a known for ga lac t i c Cephelds.

Data analysis

The appl ied Fourier decomposi t ion technique is based upon the l eas t squares method as

descr ibed in Pe t e r s en (1986). Essent ia l for i n t e r p r e t a t i o n of the resu l t s is our calcula t ion

of s t anda rd e r ro r s of individual Fourier pa ramete r s .

The LMC sample is homogeneous in quali ty. Typically, the B l ight curves are defined

by abou t 37 obse rva t ions and t he V curves by abou t 29 observa t ions . Thus, we regard the B

curves as the pr imary da ta . The same order of the Fourier decomposi t ion is chosen for all

s tars . A four th o rder decomposi t ion is found to be the opt imal one, using the unit- lag

au to -co r re l a t ion . In the fol lowing we r e s t r i c t the analys is to an edited sample consis t ing of

128 s t a r s wi th a t leas t 30 B observa t ions per curve and with a re la t ive ly low s tandard

er ror of the f i t compared to the ampli tude (~ < 0.175 A). For each s t a r we ca lcu la te the

195

s tanaara e r rors ot the individual Fourier parameters . The average accurac ies obtained for

the ed i ted sample are: ~ = .m13 -* .m05, a/A = .I0 + .03, a(R21) = .08 ± .03,

~(R31 ) = .07 ± .03, a(¢21) = .27 -+ .22 (tad), a(¢o31) = .55 + .67 (tad).

The LMC point dis t r ibut ion in the (rl, ~21 ) and (11, ~31 ) - p lanes is shown in Fig. l

t oge the r wi th a schemat ica l represen ta t ion of Simon and Lee ' s da ta . The LMC data are

cons is ten t with the ga lac t ic represen ta t ion taking into account (i) the larger sca t te r

associa ted with the use of photographic data instead of photoelectric measurements, and

(ii) a per iod shif t of the sharp break of the Hertzsprung progression from -9.d5 to ~10.%.

In the (rl, Rkl)-<iiagrams we find a considerably larger spread and somewhat larger values

for n < 9 d than seen by Simon and Lee 's data.

8

o o v ~ r t o n ~ c o n d i d o t ~

2

0

6

4

[3 2 4 6 a 1O 12 14 16 18

II ( days )

Fig. I Comparison of the Hertzsprung sequenees in LMC and galactic Cepheids. The plotted

point distribution on the (rl, w21 )- and (n, ~31 )- plane represent LMC stars. Possible

overtone pulsators are marked (o). Error bars of ~21- and ~31- points give ±1 a(0~21) and

±1 0(~31), respectively. The fully drawn lines are regression lines given by a weighted

least squares solution. The dashed, straight lines and the areas are sehematical

representations of data on classical Cepheids in the Galaxy.

196

Ste l l ingwerf phases

Recently, S t e l l i ngwer f and Donohoe (1986, h e r e a f t e r SD) have computed Fourier pa ramete r s

of ad i aba t i c one -zone pulsa t ion models in a f i r s t a t t e m p t to unde r s t and the re la t ion

be tween t h e physical p rope r t i e s and the Fourier p a r a m e t e r s of a s tar . They advoca te the

use of phases r a t h e r than phase d i f f e rences because the phases besides being more

accura te , a l low the most s ign i f i can t da t e point , ~ I , t o be r e t a i n e d - provided tha t a

Standard point on the l ight curve is defined. As a s t anda rd point they use the point a t

which the r is ing b ranch c r o s s e s the median magnitude. In the pulsa t ion cycle this point is

def ined to have the phase ~s = 0.5. SD find t h a t the Four ier phases of the model veloci ty

CUrves a lways assume one of two values. Genera l ly , the phases for success ive modes

a l t e rna t e : ~1 e 3,~/2 , ¢o 2 -~ ,~/2, ~3 -~ 3,x/2, and ~4 • ,~/2 e tc . (in our no ta t ion) , except

for t he i r very th in , low ampl i tude models.

We have c o n s t r u c t e d diagrams for SD phases ~1 to ~4 versus period, using the edited

Sample (Fig. 2). S imi lar diagrams have been p lo t t ed for ~o k values computed from Simon

and Lee ' s Four ie r p a r a m e t e r s and we find ve ry good a g r e e m e n t b e t w e e n the LMC and

ga lac t i c da ta . ~1 d e c r e a s e s l inear ly from n = 1 d to 9 d, bu t assumes roughly the value 3~/2.

Disregarding th i s va r i a t i on we find t h a t the overa l l phase p a t t e r n is cons i s t en t with the

a l t e rna t i ng p a t t e r n p r e d i c t e d by SD for per iods smal ler than abou t 9 d. Thus, i t seems that

appl ica t ion of S te l l i ngwer f phases al low a very simple Four ier desc r ip t ion for periods

1 d - 9d: Only the phase of the fundamenta l mode seem to change s ign i f i can t ly with period,

While phases of the harmonics remain almost c o n s t a n t be ing e i t h e r ~ ,~/2 or - 3 ~ / 2 .

For per iods l a rge r than 12 ~ we note re la t ive ly small bu t s ign i f i can t dev ia t ions be tween

the obse rva t ions and S D ' s predic t ions . For ~1' ~2 ' and ~3 these dev ia t ions a re <0.5, for

~4 the d iv ia t ions a r e 1.0 - 1.5.

A few s t a r s - mainly sinusoidal Cepheids - dev i a t e from the genera l p a t t e r n . Their

phase p a t t e r n can be i n t e r p r e t e d in te rms of the very thin, low ampl i tude models. Two

sinusoidal Cepbeids wi th per iods nea r 10.d5 d i f f e r s ign i f i can t ly wi th r e spec t bo th to phase

value and p a t t e r n f rom possible model resul ts , and t h a t ts cons i s t en t wi th resonance as

driving these e f f ec t s .

197

P'd

32 CL

6 . 0 8

t o:v0r o o .......... ] • 5 . 5 6 ,

4 . 0 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . _

3 . 5 0 . . . . : ,~ : ; : ~ , ; ~ ~

4 t . ! : ! . - . ! .: " [ , t ! :..t ......... L.:.

tf 3

*-i - ....

o + . . . . . . . . . . . . . .

13 2 4 6 8 l 0 12 1 4 1 6

i1 (doy~)

~4 I

0 ~ , i ~

0 2 4 6 8 10 12 14 16 8 I] (doy~)

Fig. 2 Phase diagrams (n, ~k) for the fundamental mode and the f irst three harmonics. The

horizontal lines represent ei ther the value ~ /2 or 3~/2. The fully drawn line is the

regression line for ~1 on n = 1 d - 9%

References

Becker, S.A., Iben, I., Tuggle, R.S.: I977 Astrophys. J. 218, 633

Connolly, L.: 1980, P.A.S.P. 92, 165

Petersen, J.O.: 1986, Astron. Astrophys. , in press

Petersen, J.O., Diethelm, R.: 1986 Astron. Astrophys.. 156, 337

Simon, N.R., Lee, A,S.: 1981, Astrophys. 1. 248, 291

Simon, N.R., Moffett, T.J.: 1985, P.A.S.P. 97, 1078

Stellingwerf, R.F., Donohoe, M.: 1986, Astrophys. J., in press

Wayman, P.A., Stift, M.J., Butler, C.J.: 1984, Astron. Astrophys. Suppl. Se t . , 56, 169

198

TIME-DEPENDENT FOURIER ANALYSIS, APPLICATION TO NONLINEAR PULSATIONS OF STELLAR MODELS

c. G. Davis Los Alamos National Laboratory Los Alamos, NM 87545

and

G. Kov~cs and J. R. Buchler University of Florida, Gainesville, FL

ABSTRACT

A time-dependent Fourier analysis is described and applied to the study

of the numerically generated hydrodynamic behavior of a stellar model.

The approach yields the temporal variation of the amplitudes and phases

of the various excited modes and sheds new light on the modal

interaction and on the approach to the final steady pulsation.

The classical radial variable stars are characterized by the fact that

the growth- or decay-rates of the low-lying modes are much smaller than

the pulsational frequency. This allows us to described their behavior

in terms of a multiperiodic signal with slowly varying amplitudes and

phases, in other words to give a time-dependent Fourier description,

K = Ak(t ) sin { (~k(t-t0) + ~k(t) } R(t) A0(t ) +~k=l

Where {~k} is a set of (constant) angular frequencies, which is obtained

from a Maximum Entropy Method (MEM) over a reasonably long timespan

(several hundred pulsation periods). The possible temporal variation of

the actual frequencies appears here as the variation of the phases. The

epoch t O is arbitrary. The Fourier fits, which assume constant {Ak} and

{~K} , are performed over successive overlapping time-bases, the lengths

of which are 2 to 4 fundamental periods of oscillation and the overlap

is 2 periods. These fits generate the 'instantaneous' amplitudes

{Ak(t)} and phases {~k(t)). The desired quality of the fit sets the

Order of The fit, K. Too small a choice of K introduces spurious noise

in the Fourier coefficients and too large a value makes the fit ill-

conditioned. Typically a value between 6 and 15 seems optimal for RR

199

Lyrae pulsations. Details of the approach will be published elsewhere

(Kov&cs, Buchler and Davis 1986).

We have applied the time-dependent Fourier analysis to an RR Lyrae

model, characterized by a mass of 0.65, a luminosity of 64, both in

solar units, an effective temperature of 7300 K and a composition of

X=0.700 and Y=0.299. The numerical hydrodynamic integrations have been

performed with the code DYN (described in Castor, Davis and Davison !977

and Davis and Davison 1978). The pseudo-viscosity is quadratic with a

coefficient CO=I.0 and with a Stellingwerf cutoff of Cc=0.1. The model

uses 72 mass-zones down to 10% of the radius and includes 15% of the

total mass.

We have started the hydrodynamic integration of the RR Lyrae model with

two different initial conditions, the first run with a velocity profile

corresponding to the fundamental LNA eigenvector and the second run with

the first overtone, both with a surface velocity of lOkm/s. We perform

our analysis on the temporal variations of the radial position of the

photosphere (optical depth 2/3).

In Fig. 1 we show the MEM spectrum obtained over 20 periods of

oscillation of some fraction of run i.

I~O]US HODEL ND. = ! F)LIEI~ LD~1%1=60 NUNW~ or ~VD~GZN5--3

I.O0

0.80

O.qO

0.80 ~ . 2

O.Oo i . . . . . . 0.00 !.00 ~.OO 3.00 q.O0 5.0O

t+I~DUCNCv IC,,'9 J

Fig. i. MEM spectrum over 20 periods.

The quality of the fit is very insensitive to the time-base in the MEM

analysis, except that the height of the peaks and thus the accuracy of

the determination of the {~k} has some sensitivity. Actually, the peaks

of the MEM spectrum are only very loosely connected with the amplitudeS,

200

in contrast to a power spectrum. The MEM analysis for run 1 yields the

frequencies {fk} in c/d

fl = 2.017 e0 = 2.0165

f2 = 2.704 ~i = 2.7044

f3 = 3.397 2¢I-¢ 0 = 3.3923

f4 = 4.034 2~ 0 = 4.0330

f5 = 4.721 ~0+~I = 4.7209

f6 = 0.688 oi-~ 0 = 0.6879

f7 = 1.370 2~0-~ 0 = 1.3286

u O = 2.004 P0 = 0.499

u 1 = 2.688 Pl = 0.372

u2 = 3.367 P2 = 0.297

v3 = 4.098 P3 = 0.244

The spectrum for run 2 is very similar, of course. These frequencies

can be identified as two basic frequencies, #0 and o I and their com-

binations as shown in the second column. Also shown are the linear

nonadiabatic frequencies (Wk} and periods {Pk } in days. All are stable

except the first overtone. The nonlinearity is seen to increase the

frequency here. It is interesting to note two resonances, 2w0~u 3 and

2Ul-o0-u 2. Such resonances are typical of RR Lyrae models.

Our 12th order fit includes, in addition to column 2, all 3rd order com-

binations of ~0 and ~i" The results are shown in Figures 2 and 3 for

the two runs, both for the amplitudes and for the phases of the funda-

mental and the first overtone.

O.03g

000g

e~0lL~ ~4A~ M~D~I NO. " I

O,O1E ~ rI-~

O.~"fO

~ - - . . - - . - - . . ~

3,1~e,~ -II

E,513

TIME (DAYS) -O.02Z

Amplitudes and phases for f u n d a m e n t a l .

Fig. 2 .

~,J3~UG

O* D93

0.0e~

o.gT&

O.o[~

O. 026

0.oi?

D, [io~

o.ool

~'~PL1 TLD£ e*4DD(3.~ ~.= I

/ ,,2

R~%D [L~ P~4e,~ PIDI~G. NO., I

"0.~3!

TIME (DAYS) - 0 , 0 2 0

Fig. 3. Amplitudes and phases for ist overtone.

201

The fits give a very clear idea of the status of the evolution of the

model, except perhaps for a short-lived transient (the first few

periods). This procedure advantageously replaces the usual simple test

based on the variation of the total pulsation kinetic energy.

For the first run we also show in Fig. 4 some higher order Fourier amp-

litudes corresponding to 2¢i-= 0 and 2e 0. Because of the resonances and

the impossibility of resolving the corresponding peak in the MEM anal-

ysis and in the fit, there necessarily appear beats (wiggles) in the

amplitudes and phases, especially in the higher order ones. The fitting

technique could cause them to appear in all amplitudes and phases, so

that it is very difficult to decide for a giwen quantity whether they

are a mathematical or physical origin. The figures show some very

short-lived, but large initial transients lasting several fundamental

periods and possibly some smaller ones, which could persist over I00

days.

Fig. 4.

O, 00g

O. 007

0,004

0.001

0.010

0. 007

O. 004

0.~32

R,~OIUq ~HPLITUOE MODEL 143. : I

64

O ~0 81 z2J t6~ a~ a4a 2~ ~3 3s3 403

T]mE(D~¥SI -0.OAT

Higher order fourier amplitudes for fundamental.

A3

The overall behavior is in line with our expectations, namely the

disappearance of all linear combinations of the decaying mode.

In summary, the method gives not only a clean definition of the instan-

taneous modal content and allows an assessment of where it is ultimately

headed to, but it also yields the very quantities, which appear in the

nonlinear, nonadiabatic amplitude equation formalism (Buchler and Goupil

1984, Buchler 1985). The latter formalism, in spite of its mathematical

appearance, gives a very physical description of the modal interaction

and the saturation mechanism (Buchler and Kov~cs 1986).

202

This work has been supported by the DOE, the NSF (AST84-10361) and the

NER Data Center at the University of Florida.

Re~@~gferences

Buchler, j. R., 1985, in Chaos in Astrophysics, Eds. Buchler, J. R., et al., NATO ASI Ser. C 161, 137, Reidel Publ.

Buchler, J. R. and Goupil M.-J., 1984, Ap. J. 279, 394.

Buchler, J. R. and Kovacs, G., 1986, "On the Modal Selection in Stellar PUlsators, II. Application to RR Lyrae Models", Ap. J. (submitted).

Castor, j. I., Davis C. G. and Davison, D. D., 1977, LASL Rep LA- 6644.

Davis C. G. and Davison, D. D., 1978, Ap. J. 221, 929.

KOV~cs, G., Buchler, J. R. and Davis, C. G., 1986, ,'Application of Time- Dependent Fourier Analysis to Nonlinear Pulsational Stellar Models", Ap. J. (submitted).

203

A STUDY OF TIME-EVOLVING HYDRODYNAMIC CEPHEID MODELS

KEIICHI UJI-IYE Astronomical Institute, Tohoku University, Sendal 980, Japan

TOSHIKI AIKAWA Faculty of Engineering, Tohoku-Gakuin University, Tagajo 985 Japan

TOSHIHITO ISHIDA Faculty of Sciences, Ibaraki University, Mito 310, Japan

MINE TAKEUTI Astronomical Institute, Tohoku University, Sendai 980, Japan

i. Introduction

Hydrodynamic simulations starting with different initial velocity

perturbations are performed for model envelopes whose stellar parame-

ters are in the range of observed double-mode cepheids. The aim of

the present study is to investigate the properties of modal coupling

in cepheid models and seek possible ways to obtain double-mode

pulsation in these models.

A general explanation of the origin of double-periodic oscillation

was suggested by Dziembowski and Kov~cs (1984) on the basis of

consideration on the synchronization between coupled oscillators.

Even though Simon et al. (1980), Takeuti (1986) and Uji-iye (1986)

tried running time-evolving cepheid models for investigating the

properties of modal coupling, long-lived double-periodic oscillations

were not found. Very recently Buchler and Kov~cs (1986) found models

of RR Lyrae stars that pulsate in double-mode. They used a code

constructed by Stellingwerf (1974) to search the growth rate of one

mode in the limit-cycle of another mode. Unfortunately, the Stelling-

werf code does not work well for the classical cepheids models because

the number of shells required to simulate extended outer envelopes is

too great. The investigation of models for classical cepheids is more

difficult than in the RR Lyrae case.

In the present study, hydrodynamic simulations of classical cepheids

starting with different initial velocity perturbations are investigated.

To release the synchronization of the first overtone (lO-mode), which

can suppress the fundamental mode (F-mode) or makes the pulsation

singly-periodic near the blue edge of the instability zone of the F-

mode, it seems necessary to weaken the 10-mode relative to the F-mode.

We increased the artificial viscosity as an agency to weaken the lO-mode.

204

2-~_~dModels and method for decomposition

We used a hydrodynamic code T-GRID (Simon and Aikawa 1986), to

Simulate stellar radial pulsations. The code ignores convection.

Chemical composition is (0.7, 0.28, 0.02).

We choose the effective temperature of the models to be 5850K

nearly the same as the observed temperatures of double-mode cepheids.

Physical properties of the models are tabulated in Table I.

To analyse the results of these hydrodynamic simulations, we used

the Maximum Entropy Method (MEM). The MEM results, obtained by

decomposing a time-evolving model, are described in a previous paper

(Uji-iye 1986).

Table I. Physical properties of models.

Model A B C

Masses(M/Me) 6.71 4.00 3.50

Luminosity(L/L®) 2279.27 2279.27 2279.27

Effective temperature 5850K 5850K 5850K

F-mode 2.4924 1.7496 1.5868 Frequency (in 10-6Hz) {iO-mode 3.3350 2.4550 2.2560

F-mode +0.0032 +0.0152 +0.0199 Growth rate {

lO-mode +0.0191 +0.0215 +0.0029

3~_Results

The model A is pulsationally unstable both in the F-mode and

iO-mode. A part of the time-evolving properties of hydrodynamic

Simulations are described in a previous paper (Takeuti 1986). The

results of model A in cases with various strength of artificial

Viscosity are tabulated in Table II.

The model has the lO-mode as a stable limit-cycle and the F-mode

as an unstable limit-cycle. Wi%h the increase of the parameter CQ, in

the artificial viscosity, the F-mode changes into the stable limit-

Cycle. The change occurs as we run the model with the parameter CQ= 6.

• he modal coupling of the model moves from Case a to Case c defined by

Simon et al. (1980), where the iO-mode suppresses the F-mode in Case a

and either the iO- or the F-mode is permitted to oscillate in Case c.

~he properties of modal coupling may change gradually, so it seems

Probable that no double-mode pulsations occurs in the midway of Case a

and Case c° It seems clear that the limit-cycle of the IO-mode will

become unstable where the artificial viscosity is very strong. The

ease is Case b of Simon et al. (1980), that is only the F-mode should

he found.

205

Table II. The results of Fourier decomposition by the maximum entropy method (MEM).

Model CQ Mode

A 1 F

i0

7 F

iO

20 F

i0

B 1 F

iO

C 1 F

iO

Time

12 60

31 16

25 00

37 45

35 25

15 30

13 00

14.45

Frequency A S

3.3220 25.4 -

2.4892 13.9 -

3.3550 13.9 -

2.4965 10.7 -

3.3420 9.8 -

1.7918 18.7 -

2.4~52 22.6 -

1.5288 16.6 -

Remarks: Properties of pulsations at the indicated time from

the initial state are tabulated. Time is indicated

in 107 sec. Frequencies are indicated in 10 -6 Hz.

A is a half amplitude of the radial velocity

expressed in km/sec. S is the stability of the

limit-cycle, where - means the limit-cycle is stable

(convergent) and + indicates the instability

(divergence).

Models with reduced total mass are constructed to investigate the

changes of modal-coupling properties with reduced surface gravities.

Model B (4 solar masses) shows a property that either the iO- or the

F-mode pulsates. With the decrease of the mass, the instability of

the iO-mode becomes smaller. The 3 solar-mass model shows another

property that only the limit-cycle of the F-mode is stable, that is

Case b of Simon et al. (1980).

It is found that the iO-mode zone, Case a, is separated from the

F-mode zone, Case b by the "either-or-zone", Case c (see Figure i).

The decrease of surface-gravities behaves in the same way as the

increase of the artificial viscosity. So that we may conclude that

there are no double-mode pulsators, Case d models, in the models with

the luminosity and the effective temperature adopted in the present

paper.

4. Concluding remarks

The increase of artificial viscosity works like the decrease of

surface-gravity, in the viewpoint of modal coupling. Anyhow

206

Figure 1.

Cq Case b

/ /

• " Case C

J / /

Case b/'" Case a

0 M

Status of modal coupling on the M(mass)-CQ diagram.

they work to dissipate the kinetic energy of the lO-mode much more

Compared with that of the F-mode. Consequently, the suppression of

the iO-mode is weakened and the F-mode changes into the master mode.

The next step in the search for double-periodic pulsations may be

to construct models of different luminosity and/or different chemical

Composition.

We have not succeeded in constructing a model pulsating in double

mode. As stated by Buchler and Kov~cs (1986), the physical properties

of the model would differ considerably from observed ones. If so, the

Present hydrodynamical code may have an important defect for reproduc-

ing classical cepheids.

R~geferences

BUchler, J.R. and Kov~cs, G. 1986, private communication.

Dziembowski, W. and Kov~cs, G. 1984, Mon. Not. R. Astr. Soc., 206, 497.

Simon, N.R., Cox, A.N. and Hodson, S.W. 1980, Astrophys. J., 237, 550.

Simon, N.R. and Aikawa, T. 1986, Astrophys. J., 304, 249.

Stellingwerf, R.F. 1974, Astrophys. J., 192, 139.

Takeuti, M. 1986, Astrophys. Space Sci., 119, 37.

Uji-iye, K. 1986, science Reports Tohoku Univ., 8th Ser., 6, 173.

207

HYDRODYNAMIC MODELS OF BUMP CEPHEIDS

Toshiki Aikawa

Faculty of Engineering, Tohoku-Gakuin University Tagajo 985, Japan

Summary By a procedure of the continuation of periodic solution combined

with the Baker-yon Sengbush-Stellingwerf algorithm, limit cycles of 135 bump models are obtained and their velocity curves are Fourier- decomposed. We compare the results with analytical theories.

1. Introduction

Since Simon and Schmidt (1976) suggested that resonant mode

interaction between the F- and 20- modes will produce the bump feature

in classical Cepheids, many authors have developed analytical theories

of the mode interaction and confirmed it as a basic mechanism of bump

Cepheids. On the other hand, Simon and Lee (1981) introduced a Fourier

decomposition technique to quantify bump features and compiled Fourier

components of observed light curves. Simon and Davis (1983) compared

the Fourier components of hydrodynamic models with the observed data.

The theoretical models qualitatively agreed with observation in terms

of Fourier components.

2. Models and limit cycles

In this report, a large number of hydrodynamical models are

constructed to derive detailed trends of the Fourier components. For

this purpose, we use a procedure of the continuation of periodic

solution combined with the Baker-Yon Sengbush-Stellingwerf algorithm

(e.g., Holodniok and Kubi6ek, 1984). The radius of the innermost

interface at a fixed mass coordinate is held motionless as a boundary

condition in the hydrodynamic modelling. Fig. 1 displays a close

correlation between the effective temperature and the radius of the

innermost interface for models with equal mass and luminosity. Thus

models with the same mass and luminosity but different values of the

radius correspond to models of different effective temperature. We

extend the Flouquet matrix to include the response of variables on the

innermost interface in a manner similar to the original BSS algorithm.

When one finds a limit cycle for the model, with a certain boundary

value, one can easily construct a limit cycle for an adjacent model.

By this continuation procedure, we obtained limit cycles for 135

models that cover the resonance. The standard models used as the

208

COordinate for the continuation are tabulated in Table I. Velocity

Curves of the limit cycles then were Fourier-decomposed. We concen-

trate on the Fourier quantities that are comparable with analytical

theories and are significant to observation.

Table I The standard models for the continuation. (M= 3 M e , L= 2546.8 L®)

Model 1 2 3 log Te 3.7546 3.7646 3.7746 R1 (1011 cm) 3.374 3.272 3.185 PF (days) 10.094 9.334 8.616 P2/PF 0.4887 0.4971 0.5054

0.065 0.024 -0.003

3.40

~3.30

o

~: 3,20

3.13o

x

I

×

I , t , 3.77 3 .76

TE 3.75

Fig. 1 The correlation between the radius of the innermost interface and the effective temperature for models of fixed mass and luminosity.

3. Results

Figs. 2a and b display trends of R21 and ~21 of the models as a

function of P2/PF. See Simon and Lee (1981) for definition of R21 and

~21. As expected from results by Simon and Davis (1983), @21 has a

~aximum and R21 has a minimum in the range of the resonance center,

P2/PF~ 0.5. It is noted, however, that the extrema of R21 and ~21

Occur at different values of P2/PF; 0.495 for the minimum of R21 and

0.502 for the maximum of ~21. This difference seems essential to an

Understanding of the minimum of R21 and it appears in hydro models as

Well as in observation.

4. Analysis

We apply analytical theories to understand the result of the

hydrodynamic models. For this purpose, we extend the analytical theory

of modal coupling by Takeuti and Aikawa (1981) to include non-resonant

209

effects in the coupling equations. According to the theory, amplitudes

of the F- and 20 modes may be expanded as follows:

q0 = A0 cos(wt) + a 0 cos(2~t + ~),

q2 = A2 cos(2~t + @) + a 2 cos(wt + e).

The amplitude A 2 becomes large near the resonance center due to modal

coupling between F- and 20-modes. The phase 0 is important for

understanding the Hertzsprung progression of bumps. At P2/PF < 0.5, it

is almost ~; at P2/PF > 0.5, it approaches 2~. The terms of a 0 and a 2

come from self coupling of the F-mode and non-resonant coupling between

the two modes. The term a 0 has been conside~red to make the asymmetry

of Cepheid light curves and we note the phase of the term is z and so

this term is in phase with the first term of q2 in case of P2/PF < 0.5;

in anti-phase in case of P2/PF > 0.5.

Figs. 3a and b demonstrate R21 and ~21 of the analytical model

computed with (solid line) and without (dashed line) the non-resonant

effects. Without non-resonant effects, R21 is a symmetric function of

P2/PF and a peak at P2/PF= 0.5, as expected. When one takes non-

resonant terms into consideration, R21 becomes asymmetric in respect

to P2/PF= 0.5 and a minimum at the region of P2/PF > 0.5. Similarly,

~21 drops to ~ at the region of P2/PF >> 0.5.

Buchler and Kovacs (1986) have obtained similar results in their

analytical models of two mode interaction.

5. Conclusions

We find good correspondence between the analytical theories and

the hydrodynamic models. We thus conclude that the maximum of ~21 and

minimum of R21 near the resonance center come from combination of the

modal resonance and the non-resonant effects.

References

Buchler, J.R. and Kovacs, G. 1986, Astrophys. J. 303, 749. Holodniok, M. and Kubi~ek, M. 1984, J. Comp. Phys. 55, 254. Simon, N.R. and Schmidt, E.G. 1976, Astrophys. J. 205, 162. Simon, N.R. and Lee, A.S. 1981, Astrophys. J. 248, 291. Simon, N.R. and Davis, C.G. 1983, Astrophys. J. 266, 787. Takeuti, M. and Aikawa, T. 1981, Science Report Tohoku Univ. eighth

ser. 2, 106.

210

7.50 BUMP MODELS

20.20

0.15

0 . 2 5

0.30 BUMP MODELS

0 4 8

7o00

W

6 ° 5 0

I , I , 6.At1 ~ , I . I 0 . 5 0 0 . . 49 ~0"~5. 0 . 5 0 0 . 4 9

P 2 / P F P 2 / P F 0 . 4 8

Fig. 2 R21 and ~21 of the hydrodynamic models as a function of P2/PF. The unit of ~21 is radians.

. 5 7.@

.4

.3

.2

.!

0.0

R21

t

/ /

I !

. 6 0 . 5 5 . 5 0 . 4 5 . 4 0

6 ,@

5 . 0

4 . 0

3 .@

P h t 2 1

.6@ .4@

i I I

! !

. 5 5 . 5 0 . 4 5

P 2 / P F P 2 / P F

Fig. 3 R21 and ~21 of the analytic model as a function of P2/PF. Solid curves were computed with non-resonant effects and dashed curves included only the resonant effects.

211

MODELING OF CEPHEID BEHAVIOR IN THE INFRARED

Robert Hindsley and R. A. Bell Astronomy Program University of Maryland College Park, MD 20742

We have calculated surface brightness in the visual and infrared from

a grid of Gustafsson-Bell flux constant static model atmospheres. The

visual surface brightnesses have been combified with observed colors to

calculate angular diameters and distances. The infrared surface bright -

nesses were used to derive synthetic infrared magnitudes which were com ~

pared with observations.

We start with the surface brightness equations

S v - V o = 51og(O) (I)

SIR - 51og(O) = (IR) o (2)

Equation 1 is a standard formulation of the visual surface brightness

equation. S V is a visual surface brightness parameter, V ° is the de-

reddened apparent V magnitude, and @ is the angular diameter. Equatio~

2 is the infrared analog of equation I, slightly rearranged. (IR)o

represents any dereddened infrared magnitude, and SIR is the surface

brightness in that filter. In practice we used equation 1 to calculate

angular diameters. These were matched to the integrated radial velocity

curve, and "smoothed" angular diameters read off the matching curve.

This smoothing procedure is why the angular diameters are not eliminated

from equations 1 and 2.

The use of model atmospheres makes possible adjustments in the re-

lationship between surface brightness and color. We assume solar met-

allicity; a Doppler Broadening Velocity (thermal broadening and micro-

turbulence) = 5.0 km/s. Gravities were determined from photometry and

the radial velocity curve. In practice we drew smooth curves by hand

through the magnitude and color curves and evaluated these parameters

at phase intervals of 0.05. We found 23 galactic Cepheids with suffi-

cient photometry to determine angular diameters. Eight of these CepheidS

were observed by Welch et. al. in the infrared filters J, H, and K of the

Caltech system.

A comparison of our synthetic infrared magnitudes with the observati °~

is shown in the accompanying figure for Eta Aql. The star-shaped symb ols

are the observations, with the size of the symbol representing the un-

certainty of 0.02 mag. The smooth curve has been drawn through the syn ~

212

thetic magnitudes and shifted vertically onto the observations. While

the synthetic magnitudes reproduce the variations very well, they are

consistently fainter than the observations by 0.15 mag in all three fil-

ters. This offset exists in the same sense (synthetic magnitudes being

fainter) in all three filters for all Cepheids.

To quantify the offsets we divided the observations of Welch et. al.

into bins of phase width 0.05 and averaged the values. This yielded 20

Pairs of observations and synthetic magnitudes per cycle~ which allowed

us to calculate twenty values of the offset. The means of these twenty

Values are given in Table I. The means are typically 0.1-0.2 mag, but

¥ Oph has notably large mean offsets. Yet this does not seem to be a

trend with period, as SV Vul has the smallest offsets in H and K.

But while the synthetic magnitudes are fainter, the offsets are nearly

constant with phase. We calculated the standard deviations of these

offsets around the means. In other words we bare quantified the spread

in the offsets. As the observations are uncertain by 0.02 mag, any value

~maller than this suggests that the offsets are constant with phase. The

Standard deviations are near this value for the short period Cepheids,

but are fairly large for the 3 longest period Cepheids. Examination of

the data shows that the light curves are well reproduced, but that the

SYnthetic magnitudes are shifted slightly in phase. For Y Oph the shift

is almost exactly 0.05 (the phase resolution of the synthetic magnitudes),

in the sense that the synthetic magnitudes have been shifted to an earlier

Phase. For X Cyg the phase shift is a little less, and it is a little

~ore for S¥ Yul. Such a phase shift, increasing with period, is probably

symptomatic of some systematic error.

~hat might be responsible for the offsets? As noted immediately above,

there is some evidence of systematic error. This might be an error in

the filter transmission curves, perhaps just a shift in the wavelength

Scale. Changing the assumed metallicity from solar values has no effect

on the offsets. Reducing the assumed microturbulence reduces the offsets

~or the blue, short period Cepheids, but has no effect on the long period

Cepheids, An obvious possibility for the source of the offsets is an

e~ror in the reddening, but changing the color excess varies the observed

and synthetic magnitudes by the same amount, leaving the offsets un-

Changed. However a problem may exist with our calculation of the visual

absorption from the B-V color excess. We used the multiplier 3.0, but

Values as high as 3.3 have been observed. Increasing this value would

~ake the apparent dereddened visual magnitudes brighter, the angular

diameters bigger, and the synthetic magnitudes brighter. Increasing

the multiplier from 3.0 to 3.2 yields the offsets in Table 2. Y Oph,

213

with the largest color excess, now has offsets s~milar to the other

Cepheids. The mean offset in H for SV Vul now is negative. T Vul has

a B-V color excess of only 0.07, so its offsets are not reduced. But

reducing its microturbulence can reduce the offsets by about 0.05. So

some combination of systematic error, error in the calculated visual

absorption, and uncertainty in microturbulence seems to account for the

offsets.

References:

Welch, D. L., Wieland, F., McAlary, C. W., McGonegal, R., Madore, B. F.,

McLaren, R. A., and Neugebauer, G. 1984, ~p. J. Suppl., 54, 547.

2 . 0 0

2 . SO

! . 7 5

2.2S

1.75

2 . 2 S - . 2Sl

I I ' I I I

,, I I .......... I I I ,

O, 0 0 0 • 2 5 0 . 5 0 0 • 7 5 0 1 , 0 0 0 1. 2 5 0

PHASE

FIGURE I: Plot of synthetic infrared magnitudes (solid line) and obser ved infrared magnitudes (star-shaped symbols) from Welch et. a]. (1984) for Eta Aql. Synthetic magnitudes have been brightened by the amount given in Table I.

2 1 4

Table 1

Mean 6m's and standard deviations, ~-m is the difference between the

synthetic magnitudes and the observations of Welch et. a]. (1984). The

sense is msy n- mWelc h. (C) means the Cepheid was analyzed with Cousins

photometry; (J) means Johnson photometry was used.

~_~p_he i d

T Vul (J)

V350 Sgr (C)

.... (j)

BR Sgr (C) vl ,, (j)

U Sgr (C)

,, ,, (j)

Eta Aql (J)

X Cyg (J)

Y Oph (C) ~w

" (J)

SV Vul (J)

AJ

0.129 + .023

0.170 .015

0.198 .018

0.131 016

0.127 .016

0.175 025

0.173 029

0.158 035

0.229 053

0.295 035

0.267 030

0.175 065

&tI

0 115 + .012

O 178 .O30

0 192 035

0 097 025

0 094 028

0 149 O17

0.148 023

0.142 020

0.148 031

0.195 022

0.182 019

O.O49 O45

aK

0.120 + .017

0.165 .014

0.179 .019

0.095 .018

0.093 .019

0 , 1 3 0 . 0 2 5

0 , 1 2 8 . 0 2 6

0 . 1 5 2 . 0 2 4

0.182 .041

0.185 .020

0.171 .017

0 087 .042

Table 2

New mean Am's after correction for visual absorption AA V.

deviations for means are the same as in Table I.

Ceq~Phe i d A_~Av A--J

T Vul (J) 0.O1 0.119

V350 Sgr (C) 0.07 0.I00

.... (j) ,, 0.128

BB Sgr (C) 0.05 0.O81

" ,, (J) " 0.077

U Sgr (C) 0.09 0.085 Ii

,' (j) " 0.083

Eta Aql (J) 0.03 0.128

X Cyg (J) 0.05 0.179

Y Oph (C) 0.13 0.165

" ,, (J) " 0.137

8V Vul (J) 0.08 0.095

aH

0 105

0 108

0 122

0 047

0 044

0.059

0, O58

O. 112

O. 098

O. 065

0. 052

-0. 031

AK

0.Ii0

0.095

0. 109

0.045

0. 043

O. 040

0. 038

O. 122

0. 132

0.055

0.041

0. 007

Standard

215

Field RR Lyrae Stars

J. Lub

Sterrewacht Leiden

RR Lyrae stars provide still one of the best ways to determine distances in our

Galaxy, and as it stands maybe even within the local group of galaxies (e.g. van den

Bergh, 1986). Apart from that they provide interestJpng insight into the study of

stellar pulsation and evolution.

In this review I will try to emphasize those results which can be reached from

observations of the RR Lyrae stars themselves and disregard most of the information

which can be got from other methods. One piece of theory will be used heavily,

however, the Pulsation~formula which is known not to depend on the details of the

stellar models used.

Recently there has again been an increased activity in the field of the study

of the RR Lyrae stars due to new observational data, of which I mention: double mode

pulsation, proper motions, radial velocity curves and infrared photometry. I will

start this review, by considering what information can be derived from the study of

multicolour photometry of field RR Lyrae stars.

Mult lco lour Photometry

Oke and co l labora tors (e.g. 1966) used scanner data in order to determine the

variation of temperature and radius over the pulsation cycle of an RR Lyrae star.

Afterwards it was reallzed that It was more economical to take observations in the

various Intermediate~band photometric systems (e.g. van Albada and de Boer, 1975).

I will take all examples in this part from my own VBLUW photometric survey of

southern field RR Lyrae stars (Lub, 1977), but I emphasize that as long as the same

spectral features or regions are observed any well-callbrated photometry will serve.

The table below gives information on the VBLUW passbands (valid prior to 1979) used:

V B L U W o

( A ) 5835 4325 3850 3630 3265 O

A~ ( R ) 690 430 225 235 135

The following three photometric indices are particularly relevant for the

discussion of the RR Lyrae photometry (units are 101og (intensity)):

Reddening free blanketing parameter: [B-L] = (B~L) - O.43(V~B)

Blanketing free Paschen continuum : (V-B) = (V~B) - 0.37(B~L)

"Blanketing" free Balmer jump : (L-U)

218

Sturch (1966) noted that for hlgh~latltude RRab stars, there exists a n apparent

Correlation between (B~V)j at minimum light and the Period. I have followed his

aPProach (Lub, 1979) but noted that it is unsatisfactory to use the period to

determine a colour excess, because the excess affects the temperatures which later

On Wlll be correlated with the period again.

A way out is to measure two blanketlng~free colour indices, one of which is

Peddenlng free e.g. the HB index, but there is a small dependence of HB on

blanketing, see e.g. Lester et al. (1986). If one takes the blue envelope in the

(V-B)* vs H~ diagram the reddenlngs can be derived independently of the period. I

have checked that the Period mlnimum~colour relation calculated from thls second set

of COlour excesses does indeed give the same "Stureh~relatlon". The advantage of

the HB method Is that It is also valid for the RRc~variables.

E(V~B) is related to the more familiar Johnson E(B~V)j by the relation, valid

for A and F ~stars only: E(B~V)j ~ 2.47 E(V~B)

Strugnell et al. (1986) have pointed out that excellent agreement exists

between the VBLUW reddenlngs and the HI~reddenings of Bursteln and Heiles (1982).

I COnclude that reddening is no longer a problem in studying RR Lyrae stars apart

from a possible uncertainty in the zeropolnt of O~O1.

Bla~nketln~Metal Abundance

The RR Lyrae stars span a range of more than two decades in abundance. Most

blanketing indicators appear to be linearly correlated with the logarithm of the

abundance [Fe/H ] , but the VBLUW relation is curved. The best known such blanketing

Indlcator Is Preston's AS ~Index; it has recently been reviewed by Smlth(1984).

In~pectlng the Kurucz models I noted that a fractional power of the abundance gave a

much better description.

I I I I I ~

1.0 0 FIBure I The VBLUW Abundance Callbratlon,

Source for [Fe/H]: Butler and Deming(197g)

~,3 {FelH]

0 5 -1

-2

0.0 ~ I 1 ~ t 1 o~ IS-LI ~ ~SO

This Is shown in Fig. I where I find: k I/3 = II.85 I[B-L]]-O.060}

Where [Fe/H ] = log k , is the iron abundance In Solar unlts. This emplrlcal

abundance callbratlon is compared to the model results in Flg. 2, where it Is

2 1 9

clearly seen that the reddening~free [B~L] index is independent of surface gravity

and almost independent of temperature

o -

. . . . . . A=O01 ~ @ . . . . A OtO . . . . . . . A : : o o o i

O2

0 005 0. I O, t5 ( v - B ) '

B l a n k e t l n g d i a g r ~ t o t t h e i n t e r v a l Tel f 7500~5500K. At e a c h a b u n d a n c e

llne~ of constant ~avlty at 1o8 g=2.5,3.0 and 3.5 are dram. The position Of the

RRab ~tar~ at mlnim~ light I~ shown as a runetlon of abundance.

The blanketing line for (V-B) has a slope of 0.37. The blanketing free index (V~B)*

= (V~B) m 0.37(BhL) is equivalent to StrSmgren's (b~y) index (Lub, 1979).

Effective Temperatures - Physical Parameters

When blanketing and reddening are known the variation of effective temperature

and gravity can be determined from one grid of model atmospheres independent of

abundance. A convincing Justification that static model atmospheres can be used was

given by Castor (1967). For most of the pulsation cycle the deviations from a freely

falling atmosphere are very small at the continuum forming depths in the atmosphere.

Only during the acceleration phase do deviations occur, probably more due to the

occurrence of convective energy transport than to the effects of llne emission from

shock waves, which normally occurs only in optically thin regions.

My temperature and gravity calibrations are based on the improved version of

Kurucz's (1979) models as presented by Lester et al. (1986). Because many authors

have used the much less detailed grid based on the Bell models (Manduca, 1980) I

have taken care to check that the Bell models give essentially the same slope of the

temperature~oolour relation, and the same gravities as the Kurucz models.

In Fig.3 I show the variations of the physical parameters for 4 representative

RR Lyrae stars each plotted in the grid corresponding to its abundance. With the

variation of effective temperature, effective gravity and thus of bolometric

luminosity and radius known it is possible to calculate the "equilibrium

temperature" of each star following van Albada and de Boer (1975). This assumes that

the mean radius (or Baade~Wesselink radius, see below) is equal to the equilibrium

radius.

220

01

02

03

04 " " "'"';

1

7 A = o ~

.... "' :~ ...... ..--'":

.-- :

.. ..,.

. .

i R~ C

e~ ~oo~ ""

.5- ..<.'

. ........ i

I

i ""

" ......

~ ' "?

""' '~

i

~" ..".."; :..."" .!.--4:~'~-"

. ......... . ......... r i ./i j-

.... - .:

..

L :':i ...... ....

"{i ....... ,:

1

0 O1

0.2 V-B

S-AUG 86

A= Ol

.. ""

X -""

---':'

o ~ I .........

;" " ........

"" ............ }"'1"

SA

ra ?mm3. ""

,i.'" .-:'"

.-"" !

. -.?

....... / .....

......~

/ ....--L"

]. .~,..

.. : ../

-.,

o.3L i: ,!/i:: >: ....

i 04 f

:" ' .....

-- J

, 0 O

l 0.~

V-B

~. ..........

-.:-..;~

io~ A = ~.~

............... }: ......... i/.---:'

VY Ser

?ooo,..'" ,~,--" "

,.-::"" - ....... i

........ /.-

}: .......... :! ........ . .......... :

I .. '

/, " ' "

-"" .-"

"2

~""} '"":

'"}: '"""~i: ....... " ........... 0

2 I

" '

" ' '

/" " ")

'"'°"" .....

" " :

....'" ':' °

-' :

..-° .i

,.-' [

: .... ....

. ..... ...- ./.~'. ....... /.. ....

I ....... : }.../,-.:.i'}": ..........

y

03 I

S-'" '

~" *." , " ~""

.-""

0,4 !-

I .

--

~

--

J

1

o o.1

0.2 V

B

~-AU

C-B

~

ol I

A = o

.ol

X A

ri

D.2

0.3

D.4 ( '...3.

--'" :; ..-"

/ ...-" ...;3

,~.. " ./. ..... ../-

. ........ :

..:

.,.. ..:

.., :

... ....-:

~." .".."",.": ........ ,i.-",.~.>

. .......... .i ..........

#.- ./..-......,::

i. ¢ .,.! .......

. ........... ...... ..-~..-".. ......

}.'. ..... .~..,- .... ..:

.........

.,<. --- - ..- ..}:: .. }: • ...- / ......

.. -"

0.1 0.2

V-B

_I

,o

o

,o

o

v

Then I can rewrite the pulsation formula (van Albada and Baker, 1971), as:

log M0"81/L = (-I.T72 + log P + 3.48 log 6500/Tell) I 0.84 ( i )

to derive physical information on the mass and luminosity of the RR Lyrae stars.

Flg. 4 shOWS the period~Itemperature diagram. Originally I studied this diagram in

order to derive Information on the helium abundance of RR Lyrae stars both from the

position of the fundamental and first harmonic blue edges and from the value

log M0"81/L . However, both methods may not be directly applicable (Stelllngwerf of

1984, and below) so that I will now first describe what can be found out directly

about the Field RR Lyrae stars from Fig. 4.

P

Fl~ume 4_ The pemlod temperature diagram. 3SO Bat°~ ° • oo xHO ~71~7

The RRc stars (trlangle~) are shown at the ~ o~p/u~ ~p

position corresponding tO their fundamentallzed ~ I AE~v:O0~ o .~ ~Oo °

period. AI,~O Bhown are the fundamental (FBE) ~ a •~• ~BL

The line oorre~pondin£ to logMO'B1/L-~1.65 is 385 _ ~ a o •

°FB E • ~ al~o ~hown, The po~itlon of AQ Leonl~ 19 t~• / ~ ~ Y:025 ~ Indlcated by the "~" Sil~n. HBE IFe/HI

• -,C lo-og

. . . . . . . I I I I -6 -2 t~l ~ 0

the sample at [Fel H] =-I.4 and -0.9 respectively, I find as shown Subdividing

before (Lub, 1977, Pel and Lub, 1978) that there are 3 natural groups of stars:

[Fe/H]<-I.~: , mean period 0d65 ,transition period roughly 0d56. at a I.

temperature (e.g. for the double mode star AQ Leo) of 6850K Reminiscent of

Oosterhoff group II (011) globular clusters.

>[Fe/H] ~ mI.4: mean period 0P55 ,transition period of roughly 0d45 2. ~'0. 9 at

a temperature of 7050K, reminiscent of Oosterhoff group I (OI).

[Fe/H] > -0.9 "metal rich" stars, not present in globular clusters. 3.

The data for the RRc~type stars are much more difficult to interpret but agree in

the mean. In Fig. 4 a clear trend with metal abundance is visible, and therefore

this is shown in more detail in Fig. 5 as a correlation between metal abundance and

the mass~to-lumlnoslty parameters.

I find:

log M0"81/L ~ -1.74 + 0.10 [Fe/H]. (2)

This of course can be recognized immediately as Sandage's (1982b) AlogP vs

abundance relation. However, In order to interpret the AlogP parameter he has to

assume a one~to~one relation between amplitude and temperature, throughout the

instability strlp. From my data I would suggest that whereas group 2 shows a period ~

amplitude and perlod-temperature relation in good agreement with the one for M3, the

222

group 1 relation is shifted towards lower temperatures, and moreover does not reach

equally high amplitudes. The position of several representative Globular Clusters in

Fig. 5 (de Bruyn and Lub, 1986; Bingham etal. 1984) shows once again that there is

no essential difference between RR Lyrae stars in the Field and in Globular

Clusters.

F l ~ r e 5 The r e l a t i o n b e t w e e n m e t a l a b u a d a n c e

a n d m a m s - t o - l u m l n o s l t y p a e a m e t e r f o r F i e l d

RR Lyrae ~tars,

Data for 3 Globular clusters are shown for

comparison. The lumlnoslty ~cale on the right

is d e r i v e d a ~ s u m l r ~ H - 0 . 6 3 H e * The p o s i t i o n o [

the doublemmode var lable AQ Leonis is indloated.

Stars wlth incomplete or deficient photometry

are ~hown as open c l rc le~ .

- 20

~-'E-

I I OXX V~R ........ ~IAG LEO

° o M15 •

• • ~ C E N o

• • •

o ° ~

[FelH)

,8

Log L

At this point it is important to consider the fundamental limitations of this

method. There are errors from 2 distinct origins:

i. Applicability of the models (especially at certain specific phases)

ii. Accuracy of the calibrations (systematic) and the photometry (random)

As an illustration I mention that an error of +O~01 in E(B-V)j leads to an error of

+0.005 in log T^f~ (7OK), +0.10 in log g, none in the abundance determination, and

~0.019 in log M~'~I/L .

The mass to luminosity parameter predicted from "canonical" Zero Age Horizontal

Branch (ZAHB) models depends linearly on the Helium abundance Y only (e.g. Renzlni,

1977), but as Sandage (1982a) pointed out, this leads to an anticorrelation between

helium~ and metal abundance. Ways out of this dilemma, the so called "non canonical"

frame are discussed by Caputo etal. (1983).

Combining (I) and (2) and taking from my analysis of the lightcurve variation

the bolometrlc correction is: BC= ~OT05[Fe/H]- . that I find:

Mv=+0~37~2.03 logM+O.20[Fe/H] (3)

taking from the ZAHB models a most likely mass of order M= 0.63M O , I then flnd

My= 0~55 at [Fe/H]= -1.2 and M = 0~43 at [Fe/H]= ~1.8 If one accepts, however, the

typical values of the masses derived from double mode pulsation (Cox etal., 1983,

Clement etal., 1986), i.e. M=0.55 M O for OI and M=O.65 M® for OII globular clusters

then I find O~65 and 0~37 respectively.

223

Baade Wessellnk - Methods

Under this heading I will include all methods which either determine the radius

variation from temperature, gravity and bolometrlc correction or from a relation

between surface brightness and colour. The radius amplitude so derived can be

combined with the radius excursion from the radial velocity curve to give the

absolute value of the radius. The surface brightness method, if calibrated

absolutely, directly determines the distance and so avoids the use of effective

temperatures.

Even though the main idea behind this method is simple in principle, many

stumbling blocks remain. For a long while very few good r.v. curves were available,

this situation has been much improved recently by the production of very high

quality curves from radlal~velocity photometers. (e.gf Burki and Meylan, 1986,

Cacclarl and Clementlnl 1986, Jameson et al., 1986). The relation between llght~ and

radial velocity amplitude (Avrad) still remains badly determined, however.

Many applications of this method must be criticized, however, because they

disregard the dependence of the surfaceabrlghtness~colour or temperature~colour

relation on surface gravity and metal abundance.

The older data indicated a kind of uniformity of the r.v. curve shapes and even

from the newer data the shapes are not wildly different. McDonald (1977) constructed

a mean r.v. curve, which with a typical converslon~factor of 4/3 (Castor, 1967)

gives rise to:

- 1.39 P fAvrad~ solar radii. (4) AR =Rma x Rml n " 55 "

The Wessellnk radius <R> is then <R> = Rma x a 0.39 AR~ combining this with a simple

linear surface brightness~colour relation restricted to a phase-interval free from

problems (i.e. 0.4~0.8) he derived as a function of AS~ (My own reworking of his

data and taking as above a mean bolometric correction BC~ ,0~05[Fe/H]. ).

N AS [Fe/H] < My> log L log M0"81/L log L

8 I -0.4 0~82 tO.08 1.57 ~1.78 1.61

16 6 -1.2 0~62 ±0.04 1.66 -1.86 1.70

9 9.5 ~1.75 0~49 ±0.06 1.74 ~1.92 1.75

A mass of M~0.63 (is assumed) in order to calculate in the last two columns what I

predict from formula (2).

These results seem very encouraging, but unfortunately it was found later by

Carney and Latham (1984) that a phase shift occurs between the radius and the r.v.

curve for the longer period stars. An example of this is shown in Fig 6.

For VY Set minimum radius coincides with maximum light. Whereas the photometric

gravity peak agrees within the error of measurements with the one predicted from the

r.v. curve the minimum of the radius curve is displaced . Fig. 7 shows this

agreement both in phase, amplitude and absolute value for the variable X Ari

224

3.90

4.5

" ' . . RR CET °%

°, ,

LOG T~,~

@ .°

• • o . . ° • • • ° * * " . . . . * , ° * , • . * * , ' , •

LOO G • , .

D . R

.* .o

.. ..° °. .o"

%" -°

RIB ,

5,S~

3.1~!

~5

1.0

VY SER

, • $ •

° • . . . . . , .o

LOG G

• " ~*°" ", , % , , ,

• , , o

• . o

B/R~

~.~}

I

I

1.o I

o.~I

o. s

.• ".. X f iR ]

% . .

L06 "i~F,

° % • • . . , , . , . * ° ' * ° ° * * ° " ° ' ° * . • • • . ° . . , . ° .

LOG G .' ..

%°

o.° , , "

B/R~

". 5 ARA

%

LOG l l~ r

o ' * ' ° . . . • ° " . . . °o . . ' . . . ' ° . . " . ° " . • "

*LOG G

Figure 6 The variation of effective temperature, gPavlty andradiu8 from the grids in

Pig. 3, The phase~s~ift between the gravity and the radius curve ie clearly visible

for the RRb star. VY Set and possibly for RR Cet.

2 2 5

30 ~ogg

25

~o

o o

o o eo o o

% ,o~.,, %

°OoooeoooOO°

"-:..::.......:.'\;"

Figure 7 CompartBon between the effeetlve gravlty

derlved from the photometry (gphot) and

calculated fPCX~ ~he mean g rav i t y , radius a m p l i t u d e

and the rad ia l ve loe t ty curve (ge f f ) . Note t h e

agreement ~n pha~e, amplitude and value.

Longmore et al. (1985) show that the phase~shlf~dlsappears if instead of going

towards the blue one goes to the infrared when deriving the surface brightness. More

study Is needed, however, to understand why the surface~brlghtness colour relation o

changes so abruptly if one includes measurements below 5000 A.Most analyses try to

avoid the phases around ri~Ing light, so It Is still of some interest to see which

relation between period and radius resulte. This Is done in Flg. 8, where I have

drawn the theoretical lines from the van Albada~Baker formula:

log R = 0.952 + 0.595 log P + 0.405 log M ~ 0.07 log 6500/Tef f. (5)

Surprisingly the data do not deviate much from what is expected on these grounds;

only the results from Burkl and Meylan (1986) deviate. A mass somewhere around

0.6 M e is indicated. -T 1 l l =

M'=lO / / / / 09 -- ~"

Figure 8 The p e r l o d ~ r a d i u m r e l a t i o n f~ o / / / 6 Fie ld RR Lyrae ~tar~. The d a t a a r e f r o m : / I / M = O 3

• Manduca e t e l . (1980)

• Caeclari and Clementtn~ (1986) ~ / ~

A Jameson et e l , (1986) - 07 / / / ~ & A ~ / ~ &

o Burki and Meylan (1986) / 4 " The drawn l ines ae~ume~ a mass of

M-O.63.M o

05

-05 -03 -O1

Statistical Parallaxes

The classical work in this field Is by van Herk (1965). His main result

of M =0~87±0.22 where ~ is the mean of the maximum and the mlnlmum magnitude over Pg cycle, gives <Mv>= 0~77±0.22 or log L = 1.60±0.09. Many questlons the remained, as

regards the quality of the photometry and the reddening model. Therefore Hemenway

(1975) produced an improved catalogue leading to a value of <My >= 0~50. Using my own

study of RR Lyrae colours it was posslble to select from her catalogue a well

226

8tudled and distance limited sample of 60 stars with selection criteria AS 5,6,7,

P> 0944 and <V>o<11~5 ; I found <My>= 0~77 again from 60 stars (no error estimate;

Lub 1978, unpublished).

Meanwhile the modern maximum likelihood solutions gave rather awkward or even

Plainly unphysical results implying brighter absolute magnitudes for more metal rich

Stars, which at the same time have much shorter periods, thus also radii, but

roughly the same temperatures (Heck and LaKaye, 1978, Clube and Dawe, 1980).

Recently a new compilation of proper motions became available, and both Hawley

et al. and Strugnell et al. used these data to derive essentially the same result:

[Fe/H]= ~1.1 . It was impossible to decide decisively upon any <Mv>=O. 75±0.14 at

dependence upon the metal abundance, because of the smallness of such subSsamples.

I am grateful to have been communicated at this meeting a recent result showing that

most of the UBV photometry used had a zeropolnt error of 0~09 (Barnes and Hawley,

1986) giving an final value of <Mv>~ 0~68±0.I0. (Log L~ 1.64±0.O6).

In combination with our relation (2) this implies a mass of M = 0.55 ±0.05 M O , in

agreement with the result from the double mode variables.

SuU~ary of Absolute Magnitude Determinations

Let me now give my own evaluation of the absolute magnitude of RR Lyrae stars

based on the discussion above. It appears best to relate M v to a mean metal

abundance [Fe/H]= -1.2 (orAS=6)

i Analysis of the llght~ and colour variation would predict log M0'81/L=-1.86 and

Mv~ 0~66 or 0~54 as either the mass for Oosterhoff I or for Oosterhoff II-cluster

Stars is taken.

iI The best analyses of the Baade~Wesselink type, suggest a value near My= 0~62.

ill The most refined statistical parallax determination gives: My= O~6B .

Since methods i and ii are closely related, the best choice is My= 0~66, with an

error of the order of 0~I0 ; this is a compromise between the errors on the

Photometrlc calibrations, on the phase matching and the errors of the radial

Velocity curves and of the statistical parallax solution.

The dependence upon metal abundance takes into account the apparent dependence

of the mass on metal abundance (lower at higher metal abundance, as also expected

from theoretical arguments, e.g. Renzlnl, 1978) and the inability to see such an

effect within the statistical parallax solutions.

Therefore I would suggest:

My= 0~66 + 0.25 ( [Fe/H] + 1.2) (6)

A~!~plicat,lon to t,he Galactic Centre

RR Lyrae stars are present in large numbers in the Galactic bulge; the maximum

of the space distribution directly determines the distance to the galactic centre

227

(e.g. Oort and Plaut, 1975). Recently Blanco and Blanco (1985) and Walker and Mack

(1986) restudied the RR Lyrae in Baade's window at b=m3 °. Taking [Fe/H]~mO.9 their

value for R o is 7.8±0.4kpc. A much larger sample of stars is available in Plaut's

field no 3 at b: ~6 ° to ~12 °, moreover the value of the reddening is much more

regular in this field. A restudy based on new photographic material in B and R was

therefore undertaken by Wesselink (1987) at Nljmegen. From the 25 plate pairs taken

at the UK~Schmldt telescope excellent amplitudes and colours can be measured with

the ASTROSCAN automatic plate measuring machine at the Sterrewacht In Leiden.

Individual reddenlngs with quality of 0~05 in (B-R) could be derived from the

minimum colours, but more important it has been possible to improve upon the

completeness analysis of Oort and Plaut (op. clt).

Using the period-amplltude diagram where it is assumed that for M3 My: 0~60 he

made a maximum likelihood solution for the space distribution giving

Ro:8.20±0.17kpe, oblateness c 0.60±0.07 and exponent of the density distribution a

law n= -2.95±0.22.

It is interesting to note that in a recent thesis on the velocity field of our

Galaxy (Brand, 1986) a distance Ro= 8±O.5kpc is determined based upon very precise

radial velocities from CO-emlssion lines in molecular clouds associated with young

stars.

Acknowledgements: My visit to the Los Alamos meeting on Stellar Pulsatlon was made

possible through grants from the Leiden KerkhovenmBosscha Foundation and the

Astrophysics Visitors Program at Los Alamos. I wish to thank Theo Wessellnk for

sharing hls prellmlnary results on Plaut's fleld no.3 with me, Jan Wlllem Pel for

moral support, EIs Zikken and Wanda van Grieken for all of the typing under high

tlme~pressure.

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van Albada, T.S., de Boer, K.S., 1975 Astron. Astrophys., 3_~9. 83

Barnes, T.G., Hawley, S.L., 1986 to be published in Astrophys. J. Lett.

Blngham, E.A., Cacclarl, C. Dickens, R.J., Fusl Peccl, F., 1984 Mon. Not. Roy. Astr.

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Blaneo, V.M. Blanco, B.M., 1985 Mem. Soc. As t r . I t a l . 5_.~6,15

Brand, J . , 1986 Thesis Un i ve r s i t y of Leiden

de Bruyn, J.W., Lub, J . , 1986 These proceedings

Burk l , G., Meylan, G., 1986 Astron. Astrophys. 156, 131

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Carney, B.W., Latham, D.W., 1984 Astrophys. J. 278, 241

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228

Clement, C.M., Nemec, J.M., Normand, R.# Wells, Th., Dickens, R.J., Bingham, E.A.,

1986 Astron. J. 9~2, 825

Clube, S.V.M. Dawe, J., 1980 Mon Not Roy Astr Soc 190, 591

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Hawley, S.L., Jefferys, W.H., Barnes, T.G., Wan, L., 1986 Astrophys. J. 302, 626

Heek, A., LaKaye, J.M., 1978 Mort Not Roy Astr Soc 184, 17

Van Herk, G. 1965 Bull. Astron. Inst. Neth. 18, 71

Hemenway, M.K°, 1975 Astron. J. 80, 199

Jameson,R.F., Fernley, J.A., Longmore, A.J., 1986 in preparation

Kuruez, R.L., 1979 Astrophys. J. Suppl., 40, I

Lub, J., 1977 Thesis University of Leiden

Lub, J., 1979 Astron. J. 84, 383

Oke, J.B., 1966 Astrophys. J. 145, 466

Longmore, A.J., Fernley, J.A., Jameson. R.F., Sherrington, M.R., Frank, J., 1985

Mon. Not. Roy. Astr. Soc. 2!6, 873

Lester, J,B., Gray, R.O., Kurucz, R.L. 1986 Astrophys. J. Suppl. 61, 509

McDonald, L., 1977 Thesis University of California at Santa Cruz

Manduca, A., 1980 Thesis University of Maryland

Manduea, A., Bell, R., Barnes, T.Go, Moffet, T.J., Evans, D., 1980 Astrophys. J.

250, 312

Oort, J.H., Plaut, L., 1975 Astron. Astrophys, 4 L, 71

Pel, J.W., Lub, J., 1979 in: IAU Symp 80, eds. Davis Philip A.G., Hayes, D.S.,

Reldel Dordrecht, p.229

Renzlni, A., 1977 in "Advanced Stages in Stellar Evolution", Bouvler, P., Maeder,

A., eds. Observatolre de Geneve, p. 149

Sandage, A.R., 1971Aetrophys. J., 244, L23

Sandage, A.R., 1982a Astrophys. J, 152, 553

Sandage, A.R., 1982b Astrophys. J. 152, 574

Smith, H.A., 1984 Publ. Astr. Soc. Pacific, 96, 505

Stellingwerf, R.F., 1984 Astrophys. J. 177, 322

Strugnell, P. Reid, N., Murray, C.A., 1986 Mon. Not. Roy. Astr. Soc. 220, 413

Sturch, C.R., 1966 Astrophys. J., 147 , 774

VandenBergh, S., 1986 in: Galaxy Distances and Deviations from Universal Expansion

eds. B.F. Madore and R.B. Tully, Dordrecht Reldel, p.41

Walker, A.R., Mack, P., 1986 Mort. Not. Roy. Astr. Soco 220, 69

Wessellnk, Th., 1981 Thesis University of NiJmegen

229

ON THE MIXED-MODE RR LYRAE VARIABLES IN THE GLOBULAR CLUSTER IC aa99

Christine M Clement David Dunlap Observatory

Department of Astronomy

University of Toronto

Toronto, Ontario, M5S IAI

Canada

James M Nemec

Department of Astronomy i05-2~ Division of Physics, Mathematics and Astronomy

California Institute of TechnoloEy

Pasadena, CA 91125, U.S.A.

R J Dickens Space and Astrophysics Division

Rutherford Appleton Laboratory

Chilton, Didcot, Oxfordshire

OXll 0QX, United KinEdom

Elizabeth A BinEham

Royal Greenwich Observatory

Herstmonceux Castle

Hailsham, East Sussex

BN27 IRP, United Kingdom

Double-mode RR Lyrae variables have been found in several Elobular

clusters, notably MI5 (Cox, Hodson and Clancy 1983 (CHC), and

references therein) and M3 (GoransMiJ 1981), and in the Draco dwarf

Ealaxy (Nemec 1985). Conversely. none at all have been found in some

other clusters investiEated exhaustively, notably OmeEa Cen (Nemec et

al 1986) and M5 (Nemec and Clement 1966). AlthouEh the incidence of

double-mode behaviour in RR Lyraes is not well understood

theoretically, the application of linear pulsation theory to

sophisticated stellar models (J~rEenson and Paterson 1967, Peterson

1978, CHC) has shown that the ratio of the first-overtone period to

that of the fundamental has been found to depend on the stellar mass.

The Oosterhoff type I southern cluster IC 2~99 has the highest frequency of RR Uyrae stars of any Elobular cluster in the Galaxy. In

an earlier study (Clement et el 1979), many of the c-type variables

were found to have large scatter in their liEht curves, motivatinE a

major effort to search for secondary periods, ueinE extensive new and existing plate material. As a result of this study, thirteen

double-mode (RRd) variables have been discovered (Clement et al 1986)

and their primary and secondary periods determined,

230

These sta~s have surprislnEly uniform properties, which a r e Considerably different from RRd stars found in Oosterhoff type II

SYstems. The mean first-overtone period of the IC aa99 REd stars is

0.357 days, with a dispersion of 0.005 days. as compared to a value

~ear 0.~0 days for Oosterhoff II REd stars. The mean ratio of

first-overtone periods to fundamental, <PI/Po> = 0.Taaa daws, with a

dispersion of 0.0002 days.

P~ 0 4 5 6 0-485 0"509 ~" 0-556 0 5 6 5 0 '590

i I I I

0 - 7 4 8

P,/Po 0 .746 -

0 ,744

0 . 7 4 2

0 - 7 4 0

• DRACO o M I 5 X M3 -~ AO LEO t, IC 4 4 9 9

o

o,~ ~ 0 " 6 5

~ - - ~ 0 ' 6 0

King Y Z Masses t - - I o 0 . 2 9 9 0 0 0 1 0 . 6 5 , 0 . 6 0 , 0 5 5

- - I [ o 0 .199 0 -001 0 6 5

0 . 5 5

I I I ............... i

O' 34 0-36 O" 38 0 -40 O-42 O" 44 P,

Figure i. Known RRd stars plotted in the Peterson diagram, where

Pl is the first-overtone period and Pc the fundamental. Also

shown are the mass calibration lines of CHC for masses of 0.55,

0,60 and 0.65 Me. The points fall into two distinct groups,

co~respondinE to clusters of each Oosterhoff type, the inferred

masses being 0.5~ Me for type I and 0.65 M® for type II.

The location of these stars in a plot of period ratio versus fundamental or first overtone period, (the Paterson diagram) is shown

in Figure i, which also ~ives the locations of other known RRd stars,

together with theoretical lines (CHC) for various masses and COmpositions. The RRd stars can be seen to comprise two quite

distinct groups, distinguished by the Oosterhoff type of thei~ parent

SYStem. The Oosterhoff I variables are particularly tightly clumped,

and could have zero intrinsic scatter on the basis of the estimated

errors. Since only two RRd stars have been found in M3, the IC aa99 results dramatically confirm the striking difference between the two

OOSterhoff types.

As indicated above, such a difference would be intempreted as beln~

Drimarlly caused by a difference in stellar mass, and this is

illust~ated in FiEuDe i. Using the Kin~ Ia models, Oosterhoff I

Clusters are required to have a mean mass close to 0.55 Me and

OOsterhoff II systems a value near 0.65 Me, essentially conflrminE the

earlier derivations using this method (CHC).

231

These results indicate that a mass difference between the Oosterhoff

types could be the underlying physical cause of the remarkable

distribution shown in Figure i. However, it has been pointed out

(Bin~ham et al 198~) that a mass of 0.55 M®, which appears formally to

be an extremely precise determination, is too low a value for standard

evolutlona~y models to be able to produce a significant population of

RR Lyraes in the instability strip. How accurate, therefore are the absolute values of these derived masses? CHC quote an uncertaint~ of

0.001 in Pl/Po, yet the agreement of linear with non-linear

(Stellingwerf 1975) calculations (Cox et al 1960) is only good to

about 0.003; ie an uncertainty of about 15M in the mass due to

computational uncertainties alone.

There are also other sources of uncertainty of a more physical nature,

as discussed by Simon (198R), who found that an increase in metal

opacity changed the location of the model lines in the Peterson diagram so as to increase the derived mess and thereby improve the

a~reement with other Cepheld mass determinations, This occurs because

the periods and period ratios depend on details Of the stellar

structure, which are altered by a chan~e in the opacity. This will also be true of RR Lyraes, although being more metal-defielent the

effects of changes in heavy element opacity would appear to be much

smaller. Of course, any change in the stellar structure, for whatever

reason, will affect both pulsation and evolution models to some

extent; further work in these areas appears necessary if we are to

si~nlflcantly narrow the range of uncertainty in these mass

determinations.

Our observations, together with previous work suggest that double-mode

pulsation is not an uncommon phenomenon, even though some globular

clusters do not appear to contain any such stems. This implies that

double-mode behavlour is a "stable" mode of pulsation, yet no RR Lyrae

model exhibiting long-term mlxed-mode behsvlOUr has yet been

calculated. At the same time, these stars do not appear to be

mode-switching because the current estimates of growth rates are much

too fast to account for the observed frequency in some clusters. It

has, however, been found (Bingham et al 198a) that in MI5 add in M3. the double-mode stars occur precisely at the transition temperature

between a and c type pulsation, strongly suggesting that

mode-swltching is involved.

Clearly, mlxed-mode behaviour in RR Lyrae variables continues to provide an intriguing challenge, to observation and theory alike.

REFERENCES

Bingham,E.A.,Cacciari,C.,Diekens,R.J.,Fusi Pecci,P.F.,198~. Mon, Not.

R. astr. Soc, ,209,765. Clement,C.M. ,Dickens,R.J. ,Bingham, E.A. ,1979. Astron. J. ,8~,217.

Clement,C.M.,Nemec,J.M.,Robert,N.,Wells0T.,Dickens,R.J.,Bingham, E.A.,

19S6. Astron. J.,in press.

Cox,A.N. ,King°D.S. ,Hodson,S.W. ,1980. Astroph~s.J. ,236,219.

Cox,A.N. ,Hodson,S.W.,Clancy,S.P. ,1983. Astrophys. J,,266,9a(CHC).

Goranskij,V.P.,1981. Inform. Bull. on Variable Stars. No. 2007.

J~rgenson,H.E.,Peterson,J.Oo,1967. Z. Astroph~s.,67,377.

Nemec,J.M.,19SS. Astron. J.,90,ROa.

Nemec,J.M.,Clement,C.M.,1986. In preparation.

Nemec,J.M.,Nemec,A.F.L.,Norris,J.,1986. AstPon. J.,in press.

Peterson,J.O.,1978, Astron. A stroph~s.,62,205. Simon,N.R.~1982. Astrophws. J.Lett.,260,L87.

Stellingwerf,R.F.,1975. Astrophws. J.,22a,953.

232

VBLUM p h o t o m e t r y o f RR L y r a e s t a r s I n ~ Cen and Mq *

J.W. de BruiJn and J. Lub

Sterrewacht Leiden.

Multicolour VBLUW photometry of RR Lyrae stars in the globular clusters

M4 and ~ Cen is used to derive information on reddening, blanketing, effective

temperatures and gravity of these stars. The methods employed in the

litterature to determine the reddening of globular clusters from the UBV

Colours of the RR Lyrae stars are in complete agreement with the results from

VBLUW photometry.

Reddening was determined from the minimum colours of ab~type stars by the

relation found earlier for Field RR Lyrae's between their period and

blanketingfree minimum colour (V~B)* = (V~B)~O.37(B~L) (Lub, 1979).

Similarly the blanketing was found from the reddening independent [B~L] =

(B~L)~O.43(V~B) index. The RRc stars have colours which are much less

Sensitive to blanketing however and therefore we have taken for these stars

the mean abundance and reddening derived for each cluster.

The physical parameters are based on the VBLUW calibrations determined

from the improved version of the model atmospheres by Kurucz (1979) presented

by Lester et al. (1986) as derived by Lub (these proceedings).

M4: Observations were made in May and June 1979 of 12 ab~ and 5 c~type

stars in M4, selected to span the whole available range of periods and

amplitudes. For each star about 7 measurements are available evenly

distributed over the light variation. A diaphragm of 11~5 was used throughout

and skies were taken at selected positions at the same radius with respect to

the cluster centre as the variable. We derive:

Reddening: E(VmB) ~ 0.148± 0.019 or E(B~V)j - 0~37 ±0.04 for a single star.

This compares very well with the values of for example:

Richer and Fahlman (1985), hereafter RF: 0~37 ±0.06 or Cacclari(1979):

0~36±0.02. Mean abundance [Fe/H] E -0.97±0.35, remarkably close to the

~0.93±0.31 of RF and the ~I.0±0.4 of Cacclari.

Then we calculate from all RR Lyrae log M0.81/L = -1.85±0.05, whereas we

would predict (Lub, these proceedings) ~1.84, from the approximate relation

valid for field stars: log MO'81/L=~I.74+0.1 [~] presented in that paper.

based on observations collected at the Dutch Telescope, ESO La Silla

233

Cen: In February, March and April of 1981 observations were made of 14 ab

and 7 c-type stars. For each star about 20 measurements are available.

As before we derive:

Reddening E(V~B) = 0.045±0.004 or E(B~V)j = O~11±O.O1.

This compares well with the value of 0~IO±O.O1 given by Butler, Dickens and

Epps, hereafter BDE, (1978). The mean abundance for our selection of stars is

[Fe/H] = ~1.38±0.18 whereas BDE found ~1.43±0.43.

The mass-to-llght ratio is found as log M0"81/L = -1.88±0.03 whereas from

the data of BDE -1.98±0.05 was derived. However they used a different way

(intensity mean) of averaging over the colours, as well as Bell~models. The

use of the intensity~averaged colours causes the effective temperature to be

overestimated depending on the amplitude if the lightcurve giving a value of

the mass~to-luminoslty parameter about 0.04 lower on the average. We would as

before predict ~1.88 from the relation valid for field stars.

In the accompanying review these 2 points are compared with the general

relation between metal abundance and ~ mass to light ratio. The agreement is

extremely good. As found before (e.g. Lub 1977) the Field RR Lyrae are shown

to have a wider range in metal abundance than the RR Lyrae in Globular

clusters.

The most important conclusions of the present work are:

first the close similarity between the RR Lyrae variables in the field and in

globular clusters, and

second the agreement between the reddenings derived for RR Lyrae in the field

and in globular clusters. This means that at least one parameter which

normally is taken as a free parameter in studying globular cluster colour

magnitude diagrams can be constrained very precisely, and cannot be varied at

will as done e.g. by Caputo et al. (1984, 1985).

References.

Butler, D., Dickens, R.J., Epps, E., 1978, Astrophys. J. 225, 148.

Cacciarl, C., 1979, Astron. J. 84, 1542.

Caputo, F., Castellani, V., Quarta, M.L., 1984 Astron. Astrophys. 138, 457.

Caputo, F., Castellani, V., Quarta, M.L., 1985 Astron. Astrophys. 143, 8.

Kurucz, R.L., 1979, Astrophys. J. Suppl., 40, I.

Lester, J.B., Gray, R.O., Kurucz, R.L., 1986, Astrophys. J. Suppl. 61, 509

Lub, J., 1977, Thesis, Unlversltelt van Leiden

Lub, J., 1979, Astron. J., 84, 79

Lub, J., these proceedings, p.

Richer, H.B., Fahlman, G.G., 1984, Astrophys. J. 277, 277.

234

A SURFACE BRIGHTNESS ANALYSIS OF EIGHT RR LYRAE STARS

Suzanne L. Hawley and Thomas G. Barnes HI Department of Astronomy and McDonald Observatory The University of Texas at Austin

Thomas J. Moffett Department of Physics Purdue University

Abstract

We have used a surface brightness, (V-R) relation to analyze new contemporaneous photometry and radial velocity data for 6 RR-ab type stars and to re-analyze previously published data for RR Lyrae and X Arietis. Systematic effects were found in the surface brightness at phases near minimum radius. Excluding these phases, we determine the s lope of the surface brightness relation and the m e a n radius for each star. We also find a zero point which includes both a distance term and the zero point of the surface brightness relation.

The sample includes stars with Preston's metallicity indicator AS = 0 to 9, with periods ranging from 0.397 days to 0.651 days. Our results indicate a log(R/R O ) vs. log P relation in the sense that stars with longer periods have larger radii, in agreement with theoretical predictions. Our radii are consistent with bolometric magnitudes in the range 0.2 - 0.8 magnitude but accurate magnitudes must await a reliable T, - color calibration.

Introduction

As part of our continuing investigation into the absolute magnitudes of RR Lyrae stars (see for example Hawley, Jefferys, Barnes and Wan, 1986), we have obtained contemporaneous photometry and radial velocity data for several RR-ab type stars. These observations were made at McDonald Observatory between the years 1977 and 1984 and will be published in a separate paper. We also use data for RR Lyrae and X Arietis taken from Manduca et al. (1981), In Section II we discuss the analysis of these data using a surface brightness, (V-R) relation. In Section III we examine the mean radius results and present a period-radius relation. In Section IV we conclude by looking briefly at the problem of determining absolute magnitudes.

Surface Brightness Analysis

We use the definition of the surface brightness

S v = 5 1 o g 0 + V 0

and assume that the surface brightness is a linear function of color (here we use the Johnson (V-R) color):

S, = A + B(V-R) 0.

235

Combining these expressions to eliminate S~ and using the following definitions:

c~ = constant + A + 5 log D (the distance) A = zero point of surface brightness relation B = slope of surface brightness relation V0,(V-R)0 = unreddened magnitude and color R o = mean radius r = displacement from mean radius

we obtain

V o = ot + B(V-R) o - 5 log R o - (5]lnl0)(rt~o).

Here we have used the fact that 0 o~ R/D (the radius divided'by the distance), and at any phase R = R 0 + r. Also we have assumed that for r << Ro, log Go + r) = log R 0 + (I/lnl0)(r/R0)+

The radial displacement, r, is found by integration of the radial velocity curve. We use a projection factor p = 1.38 which has been determined to be appropriate for RR Lyrae stars (Hindsley, private communication, 1986). We solve the set of nonlinear equations in the observed quantities V 0, (V-R) 0 and r for the parameters et, B and R0, using an iterative, nonlinear least squares technique (Jefferys, 1980). The observational uncertainties in (V-R) 0, V 0 and r were taken to be in the ratio 1:1.5:3. The adopted reddening relations were A v = 3.2 E(B-V) and E(V-R) 0.9 E(B-V). The results are insensitive to the choices for the uncertainties and reddening parameters.

At phases near minimum radius we found that the residuals from the fit were distributed in a systematic manner which had a strong effect on the determination of the parameters. Figure 1 illustrates these systematic residuals for the star TI" Lyn, the worst case. This phenomenon has been noted before (e.g. Burki and Meylan, 1986) and is probably due to a breakdown in the assumption that the surface brightness is a linear function of color during these phases, where strong accelerations are known to be occurring in the atmosphere,

Considering each star individually, we excised those phases around minimum radius which showed this systematic behavior. We also iterated on the solution for R 0 as in Coulson, Caldwell and Gieren (1986) to test the assumption that ln(l+r/R0) - r/R 0. The resulting R 0 differed by only 2-3% from the original values, always in the sense that the new values were smaller. In Table I we present our results from this analysis, including the list of stars, the period and metallicity (AS) for each, the adopted reddening, the excised phases and the values for c~, B and R o. Note that the quoted uncertainties are the formal errors from the fit and do not take into account uncertainties such as the exact choice of phases to excise, and the validity of the surface brightness - color relation. We estimate the total uncertainty on the mean radius values to be + 0.4 RE).

Table I

Name period metallicity E(B-V) excised phases 0~ B R 0 (days) (AS) (R~)__

V445 Oph 0.397 1 0.32 0.78-0.02 12.83+0.06 3.28_+0.02 5.6-+0.1 SW And 0.442 0 0.13 0.81-0.05 11.57+0.05 3.67-+0.02 4.7+0.1 DX Del 0.473 2 0.11 0.81-0.03 12.26_+0.04 3.48_+0.02 5.6_-_+0.1 UU Vir 0.476 2 0.02 0.84-0.05 13.23+0.06 3.54_+0.03 5.9_+0.2 TU UMa 0.558 6 0.05 0.65-0.04 12.49:k0.09 3.59-M).03 6.3!-0.2 RR Lyr 0.561 6 0.02 0.79-0.01 10.09+0A2 4.29_+0.07 6.1_+0.3 TT Lyn 0.597 7 0.07 0.73-0.13 12,82+0.05 3.41_+0.03 7.4_+0.2 X Ari 0.651 9 0.15 0.82-0.02 11.60!-_0.05 4.17:k0.02 6.4_-t-0.1

236

i 0.06i

0.04

oo2

"6 0

°t ~" - 0.02

- 0 0 4 [

-0 .06 [

o17 I

019 O. I Phc~se

j • *J o • o ~ ,

• ! **~

o13

Figure 1: Residuals from fit vs. phase for TT Lyn. The residuals exhibit systematic behavior at phases near minimum radius. The excised phases are indicated. Phase 0.0 corresponds to maximum light.

085

0.80

o~ 0.75

0 . 7 0

0.65

i J • i

LSO Fit

. . ' {" /

-0.4 -0.3 -0. ~' -0.1 Log P

Figure 2: The log(R/RQ ) vs. log P relation determined frm our analysis. The dots are our radius results, the line through the result is a least squares fit and the lower line is the theoretical relation with an arbitrary offset. A typical uncertainty of 5:0.4 RE) is shown,

Mean Radii

Figure 2 shows the Iog(R/RQ ) vs. log P relation for the stars in our sample. The line through the data is a straight least squares fit with the result

log(R/RQ ) = 0.59 (_+0.14) log P + 0.95 (_+0.04).

The slope is in good agreement with that in the theoretical relation given by van Albada and Baker (1971):

log(R/RQ ) = 0.595 log P + 0.405 log (M/MQ) - 2 log (Te/T Q ) - 2.071 log (6500/Te) + 1.055

shown as the lower line in figure 2, with an arbitrary offset. Although the individual radius determinations are rather uncertain, we believe that the trend to larger radii with increasing period is real, and that the theory is in satisfactory agreement with our observational result.

It is not strictly correct to use a single mass and temperature for the zero point determination. HOwever, at the request of the referee, we note that for M = 0.75 MQ (midway in the range 0.45 < M < 1.05 used by van Albada and Baker to determine this relation) and T e = 6500K (which they use as their reference temperature), the theoretical relation gives a zero point of 0.902. While b0a'ely out of the range of the formal uncertainty in our fit, this value is certainly consistent with our determination, considering the large uncertainties in the radii.

237

Absolute Magnitudes

Clearly the parameter ot determined from our analysis contains distance information and hence would allow us to determine the absolute magnitude if we knew A, the zero point of the surface brightness relation. The only theoretical work on RR Lyrae stars using the (V-R) color has been done by Manduca and Bell (1981). Comparison of our derived slopes (B) with their theoretical slopes shows that ours are much shallower, and are in fact unphysical when placed on their S V - (V-R) plot. This is possibly due to the notorious difficulty in reproducing the Johnson (V-R) color theoretically. Unfortunately, we are left without a theoretical estimate of A. The only recourse is to the set of static, LTE models of Kurucz (1979) which give (B-V) - T~ relations for stars of various metallicities and gravities. The absolute magnitudes we find from these models are very uncertain and wilt be discussed in detail in a later paper. Here we give merely an illustrative example of the range of magnitudes which may be expected. M is taken to be the bolometric magnitude, although the bolometric corrections are expected ,to be small for these stars.

Assuming MQ = 4.77 and TQ = 5780K we have:

M = 4.77 - 2.5 log ((T/5780) 4 (R/R O )2)

Consider two cases. First, if R = 6 RQ, a typical value from Table I, then for two extreme values of T~ = 7000K, 6000K we find M ff=7000K) - 0.0 and M (T=6000K) - 0.7. Second, if we fix T,=6500K and take our two extreme radii, we find M (R=7.4RQ) - -0.1 and M (R=4.SRQ) - 0.9. Theoretical models and observations both indicate that metal poor, longer period stars are larger, but probably somewhat cooler. Since the radius figures most prominently in the magnitude calculation, we expect to find a range of some half a magnitude in bolometric magnitude, in the sense that the metal Ix)or, longer period stars will be brighter. The "mean" bolometric magnitude is probably of order +0.5. However, precise individual values must await an improved theoretical (T-color) relation and more accurate radii.

This research was supported by NSF grants AST-8418748 (TGB) and AST-8417744 (TJM). SLH acknowledges the support of a ZONTA Amelia Earhart Fellowship.

References

Burki, G. and Meylan, G. 1986, Astron. and Astrophys. 156, 131. Coulson, I.M., Caldwell, J.A.R., and Gieren, W.P. 1986, Astrophys. J., 303, 273. Hawley, S.L., Jefferys, W.H,, Barnes, T.G. and Wan, L. 1986, Astrophys. J. 302, 626. Jefferys, W.H. 1980, Astron. J. 85, 177. Kurucz, R.L. 1979, Astrophys. J. Suppl. 40, 1. Manduca, A., Bell, R.A., Barnes, T.G., Moffett, T.J. and Evans, D.S. 1981, Astrophys. J. 250, 312. Manduca, A. and Bell, R.A. 1981, Astrophys. J. 250, 306. van Atbada, T.S. and Baker, N. 1971, Astrophys. J. 169, 311.

238

A UNIVERSAL PERIOD-INFRARED LUMINOSITY RELATION FOR RR LYRAES?

J.A. Fernley, Dept. of Physics and Astronomy, Univers i ty College London, London, England A. j . Longmore, Royal Observatory, Edinburgh, Scotland R.F. Jameson, Dept. of Astronomy, Univers i ty of Leicester, Leicester, England

From the observed opt ica l (V) propert ies of RR Lyraes in g lobular c lusters i t is

Straightforward to show that in a given c lus ter these stars should show a Period-

Infrared (K) Luminosity re la t ion with a slope of -2.2 and scatter ±0.05 mags. We have

obtained in f rared data on RR Lyraes in several c lusters and th is data is s a t i s f a c t o r i l y

Consistent with the above expectat ion. Providing the to ta l spread in RR Lyrae opt ica l

(V) absolute magnitudes is ~0.5 mags i t is reasonable, from the point of view of distance

determination, to consider a universal Per iod- lnf rared Luminosity re la t ion .

We have also obtained VJHK l i g h t curves fo r three f i e l d RR Lyraes. Using th is data,

Plus radia l ve loc i ty curves e i ther obtained ourselves or from the l i t e r a t u r e , we have

derived Baade-Wesselink r a d i i , and hence absolute magnitudes, for these three stars.

The in f rared, because of i t s reduced s e n s i t i v i t y to temperature, is p a r t i c u l a r l y s u i t -

able for Baade-Wesselink analysis. Using these resul ts to ca l ib ra te the zero-point we f ind

M k = -2.2(logP + 0.2) -0.53

Gl~Obul ar Clusters

At opt ica l wavelengths observations of RR Lyraes in g lobular c lusters t y p i c a l l y

Show, in the Period-m plane, an approximately hor izontal band. To transform th is to v

the Period_mk (K = 2.2um plane we take the period-mean density re la t ion

P v1~/R3 = constant ( I )

and the standard de f i n i t i on

L~ = RZTen~ (2)

Where p is the period (days), M, R, L~ and T e are respect ive ly the mass, radius, lumi-

noSity at a given wavelength and e f fec t i ve temperature (a l l in solar un i ts ) and n~

the wavelength dependent exponent. I f m v and mass are constant fo r the RR Lyraes in

any c lus ter and taking the approximate values for n~ of n v = 4 and n k = 1.5 then

combining ( I ) and (2) gives

m k m - 2.2 log P (3)

Furthermore the reduced temperature dependence of the in f rared e f f e c t i v e l y "narrows"

the i n s t a b i l i t y s t r i p and predicts the thickness of the band defined by (3) to be ~0.I

239

mags, compared to ~0.25 mags in the op t i ca l . (For the analogous resul ts wi th Cepheids

see McGonegal et al 1982).

To test these simple predict ions (slope -2.2, scat ter ±0.05) we have obtained

in f rared (K) data fo r RR Lyraes in ~ Cen, ~13, M4, M5, MI5, MI07 and NGC5466 (Longmore

et al I986a,b). For the four c lusters with the best data (m Cen, M3, M4 and M5) we find

slopes and scat ter of -2.15 (±0.04), -2.18 (±0.04), -2.48 (±0.04) and -2.12 (±0.06), in

reasonable agreement wi th the simple predic t ion. The slope in M4 is steeper, th is maY

be due to lack of data or i t may be rea l . ~he predicted slope of -2.2 is derived

assuming the period-m v re la t ion is hor izontal and the mass constant. Varying these

assumptions w i l l obviously change the period-m k re la t ion .

These resul ts are i l l u s t r a t e d in Figure 1 for m Cen. I t is in te res t ing to note

that the most metal - r ich and most metal-poor stars in the c lus te r show no systematic deviat ions from the mean l ine .

12.8

<ink>

13.2

13.6

A •

A

:~ 0 FttH • -1-0

- 0 . 4 - 0 .2 0.0 Log. P

Fig. I . Per iod- lnf rared Luminosity re la t ion fo r w Cen.

Clearly more data is needed

to examine the var ia t ion in slope

from c lus te r to c lus ter , however,

pre l iminary resul ts (using the less

extensive data on H15, MI07 and

NGC5466) suggests the var ia t ion may

be small. I f th is is so, is i t

reasonable to th ink of a universal

Per iod- lnf rared Luminosity relati0~?

The Period-m k re la t ions found in

ind iv idua l g lobular c lusters are

essen t ia l l y ridge l ines, so the

question of the un i ve rsa l i t y of

the re la t ion is e f f e c t i v e l y asking

what are the separations, in absO"

lute in f rared magnitude, between

these ridge l ines. This is related

to the opt ica l question of what is

the spread in absolute opt ica l

magnitudes of RR Lyraes. This

subject has recent ly been reviewed by Jameson (1986) and th is work suggests a to ta l

range in Mv of ~0.5 mags. The "narrowing" of the i n s t a b i l i t y s t r i p at in f rared wave-

lengths reduces th is range to ~0.2 mags. This is equivalent to a scat ter about a mean

l ine of ±0.I mags. Thus, i f we consider distance determination, and remembering that

the in f rared has the property of being r e l a t i v e l y insens i t i ve to i n t e r s t e l l a r reddening

(Ak/Av t O . l ) , then for a range AMv#O.5 a mean Period-m k re la t ion can give essen t ia l l y

reddening free distance moduli to ~0.I mags.

240

• t a r s There has recent ly been an increase in attempts to estimate RR Lyrae absolute

magnitudes using Baade-Wesselink methods (Manduca et al 1981, Siegal 1982, Carney and

Latham 1984, Burki and Meylan 1986a,b). A l l th is work has used opt ica l photometry and

then applied one or other of the several var ia t ions to the basic Baade-Wesselink method

CUrrently avai lab le in the l i t e r a t u r e . This has met wi th varying amounts of success.

Manduca et al and Siegel both obtained reasonable (though d i f f e ren t ) r a d i i , Burki and

Meylan found they had to remove the phase in te rva ls around maximum l i g h t in order to

achieve a sa t is fac tory f i t to t he i r data and f i n a l l y Carney and Latham fa i l ed to derive

any sensible resu l t at a l l . The main problem, h igh l igh ted by a l l these authors, is the

d i f f i c u l t y of "removing" the temperature cont r ibu t ion from the l i g h t curve so that the

spectroscopic (absolute) radius var ia t ion may be matched to the photometric ( re la t i ve )

radius var ia t ion . The "removal" of the temperature cont r ibut ion is achieved e i the r

empi r ica l ly , by postu la t ing some co lour- log.T e re la t i on , or t heo re t i ca l l y , by comparing

the colours to a series of s ta t i c model atmospheres. The large (h igh ly supersonic)

atmospheric accelerat ions in pulsat ing stars, which w i l l cause changes in the e f fec t i ve

grav i ty and give r ise to sources of non-LTE emission (due to shock heat ing), i nev i tab ly

introduce uncerta int ies in the assigned temperatures. Since the opt ica l l i g h t curves

for RR Lyraes are dominated by the temperature cont r ibu t ion (L v ~ Te4) errors in the

assigned temperature can eas i ly "swamp" the radius cont r ibut ion thus g iv ing poorly

determined rad i i . The in f rared l i g h t curves, however, are only weakly dependent on

temperature (L k~Te l ' 5 ) thus errors in the derived temperatures are far less s i g n i f i -

cant. This is i l l u s t r a t e d in Figure 2 where we show the V and K l i g h t curves fo r X Ar i .

1.0

AV

0.5

1.0

AK

__-I I ~ I"1 I I I I I I T ~

- - Q - - • •

0

- - • - -

o e Z

0,5 1.0 P h a s e

0.5

0,0 0.0 0.0 0.0

_ - T ~ - I

0,5 ] .0 Phase

I t l I f l ) I I I r l -

Fig. 2. The radius cont r ibu t ion to the opt ica l and in f rared l i g h t curves of X Ar i .

241

The sol id l ine in each case is the radius contr ibut ion. I t can be c lear ly seen that at

V the radius contr ibut ion is essen t i a l l ya "pe r tu rba t i on " on the l i gh t curve whereas at

K i t is much more s ign i f i can t .

Using opt ica l and in f rared l i g h t curves and the general izat ion of the old Baade

(1928) method due to Balona (1977) we have determined rad i i , and hence absolute magni-

tudes, for three f i e l d RR Lyraes and these results are shown in Table 1 ( for more

deta i ls see Jameson et al 7986). Since these radi i have typ ica l errors of ±10% we

estimate the absolute magnitudes are good to ±0.2 mags.

TABLE 1

Period M e t a l l i c i t y Mean Star (days) (AS) Radius (Ro) <My> <Mk>

V445 Oph 0.397 1 4.~ 1.08 0,06

X Ari 0.651 I0 5.9 0,30 -0.69

VY Ser 0.714 9 6.2 0.55 -0.66

Conclusion

With the zero-point from the Baade-Wesselink rad i i and the slope found e a r l i e r

we obtain

M k = -2.2(1o@ + 0.2) - 0.53 (4)

however, th is is a very prel iminary ca l ib ra t ion . In the near future we hope to f i r s t l y

examine the Period-m k re la t ion in more clusters and secondly obtain Baade-Wesselink

rad i i for several more f i e l d stars.

References

Baade W, 1928 Astron. Nach. 228 359

Balona L. 1977 M.N.R.A.S. 178 231

Burki G. and Meylan G. 1986a A+A. 156 131

" 1986b A+A. 159 255

Carney B.W. and Latham D.W. 1984 Ap.J. 278 241

Jameson R.F. 1986 Vistas in Astronomy Submitted

Jameson R.F., Fernley J,A. and Longmore A.J. 1986 M.N.R.A.S. Submitted

Longmore A.J., Fernley J.A. and Jameson R.F, 1986a M.N.R.A.S. 220 279

Longmore A,J., Fernley J.A., Jameson R.F, and Ski l len I. 1986b In Preparation

Manduca A., Bel l R.A., Barnes T.G., Moffett T.J. and Evans D.S. 1981Ap.J. 250 312

McGonegal R., McLaren R.A., McAlary C.W. and Madore B.F. 1982 Ap.J. 257 L33

Siegel M.J. 1982 P.A.S.P. 94 122

242

NONLINF2d{ RR LYRAE MODELS WITH TIME DEPENDENT CONVECTION

D.A. Ostlie Department of Physics Weber State College Ogden, UT 84408

A.N. Cox Los Alamos National Lak~ratory, T-6 Los Alamos, ~{ 87545

ABSTRACT

Results of convective, nonlinear RR Lyrae models are presented. The standard

miXing length theory has been used wit]] time dependence being introduced through the

convective velocity phase lag technique. Turbulent pressure and turbulent viscosity

are also included. Results are cc~pared with those of other time dependent convection

theories.

INTRODUCTION

One of the lingering problems in stellar pulsation theory is that of time

dependent convection. ~'~enever the time scale for convection, defined as the amount

of ~ necessary for a convective eddy to travel one mixing length, is of the same

order as the pulsation time, consideration must be given to the finite amount of time

necessary for the adjustment of convection to changing conditions. Several attempts

have been made recently to incorporate time dependence in nonlinear stellar models.

Deupree (1979) used a two dimensional simulation of convection to investigate the

red edge of the RR Lyrae instability strip and found that convection does indeed

sumpress pulsation at about the right location in the HR diagram. A 2D approach is,

hOwever, inc(mloatible with existing ]19 nonlinear pulsation codes. Stellingwerf (1982)

proposed a 1 D nonlinear, nonlocal, tLme dependent convection theory based on a phase

lagging of the convective velocity. His approach was then used to investigate

several features of the RR Lyrae ins~ility strip, including red and blue edges

(1984a). He also described, in detail, effects of time dependent convection on a

model (model 2.5) located in the center of the fundamental mode instability strip

(1984b,c). Very recently, two papers have been published that consider the effect of

time dependent convection on one zone models in an attempt to understand more clearly

243

the relationships between various convective parameters and pulsation (Pesnell, 1985;

Stellingwer f, 1986).

EQUATIONS

In the present study, modifications are made to the standard mixing length theorY

(BOhm-Vitense, 1958) to incorporate t/me dependence through a convective velocity

phase lag. It is hoped that this simplified approach will yield reasonable results.

The convective velocity" of a zone at tJune step n has been modified by setting

Vn--~n- 1 + T ( Vo-Vn_ 1 )

where

T=At*Vn/~

At is the time step, £ is the mixing length, and v ° re~resents the instantaneous

convective velocity determined from local conditions.

Nonlocal effects have also been incorporated by weighting the current convective

velocity of zone i with the convective velocities of neighboring zones from the

previous time step, i i - I + i + l + ,~ . i

Vn=ai_iVn_ 1 ai+iVn_ 1 ~±-ai_l-ai+l)V n

where the weighting factor for zone k is given by

ak: (i- I rk-ril/£)/3

with r k being the radius of zone k.

Other contributions due to the effect of convection have been included in the

form of turbulent pressure, energy, and viscosity.

RESULTS

TWO 60 zones models were calculated, both with an initial fundamental mode

velocity of 20 km/s. The first was model 2.5 of Stellingwerf (1984b) with L=63 L,

M-0.578 M o, Teff=6500 K, ~/Hp=l.5, and (Y,Z)=(0.299,0.001). The initial model was

integrated inward to 1% of the radius using 14% of the mass. In the static model

two convection zones exist, one in the hydrogen ionization region carrying 97% of

the total flux and one in the helium ionization region carrying 2% of the total

flux. The linear fundamental mode period is 0.812 d and the growth rate is 0.0926.

After the initial perturbation the model's amplitude grew rapidly over approxi-

mately 50 cycles to a limiting an~p!itude of 75 km/s and 1.2 magnitudes. During the

growth to limiting amplitude the strength of the convection zones steadily decreased,

with the hydrogen ionization region carrying a maximum of 18% of the flux shortly

before m i ~ radius. The figures show both the variation in absolute bolometric

magnitude and the ratio of maximum convective luminosity to total luminosity for a

typical period at limiting aniolitude. Maxim~n radius occurs at approximately i. 4x104

sec. and min/mum radius occurs at 5.1x104 sec. It is interesting to note that

Stellingwerf (1984b) finds a nearly saturated convective flux at approximately the

244

Same phase as is found here. Not surprisingly, his limiting amplitude is signifi-

cantly less than in the current study. He also finds a prominent dip in the rising

branch of his light curve that is not present in our calculations.

. 1 5 0 0 0 -

~ . 1 0 0 0 0 -

.ooooo '~ .00000

-- ,20000

.00000

.40000

.6OOOO

100000 .(X ~00000

I I I

/ Z00000 ~00000

! !

I 61D000 ~00000

This model was also tested for stability against other modes at limiting

an!olitude using Stel!in~qerf's (1974) periodic solution method. It was found that

none of the overtones were pulsationally unstable as would be expected for an object

in the center of the fundame/~tal mode instability strip.

The second model studied here is a convective version of a model of Hodson and

Cox (1982) located in the region of the double mode RR Lyrae variables. This model

245

has L=59 L0, M=0.65 M o, Teff=7000 K, ~/Hp=l.5, and (Y,Z)=(0.299,0.001). The static

model was integrated to 8% of the radius using 6% of the mass. The linear fundamental

mode period is 0.544 d with a growth rate of 0.0094; the period ratio of the first

overtone to the fundamental mode is 0.744. Due to the smaller growth rate and

limited ccmputing time, this model could not be followed long enough to determine if

the higher overtones present in the initial perturbation would danlo out. However, it

does appear that a limiting an~litude of approximately 40 km/s can be expected. It

was found that a similar phasing of the convective flux exists between this model and

the previous case even though only 4% of the total flux is ever carried by convection.

This small amount does seem to be suffioient to give a smaller amplitude than the

52 km/s obtained in the purely radiative model of Hodson and Cox.

CONCLUSIONS

The results obtained here for a modified version of the standard mixing length

theory do seem to give reasonable results for the models calculated. Apparently the

presence of pulsation tends to decrease the convective flux, with maximum flux

occurring during the compression phase, just at the time when the radiative flux is

at a n~inimum, thus limiting the final amplitude. Work still remains to determine if

enough damping will exist to stop pulsation conloletely at the red edge of the

instability strip.

REFERENCES

BOhm-Vitense, E. 1958, Zz.Ap., 46, 108.

Deupree, R.G. 1979, Ap.J., 234, 228.

, Pulsatlon in Classical and Cataclysmic Hodson, S.W. and Cox, A.N. 1982, in "

Variable stars", eds. J.P. Cox and C. J. Hansen (Boulder: JIIA reprint),

p. 201.

Pesnell, W.D. 1985, Ap.J., 299 161.

Stellingwerf, R.F. 1974, Ap.J., 192 139.

. . . . . . . . 1 9 8 2 , ~ J . , 262, 330.

. . . . . . . .1984a, A p . J . , 277, 322.

...... . 1 9 8 ~ , ~.a., 27 !, 3 2 7

....... .1984C, Ap.J., 284, 712.

246

THE STRUCTURE OF VARIABLE STAR LIGH T CURVES

R. F. Stellingwerf and M. Donohoe Mission Research Corporation

Albuquerque, New Mexico 87106/USA

In Stellingwerf and Donohoe, 1986 (Ap. J. 306, 183) simple models

were used to generate a set of radial velocity curves showing the

effect of varying amplitude and central condensation of a star.

Fourier analysis of these curves produce amplitudes in excellent

agreement with observations. A single parameter adequately describes

the variation in the shape of all the velocity curves considered:

the skewness of the curve.

In the present work, light curves generated by a nonadiabatic non-

linear one-zone model are considered. A set of curves with nearly

constant skewness are used to show the importance of a second param-

eter: the narrowness of "acuteness" of the curve, defined in the

same way as the skewness parameter, except usino the phase duration

of the above-average portion of the liqht curve, rather than the

phase duration of the rising branch. This property of the light

Curve shape is responsible for the smooth variation of Fourier phases

seen in RR Lyrae stars and many Cepheids. In the one zone model, the

Value of the acuteness is determined by the opacity variation in the

deep envelope below the ionization zones, and it could prove to be a

probe of interior stellar structure.

The full text of this paper will appear in the Astrophysical Journal,

March i, 1987 issue.

247

LONG-PERIOD VARIABLES

P.R. Wood

Mount Stromlo and Siding Spring Observatories

Private Bag

Woden P.O., A.C.T. 2606

Australia

I. INTRODUCTION

Advances in our knowledge of the details of long-period variability have been gained rather

slowly. Much of the difficulty which has been experienced in the study of the long-period variables

(LPVs) can be attributed to two causes: firstly, the very extended nature of the atmospheres of these stars,

which makes definitions of quantities such as radius and effective temperature rather hazy and makes

model atmospheres hard to produce; and, ~condly, the dominance of convective energy transport in the

interiors which, combined with the lack of an adequate theory of convective transport, has made accurate

quantitative studies of the pulsation of the envelopes difficult to produce. However, the LPVs occur at a

very interesting phase in the life of nearly all stars, from those of low mass (M < Mo) in globular clusters

to massive supergiants with M a 25M~, and their study is well worth pursuing. During the LPV phase,

stars are at their most luminous which means they can be used to derive distances to remote stellar

systems. They are also losing mass at such large rates that the mass loss essentially terminates the nuclear

burning life of the star at this point. It is highly likely that long-period pulsation is the driving mechanism

for the high mass loss rates. A thorough understanding of the pulsation of the LPVs would allow us to

derive quantities of great interest in the theory of stellar evolution such as mass loss rates, current masses

for the LPVs, cumulative mass loss since the main sequence, and final stellar masses. In this review, I

will describe recent work on the LPVs, much of which is aimed at the problems mentioned above.

II. OBSERVED PERIOD-LUMINOSITY LAWS

As a result of the accumulation of a large number of infrared observations of LPVs in the

Magellanic Clouds, there now exist period-luminosity laws for two groups of LPVs: (1) low mass (M ~<

1.6Mo) asymptotic giant branch (AGB) stars, and (2) massive (9 Z M/Mo ~ 30) core helium burning

supergiants, where the distinction between these two groups of stars was established by Wood, Bessell

and Fox (1983) (WBF).

An (Mbol,P) diagram of LPVs in the LMC is shown in Figure I (SMC stars were not

plotted due to the uncertainty of up to ~ 1 mag. in the distance modulus to stars in the SMC - Mathewson,

Ford and Visvanathan 1986). Mbo I was derived from infrared JHK photometry using the prescriptions in

WBF and photometry from WBF, Wood, Bessell and Paltoglou (1985) (WBP), Glass and Reid (1985),

Wood and Bessell (1985) and BesselI, Freeman and Wood (1986). An overall instability region for stars

on the AGB is marked, along with two lines of constant pulsation mass (the lines shown were derived

250

~-6

-9 I • -8 Sup,rgi,nt~l~ ~ i I i

-7 • @o + + . ..-~" @,3 @

' o @ ¢

-5 ./~. "¢ ~ql'~ '~" M = M.

- " ~ h ' t y Slrip" -~ _ , - - f _ L _ I / I i 100 200 300 400 500 600 700

P(days)

I . . . . I I I I - - ! I

• I mm

." ~- M~3SMe-- @

I 800 900

Figure 1, Mbo t plotted against period P for LPVs in the Large Magellanic Cloud. Filled squares are core helium burning

SUpergiants and diamonds are AGB stars. The filled diamonds are LPVs found by Wood, Bessell and Paltoglou (1985) near the

bar of the LMC. The crosses could be either supergiants or AGB stars (see text). An overall instability region for AGB stars

is bounded by solid lines and two lines of constant pulsation mass are shown.

assuming first overtone pulsation in the LPVs; the assumption of fundamental mode pulsation makes a

quantitative shift in the position of the constant mass lines but they still retain their general slope - see

WBF). Clearly, the complete AGB instability region represents a group of stars with a range in mass and

luminosity, as does the Cepheid instability region in the HR diagram.

Now, if a deep survey such as that of WBP is made in the bar region of the LMC, the LPVs

found occupy only a small part of the instability strip corresponding to relatively low masses - see Figure

1, A small sample of 7 LPVs discovered by Lloyd Evans 1971 near the bar of the LMC occupies a similar

region (Glass and Lloyd Evans 1981). WBP argue that most of the LPV population sampled in this way

Comes from the dominant stellar population in the LMC which consists of stars with ages ~2-3x 109 years

and current turnoff masses g1.6M®, The shorter period members of the WBP sample (P ~ 230 days)

appear similar to the Mira variables in 47 Tuc and probably have ages >10 l° years and masses <lMo.

Studies of the kinematics of LPVs in the solar neighbourhood (Feast 1963) indicate that the

local Mira variables occupy a mass range similar to that of the stars in the WBP and Glass and Lloyd

Evans (1981) samples, Hence, the period-luminosity law for Mira variables in the solar neighbourhood is

probably similar to that found for the low mass LPVs in the LMC. However, it should be noted here that

the LPVs in the LMC are more metal poor than those in the solar neighbourhood, and that most of the low

mass LPVs in the LMC are carbon stars in contrast to the situation in the solar neighbourhood where Miras

of spectral type M dominate. Since theoretical stellar evolution models predict a warmer giant branch for

more metal poor stars, we might expect that LPVs of a given period and mass would be more luminous in

the LMC than in the Galaxy. For a metal abundance in the LMC of 0.25 solar, and a giant branch slope

and Z dependence as given by Becker and Iben (1979), the LMC period-luminosity relation for stars of

spectral type M would be ~0.2 magnitudes more luminous than the Galactic relation. In the LMC, there is

good evidence that the M type LPVs lie on a sequence ~0.2 magnitudes brighter than that occupied by the

carbon stars in the (Mbo 1, P) plane (WBP). Hence, the period-luminosity law for the dominant population

of carbon stars in the LMC should be a good approximation to the period-luminosity law for M type Mira

variables in the solar vicinity.

251

In addition to the period-luminosity law described above for AGB stars, a second

period-luminosity law exists for the massive supergiant LPVs (Wood and Bessell 1985; see Figure 1 also).

The LPVs in this supergiant sequence range in period from ~400 days to >1000 days. The important point

about this group of LPVs is that they are very luminous (-9 S Mbo I ~ -7) and are eminently suitable for the

determination of distances to external galaxies.

III. DISTANCE DETERMINATIONS FROM THE PERIOD-LUMINOSITY LAWS

The LPVs have recently been used for the determination of distances in a number of cases.

Firstly, relative distances can be found for the LMC and Galactic Bulge (Glass and Feast 1982; Feast

1984) since these systems atl contain substantial numbers of LPVs. A search to faint magnitudes is being

carried out in the SMC by G. Moore (private communication) which should allow the relative distance to

the SMC to be obtained also. In order to get an absolute scale for the luminosities of the LPVs, Menzies

and Whitelock (1985) have obtained infrared JHK photometry of LI~Vs in a number of globular clusters

whose distances were obtained from the RR Lyrae stars in these clusters. The resulting period-luminosity

law was used by Feast (1984) to derive a distance modulus to the LMC of 18.50, assuming <My> = 0.6

mag. for the RR Lyraes. If the recent evaluation of the RR Lyrae absolute magnitude of <Mv> = 0.76

(Hawley et al 1986) is used, the LMC distance modulus becomes 18.34.

The period-luminosity relation for the supergiants has recently been used by Kinman,

Mould and Wood (1987) to derive a distance modulus to M33 of 24.64, assuming a distance modulus to

the LMC of 18.5. The high luminosity of the supergiant LPVs will allow these stars to be used for

distance determination to other galaxies in the local group.

IV. THE PULSATION MODE

There is still no definitive identification of the pulsation mode of the large-amplitude LPVs

(the Mira variables). Evidence that the pulsation mode of the LPVs is the first overtone has been presented

by Wood (1981, 1982) while evidence that the LPVs pulsate in the fundamental mode has been presented

by Willson (1981, 1982). Some recent work aimed at the identification of the pulsation mode will now be

reviewed.

a) LPVs in Globular Clusters

The mode question has recently been examined by Whitelock (1986) who used data for

LPVs in globular clusters to derive values for the observed pulsation constant Q = P (M/M®)ll2/(R/R~) 3r2

for comparison with theoretical Q values for the fundamental and first overtone modes. Both large and

small amplitude variables were examined. The advantage of studying the small amplitude variables is that

reliable effective temperatures should be obtained for these stars from the colour-temperature relation of

Ridgway et al (1980) which was derived for non-variable stars. For the large amplitude (Mira) variables,

the temperature scale is much more uncertain, but the few data points that exist indicate that a blackbody

colour-temperature relation is appropriate (Robertson and Feast 1981). Adopting these temperature scales,

Whitelock (1986) found that in globular clusters with a metal abundance [Fe/H] ~ -I, the LPVs (which

were all of small amplitude) pulsate in the fundamental mode. The LPVs in the more metal rich globular

clusters all pulsate in the first overtone mode. Many of the latter LPVs are large amplitude (Mira)

variables. (If the Ridgway temperature scale is adopted for the Miras, rather than the blackbody scale, then

the observed Q value lies roughly mid-way between the theoretical first overtone and fundamental values -

Fox 1982; Frogel 1983.)

252

b) Bright, short-period LPVs in the Magellanic Clouds

A group of bright (Mbo I = -6.5), short-period (P r, 200 days) LPVs identified in the SMC

and LMC by Wood and Bessell (1985) may represent the best way to determine the pulsation mode of the

LPVs; this group of LPVs is represented in Figure 1 by crosses. There are two possible explanations for

these stars. Firstly, they may be on a short-period extension of the sequence of supergiant LPVs; in this

case, continuity suggests that they are pulsationg in the same mode as the remainder of the supergiants

(and AGB stars) and nothing unique can be deduced about their mode of pulsation. However, there is a

second possible interpretation for these LPVs: they are the immediate precursors of the bright (Mbo I ~ -6.5)

AGB LPVs that have P - 600 days. In this scenario, the bright, short period LPVs are first overtone

pulsators on the AGB while all the other LPVs (including the Miras) are fundamental mode pulsators. As

they evolve up the AGB to slightly higher luminosities they switch from first overtone mode (P - 200

days) to fundamental mode (P - 600 days) where they become "normal" AGB LPVs (note that P0/Pl ~ 3

for LPVs).

In order to decide between the above two possibilities, some means of determining the

evolutionary status of the bright, short-period LPVs is required. A test for AGB status in red giants is the

presence of strong ZrO bands due to the presence of an enhanced Zr abundance caused by dredge-up of the

s-process element Zr during helium shell flashes. Wood and Bessell (1985) examined spectra of two of

the bright, short-period LPVs without finding evidence for enhanced Zr. Examination of spectra of the

other stars is clearly required as identification of even one of these stars as AGB objects would be very

strong evidence that the Miras are pulsating in the fundamental mode.

c) Atmospheric kinematics and dynamics

Another indicator of the pulsation mode of the Mira variables comes from the observed

pulsation velocities in these stars coupled with models of the pulsating atmospheres (Willson and Hill

1979; Bertschinger and Chevalier 1986). Willson (1982) claims that first overtone pu]sators are not

capable of producing the velocity amplitudes observed in Mira variables. This problem will now be

examined further.

Atmospheric pulsation velocities come from high dispersion Fourier transform spectra in the

infrared (1-31~m). These spectra (Hinkle 1978; Hinkle, Hall and Ridgway 1982; Hinkle, Scharlach and

Hall 1984; Dominy, Wallerstein, and Suntzeff 1985) have allowed us for the first time to obtain the

amplitude of the pulsation velocity in the outer layers. In particular, at phases near maximum light,

absorption lines in the infrared are doubled with a line splitting of 25-30 km s -1. The line splitting is the

result of the passage through the photosphere of a shock wave produced by pulsation in the interior. At

maximum, the shock passes through the photosphere so that absorption lines are seen which come from

both in front of and behind the shock; the difference between the two velocities gives the velocity jump

across the shock. Since the centre of mass velocity v. of many Mira variables can be obtained very

accurately from circumstellar microwave emission lines of CO, SiO and OH, the infrared spectra can be

used to obtain the outward velocity v o of the post-shock material and the infall velocity v i of pre-shock

material. These velocities are given in Table 1 for six Mira variables for which such data are available and

are shown plotted against period in Figure 2. Typical outward post-shock velocities are - I 1 km s -1 and

pre-shock infall velocities are typically -15 km s "l , giving a total velocity amplitude zxv of ~26 km s -1 .

Theoretical studies of the atmospheric dynamics of Mira variables show that during the

infall phase of a particle's motion in a Mira atmosphere, pressure gradients are very small so that the

253

Table i

Velocities in Mira variables

Star P(d) v, v o v i vi/v o

R Aql 284 30.4 10.4 8.6 0.83

R Leo 313 6.5 8.5 19.5 2.29

O Cet 332 57.0 13.0 Ii.0 0.85

Cyg 407 -8.7 11.3 16.7 1.48

R And 409 -21.1 7.9 16.1 2.04

R Cas 431 17.0 13,0 15.0 1.15

te

>

15

I0

5

O

-5

- I0

-15

-20 250

@ @

@ @

o

i q - I I 30o 350 400 450

P(days)

Figure 2. The velocity relative to the centre of the star of post-shock material in six Mira variables (solid symbols) plotted

against period of pulsation. The velocity of infalling pre-shock material is shown as an open symbol.

particle acceleration is essentially -GM/r 2. However, during outward motions, pressure gradients can be

significant and these lead to the observed result that v i > v o, The analysis of Willson (1982) neglected the

pressure forces and treated purely ballistic motions, giving v i = v o, The recent study of Bertschinger and

Chevalier (1986) included the effects of pressure forces in a detailed fashion leading to solutions which

could be obtained only by numerical integrations. Here, a simple analytic solution to the atmospheric

particle motions which allows v i > v o is obtained by assuming that the effective particle acceleration during

outward motion is given by -GM/0t2r2), where the constant ~t > 1. If the total time it takes a particle in the

stellar atmosphere to complete one cycle of motion is equal to the pulsation period, then under the above

assumptions we require P(p.[3) = 77Q/(l+~t), where P is the function defined by Equation 11 of Willson

and Hill (1979), Q is the pulsation constant in days, and [3 = vo/v e = 1.6x103Av[P/(QM/Me)]l /3/( l+~0,

where v e is the escape velocity from the photosphere (where the shock wave gives the particle its initial

outward impulse), P is the pulsation period (days), and Av is in km s -1 . With this formulation, the ratio

vi/v o = ~t. If we adopt the observed value of tz = 15/I 1 z 1.4, a typical pulsation period of 350 days, M =

Mo, and Q values of 0.05 and 0.10 days for the first overtone and fundamental modes, respectively, then

the above equations predict zxv values of 28 and 48 km s -! for the first overtone and fundamental modes,

respectively. Comparing with the observed value of Av = 26 km s -1, we see that there is quite good

agreement with first overtone pulsation. However, the observed velocity difference of 26 km s "1 given

above has not been corrected for geometric projection factors in the limb-darkened disk. If a correction

254

factor of 1.4 is applied, the resultant true pulsation velocity amplitude of 36 km s -t lies roughly midway

between the values estimated for the fundamental and first overtone modes. As usual, the result is not

definitive.

V. OBSERVED MASS LOSS RATES

Over the last few years, a set of homogeneous and reliable mass loss rates based on

microwave observations of the thermally excited CO line in the envelopes of mass-losing red giants has

become available for oxygen-rich stars (Knapp e t a l 1982; Knapp and Morris 1985); carbon stars will not

be considered here. A subset of the stars studied by these authors are local optically-discovered Mira

variables (where Mira variables are defined roughly as LPVs of large amplitude ~.V > 2.5 mag.) or

pulsating IRC sources (essentially Mira variables with modest dust shells around them). All these stars are

local to the sun and hopefully belong to an old disk population similar to the old (age ~2x109 years) LPV

population in the LMC (Feast's 1963 study of the kinematics of Mira variables in the solar vicinity

indicated masses $2Mo for these objects). It should be noted that radio-luminous OH/IR stars have not

been examined here as these stars can be seen at great distances and may be much more massive than the

dominant local LPV population.

Distances to the sample of Mira variables were obtained by assuming they lie on the

(Mbo 1, P) relation for low mass LPVs, provided P < 450 days; the luminosity was held constant for P >

450 days since mass loss rates are so large by this time (h - 10 -5 Ms yr -1) that further increases in

luminosity in the remaining lifetime of the star are unlikely (dMbol/dt ~ 10 -6 yr -1 on the AGB). For the

optical Miras, Mbo I was calculated from J and K magnitudes given in Gezari, Schmitz and Mead (1984)

using the (bolometric correction, J-K) relation from WBF; for highly reddened IRC sources Mbo I was

computed by integrating over all wavelengths using the fluxes given in Gezari, Schmitz and Mead (I 984).

The mass loss rates given in Knapp et a l (1982) and Knapp and Morris (1985) were scaled according to

the new distances derived from the (Mbol, P) relation. The resulting mass loss rates are plotted against

period in Figure 3 as filled symbols.

In a paper on the dust-to-gas ratio in mass-losing red giants, Knapp (1985) notes that there

is a good correlation between mass loss rates derived from the CO observations and a mass loss rate

derived from circumstellar dust shell models computed by Rowan-Robinson and Harris (1983) for the

same stars. This result has been used here to derive mass loss rates for a further sample of local LPVs

modelled by Rowan-Robinson and Harris (1983). The equations of Knapp (1985) were used in the

derivation, together with new distances computed for these stars as noted above. Where the wind

expansion velocity v e was not known, it was calculated from a mean relation between v e and P (eg.

Zuckerman, Dyck and Claussen 1986). The resulting mass loss rates are shown as open symbols in

Figure 3.

The mass loss rates shown in Figure 3 indicate that, for P Z 500 days, h increases rapidly

with period in the Mira variables. Indeed, at shorter periods, the mass loss rate seems to be considerably

smaller than the Reimers' (1975) mass loss law would indicate (mass loss rates -1/3 those given by

Reimers are frequently adopted for AGB evolution in order to get reasonable agreement with the estimated

mass toss of ~0.2M, for Population II stars on the first giant branch eg. Fusi-Pecei and Renzini I976;

Wood and Cahn 1977). At longer periods, the mass loss rate seems to peak at a value that is close to the

approximate maximum value fl = L/cv e for a wind driven by the action of radiation pressure on the

material being lost (see also Jura 1983; Knapp and Morris 1985).

255

- 4

- 5

.7- 0", - 6 0 ..-t

- 7

- 8 2 0 0 7 0 0

I . L I I I

~ ' J . . . . . ,i) . . . . . . . . . . . . . . . . . . . . . . ,I,o

,)

o o

o q, c,

l

- ,..* ~ o Reimers' Law x ~-

o

o

1 1 1 [ 300 400 500 600

P(days)

Figure 3. Mass loss rate (Mo yr "1) plotted against pulsation period P for a sample of local Mira variables. The solid

symbols arc mass loss rates derived from the CO observations of Knapp et al (1982) and Knapp and Morris (1985) and the

open symbols are derived from the models of circumstellar infrared emission by Rowan-Robinson and Harris (1983). Mass

loss rates according to the Reimers' mass loss law are shown together with the approximate maximum mass loss rate for

radiation driven mass loss flows.

The evolut ionary implicat ions of the (fL P) relation shown in Figure 3 will now be

discussed. If it is assumed that each low mass AGB star evolves along the (Mbol, P) relation discussed in

§11, then it will increase Mbo I by -1 magni tude for each increase in period of 250 days. This result,

combined with the rate o f evolution up the AGB of ~10 .6 mag. yr -I (Wood and Cahn 1977), means that

low mass LPVs increase their pulsation periods at the rate of ~25 days per 105 years. Now, consider an

AGB star of initial mass M - M®: it will need to lose -0 .2 M® on the AGB in order to completely dissipate

its hydrogen-rich envelope and terminate its AGB evolution. Using the (fl, P) relation in Figure 3 and the

rate of evolution derived above, it can be seen that such a star should reach P -425-450 days before its

AGB phase is terminated by envelope loss. This result is in good agreement with the (number, P) relation

for local LPVs (Wood and Cahn 1977) which shows a very rapid fall-off in the number of LPVs for P >

425 days. The sequence of low mass LPVs in the LMC (WBP) is also seen to terminate at P ~425 days.

An interesting consequence of the above results is that the mass loss rate at the termination

of the AGB phase (ie. when P -425 days) for typical low mass LPVs in the solar vicinity is only ~10 .6 M~

yr -1. This is about an order of magni tude smaller than the "superwind" mass loss rate required on the

AGB for the production of planetary nebulae (Renzini 1981). Hence, it seems that some additional mass

loss mechanism may be required at the end of the LPV phase of evolution in order to boost the mass loss

rate to typical superwind values. Jones e t a l (1981) sugges t that a switch from first overtone to

fundamental mode pulsation may be the means by which the mass loss rate is increased.

Only LPVs with initial masses significantly greater than 1Mo can attain pulsation periods

Z 500 days, by which t ime they have lost ~ 1 M , of material via the stellar wind. This result is in general

agreement with the studies of the kinematics of local Mira variables (Feast 1963) which indicate that Mi,'as

with periods 2500 days have initial masses M i ~2Mo.

VI. ORIGIN OF THE MASS O U T F L O W

The fact that the mass loss rate seems to have a max imum value in Figure 3 given by L/cv e

2 5 6

indicates that radiation pressure plays an important role in mass loss from LPVs with P ~ 500 days. Even

at shorter periods and smaller mass loss rates the results of Knapp (1985) show that t~ ~ "~dust, which hints

at the possibility that radiation pressure may be playing a role in the mass loss process here too. However,

these results do not mean that the radiation pressure actually causes the mass outflow; grains form too far

from the star for the radiative force by itself to produce significant mass loss rates (Castor 1981; Holzer

and MacGregor 1985). The effect of radiation pressure is to increase the terminal velocity in a wind

produced by a separate mechanism (pulsation?) closer to the photosphere of the star.

In the current context, the most important effect of stellar pulsation is the extension

produced in the atmosphere (eg. Klimishin 1967; Hill 1972; Fedorova 1978). The atmospheric extension

means that the gas density at the point above the photosphere where grains form is considerably enhanced

over the values that would exist in a static atmosphere. Some models of the combined effect of pulsation

and grain formation in Mira variables have been made by Wood (1979) and Drinkwater and Wood (1985).

The results of these calculations (which assumed isothermal shock waves) show that pulsation can enhance

a mass flow produced by the action of radiation pressure on grains by factors of 102 to 107, However, the

absolute mass loss rates produced by these isothermal calculations are still several orders of magnitude

smaller than observed values. More realistic models of shock waves in Mira atmospheres have recently

been constructed by Bowen and Beach (1986). In these models, parameterized forms of cooling laws for

the post-shock gas have been given so that considerable heating of the gas far from the star occurs; this

heating, in combination with the effect of radiation pressure on grains, results in mass flows with values

similar to those observed in LPVs.

Although the mass loss rate increases rapidly with period of pulsation for ~< 500 days and

it has been argued that pulsation is the determining factor in mass loss rates for the LPVs, it is important to

note is that there does not seem to be any evidence for an increase in the pulsation velocity amplitude with

period (Figure 2). This indicates that it is some parameter other than pulsation velocity that causes the

rapid increase in t~ with period. Recent work by Bedijn (1986) on mass loss from red giants highlights the

important point that there is a very rapid decrease in density with radius in the stellar atmosphere at the

point where grains form. This radius, which occurs where the gas temperature drops to the condensation

temperature T c of grains, is assumed to also correspond to the sonic point in the flow where

v = c s = (kTc/tXmH) 1/2. The radius R e at the sonic point compared to the stellar radius R is given by Rc/R

= (TeftCTc)2. With these equations and the definition of effective temperature L = 4r~R2Tef ~, the mass loss

rate is fl = 4r~Rc2pcC s ~ LPc for a given value of T c. Now the sonic point is situated at a few stellar radii,

above the region where the large amplitude pulsation occurs. In this situation, the density just interior to

the sonic point decreases nearly exponentially with radius in a manner similar to that in a hydrostatic

atmosphere. The density scale height in the neighbourhood of the sonic point is typically only a few

percent of the radius to the sonic point. Remember that the exponential atmosphere sits on top of an

atmosphere extended by pulsation in the lower atmosphere, where this pulsation enhances the density in

the outer atmosphere by several orders of magnitude over the values that would exist in the absence of

pulsation.

Because of the exponential density distribution high in the atmosphere and the fact that the

density scale height is only 2-3% of R e, a small decrease in the radius of the sonic point will cause a large

increase in Pc and thus fl. At low mass loss rates, there are two possible causes of the increase in fl which

is observed to occur with increasing period, The first is a decrease in R c caused by a decrease in Tel f as die

star moves up the giant branch. However, in the simple approximation above where fl ~ LPc , this effect is

minor and it cannot explain the very rapid increase in fl which is observed. The other method of increasing

257

fl is to increase the extension of the atmosphere caused by the underlying pulsation, thereby increasing Pc'

This must be the most likely cause of the increase in M for (1 Z 10-5M~ yr -1. Note that the atmospheric

extension seems to occur in spite of the observed constancy of pulsation velocity amplitude with period, at

least for 250 Z P(days) g 450.

At high mass loss rates, Bedijn (1986) argues that the decrease in the mass of the star is the

primary cause in the increase in (1 via a decrease in the surface gravity and hence an increa~ in the density

scale height in the atmosphere. However, this mechanism cannot work for periods Z450days as fi is too

small for significant changes in stellar mass to occur. Clearly, the work that has been done to date on the

very important question of mass loss from red giants is exploratory, and detailed studies of the mass loss

process are badly needed.

REFERENCES

Becket, S.A. and lben, I. 1979, Ap.J., 232, 831.

Bedijn, P.J. 1986, preprint.

Bertschinger, E. and Chevalier, R.A. 1985, Ap.J., 299, 167.

Bessell, M.S., Freeman, K.C. and Wood, P.R. 1986, Ap.J., in press.

Bowen, G.H. and Beach, T.E. 1986, in Workshop on the Late Stages of Stellar Evolution, eds. S. Kwok

and S. Pottasch (Dordrecht: Reidel), in press.

Castor, J.I. 1981, in Physical Processes in Red Giants, eds. I. Iben and A. Renzini (Dordrecht: Reidel),

p. 285.

Dominy, J.F., Wallerstein, G. and Suntzeff, N.B. 1985, M.N.R.A.S., 212, 671.

Drinkwater, M.J. and Wood, P.R. 1985, in Mass Loss from Red Giants, eds. M. Morris and B.

Zuckerman (Dordrecht: Reidel), p.257.

Feast, M.W. 1963, M.N.R.A.S., 125, 367.

Feast, M.W. 1984, M.N.R.A.S., 211, 51p.

Fedorova, O.V. 1978, Astrofizika, 14,239.

Fox, M.W. 1982, M.N.R.A.S., 199, 715.

Frogel, J.A. 1983, Ap.J., 272, 167,

Fusi-Pecci, F. and Renzini, A. 1976, Astr. Ap., 46,447.

Gezari, D.Y., Schmitz, M. and Mead, J.M. 1984, Catalog of Infrared Observations, NASA Reference

Publication 1118.

Glass, I.S. and Lloyd Evans, T. 1981, Nature, 291,303.

Glass, I.S. and Feast, M.W. 1982, M.N.R.A.S., 198, 199.

Glass, I.S. and Reid, N. 1985, M.N.R.A.S., 214, 405.

Hawley, S.L, Jeffreys, W.H., Barnes, T.G. and Lai, W. 1986, Ap.J., 302, 626.

Hill, S.A. 1972, Ap.J., 178, 793.

Hinlde, K.H. 1978, Ap.L, 220, 210.

Hinkle, K.H., Hall, D.N.B. and Ridgway, S.T. 1982, Ap.J., 252, 697.

Hinkle, K.H., Scharlach, W.W.G. and Hall, D.N.B. 1984, Ap.J. Suppl., 56, 1.

Holzer, T.E. and MacGregor, K.B, 1985, in Mass Loss from Red Giants, eds. M. Morris and B.

Zuckerman, p.229,

Jura, M. 1983, Ap.J,, 275,681,

Kinman, T.D., Mould, LR. and Wood, P.R. 1987, in preparation.

258

Klimishin, I.A. 1967, Astrofizika, 3, 259.

Knapp, G.R. 1985, Ap.L, 293, 273.

Knapp, G.R. and Morris, M. 1985, Ap.J., 292, 640.

Knapp, G.R., Phillips, J.G., Leighton, R.B., Lo, K-Y., Wannier, P.G., Wootten, H.A. and Huggins,

P.J. 1982, Ap.J., 252, 616.

Mathewson, D.S., Ford, V.L. and Visvanathan, N. 1986, Ap.J., 301,664.

Menzies, J.W. and Whitelock, P.A. 1985, M.N.R.A.S., 212, 783.

Reimers, D. 1975, in 19th Liege International Astrophysical Colloquium, p.369. Renzini, A. 1981, in Physical Processes in Red Giants, eds. L tben and A. Renzini (Dordrecht: ReideI),

p.431. Ridgway, S.T., Joyce, R.R., White, N.M. and Wing, R.F. 1980, Ap.J., 235, 126.

Robertson, B.S.C. and Feast, M.W. 1981, M.N.R.A.S., 196, 111.

Rowan-Robinson, M. and Harris, S. 1983, M.N.R.A.S., 202, 767.

Whitelock. P.A. 1986, M.N.R.A.S., 219, 525. Willson, L.A. 1981, in Physical Processes in Red Giants, eds. I. lben and A. Renzini (Dordrecht:

Reidel), p.225. Willson, L.A. 1982, in Pulsations in Classical and Cataclysmic Variable Stars, eds. J.P. Cox and C.J.

Hansen (Boulder: Joint Institute for Laboratory Astrophysics), p.269.

Wiltson, L.A. and Hill, S.J. I979, Ap.J., 228, 854.

Wood, P.R. 1979, Ap.J., 227, 220. Wood, P.R. 1981, in Physical Processes in RedGiants, eds. 1. Iben and A. Renzini (Dordrecht: Reidel),

p. 205. Wood, P.R. 1982, in Pulsations in Classical and Cataclysmic Variable Stars, eds. J.P. Cox and C.L

Hansen (Boulder: Joint Institute for Laboratory Astrophysics), p.284.

Wood, P,R. and Bessell, M.S. 1985, P.A.S.P., 97, 681.

Wood, P.R. and Cahn, J.H. 1977, Ap.J., 211,499.

Wood, P.R., Bessell, M.S. and Fox, M.W. 1983, Ap.J., 272, 99.

Wood, P.R., Bessell, M.S. and Paltoglou, G. 1985, Ap.J., 290, 477. Zuckerman, B., Dyck, H.M. and Claussen, M.J. 1986, Ap.J., 304, 401.

259

MULTIPERIODICITY IN THE LIGHT CURVE OF ALPHA ORIONIS

M. Karovska

Harvard-Smithsonian Center for Astrophysics

Cambridge, MA 02138/USA

i. Introduction

Alpha Ori, a supergiant star classified as M2 Iab, is character-

ized by pronounced variability encompassing most of its observed para-

meters. Variability on two different time scales has been observed in

the light and velocity curves: a long period variation of about 6 years

and superposed on this, "irregular fluctuations" having a time scale of

several hundred days (Goldberg 1984, Stothers and Leung 1971).

This paper reports the results of Fourier analysis of more than

60 years of AAVSO (American Association of Variable Stars Observers)

data which suggest a multiperiodicity in the light curve of ~ Ori.

2. Data and analysis

The light curve of e Ori presented by Goldberg (1984, Fig. 3-5)

was assembled out of the AAVSO data from 1919 to 1982. Every data

point of the light curve represents a 25-day mean value encompassing as

many as 50 individual visual magnitude estimates. Although the

standard deviation of some of the single estimates is as high as 0~2-

0~4, the data set was considered as worthy of Fourier analysis due to

its large temporal coverage and number of data points (about I000).

The power spectrum obtained as a result of Fourier analysis of

these data is shown in Fig I. Five statistically significant peaks may

be discerned in the power spectrum, corresponding to periods ranging

from 1.05 to 20.5 years (PI=20.5, P2=8.8, P3=6.5, P4=5.7, and P5=I.05

years). Similar periods have been found by applying a technique for

detecting the presence of periodic signals in unequally sampled time-

series data (Horne and Baliunas 1986) and a "clean algorithm" (Dreher

et al 1986). For each of these periods, the false alarm probability

(Home and Baliunas 1986) was calculated (FAPI=I.7-10-7, FAP2=3.5,10-5,

FAP3=3.0-10-9, FAP4=2.1-10-10,FAP5=I.7,10-1) and the results

260

Confirm that the peaks are statistically significant. The possibility

that some of the peaks are a result of an interaction of the window and

signal has also been examined and subsequently dismissed.

The results of Fourier analysis were used to synthesize a light

curve for ¢ Ori assuming sinusoidal type variations. The portion of

this curve from 1980 to 1986 was compared with the photometric measure-

ments from the same time period (Gulnan 1986). The synthesized light

curve shows good agreement with these observations (Fig 2) except at

two epochs: the end of 1980-beginning of 1981, and the end of 1985 -

beginning of 1986.

3. Discussion

Variations having a period of about 1 year have been detected

also in broadband polarization measurements (Hayes 1984)) and Mg II H

and K line fluxes (Sonneborn et al 1986). The period of these vari-

ations is about half of the orbital period of the close companion of

Ori (2.1 yrs, Karovska et al 1986), which may suggest that their origin

is in the ellipsoidal variation of the star. However, this interpre-

tation is implausible since the amplitude of the variation is too large.

The 1 year periodicity may be attributable to the fundamental

mode of pulsation of ~ Ori. The expected period corresponding to the

fundamental mode of pulsation of ~ Ori (M=20Mo, L=5-I04Lo, T=3300K,

Q=0.1) would be about 400 days (Stothers and Leung 1971, Lovy 1984).

In order to examine possible correlation between the changes of the

diameter and the one-year period variation in the light curve, the high

resolution interferometric observations of ~ Ori in the visual conti-

nuum for the interval 1980-1985 were examined. The high frequencies in

the azimuthal averages of the visibility curves obtained by Roddier and

Roddier 1983, Cheng et al 1985, Karovska 1984, Aime et al 1985,

Karovska et al 1986 (Fig 3 a, b, c, d, and e, respectively), are

consistent with 35-40 mas stellar disk (assuming no limb darkening). The

precision of these measurements is not sufficient to determine whether

the 10% differences in the angular diameter estimates are real and

related to stellar pulsation.

The substantial excess brightness detected in 1980-1981 and

1985-1986 has been interpreted as due to an increase in the amount of

scattered light as a result of dust formation. The visibility function

in Fig 3 a b and e can be represented by a sum of the Fourier

transforms (moduli) for the brightness distributions of two disks: the

261

stellar disk (with a diameter estimated from the high frequences) and a

disk with a diameter about two times greater. The larger disk-like

structure can be interpreted as brightness distribution of the scattered

light in a spherically symmetric dust shell (Lef~vre 1982, Karovska 1984)"

Assuming that the synthesized light curve corresponds to the variation

of brightness of the stellar disk, the excess of brightness due to

scattered light has been calculated for three epochs (points A, B and E

in Fig 2) from the corresponding visibility curves (Fig3 a, b and e).

Episodes of enhanced dust formation around the star may be con-

nected to mass ejections initiated by interaction between Alpha Ori and

its close companion orbiting at distance of o~ly 5 a.u. (Karovska et al

1986). Tidal interaction may influence the pulsation of the primary

and may be the mechanism exciting several different modes of pulsation.

Wood (1976) summarized some of the multiperiodic phenomena which may be

observed in red variable stars, though it is very difficult to identify

any of them in the light curve of e Ori. Future long term high angular

resolution observations together with spectroscopic and photometric

monitoring of the star may bring some new insights and allow an inter-

pretation of the observed multiperiodicity in the light curve of ~ Ori.

Acknowledgements. It is a pleasure to acknowledge generous assistance and helpful suggestions from P. Nisenson, R. Noyes, S. Epstein, C. Papaliolios, E. Fossat, J. Lefevre and L. Goldberg. In particular, I am grateful to E. Guinan for helpful discussions and data in advance of publication. Special thanks to S. Baliunas, R. Donahue and J. Leh~r for applying their techniques for calculating periodograms. I also wish to acknowledge the ongoing support of H. Radoski of the Air Force Office of Scientific Research under contract # AFOSR-81-0055.

References Aime C., Petrov, R.G., Martin F., Ricort G., and Borgnlno, J. 1985,

in Proc. Soc. Photo-Opt. Instr. Eng., 556, 297. Cheng, A.Y.S., Strittmatter, P.A., Hege, E.K., Hubbard, E., Goldberg, L.,

and Cocke, W. J. 1985, (preprint). Goldberg, L. 1984, P.A.S.P., 96, 366. Guinan, E.F. 1986, (Private Communication). Hayes, D.P. 1984, Ap. J. Suppl., 55, 179 Home, J.H., and Baliunas, S.L. 1985, Ap. J., 302, 757. Karovska, M. 1984, Thesis, Universit~ de Nice Karovska, M., Nisenson, P., and Noyes, R. 1986, Ap. J., 308,260. Karovska, M., Nisenson, P., and Noyes, R. 1986, (in preparation). Lef~vre, J., Bergeat, J.,and Daniel, J.Y. 1982, Astr. Ap.,ll4, 314. Lovy, D., Maeder, A., No, Is, A., and Gabriel, M. 1984, Astr. Ap.

133, 307. Dreher, J.,Roberts, D., and Lehar, J. 1986, Nature, 320,239. Roddier, F., and Roddier, C. 1983, Ap. J.(Letters), 270, L23. Sonneborn, G., Baliunas, S.L., Dupree, A.K., Guinan, E.F., and Hartmann, L. 1986, to appear in New Inslghts ~n Astrophysics - 8 Years

of uv Astronomy with IUE, edited by E. Rolfe, ESA-SP. Stothers, R., and Leung, K. C. 1971, Astr. Ap., i0, 290. Wood, P.R. 1976, in Multiple Perlod~c Variable Stars, edited by W. Fitch.

262

4 } .Z ,k

! r

. 0 ' .5 '4 f r e q u e n c y (Y r)" '

Fig 1 - Power spectrum obtained as a result of Fourier analysis of the AAVSO data.

0.~

lU

"13

E

>

4.§

. . . . . ~÷%*%*~*% --.-- ~,+~,%*%

A

~.. ~ ~ / ~ ~t~ ,I ~, i ~ ' ~ !~..

2 . @ 4 0 0 0 0 + =.

4.'500 5bOO ~500 6'0o0 ............ J. days Ifgsol~98~l~SZl~98~1498~l~851

Fig 2 - Measurements of Ori visual magnitude by

Krisciunas (stars) and Guinan (dots), Guinan (1986)~

V ~

V

V

L

, a

O~ , , ~ , 0 1 0 2 0 3 0 ( # j ~ l 0

V ~

d

0 1 0 2 0

k

V

C

10 2(3 30 (//)-1 0 10 20 30( ,p)-I

\ % Q

%

Fig 3

30(p, ) - I 0

( • ) Azimuthal averages of the visibility curves obtained from high reso- lution interferometric observations. ( ) Visibility function of 35-40 mas stellar disk (no limb darkening). (- -) Fit to the data points with the sum of visibility functions of two disks: the stellar disk and a disk with a diameter 2 to 2.5 times greater.

10 20 30 "p t ' ~ I

263

A PERIODIC VARIATION IN THE RADIAL VELOCITY OF ARCTURUS

R. S. McMillan, P. H. Smith, & W. J. Merllne Lunar and Planetary Laboratory University of Arizona Tucson, AE, USA 8572l

ABSTRACT

We have detected radial velocity variations in Arcturus (K] lllb) spanning a total range of at least 160 m/s, more than 8 times ~he nightly standard deviation of measurements made during the same season (often on the same nights) on the star Pollux. The velocities of Arcturus tend to alternate between two values separated by 60-100 m/s from night to night. A perlodogram of 32 nightly velocity averages spanning 100 days shows significant power (false alarm probability less than ]%) for a period of 2.]8 days, and its alias of 1.84 days. Although these periods are close to the Nyquist period of 2.00 days, there is no preference for periods of exactly 2 solar or 2 sidereal days. Another consequence of the allaslng is that periods of I/3 these values cannot be ruled out. Exhaustive checks indicate no evidence that these variations are terrestrial or instrumental in origin.

Epoch folding of the data onto a phase diagram reveals that the shape of the velocity curve is skewed and that the sense of the skewness is opposite for the two aliased peaks. The time scale of the variation appears superficially to be consistent with the "2H" or "3H" modes of radlal oscillation by a star with the properties of Arcturus. The sense of the skewness associated with the peak at 1.84 days corresponds to a short-llved outward acceleration followed by a more prolonged deceleration by the stellar atmosphere.

INTRODUCTION

We are monitoring small changes in the Doppler shifts of late-type stars with a spectrometer that is calibrated interferometrically by a tilt-tunable Fabry-Perot etalon and coupled to a 0.9-meter telescope by an optical fiber (McMillan etal. ]985,1986). We sample between 300 and 500 orders of constructive interference by the etalon; these are distributed through the profile of the stellar spectrum between 4300 and 4600 A. The instrumental resolution is about 0.05 A and the orders are separated by 0,63 A. The ultimate purpose for this instrument is a long term search for extrasolar planetary systems; however, during the first year of observations (the 1985/86 season) it was tested on bright K giants.

OBSERVATIONS

Arcturus was observed a total of 32 nights between 1985 Dec 21 and ]986 Mar 31UT (inclusive), a span of 100 days. Pollux was observed on ]8 nights between 1986 Jan 22 and Mar 3] UT, a span of 68 days. Since the exposure times required for the individual observations were only a few minutes, we were able to make several observations on each night, On I] nights for Pollux and on ]7 nights for Arcturus we made more than 30 observations per night. This allowed us to search

for variations on time scales shorter than that presented in this paper, and to enhance the accuracy of the nightly averages. Therefore, the errors in the nightly averages are dominated by uncertainties of calibration, rather than

random errors due to photon statistics and detector noise. Results of searches

264

for Intra-night variations during the longest uninterrupted data runs were presented by Smith etal. (1986).

Nightly averages of velocities of both stars are shown versus date in Fig. ]. The apparent drift of the velocity of the star due to the Earth's motion was compensated by tilt-tunlng the etalon so that the same spectral features were sampled throughout the observing season. For Pollux the deviation of the measurements is + 18 m/s ever three months; we interpret this as an upper limit to the long-term ~allbration errors.

. . . . . . . . . o i llll i

-4o~ Nightly Averages ~[ + Icr •

=. . , I O 0

o eeo eP

o • y O • ~ - . o °

@ • • - O O • •

-IOC Arcturus

n I i I i i i i ! 810 2°~20 0 20 40 6o ,00

1986 Day of Year = MJD-46430.O

Figure I. Doppler velocities of Arcturus (filled circles) and Pollux (open circles) referred to separate and arbitrary zero points, as functions of day of year (DO¥) in 1986.

VELOCITY VARIATIONS

The variance of the Arcturus observations is ]3 times greater than that of Pollux. There is less than I in a billion chance that such a sampling of Arcturus could have such a large variance if the parent population (true behavior) of Arcturus were the same as that of Pollux. In addition, the Arcturus data (unlike those of Pollux) show a systematic daily alternation with an amplitude between 60 and 100 m/s. On DOY 30 and 89 the alternation skips a step and shows a velocity offset of about twice the other days.

A periedogram of these data (Horns and Ballunas 1986) is shown in Figure 2; the two highest peaks (one is the alias of the other) are flanked by sldelobes which are a result of our two-week observing schedule. The peaks at 1.84 and 2.18 days are significant with 99% confidence.

265

Z=

O E O Z

' i ' ' I ' ' ' ' I d , , , I '

I

P = 2.18

Arcturus: PeriodocJram of RN. series ~,2 obs, spanning I00 days

o.I 0.2 0.3 0.4 Frequency (per day )

' i ' ' ' ' : ~ P =1,84 I ' !

: I

:

05 0.6

Figure 2. A perlodogram of the Arcturus data in Figure ] . Because the data are sampled nightly, the Nyqulst frequency Is at 0.5 per day.

Figure 3 Is an epoch-folded phase diagram of all our nightly averages of observations of Arcturus, folded modulo 1.842 days. Thls period was chosen somewhat arbitrarily instead of 2.]8 days or the 1/3 submultlples as an example of all such phase diagrams. The distribution of the points in this figure shows a systematic, skewed trend during the cycle. The curve for one-thlrd thls period has exactly the same shape; the phase diagram for periods on the other side of the Nyqulst limit (2.18 and I/3 that value) are mirror images of thls shape. The smooth curve is drawn by inspection to allow a comparison of the total variance of the data wlth that of the data about the trend. Thls comparison indicates that the llne Is a good flt to the data; the variance of the points about the llne Is not significantly different from that of the comparison star. In other words, subtraction of the smooth curve from the original data eliminates the peaks at 1.84 and 2.18 days in Fig. 2, and scrambling the order of the data points destroys the systematlcs of Fig. 3.

Phase diagrams such as Fig. 3 allow us to set an error estimate of + 0°005 days on our estimates of the period. The perlodogram and phase diagrams aTso show that the phase coherence of the variation holds at least as long as the total span of the observatlons (54 cycles). The slope of the long part of the skewed curve is too low for us to have seen the variation during a single night. In addition, none of our observations occurred during the short interval of higher acceleration.

Are these variations In Arcturus at least superficially consistent wlth the expected mass and radius of the star? To make a preliminary check on the possibility of a global radial oscillation, we have used the surface gravity of log g - ].8 + 0.2 (Bell et al. 1985), the angular diameter of 0,023 + 0.00]2 arcsec from A~res and Johnson's (|977) reanalysls of direct angular dlameter measurements In the literature, and the trigonometric parallax of 0,092 + 0.005 arcsec (Woolley et al. 1970). To obtain approximate agreement wlth the ~bserved period we had to use "one-slgma" extreme values of these parameters (log g = 2,00, angular diameter - 0.0219, and parallax = 0.097). A mass of 2.15 solar masses and log(mass/radlus) = -I.053 were calculated from these parameters and

used in the algorithms of Cox et al. (1972) to compute values of log Q and pulsation periods. The most relevant result Is that the "2H" mode of radial oscillation would have log Q(2) - -1.652 and a period of 1.82 days. Convenient

equations for the "3H" values of log Q are not provided by Cox et al., but It is

266

200

I00 i=

" " -tOO

' I I ~ I '

Arcturus: Dec- Mor, t986

o o o

S

I l I l I

o

o o o o

0 , 0 / ~ ~ 0 ~

0 O / ~ " 0 0 ~ 0

oS 00 % o

O

-200 ' I , I 1 I I I I I I I ' -o.2 o.o o.z 0.4 o.s o.e I.o a.2

Phase (P = 1.842 days)

FIG. 3. Velocity vs. pulsation phase for Arcturus. The period used In the epoch-foldlng procedure was ].842 days.

possible that the observational parameters would not have to be "pushed" to their limits of uncertainty to find an appropriate period in the "'3H'" mode of oscillation. Also, the "Q" algorithms of Cox et al. refer to stars with higher envelope abundance (Y) of helium and less convection than Arcturus is expected to have.

On the basis of this elementary and preliminary analysls, it is physically plausible that radial oscillations are responsible for what we observe. The skewed shape of the curve could be the result of the superposltlon of additional higher harmonics of smaller amplitude and appropriate phases. We plan to study whether such radial modes are expected to be excited, whether they would be sustained if excited, whether the phase coherence would be that which we observe, and whether the pulsation could be detected in photometry or temperature observations.

REFERENCES

Ayres, T. R., and Johnson, H. R. 1977, Ap. J., 214, 410. BelI,R. A., Edvardsson, B., and Gustafsson, B. 1985, M. N. R. A. S.~ 212, 497. Cox, J. P., Kings D. S., and Stelllngwerf, R. F. 1972, Ap. J., 171, 93. Home, J. H., and Ballunas, S. L. 1986, Ap. ~., 302, 757. McMillan, R. S,, Smith, P. H., Frecker, J. E., Merllne~ W, J., and Perry,

M. L. 1985, in Proc. of IAU Colloq. No. 88, Stellar Radial Velocities, A. G. Davis Philip and D. W. Latham~ eds. (Schenectady: L. Davis Press), p. 63.

McMillan, R. S., Smith, P. H., Frecker, J. E., Merllne, W. J., and Perry, Mo L. 1986, Proc. S. P. I. ~o, 627, (Instrumentation in Astronomy - V l), ed. D. L. Crawford, in press.

Smith, P. H., McMillan, R. S., and Merllne, W. J. 1986, In Proc. I. A. U. S)rmp. No. 123, Advances in Hello- and Asteroselsmolosy, held 1986 July 7-11 in Aarhus, Denmark~6~. Reldel, Dordrecht), In press.

Woolley, R. v. d. R,, Epps, E. A., Penston, M. J., and Pocock, S. B. 1970, "Catalogue of Stars Within 25 pc of the Sun", Ro~. 0bs. Ann., No. 5,

267

RHO CASSIOPEIAE: A HYPERBRIGHT RADIAL PULSATOR?

YARON SHEFFER

ASTRONOMY DEPARTMENT, THE UNIVERSITY OF TEXAS

AUSTIN, TX 78712

INTRODUCTION

Stars less massive than about 60M® experience a short-lasting excursion

into the top right corner of the HRD following a photospheric inflation

which is triggered by their evolution off the MS. This red hypergiant

(RHG) phase is terminated by enhanced mass loss which forces the star

into a second blue giant phase as a WR object. Hyperbrights may be obse-

rved therefore, as yellow supergiant stars on two occasions depending

, of their age. Rho Cas is probably one such specimen now containing 25M®

the original 45M® and shining at Mbo I =-9.4 with solar-like Tef f. Being

so near the upper envelope of the HRD (Humphreys 1978) this star could

be losing mass by grazing the Eddington turbulence limit as has been

described by de Jager (1984). Cyclic photometric and spectroscopic vari-

ations have been suspected and/or detected before (see Percy, Fabro and

Keith 1985, Arellano Ferro 1985, Lambert, Hinkle and Hall 1981 = LH 2)

and have been likened to both radial and nonradial oscillations.

We have conducted ground based observations with the 2.7m McDonald ref-

lector at its coud~ focus utilizing a Reticon detector at high velocity

resolution between 1979 and 1984. These recorded low- and high-excitation

lines in the red and near infrared. Our analysis of radial velocity and

line profiles, coupled to a comparison with models of supergiant pulsa-

tions (Lovy e~ al. 1984), indicates that Rho Cas could be a post-RHG

star now returning to the hotter side of the HRD.

RESULTS

The radial velocity curves for Rho Cas are amplitude and period modulated,

exhibiting a cyclic behavior which is not purely regular. Information

from high- and low-excitation lines is similar, i.e., there is no sig-

268

nificant phase difference between the two classes. The situation is di-

fferent for very high altitude photospheric lines such as Ca II and Ba II

species. There, large departures from semi-regularity are observed: in-

tervals of monotonically increasing velocity exist which do not show

CYclic changes (Sheffer 1985). This may involve motions which are unique

to the farthest reaches of the atmosphere, and/or contaminations by cir-

cumstellar matter which is no longer coupled to lower altitude cycles.

Rho Cas had committed the following velocity extrema during the observed

interval: two maxima in September 1980 and January 1982, and two minima

in May 1981 and August 1982. These correspond to times of swiftest contra-

Ction and expansion, respectively, across the equilibrium radius of the

star. According to two well-observed cycles of N I lines we find that

Rho Cas was pulsating with a quasiperiod of 520 days (Fig. I).

Two emission lines of Fe I at 8047A and 8075A (0.9 eV) present variable-

amplitude intensity curves which closely follow the velocity curves when

shifted by a certain phase. The emission is at maximum intensity at times

of systemic velocity, just following velocity maxima (Fig. I). In the

radial pulsation scenario this means that the photosphere is in a highly

excited state when the star is fully compressed. And vice versa: emission

minima occur when the supergiant's atmosphere is fully inflated. We have

found in the literature some color data which confirm that Rho Cas is

hotter/cooler at times of line emission maximum/minimum (Arellano Ferro

1985). Furthermore, an inspection of visual magnitude estimates published

by the AAVSO reveal an interesting correlation between brightness and

Photospheric excitation: the two behave in parallel so that Rho Cas ex-

Periences cyclic fading episodes wh@n its photosphere is largest and

COolest. Besides of this being an attribute common to classical radial

Pulsators, we also notice the presence of phase lags between radius cur-

ves as estimated from velocity, and brightness. Yet another feature is

the ascending velocity branch being significantly longer than its desce-

nding colleague for the best observed velocity cycles for various exci-

tation lines. All in all, there is an undeniable similarity between Rho

Cas' curves and those of classical Cepheids.

Following the report on a single line doubling episode in Fe II lines

of Rho Cas by LH 2 we have monitored the behavior of line profiles through-

out the interval of observations to further investigate this interesting

act. Many lines have been found to split on a few occasions. These in-

Volve Fe I and II species ranging from 2.2 to 4.6eV in excitation poten-

269

tial. What we have found supports the radial pulsation picture in a va-

riety of ways.

First, the splitting of lines occurs on a cyclic basis, but only during

a limited interval of the quasiperiod, This is very similar to what has

been seen in W Vir (Sanford 1952) and Mirae (Hinkle 1978), which are all

radial pulsators. The Rho Cas episodes start immediately following times

of systemic velocity and emission maximum, in other words, as the star

re-expands out of a radius-minimum state. The splitting is fully in prog ~

ress (overall it goes on for about half a year) by the time the star is

expanding at maximum speed across the equilibrium radius (Fig. I). Since

there is no splitting seen in high excitatioa lines the following model

seems to describe these episodes to first order of magnitude. The pul-

sation of the bottom of the atmosphere acts as a piston which pushes

higher layers at the beginning of each cycle (see also LH2). This is

accomplished by reversing the velocity of collapsing layers of the pre-

vious cycle, especially those seen in Fe ]~ght.

A second finding supports the radiality of the motions observed. There

is a clearly seen evolution of the doubling in the lines as they are

first accompanied by a newly formed small blue component which then is

growing in intensity to rival the older red component. Eventually, the

former passes the latter in strength and the red component decays out of

existence. This process is not simultaneous for all the lines, however.

After plotting the times of component equality versus the line's excita ~

tion potential, we have discovered that the equality instant is attained

first by high excitation lines. We infer that this splitting by velocity

reversal exhibits very clearly how the photosphere is well stratified.

High depth, high excitation layers will be reversed before low depth,

low excitation matter is being reversed. There is no obvious way this

excitation-temperature dependence could be explained by nonradial motios~'

Our data uncovers three more splitting episodes in addition to the one

already reported on by LH 2. Together they supply three intervals which

show the regularity of these episodes: their mean period is 520 days,

with deviations less than the formal timing error of 45 days due to ob-

servational gaps. This result indicates the regularity of pulsation at

the bottom of the photosphere, while the agreement with the value derived

from the radial velocity curves is very satisfying.

270

CONCLUSIONS

These data of high resolution in velocity and time are conveniently in-

terpretable in terms of classical radial pulsation of a supergiant pho-

tosphere. Based on spectra taken at McDonald during a 1400-days long

interval, we infer a dominant quasiperiod of 520 days for Rho Cas with

a formal error of less than 10%. Since this period is significantly longer

than the radial fundamental mode predicted theoretically (see Sheffer

and Lambert 1986, Lovy et al. 1984) this supergiant may well be on its

way toward the blue side of the HRD following the end of its reddest

evolutionary phase. Rho Cas is predicted to become a WR star within 7000

Years. This established radial pulsation behavior of a Ia-O yellow super-

giant hints at the possible existence of a class of radial pulsators at

the top of the HRD. Another similar supercepheid may be HR8752, but its

behavior is more difficult to interpret (Sheffer and Lambert 1987).

REFERENCES

Arellano Ferro, A, 1985, MNRAS 2!.6, 571 de Jager, C, 1984, A&A 138, 246. Hinkle, K H, 1978, Ap J 220, 210, Humphreys, R M H, 1978, Ap J Supp 38, 309. Lambert~ D L, Hinkle, K H, and Hall, D N B, 1981, Ap J 248, 638 (LH2). Lovy, D, Maeder, A, Noels, A, and Gabriel, M, 1984, A&A 133, 307. Percy, J R, Fabro, V A, and Keith, D W, 1985, J of AAVSO 14, I. Sanford, R F, 1952, Ap J 116, 331. Sheller, Y, 1985, MA Repor~t] ''' The University of Texas at Austin. Sheller, Y, and Lambert, D L, 1986, PASP 98, in press.

, 1987, in preparation.

~Rho Cas" .~

~ N ~ w

• ~\ ~ ' / i II

/ / \ : / -- ' 'd . - " - - - "

-- Line splittings /

L

z::LC 4400 480q 5200 5600 JD-2440000

-50 ~n/s -60

Fig. i - Rho Cas curves of heliocentric N I 8680~ radial velocity, of Fe I 8047~ emission equivalent width, and of AAVSO visual magnitude estimates. Three episodes of line splitting are indicated at full development phases.

271

P O P U L A T I O N I I V A R I A B L E S

Hugh C. Harris

U. S. Naval Observatory

Flagstaff, AZ 86002

U.S.A.

ABSTRACT: Statistics are reviewed for Population II pulsating variables in globular clusters and

in the field, and current research programs are summarized~ Included are Cepheids (including

Anomalous Cepheids and Dwarf Cepheids), RV Tauri stars, and red variables (Mira, semiregular,

and irregular variables), but not RR Lyraes.

I. SCOPE

In Galactic globular clusters, we can be more sure of finding Population II stars than

anywhere else. The cluster variables listed in A Third Catalogue of Variable Stars in Globular

Clusters Comprising 2119 Entries (Sawyer Hogg 1973) lie at the heart of the following discussion.

We can (and must for some purposes) also supplement our da ta base with halo field stars listed in

the Genera/ Catalogue of Variable Stars (Kholopov 1985). These two sources, and the l i terature

from which they are drawn, form the basis for what we know about variability among Population II

stars. RR Lyraes dominate the Pop II variables, comprising 80% to 90~ of the known variables

in globular clusters. This paper will include all types of pulsating variables except RR Lyraes

which are discussed in detail at this conference by Lub.

Many excellent reviews of Pop II variables can be found in the literature, including

genera] reviews (Kukarkin 1973; Rosino 1978) and more specialized reviews on Cepheids (Waller-

stein and Cox 1984; Harris 1985a), red, long-period variables (Feast 1975; Lloyd Evans 1975; Feast

1981; Willson 1982; Wood 1982, this conference), variables (primarily Anomalous Cepheids) in

Dwarf Spheroidal galaxies (van Agt 1973; Zinn 1985a), and others. I cannot hope to cover all of

this material. Instead I will t ry to emphasize recent developments in our understanding of Pop II

variability.

Virtually all Pop II variables axe evolved stars. Most are either evolved stars in the

Cepheid instability strip (RR Lyraes, Cepheids, and RV Tauri stars) or cool, evolved stars with

unstable envelopes (Mira, semiregulax, and irregular variables). Figure 1 shows the principal

sequences of a typical globular cluster (M3, taken from Sandage 1970). The instability strip

crosses the main sequence at a sufficiently high temperature that in systems older than ~4 Gyr

no stars near the main-sequence turnoff are unstable. Blue stragglers can fall in the instability

strip near the main sequence and become Dwarf Cepheids or 6 Scuti stars, but have probably

274

-4 i ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' '

SRd, Mira

RV Tauri .................. i ......... / SR, lr~,~._--

Cepheids .............. ~7 ...... i

0 - RR Lyrae ~.~ t l !

/ i ' ' 2 -- 11

11 i I Dwarf L. /

Cepheids, ""'7 / / 5 Scuti , /

4

- .5 0 .5 1 1.5

B - V Fig, 1. The color-magnitude diagram for Pop II variables.

I i l

, l 2

reached that state as a result of mass transfer in binaries (see Sec. III). White dwarfs become

ZZ Ceti variables or DB variables when they enter instability strips for their composition as

they cool; these may be quite common among Pop II white dwarfs, but at present we have little

information on Pop II white dwarf variables in the field, and no information on them in clusters.

Other nonpulsating variables (flare stars, novae, cataclysmic variables, and eclipsing variables)

Mso appear among Pop II stars, but they are beyond the scope of this conference and this paper.

IL STATISTICS OF POPULATION II VARIABLES

A discussion of the frequency with which different types of variables occur in Pop II is

COmplicated by several factors: images are severely crowded in the cores of most globular clusters;

many red variables have low-amplitude, irregular, and/or episodic variability; colors measured in

the blue and visible are misleading for red variables, but infrared colors are not always available;

membership is uncertain for many variables near globular clusters; old-disk variables and halo

variables with very similar characteristics are mixed in the field; selection effects are affecting

(Sometimes very strongly) the discovery and classification of variables, and they are affecting our

Work in clusters and in the field differently. Furthermore, the boundaries for all types of variables

(except perhaps RR Lyraes) are uncertain. Possibly all stars near the tip of the red giant branch

(RGB) or asymptotic giant branch (AGB) are low-amplitude variables, and possibly all stars

•bove Mbol ~ --3 are variable. However, most stars on the RGB or AGB and below the tip are

probably not variable (Welty 1985).

275

Variables in g lobular c lusters are s u m m a r i z e d in Table I. A b u n d a n c e s are t aken from

Zinn (1985b). The to ta ls are based on the T h i r d Ca ta logue (Sawyer Hogg 1973) u p d a t e d f rom

numerous more recent papers , a t t e m p t i n g to include only likely c lus ter member s . My l i t e ra tu re

search has no t been exhaus t ive , a n d t he an t i c ipa t ed pub l i ca t ion of the F o u r t h Ca ta logue will

undoub ted ly include different var iables as p robab le m e m b e r s and lead to revis ion of some of these

numbers . The b r e a k d o w n into types is necessari ly s o m e w h a t a r b i t r a r y because of the u n c e r t a i n

proper t ies of some types and the incomple te d a t a for m a n y stars . T h e cr i te r ia used to des igna te

each type are: (a) Miras have ampl i t udes in B and V larger t h a n 2 magn i tudes , ba lmer emiss ion

at some phases , a n d JHK colors indica t ive of H~O a b s o r p t i o n in t he H b a n d ; (b) Mira- l ike s t a r s

have smal l amp l i t udes b u t show H 2 0 abso rp t ion in the H band ; (e) SRd s t a r s have a m p l i t u d e s in

B and V larger t h a n 1 m a g n i t u d e and ba tmer emiss ion a t some phases , b u t no H 2 0 absorp t ion ;

(d) o ther SR and Irr var iables have smal l ampl i tudes , no ba lmer emission, and no H 2 0 absorp t ion ;

(e) o ther red var iables have d a t a too incomple te to define the type; (f) Cepheids have colors bluer

t han the preceding red var iables , and r epea tab le l ight curves]. (g) RV Taur i s t a r s have colors bluer

t han the red var iables , bu t l ight curves wi th a l t e r na t i ng min ima .

Different types of Pop II var iables are s u m m a r i z e d in Table II. T h e to ta l n u m b e r s of

known var iables in g lobular c lus ters are l isted first and field var iab les a t h igh galact ic l a t i tudes

are l isted second. The f ield-star to ta l s include Volumes I and II of the fou r th ed i t ion of the GCVS

(Kholopov 1985). For Cepheids , RV Taur i , and SRd s tars in Volume III (not yet avai lable) , the

t h i rd edi t ion (plus supp lemen t s ) is used, while for the o the r s ta rs the to ta l s f rom the four th edi t ion

are scaled up by 1.4. S tars likely to be m em ber s of the Magel lanic Clouds have been removed

from the to ta ls .

TABLE I. VARIABLES IN GLOBULAR CLUSTERS Cluster [Fe/H] Mira Mira SRd S'R, Red Cep. RV Cluster IFe/H] Mira Mira SRd SR, Red Cep. RV-

Like In,. (?) Tau

(,,) (b) (~) (a). (e) If) (g} 104 - 0 7 3 3 0 18 0 0 0 288 -1.4 0 0 1 0 0 0 0 362 -1.3 I 1 0 0 0 0 0 1261 -1.3 0 0 0 0 1 0 0 1851 -1.3 0 0 0 1 0 0 0 1904 -1.7 0 0 0 I 0 0 0 2419 -2.1 0 0 0 0 4 I 0 PAL 4 -2.2 0 O 0 0 2 0 0 4833 -1.9 0 0 0 1 0 0 0

5024 -2,0 0 0 0 2 0 0 0 5139 -1.6 1 1 0 6 2 6 I 5272 -1.7 0 0 1 0 3 1 0 5466 -2.2 0 0 0 O 0 I 0 5897 -1.7 0 0 0 1 0 0 0 5904 -1.4 0 0 0 0 I 0 2 5927 -0.3 1 0 0 2 2 0 0 5986 -1.7 0 0 0 0 1 0 0 6093 - I .7 O 0 0 0 0 1 0 6121 -1.3 0 O I I 0 0 0

6171 -I,0 0 0 0 0 I 0 O 6205 -1.6 0 0 2 0 6 3 0 6218 -1.6 0 0 0 0 0 1 0 6229 - 1,5 0 0 0 0 0 1 0 6254 -1,6 0 0 1 0 0 2 0

Like Irr. (?) T~u (a) (b) (c) (d) (e) (f) (g,L

6273 -1.7 0 0 0 0 0 4 0 6284 -1.2 0 O 0 0 0 2 0 6333 -1.8 0 0 0 0 0 1 0 6352 -0.5 0 1 0 I O O 0 6356 -0.6 4 0 0 0 5 0 0 6388 -0.7 3 0 0 0 6 O 0 6402 -1,4 0 0 0 0 2 5 0 TER 5 +0.2 1 0 0 0 0 0 0 6553 -0.3 1 0 0 0 0 0 0 6569 -0.9 1 0 0 6 1 0 0 6624 -0.4 0 0 0 0 2 0 0 6626 -1.4 0 0 0 O 4 1 1 6637 -0.6 2 0 0 0 5 0 0 6656 -1.8 0 0 0 3 1 1 0 6712 -I.0 1 0 0 2 2 0 0 6715 -1.4 0 0 0 0 2 1 0 6723 -1.1 0 0 0 0 2 0 0 6752 -1.5 0 0 0 0 0 1 0 6779 -1.9 0 0 0 0 3 1 I 6838 -0.6 0 I 0 0 3 0 0 7006 -1.6 0 0 0 1 1 0 0 7078 -2.2 0 0 0 0 0 3 0 7089 -1.6 0 0 0 0 0 3 1 7492 -1.5 0 0 0 0 I 0 0

276

TABLE II. FIELD AND CLUSTER VARIABLES ~G ................. RR Cepheids RV Miras SRd Other

Lyrae Tanri red var. 10bular Clusters 1500 40 6 19 6 110

I Field, Ibl > 30 ° 1400 11 2 370 17 600 [_Field, Ibl > 30 °, IzI _> 2 kpc 1200 8 2 75 ......

Dw.Cep., 5 Scuti

3 90 1

For all classes of variables, the potential Pop II field stars have old-disk counterparts

that complicate our analysis of the field populations. Metal-rich RR Lyraes, Type II Cepheids,

Miras, RV Tauris, Irregular variables, and ~ Scuti stars all exist in appreciable numbers in the

solar neighborhood. From Table II, it is apparent that the RR Lyraes at high galactic latitude

are generally not associated with the Galactic disk and must be primarily Pop II stars, but for

Miras the opposite is true and for other types the population characteristics are unclear.

Field Type II Cepheids are from a mixture of populations. Both metallicities and

kinematics show a broad range of properties, with a large fraction of stars from an old-disk

population (Harris and Wallersteln 1984; Harris 1985a; Diethelm 1986). There are a few stars

at large IZI distances with high velocities and low metallicities (Harris 1985a,b). Available data

for these stars give [Fe /H]=- I .5 and a velocity dispersion of 125 km s -1, values very similar to

halo globular clusters. Field RV Tauri stars show a siinilar range of properties (Mantegazza 1984;

Wahlgren 1985).

Field Miras in the solar neighborhood are primarily part of an old-disk population.

Their kinematics indicate initial masses from about 1 to 2 M® (Feast 1963) and their luminosities

(Clayton and Feast 1969; Foye t al. 1975; Robertson and Feast 1981; van den Bergh 1984) are

consistent with evolution of 1 to 2 M® stars up the AGB. They have a broad range of periods,

with most in the range from 250 to 450 days, and they have a IZI distribution with a scale

height of 300 pc (Wood and Cahn 1977). These characteristics show that most field Miras are

not Pop II stars. However, using magnitudes at maximum light from the GCVS and a period-

Mmax relation to estimate distances, we do find Miras at high galactic latitudes sufficiently faint

that they probably lie far from the plane (Table II). Their calculated distances depend on whose

period-luminosity relation is used; the luminosities for these short-period Miras are somewhat

uncertain (see Feast 1981). The stars AQ Aqr, DM Aqr, AL Boo, CO Boo, RT CVn, RX Corn,

AB Corn, DO Corn, HT Her, HU Her, YZ Leo, and AO Leo probably lie between 2 and 20 kpc

from the Galactic plane. They appear to be analogous to the Miras in globular clusters. No

Miras axe known at very large distances in the Galactic halo, but few searches have been done

at high latitude deep enough and with appropriate observing frequency to find and classify very

faint Miras.

The abundance distributions of the stars in clusters are shown in Fig. 2. The stars and

types are the same as in Table I. To some extent, well-studied clusters like 47 Tuc are represented

more completely. Also, stars in ta Cen are all plotted with a mean abundance (-1.59), neglecting

the abundance range within this cluster. Nevertheless, the different distributions of Cepheids

and Miras are obvious. Red variables are produced less frequently in metal-poor clusters than in

metal-rich clusters; the same is true for the subset of high-amplitude red variables.

277

15 -- All C l u s t e r s - - a) Mira 4

, i I t o , , , . . . . , , ,ITh ,rTh ,IT';'I,I, , h , - 2 . 6 - 2 - 1 . 5 - 1 - . 5 0

5 r j - 3 b) M i r a - I l k z o ,m] . . . . . I 1 [~ I~

- 2 . 5 - 2 - 1 . 5 - I - . 5 0 0 J ~ l , , ~ l , , } , , , I , , h , l h , , , I , , , - : l

-2.5 -2 - 1.5 - 1 -.5 0

, .... .... .... .... 1 [ t1' 1 .... ....

80 ~-(e)-(e) All red 17 3 F ] (e) SRd I variables I I r-~ 10 1 ~ - ~

0 " " ~ ~ ~ ~ * ~ " ~ 0 ~ l I I I -2.5 -2 -1,5 -1 -.5 0 -2.5 -Z -1.5 -1 -.5 0

15 ~ (O-(g) Cephelds, N 15 (d) 5R, I r r I rv Taurl ~ I ~ 10 10

, u , 5

0 F ' ' ' !, , - ' n u l l , . . , . . I T I . . 1 , , I . . . . I , , o , , I , , , , I,ITh .IT1, .hh,.h,ITl,,I,. -2.5 -2 -1.5 -1 -.5 0 -2.5 -2 -1.5 -1 -.5 0

[F,,/H] I re /H] Fig. 2. The abundance distributions of Pop II variables in globular clusters.

III. CURRENT PROBLEMS ON POPULATION II VARIABLES

A. Cepheids, RV Tauri Stars, and Related Variables

At low luminosities and short periods (corresponding to BL Her stars with periods of

approximately 1 to 8 days), Cepheids are evolving through the instability strip directly from the

blue horizontal branch or are on blue loops from the lower AGB (Cingold 1976). The period

changes observed for these stars (Wehlau and Bohlender 1982) are in agreement with expected

values based on the evolutionary timescales calculated for these models. Further support is found

by examining the clusters which contain BL Her stars. They all have blue horizontal branches

(Wallerstein 1970) and they often have horizontal branches with extended blue tails (Harris

1985a), but they tend not to be the most metal-poor clusters (Smith and Stryker 1986). New color-

magnitude diagrams now being prepared for three clusters reinforce the picture: NGC 6284 and

NGC 63332 both containing short-period Cepheids, have blue horizontal branches with extended

blue tails, and NGC 62932 with a possible short-period Cepheid, also has a blue horizontal branch.

These conclusions are somewhat tentative because of the small numbers of stars being discussed,

but it appears that the occurance of BL Her stars is more closely correlated with the presence of

very hot stars on the horizontal branch than with any other factor.

The fourier coefficients of short-period Type II Cepheids have been discussed by Simon

(1986), Petersen and Diethelm (1986), and Carson and Lawrence (this conference). They show

some patterns (different from classical Cepheids), but also differences among stars of the same

278

period, indicative of classes of stars with different physical parameters which are not completely

understood. Two factors are undoubtedly the different populations (old disk and halo) from which

the field Type II Cepheids arise and their wide range of metallicities. However, some differences

are also seen among cluster Cepheids alone. The light curves available for both field and cluster

Type II Cepheids are not really adequate, and current studies by several groups should help.

Both longer period Pop II Cepheids (W Vir stars) and Pop II RV Tauri stars are making

blue loops from the AGB or making a final transition toward a hot white-dwarf state. The RV

Tauri characteristic of alternating deep and shallow minima in the light curve is a result of the

very extended envelopes of stars at this high luminosity (Bridger 1985; Worrell, this conference).

Recent infrared da ta for Type II Cepheids and RV Tauri stars from IRAS and ground-based

observations has shown that a significant fraction of long-period Cepheids and most RV Tauri

stars have excess infrared emission indicative of circumstellar dust shells and mass loss (Lloyd

l~.vans 1985; McAlary and Welch 1986; Welch 1986). The mass loss must have occurred recently

(or is presently occurring) so is probably associated with the pulsation. However, emission and

mass loss are not found for several Cepheids with periods less than 10 days except the peculiar

binary Cepheid AU Peg, for which evidence for mass loss had already been identified (Harris et

al. 1984). The statistics are limited for Cepheids (only 6 Galactic Cepheids, IU Cyg, SZ Mon,

AU Peg, ST Pup, V1711 Sgr, and V549 Sco, and 3 LMC Cepheids were found to have infrared

excesses out of 20 stars detected), and mass loss may be episodic. Further study will be useful.

Several papers in the li terature have discussed UU Her stars as a separate class of

variables; however, they are similar to RV Tauri stars in temperature and surface gravity. These

are F supergiants with semiregular variations of several tenths of a magnitude. They are found

far enough from the Galactic plane or with low enough abundances to be considered Pop II

stars, although their luminosities, distances, and abundances have been matters of debate in

the literature. Their properties are reviewed by Sasselov (1984, 1986). Recent detailed studies

of HR 4912 (HD 112374), HR 7671 (HD 190390), and HD 46703 (Luck et al. 1983; Luck and

Bond 1984; Fernie 1986a) have derived values of [Fe/H] from - 1 to - 2 for these three stars.

On the other hand, UU Her, V441 Her (89 Her), and HD 161796 apparently are not very metal

poor (Fernie 1986b and references therein), shedding doubt about their Pop II nature. Both

UU Her and V441 Her are classified as SRd in the GCVS. At least some of these stars appear

to be hotter than high-luminosity Pop II Cepheids or RV Tauri stars. They may be pulsating in

nonradial modes (Fernie 1986b). None are known in clusters, at least part ly because luminous

cluster members (other than Cepheids) with temperatures near the instability strip are very rare

(Harris et al. 1983). Precise photometry suitable for detecting long-period variability for the few

Potential UU Her variables in clusters is not available. The best-observed candidate is probably

HD 116745 (ROA 24) in w Cen: it has a spectral type of F5 based on hydrogen lines (Sargent

1965) and (B - V)o=0.25, but it has the same V magnitude within 0.12 in three independent

Studies, so it cannot have a large amplitude, if variable at all.

B. Red Variables

Recent work on red variables in globular clusters has been discussed extensively (Feast

1981; Lloyd Evans 1983, 1984; Whitelock 1986). Several studies have suggested that SRd stars in

279

moderately metal-poor clusters are equivalent to the more metal-rich Miras (Feast 1981; Lloyd

Evans 1983): both classes of star are undergoing pulsation with sumciently high ampli tude to

produce atmospheric shock waves strong enough to excite Balmer emission. In the relatively

metal-rich Miras, the TiO absorption bands modulate the blue and visual light to produce the

huge pulsation amplitudes of 4 to 6 magnitudes, while in the more metal-poor SRd stars the

amplitudes are closer to a reflection of black-body temperature variations. Both classes of star

have modest bolometric amplitudes, typically 0.7 magnitudes (Menzies and Whitelock 1985).

Both classes also fall near the same P-L relation (Whitelock 1986), although the pulsation modes,

the fundamental periods, and the exact P-L relations are still being debated.

However, stars in clusters of different metailicities can appear with similar SR charac-

teristics (see Lloyd Evans 1983). The literature on S a d stars is confusing, at least part ly because

of the high temperatures often assumed for these stars. The spectral types of F, G, or K usually

quoted for these stars (and even used in defining the class in the GCVS) are derived from the

metal lines and molecular bands; they are much earlier than. in normal stars of the same (cool)

temperatures because of the low metallicities. Rosino (1978) correctly emphasized the importance

of using bolometric luminosities in a P-L relation for red variables, but he still underestimated

the bolometric corrections and bolometric luminosities for the SRd stars.

C. Dwarf Cepheids and & Scuti Stars

A few Dwarf Cepheids a in the field are believed to belong to Pop II based on their

metal deficiencies and their high velocities (McNamara and Feltz 1978; Eggen 1979; Andreasen

1983). The stars CY Aqr, XX Cyg, KZ Hya (HD 94033), DY Peg, SX Phe, and GD 428 are

the best candidates. (Several other stars are mentioned by Eggen as possible Pop II ultra-short

variables, but they are probably RR Lyraes.) Only three Dwarf Cepheids are known in globular

clusters, all in ~ Cen (Jorgensen and Hansen 1984; Da Costa and Norris 1987). However, it is

likely that numerous variables will be found among the many newly discovered blue stragglers

in the cores of the globular clusters NGC 5466 (Nemec and Harris 1987) and NGC 5053 (Nemee

and Cohen, in preparation). Other low-concentration clusters will probably also turn out to have

blue stragglers and Dwarf Cepheids concentrated toward their centers. The periods, amplitudes,

and light curves should provide a wealth of da ta on the properties of Pop 1I Dwarf Cepheids,

including masses estimated from the period-density relation.

The periods (from 0.04 to 0.14 days) of the field Pop II Dwarf Cepheids tend to be

shorter than those of Pop I stars of similar amplitude, although the period distributions overlap.

One reason for some Pop I stars having periods longer than any Pop II stars is that the Pop H blue

stragglers that are (or will become) Dwarf Cepheids are limited in luminosity (limited to about

1.6 M® if they originate from mass transfer in binaries). On the other hand, Pop I stars can have

higher masses and luminosities on the main sequence, and so can have longer periods when they

reach the instability strip. Other factors might also be important. Slow rotat ion is apparently a

I use the term Dwarf Cepheid to refer to high-amplitude 6 Scuti stars. See Breger

(1979) and McNamara (1985) for discussion about the distinction. Other names are sometimes

also used; for example the GCVS calls the Pop II stars SX Phe stars, regardless of amplitude.

280

prerequisite for the high-amplitude pulsation of Pop I Dwarf Cepheids (Breger 1969; McNamara

1985). If this is also true for Pop II stars, then Pop II Dwarf Cepheids could be either more

or less common, depending on whether Pop II blue stragglers rotate slowly or rapidly. Possible

evolutionary scenarios can be constructed for either case.

Even fewer Pop [I low-amplitude 6 Scuti stars are known. None are known in clusters,

while three candidates in the field (McMillan et al. 1976) are VW Ari (HD 15165), SU Crt

(FID 100363), and GD 210. Possibly Pop II blue stragglers having a wide range of temperatures

(wider than the temperature range of Pop H Dwarf Cepheids) will be found to vary with low

amplitudes (perhaps in nonradial pulsation modes) as with Pop I ~ Scuti stars. Presumably the

small number of candidates is a selection effect caused by the rarity among bright stars of Pop

II blue stragglers and the difficulty in detecting low-amplitude variability in faint stars. Further

searches for variability, similar to the McMiUan et al. study, will be necessary (although very

difficult!) if we are to learn about the fraction of Pop II variables and their mode behavior near

the main sequence.

D. Anomalous Cepheids

Anomalous Cepheids are observed to be anomalously bright for their periods, and

are inferred to be anomalously massive from the period-mean density relation (Zinn and Searle

1976; Wallerstein and Cox 1984). The presence of Cepheids with masses of ~1.5 M O should

not be surprising in metal-poor stellar systems with other evidence for substantial numbers of

intermediate-age stars such as Carina and the SMC. In such galaxies, the Pop I and Pop II

Populations may form a (more or less) continuous distribution of ages and abundances, perhaps

including old very-metal-poor stars, intermediate-age moderately-metal-poor stars, and younger

slightly-metal-poor stars. The confusing data on short-period Cepheids in the SMC (see Smith

1985) is probably at least partly a result of this mixture of populations. In the other dwarf

spheroidal galaxies Fornax, Draco, Sculptor, and Ursa Minor, where the fraction of intermediate-

age stars drops, the 1.5 M® stars become more "Anomalous".

Blue straggler stars have been found in the three dwarf spheroidals with deep color-

magnitude diagrams (Draco, Sculptor, and Ursa Minor; see Da Costa 1987). Their masses are

~lso near 1.5 M®. Da Costa (1987) has noted that a small fraction of intermediate-age stars in

these galaxies cannot be ruled out on other grounds. However, in NGC 5466, the only Galactic

~lobular cluster known to contain an Anomalous Cepheid, blue stragglers are also numerous

(Nemec and Harris 1987). A younger population in a sparse, halo cluster with a low escape

velocity like NGC 5466 is highly implausible. If we apply Occam's razor, whatever produces blue

stragglers in NGC 5466 is also likely to produce the blue stragglers and Anomalous Cepheids

in the dwarf spheroidals (other than Carina). The similarity of masses derived for Anomalous

Cepheids (from their pulsation properties) and for the blue stragglers in NGC 5466 (from their

dynamical segregation in the cluster) suggests a common origin. It provides support for the

binary-star hypothesis for their origin, although the exact details of the evolution with mass

transfer are complicated and uncertain. Study of velocity variations and rotation velocities of

both blue stragglers and Anomalous Cepheids can help constrain their evolution. Unfortunately,

these studies are at present impossible for stars in dwarf spheroidals and difficult in globular

281

clusters (Chaffee and Ables 1983). Studies of field stars like the candidate Anomalous Cepheid

XZ Get (Teays and Simon 1985) should be pursued.

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283

SIMULTANEOUS OPTICAL AND INFRARED PHOTOMETRY OF RV TAURI STARS

I. 2.

M.J. Goldsmith l, A. Evans I, J.S. Alblnson I and M.F. Bode 2

Dept. of Physics, University of Keels, ST5 5BG, UK School of Physics and Astronomy, Lancashire Polytechnic, PRI 2TQ, UK

ABSTRACT

We present nearly simultaneous optical and infrared photometry of RV Tauri

stars. From this we deduce stellar and (where appropriate) dust shell parameters.

Possible correlations are suggested between dust shell extent, metalllcity and type

of RV Tauri stars. Grain formation appears to be episodic, rather than continuous.

I. INTRODUCTION

RV Tauri (RVT) stars have long been known to be associated in several instances

with extensive dust shells (see e.g. Gehrz 1972). The process of dust formation is

still poorly understood however. Recent polarlmetric studies of RVTs have suggested

that in certain objects dust formation may be episodic rather than continuous (D.J.

Axon, private communication). Such episodic formation, linked to the pulsatlonal

cycle, has also been suggested to occur in long period variables (see P. R. Wood,

this volume). In this paper we briefly discuss our nearly simultaneous observations

of 25 RVTs at optical and infrared wavelengths, with the principal aim of

distinguishing between these two types of dust formation mechanism. We also attempt

to correlate derived dust shell parameters with other known or derived stellar

characteristics. A full account of this work is given in Goldsmith et el. (1987).

The dust shell properties of RVT stars have recently been discussed by

Lloyd-Evans (1985), Cardelll (1985), and Baird and Cardelll (1985). Our approach to

the problem of determining dust shell parameters differs from those of these authors

in that we use simultaneous optical and infrared photometry, and deduce both stellar

and dust shell properties with the minimum of initial assumptions.

2. OBSERVATIONS, DATA ANALYSIS AND RESULTS

The observations were made in June-July of 1985 at the South African

Astronomical Observatory. The wavelength range covered is 0.36-10 ~m. Simultaneity

was achieved to within better than 40 minutes for 23 objects. Of these, I0 were

observed simultaneously more than once. Isolated optical or infrared data were also

obtained on several of these objects, and 2 other RVTs. IRAS data were not

284

initially included in our results due to their non-simultaneity. However,

Comparison has been made with IRAS PSC data (IRAS 1985) where appropriate.

After reducing the photometry in the standard way, the resulting fluxes were

corrected for interstellar extinction using the reddening maps of Fitzgerald (1968)

except for AC Her and R Sct, where extinction determinations were from Cardelli

(1985). The validity of the resulting interstellar reddening corrections may be

Judged from the close agreement of stellar temperatures derived here and those

derived elsewhere (see below). The procedure that was followed was then: (i)

fitting a black body function to the dereddened optical and near infrared data

(taking care to avoid bands with likely spectral features); (ii) fitting a further

black body to any residual infrared excess and (iii) finally deriving the Planck

mean absorption optical depth of the dust shell (<z>); clrcumstellar colour excess

((EB_v)cs); effective stellar temperature (T,); inner dust shell, and condensation

radii (R 1 and Re); and dust shell mass (Mgr), as appropriate, from expressions given

in Goldsmith et al. ( 1 9 8 7 ) .

Table I lists the stars in which we found no evidence of dust shells. With the

exception of E1 Peg and R Sct none of these stars were detected by IRAS, which

SUggests that our failure to detect them at shorter wavelengths was not necessarily

due to them having low temperature shells. Both here, and in table 2,

elassiflcation of the spectrum was from Lloyd-Evans (1985) and Preston et

al. (1963), and classification of the light curve from Kukarkin et al. (1969).

Table I

RV Taurl Stars With No Evidence of Dust

Spectral E(B-V)~s T, Spectrum Light Notes

Type (K) Curve

DS Aqr F2 O.00 5700 C RV i AD Aql F 0.15 4000 B? RV 1,3 DY Aql GB-M3 0.15 3500 A RV I RY Ara G5-K0 0.45 5800 A RV 1 RX Cap FS-GO 0.I0 5800 A RV 1 W Cen M3-M8 0.45 2700 - M i V385 Cra F4 0.15 7300 - - 2,3 TTOph G2-K0 0.25 5050 A RVa 1,3 UZ Oph G2-K8 0.45 4000 A RVa 1 V453 Oph F 0.45 6000 C Cep? 1 V564 Oph GS-K2 0 . 0 5 3330 A RV 1 El Peg M5 0.00 2500 gM RV I V760 Sgr G5 0.75 6500 A RV 1 R $ct G0-KO 0.20 4200 A RVa 1,3

NOtes t o T a b l e 1 I. Spec-trai- ~pe from Kukarkln (1969) 2. Spectral type inferred from present work 3. No photometry beyond 5 Bm

285

In table 2 those stars with dust shells, together with derived parameters, are

listed. The dust shell temperatures for BU Cen, BT Lib and AR Sgr are uncomfortably

high. The main contributor to this anomalous result is the very low value of <T>,

and hence large error in the derivation of T D. We may note that we were unable to

distinguish photometrically between carbon and oxygen rlch objects, but that clearly

a large contribution from some non-dielectric condensate must be present. Derived

masses ranged from approximately lo-ll-lo-SH o. Figure I dust illustrates the

fitting of blackbody curves to stellar and dust shell emission at two epochs for

SX Cen.

Table 2

RV Tauri Stars With Dust Shells

Star Spectral E(B-V)is E(B-V)c s T~ <T> ~D Spectrum Light Notes

T~pe (K) (K) Curve

UY Are G 0.15 0.05 6100 O.17 Ii00 B RV 1 RU Cen A7-G2 0.15 0.03 5110 0.07 1450 B RV I SX Cen F5-G 0.15 0.12 6450 0.44 1200 B RVb I SX Cen 490 BU Cen G9 0.15 0.03 4290 0.08 2000 A RV? 2 AC Her F2-K4 0.I0 0.04 5680 0.11 1800 B RVa i BT Lib F4 0.25 0.02 6750 0.08 2050 - RV 2 VV Mus F5.5 0.45 0.18 6320 0~63 690 - RV 2 TX Oph F5-G6 0.25 0.02 5570 0.06 970 A RVa 1 AR Sgr F5-G6 0.15 0.02 5610 0.07 2400 A? RV 1 R Sge GO-G8 0.05 0.12 5630 0.36 585 A RVb 1 AI Sco GO-K2 0.25 O.18 4540 O.44 1000 A RVb 1

Notes to Table 2 I. Spectral type from Kukarkin (1969) 2. Spectral type inferred from present work

3. DISCUSSION AND CONCLUSIONS

Table 2 contains both oxygen and carbon rich (A and B type) RVTs, whereas table

1 has only one questionable B type (AD Aql). Metal poor C types do not appear at

all among the dusty RVTs. Thls suggests that although RVT's that are either oxygen

or carbon rich may give rise to dust shells, dust formation may proceed more readily

in carbon rich objects.

There may also be some correlation with light curve type. Table I contains no

stars definitely classed as RVb (where long term oscillation is superimposed on the

normal shorter period), whereas table 2 contains three such objects, each of which

has an extensive dust shell, judging from the values of <~> deduced here. Indeed~

the stars in table 2 may be divided into two groups on the basis of <~>, a division

which also correlates with the space velocities found by Joy (1952); the high

286

-29"5

-300 x 3

~ -30"5

-310

U B VR ] J H K L M N = i , i i = ! ] = , =

5% C E N

l (pro)

F i g u r e ] . Changes i n the f l u x d i s t r i b u t i o n o f SX Cen between JD 2446244.26 (open squares and full eurves)2and I JD 2446248.30 (filled squares and dashed curves). Flux in W cm- Hz- ; errors in photometry are less than the size of the plotted points.

velocity (population II) stars having systematically less extensive dust shells. It

is not clear whether large grain formation rates stem purely from enhanced

metalllclty, and the double mode pulsation is a blproduct of the latter, or whether

a combination of the two effects may be responsible. Obviously more observational

data need to be obtained.

In most cases the SAAO results are consistent with emission from a single dust

shell, if dust Is present at all. However, in the case of AC Her fitting a single

dust shell results in a significant residual excess at N. Similar results apply to

SX Cen (see figure l), and marginal evidence for multiple shells (based on IRAS PSC

data) is found for RU Cen and UY Arao

Comparison of T c (1300-150OK for silicates) with T D shows that for several

Stars (e.g. R Sge~ VV Mus) the current dust temperature is less than one would

expect if condensation were a continuous process. Also, in SX Cen photometry was

obtained over 4 days of its 16.4 day period during which the V flux dropped'by 0.2

magnitudes and T, from 6400 to 6200K (see figure I). The fall in the inner and

outer dust shell temperatures (~1400 to ~I000 K and ~700 to ~300 K respectively) is

however rather greater than one would expect purely from the stellar changes, and

discrete outflowlng dust shells seem to be required. All these results appear to

287

confirm that episodic, rather than continuous, grain forma~on is at work in RVT

stars.

ACKNOWLEDGEMENTS

We thank the staff at SAAO for their invaluable help, and SERC for the

provision of travel funds. MJG is supported by the University of Keele, and bIFB and

JSA by SERC. MFB is grateful for the hospitality of the Earth and Space Sciences

Division, Los Alamos National Laboratory, where this manuscript was prepared.

REFERENCES

Baird, S.R., end Cardelli, J.A., 1985, Astrophys. J., 290, 689 Cardelll, J.A., 1985, Astron. J., 200, 364 Fitzgerald, M.P., 1968, Astron. J., 73, 983 Gehrz, R.D., 1972, Astrophys. J., 178., 715 Goldsmith, M.J., Evans, A., Alblnson, J.S., and Bode, M.F., 1987, Mon. Not. R. astron. Soc., submitted. IRAS, 1985. IRAS Point Source Catalog, JPL D-1855, eds. Belchmann, C.A., et al.. Joy, A.H., 1952, Astrophys. J., 115, 25 Kukarkln, B,V., et a!,, 1969, General Catalogue of Variable Stars, Moscow Lloyd-Evans, T., 1985, Mon. Not. R. astron. Soc., 2!7 , 493 Preston, G.W., Krzemlnski, W., Smak, J., and Williams, J.A., 1963, Astrophys. J., 137, 401

288

RV Tauri Stars : The Resonance Hypothesis

John K. Worre l l

U n i v e r s i t y Observatory, Buchanan Gardens,

St. Andrews, F i f e , U.K.

I-D~rodqction

Ch r i s t y (196b) found t h a t a n o n l i n e a r p u l s a t i o n model intended to

represent W V i r e x h i b i t e d RV T a u r i - l i k e l i g h t v a r i a t i o n s , i . e .

a l t e r n a t e l y deep and sha l low minima. The l i n e a r per iods of the

fundamental and 1 - t - o v e r t o n e modes were found t o s a t i s f y rough ly

P*/Po=2/3 and Ch r i s t y proposed t ha t t h i s "resonance" was the cause of

RV Tauri l i g h t v a r i a t i o n s . Since t h a t t ime the same coinc idence of RV

T a u r i - l i k e l i g h t curve and "resonance" have been found by a o ther

authors. Note t ha t in t h i s exp lana t i on of RV Tauri behaviour the t ime

between successive l i g h t minima i s Po. Using l i n e a r , a d i a b a t i c

Pu lsa t ion models Takeuti and Petersen (1983) i n v e s t i g a t e d a s i m i l a r

hypothesis in which the t ime between succesive minima was taken to be

1/2XPo and the RV Tauri behaviour was the r e s u l t of a two-mode, e .g .

P* /Po=I /2 , or a three-mode resonance. They found a nega t i ve r e s u l t .

However, i f , as now seems l i k e l y , the RV Tauri s t a r s are very c l o s e l y

r e l a t e d t o , but r a t h e r b r i g h t e r than, the W Vi r s ta rs nonad iabat ic

e f f e c t s should be very impor tan t . I t has been shown, e .g . by Aikawa

(1985>, t h a t at high L/M when nonad iaba t i c e f f e c t s are impor tan t PI/Po

is much g rea te r than the va lue obta ined from a d i a b a t i c c a l c u l a t i o n s .

Mo__od_d el s

L inear , nonad iaba t i c p u l s a t i o n models were c a l c u l a t e d using the

methods descr ibed by Castor (1971) t o which convec t ion , t r e a t e d using

&L=/L==O, has been added. In a l l models M=O.6M=, (X,Z)=(0.745,0.O05)

and the o p a c i t y has been c a l c u l a t e d using S t e l l i n g w e r f ' s (1975)

formula. The models cover par t of the HR diagram def ined by

2 .8< log (L /L=)<4 .0 and 3 . 5 5 < l o g ( T . ~ ) < 3 . 8 2 . Exper ience shows t ha t t h i s

g ives t o l e r a b l y good per iods over the range of parameters under

cons ide ra t i on but t h a t the g r o w t h - r a t e s , q, are b e l i e v a b l e on ly f o r

the h o t t e s t s ta rs .

289

Resul ts

I t was found t h a t f o r l og (L IL= )~3 .80 PI becomes g r e a t e r than P,:,

at s u f f i c i e n t l y h igh tempera tures . Looking f u r t h e r i t was found t ha t

the same e f f e c t e x i s t s f o r h igher over tones , i . e . P~+I,-P~ in

s u f f i c i e n t l y hot models. The lowest l u m i n o s i t y a t which the e f f e c t

appears, f a l l s as the mode number, j , r i s e s . This i s in accordance

w i th the work of Sa io , Wheeler and Cox (1984) who found t ha t

nonad iaba t i c e f f e c t s set i n a t a lower degree of n o n a d i a b a t i c i t y f o r

h igher over tones . An i m p l i c a t i o n of the pe r iod " c r o s s - o v e r " i s t h a t a

luminous s t a r e x h i b i t i n g p u l s a t i o n s w i th two very s i m i l a r pe r i ods need

not n e c e s s a r i l y be a nonrad ia l p u l s a t o r . C l e a r l y a l so t he re w i l l be

r a the r more resonances than would be the case i f the pe r i od c ross ing

e f f e c t was absent. The l o c i f o r PI /Po=1/2 (or 2 f o r s u f f i c i e n t l y hot

and b r i g h t models) and 2/3 (or 3/2) are p l o t t e d in f i g . I .

4.0

3 . 9

3.8

3.7

d 3 . 5 03

o 3 . 4

3.3

3.2

3.1

5.0

2.9

2.8

P=IP,=ll2 v ,'

," 13 /.;/ V P,/P,=I/2 + +

=21 v v g

& V

I I I I I

3. 80

I I ! I I

3, 70 Log (T~ r)

o Pjp.= 112

cb ~ :S I I I I

3.60

Field s t a r s

M = O. 61'1

Fundement a l

Type II cephelds

v RV (A]

+ RV (B1

x RV (C)

o SRd

Fig. 1 T h e o r e t i c a l " resonance" l o c i in the HR diagram superposed

on the o b s e r v a t i o n a l data f o r t ype I I cepheids (Demers and

H a r r i s 1974) and the RV Tauri and SRd s t a r s (Dawson 1979).

29O

The resonance Pi/Po=l/2 described by Cox and Kidman (1984> lies at

Iog(T.~)~3.75. Note, however, that for Iog(L/L~,)>3.2 the role of the

l~t-overtone is usurped by the 2 ~ because of the period "cross-over"

effect. At higher luminosities the loci resulting directly from the

Period "c ross -overs " appear enc los ing the l i n e along which the

fundamental and l - t - o v e r t o n e per iods are equal . Loci f o r three-mode

resonances become b e w i l d e r i n g l y compl icated because of the i n f l u e n c e

of the per iod "c ross -ove rs " and so are not p l o t t e d . More work needs t o

be done before any th ing can be said about them.

Also p l o t t e d on f i g . I are l u m i n o s i t i e s ca l cu l a ted f o r f i e l d RV

Tauri and SRd ( " y e l l o w " semi - regu la r v a r i a b l e s ) s t a r s assuming t h a t

the t ime between successive l i g h t minima i s the fundamental p e r i o d ,

and using the e f f e c t i v e temperatures de r i ved by Dawson (1979). (The RV

Tauri s t a r s are d i v i d e d by Preston et a l . (1962) i n t o t h ree c lasses ;

A - r e l a t i v e l y s t r o n g - l i n e d , C-weak- l ined , and B-wea~- l ined w i th

anomalously strong CH and CN features.> The luminosities for the

hottest of the RV(B> and RV(C) stars are not unique because Pc, for

log(L/Lo)>3.8 can fall below the value for lower luminosities. Three W

Vir (type II cepheid) stars are also plotted using data from Demers

and Harris (1974). It can be seen clearly that only a few of the stars

lie close to any of the loci for the period ratios. The majority of

the RV Tauri stars lie in the region where P~/Po<I/2.

C__~onclusion

The conc lus ion t o be drawn i s t ha t i t seems ve ry u n l i k e l y t h a t

the RV Tauri type l i g h t curves can be caused by occurence of the

c r i t i c a l per iod r a t i o s in a l l the s ta rs . I t might be, though, t h a t a

subclass of RV Tauri behav iou r , perhaps the r e l a t i v e l y r a r e s t a r s in

which the a l t e r n a t i o n of deep and sha l low l i g h t minima con t inues

w i thou t r eve rsa l over many cycles~ i s associated wi th the occurence of

Such a per iod r a t i o . Even though the per iods c a l c u l a t e d i n c l u d i n g

nonad iabat ic t heo ry show behav iour q u a l i t a t i v e l y d i f f e r e n t from t h a t

seen f o r the pu re l y a d i a b a t i c pe r iods , the end r e s u l t , so f a r as RV

Tauri s ta rs and "resonances" i s concerned, i s on ly s l i g h t l y changed.

A f u l l d e s c r i p t i o n of t h i s work w i l l be publ ished e lsewhere.

A.cc knowl edqements

I thank Dr. T.R. Carson f o r h i s help and encouragement over the

past years during which t h i s work was ca r r i ed out, and Dr. A.N. Co~:

f o r g i v i ng me an oppor tun i t y to t a l k about t h i s Work. Thanks also to

the SERC of the United Kingdom fo r f i n a n c i a l support in the form of a

Research Studentship and Ass is tan tsh ip .

References

Aikawa,T., (1985), The Observatory, 105, 46

C a s t o r ~ J . I . , (1971), Astrophys. J . , l & 6 , 109

Ch r i s t y ,R .F . , (1966), Astrophys. O., 145, 337

Cox,A.N. & Kidman,R.B., (1984), in "Cepheids: Theory and

Observations"~ ed. B.J. Madore

Dawson,D.W., (1979) i Astrophys. O. Suppl . , 4211 97

Demers,S. & Harr is ,W.E. , (1974), As t r . J . , 79, 627

Preston,G.W., Kreminski,W.v Smak,J. & W i l l i a m s , J . A . , (1962),

Astrophys. J . , 137, 401

Saio ,H. , WheelerlJ.C. & Co~,J.P., (1984)~ 281, 318

S t e l l i n g w e r f , R . F . , (1975) i Astrophys. J . , 195, 441

Takeuti iM. & Petersen~J.O., (1983) 1 Ast r .&Ast rophys. , 117, 352

292

COMPARISON O F ' OBSERVATIONAL AND THEORETICAL

SHORT-PERIOD TYPE II CEFHEID VARIABLES

PARAMETERS FOR

T. R. Carson and S. P. A. Lawrence

Department of Astronomy and Astrophysics

University of St Andrews, Scotland, U.K.

Abstract

A comparison is made of obser vati onai and theoretical data f~Ir

short-period Type II Cepheid variable stars. The technique of Fourier

decomposition is applied to both a set oF observed light curves and a

set of theoretical light curves from non-linear hydrodynamical models.

Comparison of the variation with period of Fourier parameters f t~ :~ r ~ the

two sets indicate broad agreement between them~ with parti r:ul ar

agreement on the abrupt changes which occur near a period of 1.6 days.

Linear models are then used to s~udy the fundamental a~d higher

harmonics and to demonstrate the coincidence of this feature with the

P~/Po = 0.5 reson~nce condition. Both linear and non-linear models

also enable tlne construction of period-mass-luminosity-temperature

relations~ with possibilities for the determination of the masses of

these variables.

In t roduc t ion

The comparison of theoretical models of pulsating variable stars with

observations has usually been confined to such parameters as periods,

amplitudes~ asymmetries and phase lags applied to both light and

velocity curves. The introduction of the method of Fourier

decomposition (Simon and Lee, 1981) enlarges the set of parameters

Which may be used in t h i s comparison. Such an analys is has been

car r ied out on observed and t h e o r e t i c a l l i g h t and v e l o c i t y curves for

the c l ass i ca l Cepheids by Simon and Davis (1983) and by Carson and

Stothers (1984). In the present paper we extend the method to short

Period (1-3 day) Cepheids of Type I I ( BL Hercu l is var iab les) and

Couple i t with a study of mode l per iod r a t i o s to ass is t in the

determinat ion of the p roper t i es of these va r i ab lee .

293

Fourier decompos i t i on

The observational data is taken from the UBV photometry of Kwee and

Diethelm (1984). The theoretical data is from the set of eleven

non-linear hydr odynamical models of Carson and Stothers (1982)

supplemented by t.hree additional models computed by the present

auU1ors using the Carson non-linear code incorporating mass zones o~

equal sound travel times and a dynamical phbtosphere. Each

time-dependent periodic quantity f(t) is expanded as a series

N f (t)==f~. + ~ [a.. c o s ( n w t + ~ ) ]

1

where w is the angular frequency, fo is a~constant, N is the order of

f i t ~ a,, i s a F o u r i e r amp l i t ude and @~ the c o r r e s p o n d i n g phase, a l l

as de termined by l e a s t squares f i t . Of p r a c t i c a l i n t e r e s t a re the

r e l a t i v e amp l i t udes and r e l a t i v e phases

A .... = an / a~ and ~ L = ~r. - n ~

P e r i o d - m a s s - l u m i n o s i t y - t e m p e r a t u r e r e l a t i o n s

Using t he l i n e a r n o n - a d i a b a t i c p u l s a t i o n code o f Wor re l l (1985) , and

the same p h y s i c s as i n the n o n - l i n e a r p u l s a t i o n s t u d i e s , f rom a l a r g e

number of models we o b t a i n a l e a s t squares f i t f o r any p e r i o d or

p e r i o d r a t i o as a s imp le monomial power law in M, L and T . . Thus, f o r

example, we may w r i t e

log (P=/Po) = A log (M/Mo) + B log (L/Lo) + C log (T. ) + D

Since the r e l a t i o n w i l l be a c c u r a t e o n l y ove r a l i m i t e d range, we

r e s t r i c t M t o d i s c r e t e va l ues and d e r i v e f o r m u l a e a p p l i c a b l e t o each:

M/Mo A B C D C/B

0.5 0 .0 -0 .0345 0.9169 - 3 . 7 2 8 0 -26 .569

0.6 O.O -0 .1010 0.8448 - 3 . 2 9 8 4 -8 .364

0.7 0 .0 -0 .1100 0.7659 -2 .9643 -6 .962

The r a t i o C/B g i v e s the s l o p e o f l i n e s o f c o n s t a n t P~/Po in the

t h e o r e t i c a l Her tzsprung diagram.

294

L ~

t , i

e,?

~ u

~ u

L I

C~SI:P.VATX OFll

I I

o

o

@ o

o

k e I , i l t ,~ I,t< L e l e *l,.T 1 4 l i I I I I

f ~ l O O ~ T S I

l e

L ; o

o L * o

I I o a

L ~

f" ~ l i . l I+~ l ~ l i . I I I i T I I I I I L l L i

PI~I~ l ~ (~AYS)

(a)

O0~EI~VATZCt~fil l . I

L S

L t

o L e o

L,e • o

~ LS

ILl

L I

L I

' l . I L i ~ I , i I , II l e I l l I . i l i . I i e I L l

P~R ] O0 ~ A T S )

7,11

L I

0 L |

L S

(b)

L I

L I

L S

~ i

t * l

L !

o o

o

o

o •

o • * o

Y £ ; O E U

Jo

i , e I . I L L i .e i .e ~,e l ; ~ * i l LS i I

~IItI.OC~TY

L 9 - - -

~ E T . I L e

o L t LD

0

i L S • •

• .

• • • •

L |

L I

~ l u i i , ,

L e L ! I . l t . 4 *,,e I A J*J L ~ L I I L l L I

P l ~ l O 0 I:U~TS)

MOOLS.I

i S

L I o

L S •

~ 8 • a L e ~ O e •

e L | u

l . e . i i , - -

L e L T L ~ ~ 4 L I L e L ! L ~ i 4 L s i e

F ~ A ] O 0 IO~TSI

(c>

Figure I . Re la t i ve ampl i tudes and phases versus per iod .

295

FigL~re 2.

P~r iod - r -a t io v~.r .~,us p~,riod.

='~'~" P2/P8 v. P(]

18 11 12 ;3 t4 t5 IG ;7 ~8 IS 2~ 21 22 23 24 XIO- I

Resul ts and d iscuss ion

The v a r i a t i o n s of the r e l a t i v e ampl i tudes and r e l a t i v e phases wi th

per iod are shown in Figures l a , b , c . A d d i t i o n a l d e t a i l s w i th t a b l e s are

given by Lawrence (1985). In the case of the l u m i n o s i t y both the

ampl i tudes A=~ and the phases 0 ~ f o r both obse rva t i ona l and

t h e o r e t i c a l data agree in e x h i b i t i n g a d i s c o n t i n u i t y near a per iod of

1.6 day. In the case of the v e l o c i t y the lack o÷ obse rva t i ona l data

does not a l l ow a s i m i l a r comparison, but again both the ampl i tudes and

the phases f o r the t h e o r e t i c a l data e x h i b i t the same d i s c o n t i n u i t y .

The hypothes is t h a t the d i s c o n t i n u i t y i s assoc ia ted wi th the resonance

cond i t i on P~/Po = 0.5 may be exp lored by the use of the p e r i o d - r a t i o

mass- luminos i t y - tempera tu re r e l a t i o n de r i ved above. Transforming from

the v a r i a b l e s L and T. to the v a r i a b l e Po using n o n - l i n e a r models we

ob ta in P~/Po as a f unc t i on of Po. The r e s u l t i s shown in F igure 2. I t

can be seen t h a t the resonance c o n d i t i o n i s s a t i s f i e d near the per iod

Po = 1.b day f o r masses in the range O.b < M/Mo < 0.7 in accord wi th

es t imates de r i ved from o ther cons ide ra t i ons .

Re~erences

Carson,R. and Stothers ,R.~ 1982, Ap. 0 . ,259 ,740 . Carson,T.R. and Stothers ,R.B.~ 1984, Proceedings of the 25th L iege I n t e r n a t i o n a l As t rophys ica l Col loquium, p.29. Kwee,K.K. and D ie the lm,R. , 1984, As t r . Ap. Suppl. Se r . ,50 ,77 . Lawrence,S.P.A. , 1985, M.Sc. Thesis~ U n i v e r s i t y of St Andrews. Simon~N.R. and Dav is ,C.S. , 1983, Ap. J.,2b&~787. Simon,N.R. and Lee,A.S. , 1981, Ap. J . ,248 ,291 . W o r r e l l , J . K . , 1985, Ph.D. Thesis, U n i v e r s i t y o~ St Andrews.

296

S O M E C U R R E N T P R O B L E M S I N H E L I O S E I S M O L O G Y

Timothy M. Brown High Alt i tude Observatory/Nat ional Center for Atmospheric Research*

P.O. Box 3000 Boulder, CO 80307

A b s t r a c t . Helioseismology is enjoying a tremendous surge of activity, spurred by the com- bination of reliable data and effective interpretat ion methods. Since I cannot do justice to the entire field, I a t t empt in this review to describe two current topics tha t I find interest- ing. (1) Several workers have now made measurements relating to the variat ion .of rotation with depth and lat i tude inside the Sun. Most of the observations agree fairly well on the depth dependence, but not so well on the lati tude dependence. I explain how such measure- ments are made, and discuss the current s ta te of this controversy. (2) The driving mechan- ism for solar p-modes remains a mystery. The best (in my view) explanation involves sto- chastic driving of the modes by turbulent convection. This theory (proposed by Goldreich and Keeley) has recently been extended by Goldreich and Kumar in a way that illuminates some issues and obscures others. I a t tempt to provide a simple introduction to these ideas.

I. I n t r o d u c t i o n

Leighton et al. (1962) discovered that the Sun oscillates on small spatial scales with periods near 5 m. However, i t was ten years before the oscillations were identified with p- modes t rapped in a subphotospheric cavity (Ulrich 1970, Stein and Leibacher 1971), and another five years before Deubner 's (1975) observations showed this identification to be correct. Since then, progress has been rapid. Improved theoretical understanding of the oscillations has been accompanied by advances in observational methods, with the result that we are s tar t ing to paint a detailed picture of the Sun's internal structure and dynamics, and of the physics of the oscillations themselves. Here I describe two topics tha t I find par- t icularly interesting. I have been closely involved in one of these, while in the other I am merely an interested bystander. In the la t ter case, especially, credit for good ideas goes to the original authors, while blame for misrepresenting them rests with me.

Like most forms of stellar pulsation, Solar p-modes are resonant acoustic waves that propagate wlthin a certain range of depths and are evanescent outside tha t range. The

• T h e National Center for Atmospheric Research is sponsored by the Nation=] Science Foundation

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individual p-modes are global resonances, and each is tradit ionally characterized by three numbers: the radial order n , the angular degree l , and the azimuthal order m. Their fre- quencies reflect the variation of parameters affecting sound wave propagation (such as sound speed and flow velocity) within the Sun. Solar p-modes are different from more familiar stellar pulsations in several respects. The most important of these is that very many modes are excited on the Sun, each to very small amplitude. Thus, while the oscillating velocity averaged over a small region of the solar surface is typically 400 m s -~, this signal represents

the incoherent superposition of some 107 distinct oscillation modes, each with a period near 5 m and a velocity ampli tude of 10 cm s-' or less. Almost equally important is that most p- modes have relatively long lifetimes (days, at least), so tha t their oscillation frequencies are

well determined.

The significance of these characteristics is that one can observe a vast number of extremely accurate oscillation parameters, and that (by virtue of the small amplitudes) one can then use linear theory to relate those parameters to the structural and dynamical pro- perties of the Sun. The oscillations thus have the potential to tell us a great deal about the Sun's internal structure and motions. One of the best current examples of this capability

concerns the Sun's internal rotation.

II . Measur ing Solar I n t e r n a l R o t a t i o n w i t h p - M o d e s

In the simplest approximation (Brown 1985, 1986), solar oscillation frequencies depend

on the solar rotation frequency ~ ( r ,0) via

1

f '~( O)fp,~(eosO)12 d eosO -~ , ( : ) v(n ,l ,m ) - v (n ,1,0) - - - ,n . 1

f [Pl~(cose)]2d cos0 -1

where 0 is the colatitude, ~ is a suitable depth average (which itself depends on 1 and v) of

Q, and Pt m is the associated Legendre function. Inspection of Eq. (1) shows tha t for solid

body rotat ion (i2(8) ~-- no), u is merely a linear function of m :

~,(n , l , m ) - ~ n ,t ,0) = - m a o • (2)

However, if 12 ---- f~(8), then more terms (but only odd ones) appear in the series:

v ( n ,l ,m ) - ~ (n ,1 ,0) = - m a 0 + m 3 ~ + . . . (3)

A convenient device is to represent this power series in m as a series in Legendre polynomi-

als (Duvall et al. 1986)

i = N v ( n , l , m ) -b<n, l ,O) = L Z aiPi(-rn lL ) , (4)

i To

where L ~ ~ . This expansion minimizes correlated errors between the coefficients, and using - r a / L as the argument of the Legendre functions factors out most of the l -

dependence of the coefficients.

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One can s tar t with this formalism and devise a fairly simple program for measuring the Sun's internal rotat ion: (1) Measure t/(m ) for a range of n and l values. Parameterize the results using the series expansion of Eq. (4). (2) Use the linear part of u(m ) to determine the par t of f~ that is independent of latitude. (3) Use the cubic and higher odd terms in the series to determine the latitudinal differential rotation. (4) Use the variation of these parameters with l to determine the way in which the differential rotat ion varies with depth.

It is worth noting that this process is relatively model-independent. One is concerned only with frequency differences at constant n and l -- differences that vanish identically for a stellar configuration with complete spherical symmetry. Thus, inferences about the solar rotation are almost completely insensitive to errors in the structure of the solar model used. For determining the lat i tudinal dependence of rotatlon,~one need only know tha t spherical harmonics are good angular eigenfunetions for the p-modes -- an assumption that seems quite safe. To determine the depth dependence, one must know the radial eigenfunctions as well as the angular ones. Even in this case, however, the worst error that one is likely to make corresponds merely to an erroneous depth scale. Model-independence does not, how- ever, assure simplicity, or accuracy, or even uniqueness; i t will become apparent that the observational da t a are adequate to make some inferences about rotation in the solar interior, but that these are, as yet, rather crude.

The most difficult part of the program just outlined is measuring the required u(m ) relations. Several techniques have been employed to do this. These differ considerably in their details, but are quite similar in their fundamentals. In all cases, one measures some oscillating quant i ty (usually Doppler shift, but sometimes simply the intensity) with spatial resolution over as much of the visible solar disk as possible. By forming linear combinations of the signal at different points on the solar disk, one can select some spherical harmonics and exclude all the others. A time series of each linear combination then contains signals from a small number of distinct oscillation modes; the power spectrum of this signal consists of a relatively small number of narrow peaks, each arising from a single mode. In this con- text, "relat ively small" may mean 50 or so peaks, each flanked by sidelobes tha t arise from gaps in the observations, and all lying within the frequency range 2.0 mHz < v < 4.5 mHz. Separating these peaks from one another requires high frequency resolution, and therefore long observing runs. Da ta sets spanning weeks or months are the current norm, and the Global Oscillations Network Group (GONG) plans essentially continuous observations for

three or more years.

Once one has power spectra, one must identify modes (or use some other means) to extract the needed ~ m ) relations. Two principal methods are used to do this. The more common is a cross-correlation technique, which gives ~ m / l ) averaged over a range in both n and I (Brown 1985, 1986, Duvall et al. 1986). This method has good noise properties and provides a convenient way to deal with the very large amounts of da t a involved, but its intrinsic averaging is detr imental to depth resolution, and it may produce systematic errors in ways tha t are difficult to understand. A more direct (and laborious) method is to identify peaks individually, and determine their frequencies by least-squares fitting to (say) Gausslart profiles. The tables of frequencies thus produced can then be used to derive t~m) for the

300

desired range of n and t (Libbrecht, 1986). This is certainly the best way to analyze the data, but so far the amount of work required to do so has ]imited this technique to a rela- tively small range of l values.

The first measurements of solar internal rotation by these means were done by Duvall and Harvey (1984), Dural], et al. (1984). They measured the frequency difference ~,(t ,I ) - v ( l , - l ) , allowing them to estimate the equatorial value of ~ . This frequency difference proved to be very nearly constant with l , implying that f~ is only a weak function of depth within the Sun. The depth variation they saw was mildly surprising: [2 appeared to decrease by a few percent between the surface and 0.4 R®, and then perhaps turn up

again near the solar center.

Hill and various coworkers (Hill et al. 1982, Hill 1984, 1985) have a t tempted to identify oscillating signals over a wide range of frequencies in time series of the apparent solar diam- eter. The rotational splitt ings tha t they identify in these da ta for I <: 6 are typically 4 times what one would expect from the surface equatorial rotat ion rate, implying that most of the inner half (by radius) of the Sun rotates at 5 or more times the surface rate. These results differ irreconcilably with all other measurements of rotational splittings. In my opin- ion, the discrepancy is best explained as a misidentification of modes in the extremely dense collection of peaks tha t comprise the solar diameter power spectra.

Brown (1985, 1986) first measured frequencies of M1 the azimuthal orders, covering 8 <~ I < 50. This allowed an estimate of the way in which the rotat ion varies with both depth and lati tude. For the entire range of degrees measured, the cubic contribution to u(m ) was found to be only about one-thlrd of what one would expect from the lati tudinal differential rotation observed at the solar surface. Since modes with I = 50 penetrate only as far as the bot tom of the convection zone, this result implied tha t the lati tudinal differential rotat ion was confined to a relatively small fraction of the convection zone depth. This conclusion was interesting because it was in conflict with theorist 's expectations, namely tha t ~ should be roughly constant on cylindrical surfaces aligned with the rotation axis, at least within the convection zone (Gilman and Miller 1986, Glatzmaier 1985).

Libbrecht (1986) soon obtained similar observational results for the degree range 5 < l <~ 20, and i t began to look as if convection theory was in serious trouble. This enter- tainlng picture was challenged, however, when Duvall et aL (1986) made similar frequency measurements by completely different methods, and found the cubic term a 3 to be just equal to the value expected from the surface rotation, for all degrees between 20 and 98. This result implies tha t the surface differential rotation holds at all depths in the Sun, even well into the radiative interior. This also is at variance with theoretical expectations, but much more importantly, it raised the question whether anyone was measuring the rotational

frequency shifts correctly.

The answer to this question is not yet known, but the most recent observational results (Brown and Morrow 1986) at least narrow the gap between conflicting views. These obser- vations span 15 days (giving bet ter frequency resolution than any previously reported), and cover degrees 15 ~ I ~ 99. The odd-indexed Legendre coefficients a,- of Eq. (4) obtained

from these observations are shown in Fig. 1. The dashed lines in Fig. 1 are the coefficients one would expect if the surface differential rotation prevailed at all radii within the Sun. a I falls 1-2% below the surface value, more or less independent of l , suggesting that the

mean rotat ion rate in most of the solar interior is slightly lower than at the surface, a 3, on

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the other hand, falls substantially below the surface value for I :~ 40. This deficit implies that the latitudinal differential rotation decreases with depth, though not by as much as Brown's (1985, 1986) original observations indicated.

45°1_ l" , , , , , , , , ,

¢1 ! 4 3 0 ~ - - , ~ ! I i I I I i I ! I I I l I

3 O .

2 0

I I I I I I' | I | t I 1 I I | f l II

,o ? 0 ll-, , , T . . . . . - ~ -

- 2 0 ! , i m I0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 ! 0 0

Figure I. Frequency splitting coefficients al , a3, as in sidereal nHz. Dashed lines show the variation o n e

would expect for as based on several models, as computed by Christensen-Dalsgaard (1985}. For details, see the text.

A comparison of the observed a 3 values with forward modeling results by

Christensen-Dalsgaard (1985) gives a good idea of the form that ~(r ,0) may have. The dot- ted line on the a3 plot of Fig. 1 shows the expected values for a model in which the sur-

face differential rotation holds throughout the convection zone, while the radiative interior

rotates as a solid body at the surface equatorial rate. The long-dashed line on the same plot corresponds to a model that is similar to the first, except that ~ within the convection zone is constant on cylindrical surfaces aligned with the rotation axis, and matched to the observed surface values. Both of these models are evidently reasonable fits to the data.

Moreover, it is clear that the data do not allow one to distinguish between the models. That two models with such different rotation profiles can yield similar coefficients attests to

3 0 2

the importance of the regions near the solar surface in determining the frequency perturba- tions, and to the difficulty of doing accurate inversions of helioseismological data . Work on these problems is continuing at a rapid pace, so perhaps we will soon be able to answer some of the questions tha t current efforts have raised.

HI. Est imat ing p-Mode Ampl i tudes

An obvious question is whether the Sun is linearly unstable to p-mode oscillations in the same sense tha t other pulsating stars are. Computat ions so far appear unable to decide this question; the margin of s tabi l i ty is evidently very small, and whether i t is positive or negative depends on the detailed t reatment of such matters as the modulation of convective flux by the oscillations (Ando and Osaki 1975, Goldreich and Keeley 1977a, Gough 1980, Christensen-Daisgaard and Frandsen 1983, Dolez et al. 1984). Regardless of the result of a stabil i ty calculation, one is faced with a dilemma: If the p-modes are stable, then why do we see them at all? If they are unstable, why are their ampli tudes so small, i.e., what process could cause such a linear instabil i ty to saturate at mode amplitudes of only 10 cm/s?

Goldreich and Keeley (1977b) suggested a solution to this difficulty. They proposed that the p-modes are in fact linearly stable, but tha t they are excited to observable ampli- tude by acoustic noise from the convection zone. This process, known as stochastic excita- tion, has become the subject of renewed interest because of further work done recently by

Goldreich and Kumar (1986). The fundamental observation underlying stochastic excitation is that the solar convec-

tion zone is in constant turbulent motion. The fluid speeds involved are typically not very large: even near the top of the zone, where the overturning motions are fastest and the sound speed is smallest, the Much number is only about 0.15. Nevertheless, some of the energy in the convective motions appears as acoustic radiation. This acoustic energy can excite the resonant p-modes of the Sun, in much the same way that one can sound all the notes on a piano by clapping over the open strings. One can show that in a slowly-moving unstratified fluid with no external forcing, monopole and dipole radiat ion cannot occur (Lighthill 1952, Stein 1968). The first available source of acoustic noise is therefore quadru-

pole radiation, which emits energy at a rate proportional to M s, where M is the flow Much number. For small M , one can therefore expect this to be an inefficient process.

It is impor tant to realize, however, that the ampli tude of the p-modes depends both on how fast energy is being put into them and on how fast it is being removed. If energy put into the oscillations can remain there for a long time without dissipating, then the mode amplitudes could grow large regardless of the strength of the driving. Goldreich and Keeley (1977b) determined tha t the mMn process damping the oscillations is turbulent viscosity. One therefore expects that input from quadrupole acoustic radiation should cause the mode ampli tudes to ~row until drivin$ from this source is matched by turbulent viscous damping. The intriguing point about this picture is that quadrupole acoustic radiation and turbulent viscous damping are inverse processes, in the same sense as absorption and spontaneous emission of photons. This leads to an unusual circumstance, by stellar pulsation standards: the driving and damping for the oscillations both occur within the same part of the star,

and come close to balancing each other locally.

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Brownian motion provides perhaps the closest analogy to this situation. The contin- uum equations of fluid mechanics say tha t a small, neutrally buoyant tracer particle (e.g., a pollen grain) placed in a motionless viscous fluid should remain at rest. Such is not the case, however; the pollen grain moves because it is continually jostled by random molecular impacts. In the continuum limit, as molecules become infinitely small and numerous, this motion would vanish. We know, however, that if this jostling were the whole story, the grain would perform a random walk in velocity, reaching arbitrari ly large speeds after a long time. The process tha t prevents this is viscosigy, which arises from the abil i ty of free-flying molecules to t ransport (say) x-momentum in the y direction. This process requires a nonzero mean free path, and so viscosity also vanishes in the continuum limit. Molecular j i t te r and viscosity are therefore inseparable, arising from the same physical causes. Given this understanding, one can show that the average kinetic energy of the pollen grain is equal to that of a single molecule of the surrounding fluid. The importance of this result is that it holds independent of the details of the interactions involved, and tha t the kinetic energy depends only on the temperature of the fluid.

This digression suggests a pret ty and powerful analogy between stochastic excitation and thermodynamics, namely that the oscillations are, in some sense, in thermal equilibrium with the convective motions. Exactly what one means by "temperature" in this context is not clear, but presumably it has to do wlth the mean kinetic energy of the convective motions. The advantage of adopting this viewpoint is tha t one can then make some useful

predictions about the mode amplitudes, viz.

(1) Each p-mode should contain the same average energy. (Actually, the energies may be a weak function of frequency, since modes with different frequencies have different upper turn- ing points, and therefore interact with different parts of the convection zone.) (2) The instantaneous mode energies shoutd have a Boltzmann distributions, i.e., exponen- tial in the mode energy. (3) The mean energy in each mode should equal (within small numerical factors) the tur- bulent kinetic energy in one convective eddy with a turnover time equal to the mode period. (This result is not obvious, but follows from the scaling arguments of Goldreich and Keeley 19775.)

How do these predictions compare to the observations? (1) agrees rather well. Typical mode energies vary by only about a factor of 3 over the observed range of frequencies, and the variat ion at low frequencies is qualitatively what one would expect from the thermo- dynamic analogy (Libbrecht e ta l . 1988). At high frequencies, the observed variat ion prob- ably arises from a breakdown in the assumption that the modes lose energy only to tur-

bulent viscosity.

Nobody knows about (2).

(3) agrees poorly: the observed mode energies are larger than predicted, by a factor of about 100. Some of this factor may come from approximations in the scaling arguments or from our ignorance about solar convection parameters, but the discrepancy is large enough that one suspects most of the problem lies with the theory.

Recently Goldrelch and Kumar (1986) have proposed a modification to the stochastic excitation theory, which causes it to predict larger mode amplitudes. They noticed that the stricture against dipole acoustic radiation does not apply in a stratified fluid; the stratification provides a preferred direction, which in combination with convective entropy

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fluctuations can produce dipole radiation. Since dipole radiat ion depends much less strongly on the Mach number than does quadrupole radiation, its presence might have a significant effect on the oscillations. Increased radiating efficiency alone is not enough to increase the mode amplitudes, however. If the dipole emission is balanced by equally efficient dipole absorption, then one would not expect the mode amplitudes to change. The really significant point about the dipole emission is that there appears to be no effective corresponding absorption process: the dipole absorption is exactly canceled by dipole stimu-

lated emission.

If this is true, then the picture of stochastic excitation changes substautially. The mode amplitudes now grow until the energy input from dipole emission is matched by the losses from quadrupole absorption (i.e. turbulent viscosity). This leads to a much larger estimate of the energy in each mode. The energy now becomes the thermal energy in a sin- gle convective eddy with the correct turnover time, rather than the convective energy. Assuming a typical convective Mach number of 0.1 in the region where these short-lived eddies predominate, one predicts larger mode energies by the required factor of 100.

This modification of the stochastic excitation theory evidently solves the problem of small predicted amplitudes, but it raises a few difficulties of its own. Firs t , no one has been able to explain to me (or to anyone I know) why, or under what conditions, dipole stimu- lated emission cancels absorption. This is presumably my problem and not the theory's, but it lends more weight to the various other objections.

A more significant difficulty is that the mode ampli tudes in the new view are indepen- dent of the turbulent velocities, except insofar as these velocities determine the size of the convective eddies. Perhaps this is correct, but it certainly seems counterintuitive.

Finally, one must consider how the inclusion of dipole emission affects the notion of thermal equilibrium between the oscillations and the convection. Since the energy in the oscillations is larger with dipole emission than without it, the analog of temperature for the oscillations must be higher if dipole emission occurs. But when dipole emission is absent, detailed balance between the quadrupole emission and turbulent viscosity is presumed to bring the oscillations and convection to the same temperature. Thus, dipole emission must raise the temperature of the oscillations to a higher value than that of the convection. This leads to a problem: If one takes the thermal equilibrium analogy seriously, then the net effect of dipole emission is to transfer energy from a low temperature to a high one, which (analogically, at least) conflicts with the second law of thermodynamics. If one does not credit the thermal equilibrium analogy, then one has difficulty justifying the other predic- tions of the early stochastic excitation theory, ~z. equiparti t ion of energy between the

modes and a Boltzmann distribution for the mode energies.

There are, of course, several possible resolutions to this dilemma. Perhaps "thermal equilibrium" is an inappropriate idea in this context, but the behavior of a thermal equili- brium model follows anyway, for other reasons. Perhaps both dipole emission and dipole absorption do occur, but some error in Goidreich and Keeiey's (1977b) scMing arguments caused them to estimate the equilibrium energy level incorrectly. In this case one would still expect the thermal equilibrium analogy to hold, regardless of the detailed nature of the interaction between convection and the oscillations. Or perhaps there is more to the system than just the oscillations and convection, so tha t the second law argument is irrelevant. One could imagine, for example, that dipole radiation is a relatively small side-effect of some

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process that transports convective velocity fluctuations from regions where the rms velocity is high to regions where it is low. What is clear is that speculation is easy, while arriving at correct answers is not. My hope is that these recent developments in stochastic excitation may induce others to work on this fascinating and important problem.

R e f e r e n c e s

Ando, H. and Osaki, Y. 1975, Publ. Astron. Soc. Japan 27, 581. Brown, T.M. 1985, Nature 317,591. Brown, T.M. 1986, in Seismology of the Sun and the Distant Stars, D.O. Gough (ed.), D.

Reidet, Dordrecht, p. 199. Brown, T.M. and Morrow, C.A. 1986, Astrophys. J. (submitted). Christensen-Dalsgaard, J. 1986, in Cool Stars, Stella~ Systems, and the Sun, M. Zeilik and

D.M. Gibson (eds.), Springer-Verlag, Heidelberg, p. 145. Christensen-Dalsgaard, J. and Frandsen, S. 1083, Solar Phys. 82, 165. Deubner, F.-L. 1975, Astron. Astrophys. 44, 371. Dolez, N., Legalt, A. and Poyet, J.P. 1984, Mere. Soc. Astron. Italiana 55, 293. Duvall, T.L. and Harvey, J.W. 1984, Nature 310, 19. Duvatt, T.L, Dziembowski, W.,

Goode, P.R., Gough, D.O., Harvey, J.W. and Leibacher, J.W. 1984, Nature 3109 22.

Duvall, T.L., Harvey, J.W. and Pomerantz M.A. 1986, Nature 321,500. Gilman, P.A. and Miller, J. 1986, Astrophys. J. Suppl. 619 585. Glatzmaier, G.A. 1985, Astrophys. J. 201,300. Gol.dreich, P. and Keeley, D.A. 1977a, Astrophys. J. 211,934. Goldreich, P. and Keeley, D.A. 1977b, Astrophys. J. 212, 243. Goldreich, P. and Kumar, P. 1986, in Proc. IAU Symp. 125, J. Christensen-Dalsgaard (ed.),

(in press). Gough, D.O. 1980, in Lecture Notes in Physics, H.A. Hill and W. Dziembowski (eds.),

Sprlnger-Verlag, Berlin, p. 273. Hill, H.A. 1984, International J. Theor. Phys. 231 683. Hill, H.A. 1985, Astrophys. J. 2909 765. Hill, H.A., Bos. J.R. and Goode, P.R. 1982, Phys. Rev. Letters 499 1794. Leibacher, J.W. and Stein, R.F. 1971, Astrophys. Left. 7, 191. Leighton, R.B., Noyes, R.W. and Simon, G.W. 1962, Astrophys. J. 135, 474. Libbrecht, K.G. 1986a, Nature 319, 753. Libbrecht, K.G. Popp, B.D., Kaufman, J.M. and Penn, M.J. 1986, Nature 323, 235. Lighthill, M.J. 1952, Proe. Royal Soe. London A2119 564. Stein, R.F. 1968, Astrophys. J. 1541 297. Ulrich, R.K. 1970, Astrophys. J. 162, 933.

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The Intermediate-Degree f-Mode Multiplets of the Sun

O. F. Rabaey, H. A. H i l l , and C. T. Barry

Dept. of Physlos and Arizona Research Laboratories,

gnSversity of Arlzona, Tucson, AZ 85721

Abstract

Data from t979 solar differential radius observatlons (Bos 1982) has been

analyzed for evidence of intermedlate-degree f-modes. A set of 18 intermediate-

degree f-mode multiplets has been identified and classified based on more than 300

Classified modes of oscillation. The degree of the multlplets ranged from 19 to 36.

The detection and classification of these modes have been confirmed by testing the

horizontal spatial properties of the elgenfunctions. The m = 0 elgenfrequency

Spectrum was found to be on the average ~ 10 ~Hz greater than that predicted by the

Standard solar model of Saio (1982). Rotational splitting effects up to fifth order

in m were obtained.

I. Introduction

During the course of the last five years we at SCLERA have undertaken an

ambitious mode classification program. This mode classification program has been

Presented in numerous works using both observations taken at SCLERA as well as those

taken at other observatories and has culminated in the identification and

classification of 162 multlplets and over 1200 modes. These works include: Hill

(1985a), which used 1979 data (Sos 1982) taken at SCLERA and the results of Claverie

et al. (1981), Grec, Fossat and Pomerantz (1983), Duvall and Harvey (1983) Scherrer

e~t al. (1983), Duvall and Harvey (1984) to identify and classify a set of 83

multlplets in the five minute region of the elgenfrequency spectrum; Hill (1984)

Used the 1979 data to identify and classify a set of 30 low-order, low-degree

acoustic modes with periods of - 27 mln; Hill (1985b) analysed the 1979 data and

the observations of Kotov et al. (1983) to identify and classify a set of 31

multlplets with periods of ~ 160 min; Hill (1986b) which, again, used 1979

Observations to identify a set of 4 intermediate order, g-mode multiplets (these

modes were observed to be coupled).

In the work presented here, the 1979 data (Bos 1982) was examined and a set of

18 intermediate degree f-modes (radial order n = O) were iden t i f i ed and c lass i f ied.

This work was motivated by the works of H i l l , Rosenwald, and Rabaey (1985) and H I l l

307

(1986a). In the Hill et al. (1985) work, preliminary evidence of intermediate-

degree f modes was found during an analysis of the validity of a uniformly rotating

convection zone model. In the subsequent work of Hill (1986a), which extended the

analysis of Hill et al. (1985), properties of intermedlate-degree f-mode multiplets

of angular degree ~ - 33 were identified. With these properties as a starting point

and the work of Hill (1984), which uses the horizontal spatial properties of of the

oscillations as an independent means of multlplet classification, the study of the

intermedlate-degree f modes in the 450 - 650 gHz region of the spectrum was begun.

The importance of measuring and using the horizontal spatial properties cannot be

over emphasized in this study, since with this technique we have an independent

means of testing for the detection of multiplets.

ZZ. Mode C1asslflcation Proced~e

As mentioned in the Introduction, mode classification was started with an £ - 33

Intermedlate-degree f-mode multlplet using the properties identified by Hill

(1986a), The corresponding horizontal spatial properties of the multlplet as

expressed by the Dij (see Hill 1984) were computed and used to identify the m = 0

mode of the multlplet. Next, the frequency pattern of the multiplet was fit, using

a least-squares analysis, to a power series cubic in m. The polynomial for the

elgenfrequency Vn£ m is written as

- v' m * v~m2/2 * ~m3/3! (I) Vn~m Vn~ * n~

The final selection of members of the multlplet was based upon the deviation, Av,

between the frequency of a peak under consideration and the frequency determined by

the polynomial fit. After the analysis of this multiplet was completed, the

analysis of contiguous multlplets in ~ was undertaken. Information about the

rotational splitting effects llnear~ quadratic, and cubic in m obtained from the

least-squares analysis of the multiplet was used in conjunction with theoretical

elgenfrequency spectrum of Salo's (1982) model to predict the parameters of the

multiplets contiguous in ~. With these parameters the classification process for

the first multlplet was repeated with one added constraint -- the composite llst of

observed properties for the set multlplets must be internally consistent as measured

by criteria established in Hill (1984).

As shown in Hill (1984), DIj is a quasi-perlodlc function of m which allows one

to confirm the m = 0 mode of the multlplet. In the process of following the above

prescription it was discovered that the periodic nature of the Dij deteriorated for

m values greater than 22. An analysis of the relative magnitude of the flfth-order (5) 4

term (Vn~ IL /SJ) was done. It was found that the magnitude of the flfth-order term

was a significant fraction of the magnitude of the cubic term. With this insight,

Equation (I) was amended to include fourth and fifth order terms in m ang with the

308

above classification procedure appropriately extended to handle up to fifth-order

terms in m, a set of 18 multiplets and 366 modes were identified and classified with

values from 19 to 36.

I l l . Results and Diseusslon

The observed values of m = 0 eigenfrequencles were found to be about 10 ~Hz

larger than the theoretical values computed using Saio's equilibrium model (Salo

1982). In Hill (1984), for low-order, low-degree acoustic modes, the observed

frequencies were typically found to be 10 ~Hz larger than the theoretical values as

Well. By exploiting the local properties of the f- and p-modes it may be possible

to identify the source or sources of this departure.

The rotational splitting effects linear and cubic in m are shown in Figures I

and 2 respectively. The errors shown are the formal errors based upon the least-

Squares analysis. Since the rotational splitting kernels for the intermediate-

degree f-modes are well localized in the outer ~ 10% of the Sun (see Hill 1986a),

One can easily verify that these values of v~ and v8~2/3! are consistent with the

rotation curves obtained by Hill e_tt a_~l. (1985) and Hill e_tt a l. (1986) respectively.

The rotational splitting terms of second and fourth order in m are small with 2

the second-order terms having a weighted average of v~o~ /2! - -6.35 ± 24.00 nHz and

~(4) n414' 6 the fourth-order terms having a weighted average of ~0£ ~ " " = .04 + 26.57 nHz.

The fifth-order terms in m were positive definite and increased as ~ increased.

What is the probability that the given set of peaks identified as a multlplet

are in fact due to a coincidental alignment of unrelated peaks? To answer this, a

Comparison of No~/Nma x (where NO~ is the n~ber of modes identified for a given

and N is the maxlm~ number expected for that multiplet) and p(v) is made. The max

quantity p(v) is the probability of finding one or more peaks from a random

distribution within +0.065 ~Hz of v0~ m obtained in the polynomial fits (see Hill

1984). The values of p(v) range from a low of .295 to a maximum of .357, and the

Values of No~/Nma x range from .321 to .571. Computing d, which is the weighted

average of the difference between the observed ratio No~/Nma x, and p(v), we find

- .1150 + .0164. This difference iS a 7.01 standard deviation number. Using this

number we can compute the probability that the 18 Zeeman-llke frequency patterns

were obtained from a randomly distributed set of peaks in frequency. This

Probability is less than 10 -9 . This sufficient-conditlon test demonstrates that

multiplets have been identified in the power spectra.

This work was supported in part by the Air Force Office of Scientific Research

and the Astronomy Division of the National Science Foundation.

Referenoes

Bos, R. J. 1982, Ph.D. thesis, University of Arizona.

309

Claverie, A., Isaak, R. F., McLeod, C. P., and Van der Raay, H. B. 1981, Nature, 293, 443.

Duvall, Jr., T. L. and Harvey, J.W. 1983, Nature, 302, 24. • 1984, Nature, 310, 19.

Grec, G., Fossa-t; E~., and POmerantz, M. A. 1983, Solar Phys., 82, 55. Hill, H. A. 1984, SCLERA Monog. Set. Astrophys., no. I.

1985a, Ap. J . , 290, 7--'6"5. -- 1985b, SCLERA Monog. Set. Astrophys., no. 3. 1986a, Pub. A.S.P., submitted. 1986b, TO be published in Neutrino '86: The 12th International Conference on

Neutrino Physics and Astrophysics (Sendla, Japan, June 3-8 1986) Hill, H. A., Rabaey, G. F., Yakowitz, D. S., and Rosenwald, R. D. 1986, Ap. J.,

310, In press. Hill, H. A., Rosenwald, R. D., and Rabaey, G. F. 1985, in Proc. 4th Marcel

Grossmann Meeting on General Relativity, ed. R. Ruffinl (Amsterdam: North- }~0iland), in press.

Kotov, V. A., Severny, A. B., Tsap, T. T., Moseev, I. G., Efanov, E. A., and Nesterov, N. S. 1983, Sol. Phys. 82, 9.

Saio, H. 1982, private communication. Scherrerp P. H., Wilcox, J.M., Chrlstensen-Dalsgaard, J., and Gough, D. 0., 1983,

Solar Phys., 82, 75.

N -1-

- 4 3 0

- 4 4 0

- 4 5 0

- 4 6 0

" 4 7 0

+80

A + 4 0

3 : C

~ o

, . , . ~

"~' -40

- 8 0 16

'1 . . . . . . . . I I

i 2O

" l i

L

2O

i I

1 I I I 2 4 2 8 5 2 3 6

I I I I

I I I 2 8 5 2 5 6

I 24

40

40

Fig. I. The observed

rotational splltting

linear in m shown as a

function of £ for

intermediate-degree f-

modes. The point plotted

with the square box is

the value of v~£_ found

for ~ - 33 by Hill

( 1986a ).

Fig. 2. The observed

rotational splitting

cubic in m shown as a

function of ~ for

intermediate-degree f-

modes. Since the

theoretical rotational

splitting cubic in m

varies like

I/[£(g + I) - 3/4], we

present the observed

splitting plotted as

v~3/31 to project out

any residual £-

de pen den ce.

310

Comparison of 1983 and 1979 SCLERA Observations

Leon Yi and W. M. Czarnowski Department of Physics and Arizona Research Laboratories

Unlverslty of Arizona, Tucson, AZ 85721

Abstract

Solar equatorial differential radius observations obtained between 30 May and

20 July 1983 are analyzed for evidence of solar oscillations. Previous work has

Produced acoustic mode classifications based on 1979 differential radius

Observations (Hill 1984). By applying small frequency shifts to the n~1, ~=4,...,22

multlplets from Hill (1984), we find a statistically significant correspondence

between 182 frequencies of the classified multiplet spectrum and the 1983 power

Spectrum. The measures Dij(m) (defined by Hill 1984) of the azimuthal dependence of

the eigenfunctions are also used to compare the 1983 and 1979 observations.

Weighted average Dlj'S calculated for the peaks in the 1983 power spectra that

matched previously classified modes were found to be consistent, at a statistically

Significant level, with those obtained for the 1979 observations. Modes with

~8,...,12 were used to calculate a weighted average value for the angle between an

internal axis of symmetry and the pole of the Sun of 072 ± 0~8.

I. Introduction

Helloselsmology is an important tool for the investigation of the solar

interior. In recent years, researchers at SCLERA I have endeavored to develop new

techniques of data analysis which will lead to the identification and classification

of oscillatory modes represented in power spectra of solar data obtained at SCLERA.

Work at SCLERA has already produced a set of solar acoustic mode eigenfrequencles

based on an analysis of the 1979 SCLERA data (Hill 1984). New data was acquired at

SCLERA in 1983, and this data has been examined for evidence of the previously

Classified modes. The objectives of this examination were to test the mode

Classification program and to investigate the stability of the elgenfrequencies of

the low-order, low-degree acoustic modes over the four-year period between 1979 and

1983. The analysis of the new data is presented here.

If. Observational Techniques and the 1983 Data

I. SCLERA is an acronymn for Santa Catalina Laboratory for Experimental Relativity by Astrometry, a facility jointly owned and operated by the University of Arizona and Wesleyan University.

311

The 1983 data was obtained with the same detector used in 1979 at SCLERA (8os

and Hill 1983). However, the detector was rotated periodically so that the edge of

the solar disk could be sampled at eight different positions. These eight positions

were chosen to lle on four solar diameters oriented at -90 °, -45 ° , 0 a, and 45 ° from

the projected solar axis (Czarnowskl et al. 1984). The 1983 data spans 52 days,

from May 30 to July 20. The actual analysis reported here used a total of 319.5

hours of data drawn from 44 days.

The detector consists of six slits arranged in a pattern symmetric with respect

to the detector arm (Bos and Hill 1983); it was rotated to a new diameter every 64

seconds. At each diameter, llmb profiles were recorded for each of the six slits.

An optical bandpass filter restricted the wavelengths observed to a range of 8 nm

centered at 550 nm.

Before 1983, no attempt had been made at SCLERA to obtain data on diameters

other than the equatorial set (-90°). Since we were interested in comparing the

1983 data with the 1979 data, we analyze here the 1983 equatorial data.

The analysis of the 1983 data uses the Finite Fourier Transform Definition

(FFTD) of an edge on the solar disk (Hill, Stebbins, and Oleson 1975). The

techniques used at SCLERA to obtain the FFTD lock-on points for each llmb have been

extensively discussed elsewhere (Bos 1982) and will not be discussed here. The

spectra which were analyzed were obtained with a standard Fast Fourier Transform

(FFT) program applied to a linked set of FFTD lock-on points.

III. Shifts of the Low-order, Low-degree Acoustic Mode Elgenfrequencies

In the examination of the 1983 data, the previously classified multlplets

(i.e., those based on the 1979 observations) which had values of £-4,...,22 were

divided into three groups: I) ~-4,...,9; 2) £-IO,...,16; 3) £=17,...,22. It was

found that frequencies obtained by applying small shifts to these elgenfrequencles

match, at a statistically significant level, a set of peaks in the 1983 spectra.

The shifts in frequencies whleh were found effective are -0.02, 0.03, and 0.05 ~Hz

for groups I, 2, and 3, respectively. The accuracy to which these shifts have been

determined is yet to be estimated, but they are in general agreement with results

from analysis of 1985 SCLERA data (Oglesby 1986). A tolerance of ±0.06 ~Hz was used

to determine whether a peak in a spectrum matched a given elgenfrequency.

IV. Test of the Assigned Multlplet m Values

A further comparison of the 1983 and 1979 observations is based on the measures

Dij(m) of the azimuthal dependence of the elgenfunctlons (Hill 1984). The Dijwerev

calculated for the peaks in the 1983 power spectra that matched the frequencies of

the olasslfled modes based on the 1979 observations. These were then divided into

312

t h ree groups: a) Iml ~ 0 . . . . . 7 ; b) lml = 8 . . . . . 14; c) Iml i 15 . . . . . 22 . The

Weighted average <Dij> for each of these groups was then calculated. The values of

these <DI~> are 0.125 ± .043, -0.0079 ± .074, and 0.476 ± .034 for groups a, b, and

c, respectively.

The values of these <Dij> are in good agreement with those obtained for the

1979 data. The magnitude of the Dij for group b above is smaller than the

Corresponding 1979 quantity but still has the appropriate sign. Therefore, since

<Dij>'s obtained for groups a and c deviate from zero by approximately 3a and 14o

With the correct signs, the hypothesis that they represent no correlation is highly

unlikely.

V. Orientation of the Internal Axis of Symmetry

In previous work at SCLERA (Hill 1984) it has been shown that another quantity,

Rij , can be used to determine the orientation of an axis of symmetry in the Sun.

Those acoustic modes identified in the 1983 data with values of ~-8,...,12 were used

to calculate the orientation of the internal axis of symmetry. A total of 29 modes

Wlth these ~ values were found. The weighted average angle of the axis of symmetry

from the pole, calculated from these modes, is 0~2 ± 0~8.

VI. Acknowledgements

This work was supported in part by the Astronomy Division of the National

Science Foundation and the Air Force Office of Scientific Research.

We thank T. P. Caudell, R. J. Bos and B. J. Beardsley for their help in

gathering the data, and H. A. Hill for his contributions to the interpretation and

analysis of the data.

Vll. References

Bos, R. J. 1982, Ph.D. Dissertation, Univ. of Arizona.

Bos, R. J., and Hill, H. A. 1983, Solar Physics, 82, 89.

Czarnowskl, W. M., Yi, L. Beardsley, B. J., Hill, H. A., and Caudell, T. P. 1984, Bull. A.A.S., 16, 1001.

Hill, H. A., 1984, The Observed Low-order, Low-degree Solar Acoustic Mode Elgenfrequenc Z Spectrum and Its Properties (SCLERA Monograph Series in Astrophysics, no. T~

Hill, H. A., Stebbins, R. T., and Oleson, J. R. 1975, Ap_~ J., 200, 484.

Oglesby, P. H. 1986, "Confirmation of Detection and Classification of Low-Order, Low-Degree Acoustic Modes with 1985 Observations," in these Proceedings.

313

Confirmation of Detection and Classl~Ication of Low-Order. Low-DeEree,

Acoustic Modes with 1985 Observations

Paul H. OElesby, Dept. of Physics and Arizona Research

Laboratories, University of Arizona, Tucson, Az. 85721

A b s t r a c t

Recently SCLERA has been involved in expanding its observing program with the

introduction of a new technique to include the search for global solar oscillations

in the visible to near infrared continuum. In 1985, data was collected by Oglesby

(1986) using this new technique. To a high de~ee of statistical siEnlflcance, the

classification of the low-order, low-de~ee solar acoustic modes by Hill (1984) has

been confirmed with the 1985 data. The n - I eigenfrequency spectrum has been

divided into three bins according to 7 ~ £ ~ 12, 13 ~ £ & ]7, and 18 ~ £ ~ 22 and

the confirmation for each bin is at the 4.70, 3.60, and 3.80 levels, respectively.

I . Introduction

The solar low-order, low-degree acoustic modes of oscillation play a vital role

in the understanding of the sun's interior. By inverting the observed rotational

splitting results for these modes, the approximate outer half of the sun's internal

rotation may be ascertained (Hill, Bos, and Goode 1982, Campbell et al. 1983, and

Duvall et al. 1984). However, there is little agreement about the rotational

effects. Conflicting results are being reported by groups using different

observational techniques. Therefore, the significance of any agreement found

between observations obtained with two different observational techniques should not

be understated.

Using data obtained in 1979 by Bos (1982), Hill (1984) has classified low-

order, low-deE~ee acoustic modes for n-l, £-4-22; n-2, £-3-40; and n-3, £-2-4.

Another observational technique to detect solar oscillations in the continuum has

been developed at SCLERA (Santa Catalina Laboratory for Experimental Relativity by

Astrometry) and observations were obtained in ]985 by Oglesby (1986). These

observations have been analyzed and the results are compared in the following

sections with the 1979 differential radius observations for the n-l, £~7-22 acoustic

modes.

314

If. Observations

The observations consist of 35 days spanning a 70-day period from March 23 to

Hay 31, 1985. A 200 X 200 arcsec aperture was centered on the solar disk and the

intensity at 48 different wavelengths was measured within the spectral range of 0.5

to 1.? microns. Changes in the radiation Intensity at different wavelengths were

Fourier analyzed and the resulting frequency spectrum was examined for evidence of

the multiplet structure observed by Hill (1984) in the low-order, low-degree

acoustic modes.

I I I . R e s u l t s

The eigenfrequency spectrum for n=1, ~-7-22 was divided into three bins

containing five or six multiplets each as indicated in Fig. 1. The multiplets

COntained in the three bins were shifted in frequency as a group in the region of

interest to search for a maximum coincidence rate between peaks in the 1985 power

Spectrum and the frequencies given by Hill (1984).

The number of expected coincidences within a window of may of u is estimated

by empirically comparing 105 randomly generated frequencies to the actual data.

Next, the ratio of the number of peaks that coincided (within ,07 BHz) with the

Predicted multlplet member to the number of maximum available states was determined

(Nn~/Nmax). This ratio is plotted as a function of ~ in Fig. 1.

For the three groups of multiplets the mean difference between the observed

COincidence rate and the expected rate due to s relative random distribution yields

dl = 0.214 ± 0.046; 7 $ £ $ 12,

d2 " 0.145 ± 0.040; 13 $ £ 5 17,

d3 " 0.132 ± 0.035; 18 $ £ 5 22,

( I )

Where the mean difference is weighted by the standard deviations of the Nn~/Nma x.

These values are 4.7, 3.6, and 3.8 standard deviations away from zero, respectively.

We thus find that the probability of obtaining the above quoted d/o's for each of

the group of multiplets is estimated to be

~5 I I x 10 ; ? :; ~. :; 12

p ~ 2 x ]0 -3 ; 13 < ~. < I? (2)

x 10 -4 ; 18 ~ ~ ~ 22

Therefore, these results indicate that the hypothesis that peaks in the 1985 power

~Pectrum are randomly distributed in frequency with respect to the elgenfrequencies

of the multiplets identified by Hill (1984) is very unlikely.

315

IV. Summary

One of the more important consequences of the confirmation of the detection and

proper classification of the low-order, low-degree acoustic modes concerns the role

that these modes play in inferring the internal rotation of the sun. By inverting

the observed flne structure linear in m for 137 acoustic and gravity mode

multiplets, a differential rotation curve has been obtained (Hill, Habaey, and

Rosenwald, 1985). The modes under study in this work reside primarily in the

convection zone and, therefore, confirm the approximate I/r 2 behavior of the

rotation curve found in that region by Hill, Rabaey, and Rosenwald (1985).

Furthermore, the results for n-1 have been confirmed with 1978 observations (Hill

and Caudell, 1985) and 1983 observations ~Yi and Czarnowski, 1986). Thus these type

of results have been reproducible from three independent data sets spanning a seven

year period.

Based on a comparison of 1979 and 1985 observational results, slight frequency

shifts of the multiplets were observed. These shifts are given in Fig. 2. The

study of the temporal behavior of the eigenfrequencies of these modes may provide

useful information about the convection zone.

I would llke to acknowledge the valuable scientific dlcussions with Henry Hill,

and his critical reading of this manuscript. I would also like to thank Kim Vlvier

for editing and John Sugamelli for preparing the figures. This work was supported

in part by the Astronomy Division of the National Science Foundation and the Air

Force Office of Scientific Research.

V, Refer@noes

Bos, R.J. 1982, Ph.D. t h e s i s , U n i v e r s i t y of Ar izona.

C a m p b e l l , L . , McDow, J . C . , M o f f a t , J .W. , and V i n c e n t , D. 1983, N a t u r e , 305,

508.

Duvall, T.L., Dzlembowski, W.A., Goode, P.R., GouEh, D.O., Harvey, Lelbacher, J.W.

1984, Nature, 310, 22.

Hill, H.A. SCLERA Monog. Ser. Astrqph~slcs., no. 1.

Hill, H.A., BOS, R.A., and Ooode, P.R., 1982, Phys. Bey. Let., 49, ]794.

Hill, H.A., Caudell, T.P., Ap. J., 299, 1985.

Hill, H.A., Rabaey O.F., and Rosenwald R.D., IAU Symposium, 114, 1985.

Oglesby, P.H. 1986, Ph.D. thesis, in progress, University of Arizona.

Yi, L. and Czarnowski W.M., in these proceedings.

316

o Observed coincidence rate × Expected coincidence rote

for a random d=stribulion

.... ,,,,! ,!,,,i '! '!- I. 0 0 Group I Group 2 .I Group 3

r;

~ °-~ I ' 2 o,~o[,,,,, ,

0"250 4 8 12 16 20 l

The ratio of the number of peaks in a given multlplet that coincide with the predicted multlplet frequencies to the maximum number of available states d i s p l a y e d a s a f u n c t i o n o f Z .

+0 .10 ! , ,

- - - +0.05 N

= . 0 . 0 0

{ ~o-0.05

t I

Bin l

I

t I 1 -0 .10 Bin 2 Bin 3

The f r e q u e n c y s h i f t o b t a i n e d when c o m p a r i n g be tween 1979 and 1985, displayed as a function of £. Up is the predlcted multlplet frequency an~ Yah

is the observed value. Bin I: 7 ~ £ ~ 12; Bin 2:13 ~ £ ~ 17; Bin 3: 18 ~ t ~ 22.

317

R-MODE OSCILLATIONS IN THE SUN

Jane B. Blizard University of Colorado

Boulder, CO 80309, U.S.A.

Charles L. Wolff NASA Goddard Space Flight Center

Greenbelt, MD 20771, U.S.A.

Solar p-modes and g-modes have been actively studied for over a

decade. But there are other global modes, the r-modes, rarely

mentioned in a solar context. These modes have long periods,

comparable to the rotation period, and their motion is dominated by

the Coriolis force. The p- and g-modes have periods of minutes or

hours and their restoring forces are pressure and gravity. The

oscillatory velocity field of low harmonic solar r-modes is almost

purely toriodal (Saio, 1982) while comparable p- and g-modes are

mainly spheroidal. In a spherical star, the toroidal and spheroidal

fields make up the complete mathematical set necessary to express an

arbitrary initial displacement (Aizenman & Smeyers, 1977), and are

mutually orthogonal. The inverse is very important: any complex

disturbance in the Sun would be expected to excite both toroidal and

spheroidal components.

Convection is likely to be modulated by a flux of r-modes. The

overturning times of large cells in the lower half of the convection

zone are comparable to the oscillation periods of all r-modes with low

I values. Also, the length scales of these deep seated cells are

similar to those of r-modes with moderate angular harmonic 1 and low

radial number n. Finally, convection has vorticity in the proper plane

to couple with the toroida] velocity field. Thus, the physical

conditions for coupling are close to ideal. If convection is modulated

on large horizontal scales and at times about one month, one should

expect similar modulation in solar activity such as sunspots. Wolff

(1974) suggested such a connection between large scale convection and

solar activity, and provided observational support (Wolff, 1983) based

on g-modes in the deep interior. Surface variables would be modulated

efficiently by envelope r-modes, if they are indeed excited. This is a

reason to search the solar observational record for evidence of

modulations at the repetition rates of envelope r-modes.

If solar r-mode oscillations are also represented in solar

activity indices such as sunspot number and area, then rotation of the

mode becomes a key observational variable. Table I shows the range of

318

Table I. Sidereal rotation of r-modes in the convection zone and the repetition of identical flow fields as viewed from Earth (Wolff & Blizard, 1986).

1 m ROTATION Rate Period (nHz) (days)

REPETITION, Vim Rate Period (nHz) (days)

1 1 0. OO -31.7 -365.3

2 1 300.0 38.6 268.3 43.1 2 2 300.0 38.6 536,6 21.6

3 1 375.0 30.9 343.3 33.7 3 2 375.0 30.9 686.6 16.9 3 3 375.0 30.9 1030. 11.2

OO 1 450.0 25.7 418.3 27.7 OO 2 450.0 25.7 836.6 13.8

these rates for the lowest and highest angular harmonics, 1 and m.

Negligible fine splitting (Wolff and Blizard, 1986) is assumed.

The most evident periodicity of the Sun's activity is its

rotation period. Here the question discussed is: which periods between

16 and 150 days occur in sunspot number and/or area, that are similar

to periods of the r-modes. Consistent time series of solar parameters

Were taken from the time interval 1904-1984. A particular problem for

a time series analysis of solar parameters is that the periods

detected are often not constant, but are quasi-periods varying in

amplitude (and frequency), sometimes disappearing altogether. This

Would be expected if their cause were r:modes suffering decay and phase

Change upon restimulation during the interval studied.

Daily Zurich sunspot numbers (R) were obtained for the high

activity years of solar cycles 14 through 21. Fast Fourier transforms

(FPTs) were taken of the 2048 day periods starting Jan 1 for the

Years: 1904-1909, 1915-1920, 1925-1930, 1935-1940, 1945-1950, 1956-

1961, 1966-1971, and finally, 16 Nov 1978-30 Sep 1984 (when comparable

Solar irradiance values from Nimbus 7 were available for comparison).

Daily projected whole sunspot areas (A) were obtained from the

Greenwich Photoheliographic Results (supplemented by data from D. V.

Hoyt for 1982-1984), and FFT power spectra taken of 4 day averages.

FFT power spectra were obtained of R and of four-day averages of

R. A large number of periods in the range 16-150 days were displayed.

The synodic rotation period of persistent sunspots should vary between

26.7 days (equator) and 29.1 days (30 ° latitude). The FFTs showed high

aCtivity in a narrower range, due to the selection of high activity

Years, excluding the beginning and end of solar cycles.

319

Table If. R-mode synodic repetition periods apparent in the power spectra of daily sunspot area A (four-day ave), solar cycles 14-21.

Solar Cycle 14 15 16 17 18 19 20 21

Period, I=2 43,22 43 22 43,22 43,22 43,22 43,22 43,22 days 1=3 34 34 34 34 34 34

A period of 34!1.5 days is evident in the daily and 4-day

averages of R. This could correspond to the r-mode of angular

harmonic 1=3, m=l, with period 33.7 days (Fable I). Also apparent is a

43~2 day period (2,1), and a 22!i day period (2,2). Some peaks are not

very significant statistically.

Four-day averages of the projected whole sunspot area were also

compared by FFT to r-mode synodic repetition periods over the same

eight solar cycles. Noteworthy are the same three periods appearing in

most of the solar cycles (Fig.l) (Table II). Of the two data types

considered, the sunspot projected area (A) displays the possible r-

mode periods more strongly. This indicates that r-modes are

concentrating solar activity in longitude as well as modulating its

total magnitude, since A senses both effects while R is affected more

by the latter.

A wide range of simple periods and beat periods should be caused

by r-modes and g-modes. A search for the rest of these is underway.

Evidence for some of the lowest harmonic r-modes was shown in this

first report. A prominent peak in the FFTs (34~1.5 days) corresponds

to the r-mode of angular harmonic (3,1) of period 33.7 days; it is

outside the range expected from differential rotation of sunspots

between 0 ° and 30 ° . Other periods in sunspot number and area

corresponding to r-modes are 22~i days (2,2), and 43!2 days (2,1).

Acknowledgements. One of us (JBB) acknowledges support from NASA contract NAS5-28196. It is a pleasure to thank D. V. Hoyt for projected whole spot areas from 1982-1984 to supplement the Greenwich Photoheliographic Results.

References Aizenman, M.L. & Smeyers, P., Astrophys. Spa. Sci 48 123 (1977) Saio, H., Astrophys. J. 256 717 (1982) Wolff, C. L., Astrophys. J. 194 489 (1974) Wolff, C. L., Astrophys. J. 264 667 (1983) Wolff, C. L. & Blizard, J. B., Solar Phys. 105 1 (1986)

320

a

Power Spectrum, Daily Whole Spot Area: 4-day averages

Cycle 14: Jan 1904-Aug 1909 |

r-mode |

| I r - m o d e # r - m o d e I[

16 18.3 21.3 25.6 32 42.7 6& 128

Power Spectrum: Daily Whole SIx)t Area: 4-day averages

Cycle 18: Jan 1945-Aug 1950

r-m0de r-mode

22 &3 I l

, 1

16 18.3 21.3 25.6 32 42.7 64 128 P e r t c x i D a y s

Fig. 1. Power spectra, daily projected whole spot area (4-day averages) for solar cycles 14 (a) and 18 (b).

321

Inverse Problem of Solar Oscillations

Takashi Sekii and Hiromoto Shibahashi Department of Astronomy, University of Tokyo,

Bunkyo-ku, Tokyo 113, Japan

Abstract

We present some preliminary results of numeric~l simulation to infer the sound velocity distribution in the solar interior from the oscillation data of the Sun as the inverse problem. We analyze the acoustic potential itself by taking account of some factors other than the sound velocity, and we can infer fairly well the sound velocity distribution in the deep interior of the Sun.

I. Introduction

The most important and unique aspect of solar oscillations is the possibility of a seismological approach. The so-called inverse methods yield functional forms of certaiP physical quantities such as the sound velocity distribution c = c(r) as solutions of in" tegral equations for eigenfrequencies. In this respect, Gough (1984) presented a useful method based on an asymptotic expression of eigenfrequencies of p-modes, and Christensen-Dalsgaard et al.'s (1985) numerical simulation using solar models showed that the sound velocity in the outer parts ~f the Sun (r/R ~ 0.4) are well reproduced by Gough's (1984) method. However, for the deep inside of r/R ~ 0.4, their solution of the integral equation significantly deviates from the original values. Thus, more effective and mathematically accurate inverse methods are desirable. One of us (Shibahashi 1986) recently presented an improved method, in which some factors other than the sound velocity are taken into account in the WKB expression of eigenfrequen- cies. These terms significantly contribute to eigenfrequencies of low degree p-modes. Since it is such low degree modes that extract information from the deep interior of the Sun, we may infer more accurately the solar deep interior by using Shibahashi's (1986) method. In this paper, we show some preliminary results of numerical simulation using a solar model to examine validity of his method.

2. Equilibrium Model and Eigenmodes

The solar model used in this paper is the model 1 of Shibahashi et al. (1983). We have calculated the eigenfrequencies of p-modes in the range of 0~£~I,000 and 1 ~n ncrit , where ncrit means the highest overtone whose frequency is limited by the cut-off frequency of the model. The computation method of eigenfrequencies is the same as that used by Shibahashi and Osaki (1981). Although the modes of the observed five-minute oS" cillations are restricted to only some range, we use all the calculated 2,754 modes but for radial modes of £ = 0 in the following analysis to know the validity of the inver sion method under idealized conditions.

3. Acoustic Potential and Acoustic Length

The wave equations for nonradial p-mode oscillations are reduced to a form similar to the Schr~dinger equation in some limiting cases, which is written as

d2v/dr2 + c-2(r)[~2 - @4 (r)] v = 0 , (i)

where v denotes an eigenfunction, m is the eigenfrequency. Here, ~£(r) is the 'acoust ic potential', which consists of the Lamb frequency, £(£+i)c2/r 2, and the k-independent

322

term ~(r) :

¢£(r) = £(£ + I) e2/r 2 + V(r) . (2)

Figure i shows the acoustic potential, the Lamb frequency and ~ (r) as functions of the

acoustic radius T, which is defined as

s; • r - c - 1 d r , ( 3 )

for our solar model. The i-independent term ~(r) is related to the inverse of the den- sity scale height and then it is large near the surface. The Lamb frequency term is dominant in the acoustic potential in the deep interior, while the i-independent term ~(r) dominates in the outer part. As the degree £decreases, ~(r) becomes to contribute significantly to the acoustic potential. It should be noted here that the detailed ex- pressions of v and ~(r) are not needed in the following analysis.

Based on the WKB asymptotic method, the quantization rule leads to

I/ n~ = [m2 _ @£(r)]I/2 c-1 dr , (4)

1 where n is the radial quantum number corresponding to the number of nodes in the radial direction and r I and r 2 are the turning points at which @£(r) = 92 . Strictly speaking, equation (4) gives only a relation between discrete eigenvalues m2 and the corresponding integers n and £. But hereafter we extend this relation to non-integers n and £ by in- terpolation and treat equation (4) as if it were a continuous function of w 2 and £ giving continuous variables n. Then, for a fixed £, equation (4) is regarded as an in- tegral equation to give c -I dr/d@£ and its solution gives the acoustic length, s(m 2, £), as a function of 2, which is the distance between two turning points measured with the

SOund velocity (Shibahashi 1986):

[ I @~ ~n/~ 2 m2)-I/2 r2c-I = d~ 2 (5) s(~ 2, £) ~ dr 2 (@£- •

-r I ~2mi n

2 Here ~min corresponds the minimum of the acoustic potential, and it is obtained by the

extrapolation

~2mi n = lim ~2 (n, £) . (6) n~0

Figure 2 shows the acoustic lengths thus obtained for various degrees £ as functions of

" ! .o 2,o 3*0

ACOUSTIC RADIU5 / 5£C

Fig. i. The acoustic potential (thick curve), the square of the Lamb frequency £(£+i)c2/r 2, and ~ (thin

curves) as function of the acoustic radius in the case of £ = !0.

3 2 3

XtO l 4 .0 r

Fig. 2. The acoustic lengths for various values of £:£ = I, 5 to i00 with a step of 5, I00 to 200 with a step of 25, and 200 to 650 with a step of 50.

LOGI0(BHG~/5£C -z )

4. Sound Velocity Distribution

Once we get the acoustic lengths for various values of £, since the inner turning point r I of high order modes is approximately given by

C2 (rl) / rl 2 = ~2 / £ (£ + i) , (7)

by differentiating them by £, we obtain (Shibahashi 1986)

d[c(rl)/r I ] / dlnr I ~ (2£ + i) / [2£(£ + l)].(~s/~£) -I . (8)

The right-hand side of equation (8) is evaluated at a given ~2, and then by using eq ua" tion (7), we should regard equation (8) as an equation to give d[c(rl)/rl]/dlnr I as a function of c(rl)/r I. By using a wide range of £, we eventually obtain d[e(r)/r]/dlnr as a continuous function of c(r)/r as shown in figure 3. It eventually gives the sound velocity distribution in the solar interior. The solid curve in figure 4 shows the square of the sound velocity thus solved by using the acoustic lengths for £ = i - 660, while the two thin curves show the true value and the solution by means of Gough's (1984) method. As seen in figure 4, the solution based on the present method reprod~ ce~ well the sound velocity of the model. In order to obtain more accurately c 2 in the deeper interior (r/R® ~ 0.15), we will have to use radial modes of £ = 0 and such an at" tempt is now in progress.

5. Concluding Remarks

We have extended the relation between discrete eigenvalues and the corresponding integers n to a continuous function of n = n(~ 2) for a given £ by the interpolation. 50 the case of a low degree £, we can easily do such an interPolation. However, in the case of a high degree, since the number of overtones is limited to only a few, such an interpolation is practically inaccurate. Therefore, equation (5) is inappropriate t o De applied in order to infer accurately the sound velocity distribution in the very outer part of the Sun by using high degree modes. Instead, we had better use the following equation

Ir "2 (~nl~£)~ = - (2£ + 1)12 [~2 _ ~£(r)]-I/2 clr 2 dr , (9)

1 which is obtained by differentiating equation (4~ with respect to £. For a fixed £, tD~ left-hand side of quation (9) is a function of ~ , and then equation (9) is now rega rde

u

as an integral equation to give c/r 2 dr/d@£. Its solution is given by

3 2 4

WIO |

\ ~0.s

-l.s~

Fig. 3.

LSG,0(A/SEC} .I

dlnr/da(t) as a function of lOgl0a(r), where a(r) ~c(r)/r. The thick and thin curves show the solution of the inverse problem based on the present method and the equilibrium structure, respectively.

°I

<

0.1

FRACTISNAL RADIU5

Fig. 4. The square of the sound velo- city c2(r). The thick curve and the zigzag thin curve show the solutions based on the present method and Gough's method, res- pectively. The monotonic thin curve show the equilibrium model.

~(~2 £) £ rrpi_ el r2 dr = - 2/(2£ + i) r|@£ $n/B£ (@£ - ~2)-I/2 dw2 • (i0)

~2ml n

By differentiating O by £, we obtain

d[rl/c(rl) ] / dlnr I - - (2Z + I) / [2£(£ + l)].(Bo/B£) -I , (Ii)

which eventually gives the sound velocity distribution in the solar interior. Equations (5) and (i0) [or the corresponding equations (8) and (ii)] are complementary each other, and we should use both of them in order to solve efficiently the inverse problem of the Solar oscillations.

Anyway, the introduction of a more accurate dispersion relation than that used by Christensen-Dalsgaard et al. (1985) has improved fairly well the solution of the inverse problem, and, the sound velocity in the deep interior of the Sun can now be inferred well. In this respect, it should be noted that Brodsky and Vorontsov (1986) recently also presented a nice technique to solve the integral equation given by Gough (1984) with higher accuracy. Their dispersion relation and ours are different but the final solutions are quite similar. Further comparisons are to be done.

References

Brodsky, M. A. and Vorontsov, S. V. 1986, in Advances in Helio- and Asteroseismology, IAU Symp. No.123, ed. J. Christensen-Dalsgaard (Reidel, Dordrecht), in press.

Christensen-Dalsgaard, J., Duvall, Jr., T. L., Gough, D. O., Harvey, J. W., and Rhodes, Jr., E. J. 1985, Nature, 315, 378.

Gough, D. O. 1984, Phil. Trans. Soc. London, 313A, 27. Shibahashi, H. 1986, in Advances in Helio- Asteroseismology, IAU Symp. No.123, ed. J.

Christensen-Dalsgaard (Reidel, Dordrecht), in press. Shibahashi, H. and Osaki, Y. 1981, Publ. Astron. Soc. Japan, 33, 713. Shibahashi, H., Noels, A., and Gabriel, M. 1983, Astron. Astrophys., 123, 283.

325

NONADIABATIC, NONRADIAL SOLAR OSCILLATIONS

R. B. KIDMAN and A. N. COX

Los Alamos National Laboratory

Los Alamos, New Mexico 87545

PURPOSE

Solar gravity mode (g-mode) oscillations are not easy to detect and identify because

their surface amplitudes are very small. However since their largest amplitudes occur in

the deep interior of the sun, their correct interpretation could be invaluable in unraveling

the interior structure of the sun. Toward this end, we present some of our nonadiabatic,

nonradial solar g-mode calculations, using the Lagrangian-based eigensolution program

of Pesnell.

Asymptotic theory predicts that the gravity-mode period spacing (Po) should

approach a constant value as the order n increases, independent of the 1-value. But how

does Po vary with n before it reaches this constant value? We explore this question and

examine our theoretical eigenperiods to see if they indeed give an asymptotic Po

independent of 1.

We also provide some g-mode growth and decay rate predictions that explain why

independent observations give the same 160.01 minute pulsation mode exactly in phase

over many years.

MODEL

A detailed self-consistent un-mixed current sun model was evolved by Becker

using the Iben stellar evolution program. We have reconstructed this model at the solar

age in our pulsation program, and to obtain a complete model we have had to increase the

hydrogen content by 0.0019 and increase the luminosity by 0.2 percent over the Becket

value. The reason for these changes is that we introduce into the model construction

program a composition versus mass table, and the interpolated hydrogen mass fraction

for each mass shell is not exactly that obtained by Becker in his evolution calculation.

Table 1 presents some salient parameters of our final 1700 zone solar model.

One of our chief concerns in constructing a model was zoning. Figure 1 shows

326

how the t=5, n=18 g-mode period changed as we varied the number of zones (and core

radius). Since the period has leveled off we feel our 1700 zones, which corresponds to

our smaUest core radius, removes zoning effects from our g-mode results. Also, our

1700 zones gives a finer central zone structure than apparently anyone else has used.

RESULTS

Figure 2 displays our Po results as a function of order n and degree 1. Our

Lagrangian code results appear to be heading toward a constant Po of about 38 minutes.

Unfortunately Po fluctuates approximately +-16% about the mean, due to our slightly

non-smooth composition structure. Although the fluctuations tend to mask the point, it

appears the Lagrangian approach yields an asymptotic Po independent of 1, as expected.

Table 2 shows a small sample of the g-mode growth rates we obtain. The growth rate

units are better understood with an example: The reciprocal of our growth rate is the

number of cycles it takes to change the mode energy by a factor of e. Thus for order t 7 it

takes (15845.38)/(2.36e-8) = 6.7el 1 seconds = 21000 years for its energy to decrease by

a factor of e. The gl, 1=2 mode (at 56 minutes) is driven in this calculation by the kappa

and gamma effects in the subphotosphere layers and by periodic convection luminosity

blocking at the bottom of the convection zone. Deep radiative damping is dominant for

modes higher than g2 at this 1 value.

COMMENTS

Past measurements of Po have been suggested to be in the range 36-41 minutes.

Figure 2 shows how Po varies with order n as it approaches its constant asymptotic value.

It is obvious that one must be at radial order 40 or above before one is relatively close to

the constant asymptotic value of Po" If one determines a Po from orders around 10 he

may get about 32, or from around n=20 he may get about 35, or around n=30 he may get

36! This could lead to some confusion. An observer may determine Po from orders less

than 10 (the only data he has) and fail to correct it to an asymptotic value.

The Table 2 decay rates (up to a million years) suggest that independent observers

Over several years can detect the same pulsation mode at exactly its predicted phase.

327

141.6

Parameter

Luminosity (1033erg/sec)

Mass (10 33gin)

Radius (1010cm)

Surface temperature (103 K)

Central temperature (107 K)

Central density (gm/cm 3)

Surface X, Y, Z

Central X, Y, Z

Depth of convection zone

Temperature at bottom of

convection zone (10 6 K)

Opacities and EOS

Value

3.6474

1.9910

6.9001

5.7264

1.4596

154.18

.750 .230 .020

.421 .559 .020

.24R~ (.014M (~

1.7860

Iben Fit

G-MODE ~ N=I8 f

141.4

141.2

I 141

140.8

TABLE 1

Solar Model

140.6 t l I 0 1 2 3 4 5

Core R a d i u s (lO B cm)

Figure 1: The period for the 1=5, n=18 g-mode decreases and levels off as the number of modeling zones increases from 1000 to 1200 to 1400 to 1700 zones,

328

G-mode Period Coefficient (Lagrangian) 45 ] , ~ ' 1 '

, ^l, ^ 40 t £ ' ~ - ~ ;!;i~/'i, , ~,, ,'~

'4 • ' " • i, v , y ! V 35

~" 3O

20 ::::::::::: I / - L = 4

l O I I t I I I

0 10 20 30 40 50

Order n

Figure2: The gravity mode period coefficient (Po) as a function of degree (L) and order (n). Po for any point is computed from SQRT(L(L+I))*(P(L,n+I)-P(L,n)) and is plotted at n+l]2,

Order

TABLE 2

Growth Rates For L=2

Non- Predicted Adiabatic Growth

Period Rate (seconds) re/cycle)

17 15845,38 -2.36E-08 13 12486.69 - 1.14E-08

8 8231.49 -3.33E-09 7 7457.66 -2.43E-09 6 6675.15 - 1.73E-09 5 5915.57 -1.18E-09 4 5181,65 -7,26E-10 3 4495.54 -3.3IE-I0 1 3385.21 4.86E-10

3 2 9

Pulsational Analyses of Post Planetary Nebula Central Stars

and Degenerate Dwarfs +

Sumner Starrfield*

Theoretical Division, Los Alamos National Laboratory

Los Alamos, NM, 87545, and

Joint Institute of Laboratory Astrophysics

University of Colorado, Bould)r, CO, 80309

+Supported in part by National Science Foundation Grants AST83-14788 and

AST85-16173 to Arizona State University, by NASA grant NAG5-481 to Arizona State

University, and by the DOE.

*Permanent address: Department of Physics and Astronomy, Arizona State

University, Tempe, AZ, 85287

ABSTRACT

Recent observational and theoretical studies of the ZZ Cetl variables (DA

degenerate dwarfs), the DBV variables (DB degenerate dwarfs), and the GW Vlr

variables (DO degenerate dwarfs) have shown them to be pulsating in nonradial

g-modes. The pulsation mechanism has been identified for each class of variable

star. For the ZZ Ceti and DBV variables i t is both the kappa and gamma effects in

the partial ionization regions of either hydrogen or helium and also a recently

identified pulsation driving mechanism called "convection blocking." For the GW

Vlr variables, i t is the kappa and gamma effects in the partial ionization region

of carbon and oxygen. The ZZ Ceti variables must have pure hydrogen surface

layers, the DBV stars must have pure helium surface layers, and the GW Vir stars

must have carbon and oxygen rich surface layers with only a small amount of helium

present. The accuracy of the prediction for the GW Vir stars is limited by the

lack of observational determinations of their ]umlnoslty, effective temperature,

and composition.

I . Introduction

In this review I wi l l present and discuss our current knowledge about the

nonradially pulsating degenerate and predegenerate variable stars. The three

classes of these stars are the ZZ Ceti stars (DA degenerate dwarfs wlth pure

332

hydrogen atmospheres), the DBV stars (degenerate dwarfs with pure helium

atmospheres), and the GW Vir variables (formerly known as the PG1159- 035

variables, they are also referred to as the DOV stars). The ZZ Ceti variables

inhabit a narrow instabi l i ty strip with an effective temperature around 11,000K

and a spread of about 1,000K (Greenstein 1982). The temperature boundaries of the

instabi l i ty strip are s t i l l in dispute. Their gravities are ~v10 B cm/s 2 implying a

mass of O.6Me. The DB variables inhabit a hotter instabi l i ty str ip with an

effective temperature of~25,000K and, again, the boundaries of the instabi l i ty

strip are not well known. The GW Vir variables are much hotter with effective

temperatures~1OO,OOOK. Their atmospheric composition has not, as yet, been

determined. They do not show evidence for any hydrogen in their atmospheres but

they are so hot that one cannot set an upper l imit to their hydrogen abundance.

They show lines of helium, carbon, and oxygen and their ultraviolet spectra show

numerous metal lines. One member of this class is the central star of a planetary

nebula (K1-16) and al l of the evidence indicates that the other members of the

class recently were planetary nebula central stars.

There have been a number of reviews of the properties of these variable stars

[Van Horn 1980; Winget and Fontalne 1982; Van Horn 1984; Cox 1986; Winget 1986] so

that in this paper I w i l l mainly discuss the recent results.

2. The ZZ Ceti Variables

The f i r s t of the nonradially pulsating degenerate stars to be discovered was

HL Tau-/6 [Landolt 1968]. I t had a period of~750 sec and an amplitude of~O.3

mag. This period was much longer than the radial pulsation periods predicted for

degenerate dwarfs (a few seconds or shorter) and this result went unexplained for

some years. In 1972, Chanmugam [1972] and Warner and Robinson [1972] suggested

that these stars must be pulsating in nonradla] g modes since these were the only

(either nonradla] or radial) modes that had periods long enough to agree with the

observations. Over the next few years, the Texas group came to the realization

that there was an instabi l i ty strip for DA dwarfs in a color interval around

B-V~O.2. As a result of searches of DA dwarfs in this regime, there are currently

18 known ZZ Ceil stars which makes them one of the most numerous classes of

variable stars in the galaxy [Winger 1986].

Nevertheless, i t was not unti l the late 1970's that any progress was made in

identifying the cause of the pulsations. In an attempt to understand the

structure and excitation of white dwarf envelopes, Starrfield, Cox, and Hodson

~979] used modern opacity tables and the linear, nonadlabatic, radial pulsation

Code of Castor [1971] to investigate DA envelopes for instabi l i t ies. Although they

333

were successful and found that the envelopes were unstable to radial pulsations of

very short periods, they used unrealistic compositions. They later redid these

studies with a pure hydrogen composition and again found that the stars were

unstable to radial pulsations [Starrfleld, Cox, Hodson, and C1ancy 1983; see also

Saio, Winget, and Robinson 1983]. They attributed the excitation mechanism to the

well known kappa and gamma effects in the hydrogen partial ionization zone which

11es close to the surface of these stars.

Meanwhile, Saio and Cox [1980] had developed a new numerlcal technique for

the rapid analysis of the effects of llnear, nonradlal, nonadiabatlc perturbations

on stel lar envelopes. Winget and hls collaborators [Winget, et at. 1981, 1982]

applied this technique [see also Dolez and Vauclatr 1981 and Starrfield, et al.

1982] to a variety of stel lar models. A11 of the above authors chose a mass of

0.6M e and effective temperatures and luminosities in the range of the ZZ Ceti

instabi l i ty strip [See Winget and Fontaine 1982, Van Horn 1984, and Winget 1981,

1986 for discussions of these results]. As summarized in Winget [1986]: they

found that the pulsations of the ZZ Ceti varlables were caused by the partlal

ionization of hydrogen near the stellar surface and attributed the basic physical

mechanism to the kappa and gamma effects operating near the base of the surface

convective zone. They also found that there was an upper l imit to the amount of

mass of the hydrogen surface zone of lO'8Me. I f the surface hydrogen zone was

more massive than this, the models were stable. This, of course, produced a

strong disagreement with evolutionary calculations which predict surface hydrogen

masses of order 10"4Me or larger for al l white dwarfs [ c . f . , Iben and Tutukov

1984]. An attempt to resolve this controversy was made by Michaud and Fontaine

[19841 who proposed that chemica~ diffusion of hydrogen into the deep interior

could burn the hydrogen and reduce i ts abundance significantly below the

evolutionary value. Further work on this subject has been done by Iben and

MacDonald [1985,1986] who disagree with the results of Michaud and Fontaine

[1984].

Finally, I note that the same theoretical analysis also predicted the

existence of pulsators with pure helium atmospheres: the DBV variables. This

prediction was borne out by the discovery of Winger, et aI. [1982] that GD 385 was

a pulsating variable star. Since that time three more such pulsators have been

discovered [Winger 1986].

Nevertheless, the disagreement between pulsation and evolution theory was so

severe that i t seemed important to redo the work of Winget, et al. Such a study

has now been done by Cox, et al. [1986] who find both agreement and disagreement

with Winger, et a1. First, they noted that Wlnget [1981] assumed very ineff icient

convection when he constructed his stellar envelopes. What was done was to choose

the mixing length to be the smaller of the pressure scale height or the distance

to the surface. The net effect is to reduce alpha, the ratio of mixing length to

334

scale height, to very small values near the surface. However, Cox, et al. ,

assumed standard Bohm-Vitense theory [1958] with no modifications and found

shallower surface convective zones. Second, Cox, et a_.__L1, used both the Salo and Cox [1980] code and a new

Lagrangian Code developed by Pesnell [1986]. Although the quanltative agreement

between the two codes is not ter r ib ly good, the qual i tat ive results are in good

agreement. They find a blue edge at about 11,500K for models which assume very

ef f ic ient convection: I/hp~-2 to 3. However, the blue edge does not depend on the

am_ount of h~drp~en envelope mass and, in fact, s te l la r models with Me=IO-4M e are

pulsationally unstable. This completely removes the theoretical discrepancy

between the evolution and pulsation calculations.

Another important result of Cox, et aT. is that one of the causes of the

Ins tab i l i t y iS not the kappa nor the gamma mechanism resulting from the part ial

ionization of hydrogen but is a new physical effect which they call "convection

blocking." In essence, the interaction of pulsation and convection can act to

block the flow of energy in a compression or release i t in an expansion just l ike

the normal kappa and gamma mechanisms. The strongest evidence for the existence

of this new mechanism is that the pulsation driving in the envelopes always

occurred at the bottom of the convective region even when the temperature there

exceeded 105K, which is much too hot for hydrogen pulsation driving. However, near

the blue edge both convection blocking and the kappa and gamma effects in hydrogen

are operating. Unfortunately, this means that a correct theory of the ZZ Ceti

i ns tab i l i t y str ip awaits a time dependant pulsation-convection theory. They also

found that this same mechanism was present in al l of the previous calculations but

was not interpreted correctly. Final ly, I note that Cox, et al. redid their analysis of the radial

i ns tab i l i t y in the ZZ Ceti variables and again found that these stars were

unstable to high order radial modes with periods of less that a second. As for the

nonradial modes, driving was caused both by convection blocking and also by the

kappa and gamma mechanisms. As mentioned previously, Saio, Winget, and Robinson

[1983] also found these stars to be unstable to radial pulsations but, as yet, no

star has been found to be pulsating in radial modes [Robinson 1985]. I t now

appears that the interaction between convection and pulsation may be responsible

for s tabi l iz ing these stars.

3. The Pulsating DBV Stars

Currently there are 4 known pulsators with pure helium atmospheres [Winger

1986]. They are called the DBV stars and thei r discovery is a direct result of

335

the theoretical predictions of Winget [1981]. Both u l t rav io le t and optical

atmospheric analyses have been performed on these stars and the ins tab i l i t y str ip

ranges from an effective temperature of 24,000K to about 28,000K [Liebert, et al.

1986]. However, the boundaries are rather uncertain and probably could vary by as

much as 2,000K [Liebert, et al. 1986]. Koester, et al. [1985] f ind a somewhat

cooler i ns tab i l i t y s t r ip .

Theoretical analyses of these stars have been performed by Wlnget and his

collaborators [see Winget 1986] and by Cox, et al . [ IgB6]. These two groups are in

essential agreement but with some of the same differences in interpretation as

already mentioned for the ZZ Cetl variables. As for the ZZ Ceti variables, they

found that driving occurred at the bottom of the Convective region and attributed

the mechanism to "convection blocking." The cause of i ns tab i l i t y is ult imately

the part ial ionization region of helium, and hydrogen cannot be present in the

driving region to rather stringent l imi ts . This is because hydrogen can easily

"poison" the pulsations in this temperature range and, in addition, i f there were

any hydrogen i t would f loat to the surface on a rather rapid time scale. In fact,

Cox, et al . have proposed that some DAV appearing stars might be found in the DBV

ins tab i l i t y s t r lp . These would be stars where a very thin layer of hydrogen lay

on top of a deeper layer of helium and the driving was occurring in the helium

layers.

I t was also found for these variables that very e f f ic ient convection was

necessary in order for the observed ins tab i l i t y str ip to agree with the

theoretical i ns tab i l i t y s t r ip . In fact, Cox et al. had to assume I/hpM3 in order

to obtain a blue edge as hot as 27,000K.

4. The GW Vir (PG1159-035) Variable Stars

The f i r s t member of this class of variable stars was discovered at the MMT by

McGraw and Starrf ield [McGraw, et al. 1979]. I t was found to be pulsating in a

number of modes with periods around 500 seconds. Spectroscopic studies showed no

evidence for any hydrogen in the atmosphere, log g . 7 . to 8., and that i ts

effective temperature exceeded IO0,O00K. This estimate of i ts temperature was

later confirmed by Exosat studies [Barstow, e t a ] . 1986]. In other studies,

Winget, et al . [1985] have measured a period change in GW Vir of -2.34 x 10 "14

s/s. This has now been interpreted by Kawaler, Hansen, and Winget [1985] as caused

by a shrinking, rotating star pulsating in a low order I=3 mode (] is the number

of node lines on the surface). They obtain a rotation velocity of ~35 - 50 km/sec

which does not seem unreasonable for a white dwarf. Kawaler [1986] has done an

analysis of the period spectrum of GW Vir and finds 8 periods are present and that

336

Po is either 8.8sec or 21.1 sec. Since Po depends on l( l+1), his results imply

that this star is pulsating in either I=I (21.1 s) or I=3 (8.8s). I f correct,

this wi l l be the f i r s t mode identification for any degenerate pulsator. Other

pulsating members of this class have been discussed by Grauer and Bond [1984] and

Bond, et al. [1984]. The most luminous member of the class is the central star of

the planetary nebula K1-16 and i t has a lower gravity than the other members of

the class. In addition, i t is pulsating at periods of ~1700 sec which are much

longer than those found in GW Vir. Kawaler, et al. [1986] have investigated KI-16

for the epsilon mechanism but find that although an instabi l i ty exists, the

periods are too short to agree with those observed in K1-16 or the other GW Vir

variables. The GW Vir stars have been investigated for instabi l i ty by Starrfleld, et a1.

[1983, 1984 1985]. In addition, studies of planetary nebula central stars similar

to K1-16 have found them not to be pulsating. An abundance analysis of one such

Star, the central star of NGC 246 [Hussfeld 1986, unpublished Ph.D. dissertation],

has shown that helium is ~70% in i ts atmosphere and theoretical analyses predict

that this much helium is sufficient to prevent driving and cause the star to be

PulsationaIly stable. Starrfield, et al. [1983, 1984, 1985] identified the pulsation driving

mechanism as the partial ionization of the last two electrons of both carbon and

Oxygen. Both the Salo and Cox [1980] and the Pesnell [1986] codes were used to

analyze stel lar envelopes in the effective temperature range from 70,O00K to

150,000K (and hotter). The mass of the star was assumed to be O.6M e and the

COmposition of the envelope was assumed to be either half helium and half carbon

(by mass), pure carbon, half carbon and half oxygen, or ninety percent oxygen and

ten percent carbon. They found instabi l i ty strips for these stars in the above

temperature range. They also predicted that i f GW Vir (PG1159-035) was as hot as

suggested by the X ray observations, then a significant amount of oxygen was

required at the surface in order for i t to pulsate. This prediction was confirmed

by Sion, Liebert, and Starrfield [1985] who obtained spectra in the optical

ultraviolet for GW Vir and two other GW Vir stars and found oxygen absorption

lines present in the spectrum. However, the actual abundance of oxygen is s t i l l

Unknown since no abundance analysls has been done for these stars. A recent study

of KI-16 has found these same oxygen lines present in the ultraviolet spectrum but

these lines are in emission as one would expect for the central star of a

Planetary nebula that is s t i l l (possibly) losing mass.

Starrfleld et al. [1985; and in preparation] have also done a linear,

nonadiabatic, nonradial analysis of K1-16 and found instabi l i ty strips for this

star at high luminosity. They used the same compositions and stel lar mass as in

the GW Vir studies but assumed that the star was on the high luminosity part of

the evolutionary track to the white dwarf region of the HR diagram and that i t was

337

evolving rapidly to hotter effective temperatures. In order for KI-16 to be

pulsating at periods of~1700 sec I t must have an effective temperature around

130,000K. Unfortunately, the temperature range of the instabi l i ty strip is very

broad and i t wi l l be impossible to predict the effective temperature from the

theoretical analysis. They have also studied models with different masses and

find regions of instabi l i ty ranging from~120,OOOK to more than 2DO,DOOK.

Finally, there is no overlap in the periods of the unstable modes for models with

different luminosities. The high luminosity models are unstable at long periods

and stable at the shorter periods while the low luminosity models are unstable at

short periods and stable at the longer periods. Therefore, we do not expect to

find any short periods present in the power spectra of KI-16 and we do not expect

to find GW Vir unstable at the long periods observed in K1-16.

5. Conclusions

The observational studies of these stars have shown both that they are

pulsating in nonradlal g modes and also that these modes are of low order in l

(the number of node lines on the surface) and high order in k (the number of nodes

In the radial elgenfunctlon). The principal argument In favor of these

conclusions is that the periods calculated for stellar models in the observed

temperature range are quite close to those that are observed. These conclusions

will be strengthened when there has been a successful mode identification.

The discovery and analysis of these stars has markedly improved our

understanding of the last stages of evolution of stars llke the sun. In order to

analyze these stars and demonstrate that they are pulsating in nonradlal modes, it

was necessary to develop new numerical techniques and use the latest stellar

opacities and equations of state. In addition, in order to improve the

correspondence between theory and observations it was necessary to apply diffusion

theory to the outer envelopes of DA white dwarfs. Now it appears that a time

dependant theory of the interaction between convection and pulsatlon will have to

be developed in order to accurately determine the theoretical boundaries of the ZZ

Cetl and DBV instabi l i ty strips. Finally, i t is already clear that the existence

of these variables in the observed temperature range requires that convection be

very eff ic ient.

The theoretical analysis of these stars has provided us with two new

pulsation driving mechanisms, In the case of both the ZZ Ceti and DBV variables

It is "convection blocking" which occurs as a result of the interaction between

convection and pulsation. Detailed analysis of the driving regions In both

classes of variables shows that convection cannot adjust instantaneously to either

338

a compression or an expansion and the result is a blocking or release of energy

out of phase with the envelope motions, This is analogous to the normal kappa and

gamma mechanisms which operate in the Cepheids or RR Lyrae variables and also in

the ZZ Ceti and DBV varialbles but in their case i t is the part ial ionization of

hydrogen and helium that excites the pulsations.

In the case of the GW Vir variables i t is the action of a kappa and gamma

mechanism that drives the pulsations, but for these variables i t is the partial

Ionization of carbon and oxygen at very near the s te l lar surface that causes the

Pulsational i ns tab i l i t y . In fact, as a direct result of this prediction,

observational evidence has now been obtained that shows that these stars have

oxygen present at the surface. This implies that these stars have probably

suffered a great deal of mass loss in order for them to have eliminated their

entire hydrogen and helium burning layers. I t is interesting to note that recent

studies of the 12C(~,¥)160 reaction have found i t to be a factor of about three

higher then previously thought and, therefore, we would expect more oxygen to be

made in helium burning regions. The pulsatlonal analyses have placed these stars in regions of the HR diagram

Where evolution is proceeding very rapidly. In fact, a period change has already

been measured for GW Vir and the value is as predicted for post planetary nebula

stars that have just evolved onto a white dwarf cooling curve. The observed value

of the period change can be explained by a rotating, cooling star with an

effective temperature around IOO,ODOK. I also note that the central star of the

planetary nebula, KI-16, should be evolving more rapidly than GW Vir and efforts

to measure a period change in this star are in progress but are hampered by the

fact that the period is changing so rapidly that i t may not be possible to match

observations from one season to another [Grauer, et al . 1986].

Final ly , the observations of these stars show that there is helium present in

the surface layers. The fraction of helium has yet to be determined but is

probably small; otherwise, i t would poison the pulsational i ns tab i l i t y .

Nevertheless, i t seems l i ke ly that with time i t w i l l f loat to the surface and

f ina l l y poison the driving and halt the pulsations. As the star cools, however,

i t w i l l pass through the DBV ins tab i l i t y str ip and again become a pulsating

variable star.

I t Is a pleasure to thank H, Bond, A. Cox, A. Grauer, C. Hansen, S. Kawaler,

J. Liebert, E. Nather, D. Pesnell, E. M. Slon, H. Van Horn, and D. Winget for

valuable discussions. I am grateful to the Association of Western Universities

and the Fellows of the Joint Inst i tute for Laboratory Astrophysics for Sabbat4cal

Leave Fellowships during the time this paper was being prepared. I am also

grateful to G. Bell, S. Colgate, A. N. Cox, and J. Norman for the hospital i ty of

the Los Alamos National Laboratory and a generous grant of computer time.

339

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I. Barstow, M. A., Holberg, J. B., Grauer, A. D., and Winget, D. E. 1986,

Ap. J. Lett . , 306, L25.

2. Bohm-Vitense, E. 1958, Zs. Ap., 46, 108. Bond, H. E., Grauer, A. D.,

Green, R. F., and Liebert, J. W. 1984, Ap. J., 279, 751.

3. Castor, J . I . 1971, Ap. J., 166, 109.

4. Cox, A. N. 1986, in "Highlights of Astronomy", ed. J.-P. Swings

(Dordrecht: Reidel), p. 229.

5. Cox, A. N., Starrf ield, S,, Kidman, R. B., and Pesnell, W.D. 1986, Ap.

J., submitted.

6. Chanmugam, G. 1972, Nature Phys. Science, 236, 83.

7. Dolez, N., and Vauclalr, G. 1981, Astr, Ap., 102, 375.

8. Grauer, A. D., Bond, H. E., Liebert, J. W., Fleming, T., and Green, R. F.

1986, in "Stellar Pulsation: A Memorial to John P. Cox", ed. A. N. Cox,

W. M. Sparks, and S. Starrf leld, (Springer Verlag), in press.

9. Grauer, A. D. and Bond, H. E. 1984, Ap. J., 277, 211.

10, Greensteln, J.L. 1982, Ap. J,, 258, 661.

11. Hansen, C.J., Winget, D.E., and Kawaler, S.D. 1985, Ap. J., 297, 544.

12. Iben, I. and MacDonald J. 1985, Ap, J., 296 540.

13. Iben, I. and McDonald, J. 1986, Ap. J. 301, 164.

14. Iben, I. and Tutukov, A.V. 1984, Ap.J., 282, 615.

15. Kawaler, S. D., Hansen, C. J., and Winget, D. E. 1985, Ap. J., 295, 547.

16. Kawaler, S. D., Winget, D. E., Hansen, C. J., and Iben, I. 1986, Ap. J.

Lett. , 306, L41.

17. Koester, D., Vauclalr, G., Dolez, N., Oke, J.B., Greenstein, J.L., and

Weidemann, V. 1985, Astron, Ap., 149, 423.

18. Landolt, A. 1968, Ap. J., 153, 151.

19. Leibert, et al. 1986, preprlnt.

20. McGraw, J. T., Starrf leld, S., Angel, J.R.P., and Carlton, N.P. 1979, in

Smithsonlan Ap. Obs. Spec. Report No. 385, p. 125.

21. Michaud, G., and Fontaine, G. 1984, Ap.J., 283, 787.

22. Pesnell, W.D. 1986, in "Stellar Pulsation: A Memorial to John P. Cox",

ed. A. N. Cox, W. M. Sparks, S. Starrfleld (Sprlnger-Verlag), in press.

23. Robinson, E.L. 1984, Astron. J., 89, 1732.

24. Saio, H., Winger, D.E., and Robinson, E.L. 1983, Ap. J., 265, 982.

25. Salo, H. and Cox, J.P. 1980, Ap. J., 236, 549.

26. Sion, E.M., Liebert, J. W., and Starrf ield, S. 1985, in Ap. J., 292, 477.

27. Starrf ield, S. 1986, in "Stellar Pulsation: A Memorial to John P. Cox",

ed. A. N. Cox, W. M. Sparks, and S. Starrfield (Springer Verlag), in

340

press.

28. Starrfield, S.G., Cox, A.N., and Hodson, S.W. 1979, in IAU Colloqulum 53,

"White Dwarfs and Variable Degenerate Stars," ed. H.M. Van Horn and V. Weidemann (Rochester: University of Rochester), p. 382.

29. StarrfieId, S.G., Cox, A.N., Hodson, S.W., and Clancy, S.P. 1983, Ap. J.,

269, 645. 30. Starrfield, S.G., Cox, A.N., Hodson, S.W., and Pesnell, W.D. 1982, in

"Pulsations in Classical andCataclysmlc Variable Stars", eds. J.P. Cox and C.J. Hansen (Boulder: University of Colorado), p. 78.

31. Starrfleld, S., Cox, A.N., Hodson, S. W., and Pesnell, W. D. 1983, Ap.

J., 268, L27. 32. Starrfield, S., Cox, A. N., Kidman, R. B., and Pesnell, W. D. 1984, Ap.

J.,281, 800. 33. Starrfield, S., Cox, A.N., Kidman, R. B., and Pesnell, W. D. 1985, Ap. J.

Lett., 293, L23. 34. Van Horn, H.M. 1980, in "Current Problems in Stellar Instabi l i t ies", ed.

D. Fischel, J. R. Lesh, and W. M. Sparks (NASA Technical Memorandum

80625), p. 453. 35. Van Horn, H.M. 1984, Proceedings of the 25th Liege International

Astrophysics Colloquium, 1984, eds. A. Noels and M. Gabriel (Liege:

Universite de Liege), p. 307. 36. Warner, B. and Robinson, E.L. 1972, Nature Phys. Sci., 329, 2.

37. Winget, D.E. 1981, Thesis Unversity of Rochester.

38. Winger, D.E. 1986, in Highlights of Astronomy (Dordrecht; Reidel Press) 39. Winget, D.E., and Fontalne, G. 1982, "Pulsations of Classical and

Cataclysmic Variables," ed. J.P. Cox and C.J. Hansen (Boulder: JILA)

p.46. 40. Winget, D. E., Kepler, S. 0., Robinson, E.L., Nather, R.E., and

O'Donoghue, D., 1985, Ap. J., 292, 606. 41, Winget, D.E., Van Horn, H.M., and Hansen, C.J. 1981, Ap. J. Lett., 245,

L33. 42. Winger, D.E., Van Horn, H.M., Tassoul, M., Hansen, C.J., Fontaine, G.,

and Carroll, B.W. 1982, Ap. J. Lett., 252, 65. 43. Winget, D.E., Van Horn, H.M., Tassoul, M,, Hansen, C.J., and Fontalne, G.

1983, Ap. J. Lett., 268, L33.

341

CONSTRAINTS ON THE ATMOSPHERIC COMPOSITIONS OF PGI159-035

AND SIMILAR PULSATING STARS

James Liebert

Steward Observatory

University of Arizona

Tucson, Arizona 82721

1. Introduction

The pulsation properties of PGI159-035 (GW Vir), the central star

of the planetary nebula KI-16, and related pre-white dwarf stars are

very much dependent on the chemical composition of the outer

envelopes. Of course direct information on the atmospheric

composition-- as well as on the temperatures and surface gravities--

can come from careful analysis of the line spectra and energy

distributions. Guided by these parameters, pulsation studies can then

be used as probes of the layers beneath the surface, including the

composition structure.

The only model atmospheres analyses applied to PGI159-035 and

related objects have utilized the two simplifying assumptions of (i)

local thermodynamical equilibrium and (2) the inclusion of only the

elements hydrogen and helium (c.f. Wesemael, Green and Liebert 1985;

Nousek et al. 1986). The derived surface temperatures and gravities

assume that helium is the dominant atmospheric constituent. While the

LTE assumption is normally safe at the high surface gravities

appropriate for these objects (Wesemael 1981), the high temperatures

and hydrogen-poor atmospheric compositions give considerable reason to

question its use here. Likewise, it has been proposed that the

pulsational instabilities are driven by ions of the CNO species,

oxygen in particular (Starrfield et al. 1983,1984), and would

apparently require considerable abundances of such species in the

outer envelopes. Moreover, these ions appear prominently in the

optical and ultraviolet spectra (c.f. Sion, Liebert and Starrfield

1985).

342

It is clearly of crucial importance to know whether these

atmospheres and envelopes are dominated by helium or by CNO species.

However, the limitations in the existing model atmospheres analyses

therefore do not permit a definite answer to this question. More

comprehensive, non-LTE model calculations are urgently needed; there

is also room for improvement in the observational data base. In

Section 2, we comment on some recent observational work and ongoing

theoretical analyses which promise to remedy the situation. However,

in Section 3, consideration of the likely evolutionary state of the

Pulsating stars leads to arguments unrelated to the direct analysis of

the spectrum which suggest that the dominant atmospheric constituent

of the stars is helium.

2. Inferences from New Spectroscopy and Spectroscopic Analyses

The International Ultraviolet Explorer telescope and echelle

Spectrograph is just sensitive enough to attempt a high dispersion

Spectrum of PGI159-035. Putting together two U.S. and one European

shift, E. Sion, F. Wesemael, R. Wehrse and I obtained an 18 hour

exposure of the prototype using the IUE SWP (1200-2000A) echelle

camera. This will complement the published optical and ultraviolet

Spectrophotometry (Bond et al. 1984; Wesemael, Green and Liebert 1985;

8ion, Liebert and Starrfield 1988) and X-ray observations (Barstow

e_.tta!. 1986).

Interpretation of these observations is underway using non- LTE

Stellar atmospheres calculations incorporating C IV and 0 VI ions as

Well as the usual helium and hydrogen. This effort at the University

of Munich Observatory involves D. Busfeld, R. Kudritzki and K. Butler

and extends basically the atmospheres code developed by this group and

K. Hunger's group in Kiel for the study of planetary nebulae nuclei

(PNNs) and subdwarf 0 stars. Since the pulsating stars include at

least one PNN, it is appropriate to consider recent analyses of the

central stars.

An upper limit of 30~ of the PNNs have hydrogen-poor and probably

helium-rich atmospheric compositions (Mendez et al. 1986). These have

traditionally been classified into several types, ranging from those

With subdwarf 0 absorption line spectra to those showing strong, broad

emission lines (the Wolf-Rayet and 0 VI types, Heap 1982). In

S43

particular, some of the hotter of the O VI type show He II, C IV and O

VI emission and absorption features; these are also identified, but

the emission is generally much weaker in the pulsating stars

PGl159-035, two other PG Survey stars, and the KI-16 nucleus.

Husfeld's (1986) analysis of the PNN of NGC246 is of particular

relevance since this central star shows many of these same helium,

carbon and oxygen features and is clearly very hot. Moreover, the

observational data on this bright nucleus are excellent, and the

photospheric absorption spectrum is not badly contaminated by nebular

radiation or by ongoing mass loss. It shows weak O VI emission. C IV

absorption lines are prominent, and often nearly coincide with the

positions of He II transitions, since both ions are hydrogenic. This

star provided a strong motivation for including carbon in the model

atmospheres code. Husfeld (1986) derives Teff=130,000 K, log g = 5.7,

in good agreement with results in Heap (1982). This is similar to the

current temperature estimate for PGI159-035, but is at least an order

of magnitude lower in surface gravity. However, there were clear

difficulties in incorporating the physics for the C IV profiles, and

the derived carbon abundance was believed to be quite uncertain. The

best fit suggests an atmospheric carbon abundance of 30~!

I f basically correct, this result means that one object which may

well be a predecessor to the KI-16/PGII59-035 pulsating stars has a

carbon abundance orders of magnitude higher than the "trace" values

derived for hot, helium-rlch white dwarfs and for the 0 subdwarfs.

Likewise, the very hot star H1504+65 (Nousek et al. 1986) has no

detected helium, while showing C IV and 0 VI lines. It is unclear

whether its unique spectrum is due to a temperature or an abundance

difference. It should be noted that Grauer et al. (1987--see also

H. Bond, this conference) have established that both H1504+65 amd the

NGC246 nucleus are not rapid photometric variables.

3. Other Inferences about the Dominant Atmospheric Constituents

The results available from direct spectroscopic analysis of the

pulsating stars and objects possibly related to them leave the issue

of the dominant atmospheric compositions in considerable doubt. It is

appropriate, however, to consider broader inferences following from

the likely evolutionary states of these objects. Three kinds of

344

arguments outlined below suggest that most or all of the hot pulsators

should have helium-dominated atmospheres and envelopes:

(I) It is difficult theoreticallZ t__oo ~et rid of most of the outer

helium iR~er , b__[ helium she!! flashes or otherwise. The published

Calculations of double shell-source burning / asymptotic giant branch

(AGB) models leave a considerable buffer of helium -- of 10 -4 to

IO-2M 8 -- between the carbon-oxygen core and the stellar surface or

hydrogen layer. The pulsation calculations of Starrfield et al.

(1984) suggest that CNO species should be prevalent at a depth of

IO-12M. for the driving to occur, and it would then be expected that

any outer helium envelope would have a smaller mass than this.

It is quite another thing, however, to argue that all physical

Processes including the mass loss mechanisms in AGB and post-AGB

evolution are sufficiently well understood to preclude the loss of an

entire helium envelope! For example, D'Antona and Mazzitelli (1979)

state that no definite conclusions concerning the helium layer masses

may be derived from existing calculations. Observationally, we have

seen in the previous section that one cannot rule out the existence of

hydrogen-poor, post-AGB stars with substantial surface abundances of

Carbon. Hence, the standard model results and the observations of

evolved, hot stars provide only a weak argument against the existence

of exposed carbon-oxygen cores in hot evolved stars.

(2) The space . densities of the PGI159-035 stars are consistent

With their beln K i__nn ~ hellum-shell-burnlng phase. Though poorly

known, the space densities estimated in Wesemael, Green and Liebert

(1986) are more consistent wlth the evolutionary models of Iben and

TUtukov (1984) than with a phase of rapid, gravitational contraction.

Note that the Iben and Tutukov (1984) Case B model requires 105 years

to decline in luminosity to log L = 2. This may also account, at

least in part, for the relatively large numbers of hydrogen-poor stars

of high surface gravity near I00,000 K, despite the paucity or absence

of similar remnants with hydrogen-rich atmospheres (Fleming, Liebert

and Green 1986; Holberg, Wesemael and Basile 1986). As noted, if too

little envelope helium were retained for helium-shell burning, the

evolution from the AGB to the white dwarf cooling phase (for a given

Core mass) would be much more rapid.

(3) It is difficult to ident!f ~ white dwarf successors to such

ho~p~t precursors wlth nearly-bared 9arbon/oxygen cores. This question

345

must be approached with care, however. Even the retention of a tiny

mass fraction of helium, spread throughout the outer layers of the

star in the pulsating (PGI159-035) stage may be sufficient for the

star later to become a white dwarf with a helium-dominated atmosphere.

Gravitational settling may bring the helium rapidly to the surface,

and the object may cool as a spectroscopic DO and DB white dwarf. Of

course, after the application of gravitational and thermal diffusion

processes, the surface mass of helium may be truly miniscule--above

the IO-16M. or so to more than include a stable atmosphere, but many

orders of magnitude below the IO-4M. predicted by the AGB/post-AGB

models discussed earlier.

As the white dwarf cools towards I0,000 K, however, convective

mixing spreads very deep in an outer helium envelope, and would reach

the carbon boundary quickly if the helium envelope were thin. It

would then mix and dilute the thin helium layer, turning the

atmosphere into a carbon/oxygen-domlnated composition (see Koester

et al. 1982, Wegner and Yackovich 1984, Fontaine et al. 1984, and

references therein). The problem is that no such cool white dwarf has

ever been found, and hundreds of cool white dwarfs with helium-rich

compositions have been catalogued. In fact, the prevalence of tiny,

trace abundances of carbon observed in the atmospheres of these stars

(the DQ white dwarfs) now appears to be consistent with helium

envelope masses expected from the basic AGB and post-AGB evolutionary

models.

None of these arguments favoring thick outer envelopes dominated

by helium preclude these envelopes from having substantial enrichments

of CNO elements, as observed in the spectra of the hottest helium-rich

PNNs and pulsating stars. These enrichments may be caused by mixing

events during helium shell flashes on the AGB or afterwards. It is

also true that such elements may be pushed towards the surface by

selective radiative acceleration processes (c.f. Vauclalr, Vauclalr

and Greenstein 1979). Mass loss featuring the 0 VI and C IV ions is

certainly observed in the hotter Wolf-Rayet or 0 VI nuclei, and may

result in a significant amount of the helium envelope being lost. It

is unclear as to what abundance distribution is expected near the

surface.

Likewise, it is obvious from the earlier remarks--and from the

prior presentation by A. N. Cox--that a conclusion that the KI-16/

PGI159-035 pulsating variables have predominantly helium-rich

346

atmospheres may lead to a serious problem in accounting for the

pulsations. Helium itself cannot provide the driving at these high

surface temperatures, nor can the CNO species unless they exist close

to the surface in substantlal--though as yet unspecified--abundances.

Kawaler et al. (1986) have suggested that nonradial g-mode

instabilities may be driven by a helium shell source, but the

calculated periods are too short.

On b a l a n c e - - l o o k i n g t h r o u g h a c l o u d y c r y s t a l b a l l a t how t h e s e

questions might be answered--these post-AGB stars may retain m u c h of

their outer helium envelopes, but these must be enough enriched in

some combination of oxygen, carbon and perhaps nitrogen to cause

instabillty in this hottest known group of pulsating stars.

References:

Barstow, M. A., Holberg, J. B., Grauer, A. D. and Winget, D. E. 1986, Ap. J. (Letters), 306, L25.

Bond, H. E., Grauer, A. D., Green, R. F. and Liebert, J. 1984, Ap. J . , 279, 751.

D'Antona, F. and Mazzitell i , I. 1979, Astr. Ap., 74, 161. Fleming, T., Liebert, J. and Green, R. F. 1986, Ap. J . , 308, 176. Fontaine, G., Villeneuve, B., Wesemael, F. and Wegner, G. 1984,

Ap. J. (Letters) , 277, L61. Grauer, A. D., Bond, H. E., Liebert, J . , Fleming, T. and Green, R. F.

1986, preprint . Heap, S. R. 1982, in IAU Symp. 99, Wolf-Rayet Stars: Observations,

Physics, Evolution, ed. C. W. H. de Loore and A. J. Wil l is , Dordrecht: Reidel, p. 423.

Holberg, J. B., Wesemael, F. and Basile, J. 1986, Ap. J . , 306, 629. Husfeld, D. 1986, Ph.D. disser taion, the Ludwig-Maxlmilians

University, Munich. Iben, I. and Tutukov, A. V. 1984, Ap. J . , 282, 615. Kawaler, S. D., Winget, D. E., Hanson, C. J. and Iben, I. 1986,

Ap. J. (Letters), 306, L41. Koester, D., Weidemann, V. and Zeidler-K.T., E. M. 1982, Astr. Ap.,

116, 147. Mendez, R., Miguel, C. H., Heber, U. and Kudritzki, R. P, 1986, IAU

Coll. 87, Hydrogen Deficient Stars and Related Objects, eds. K. Hunger, D. Schonberner and K. Rao, Reidel: Dordrecht, in press.

Nousek, J . , Shipman, H. L., Holberg, J. B., Liebert, J . , Pravdo, S. H., White, N. E. and Giommi, P. 1986, Ap. J., in press.

Sion, E. M., Liebert, J. and S ta r r f i e ld , S. 1985, Ap. J . , 292, 471. S ta r r f ie ld , S., Cox, A. N., Hodson, S. W. and Clancy, S. P. 1983,

Ap. J . , 269, 645. Starrfleld, S., Cox, A. N., Kidman, R. B. and Pesnell, W. D. 1984,

Ap. J., 281, 800, Vauclair, G., Vauclair, S. and Greenstein, J. L. 1979, Astr. Ap.,

80, 79. Wegner, G. and Yackovich, F. H. 1984, Ap. J., 284, 257. Wesemael, F. 1981, Ap. J. Suppl., 45, 177. Wesemael, F., Green, R. F. and Liebert, J. 1985, Ap. J. Suppl., 58,

3 7 9 .

347

P G 1346+082: An In terac t ing Binary White Dwar f Sys tem

M. A. Wood, D. E. Wingett, and R. E. Nather Department of Astronomy and McDonald Observatory University of Texas at Austin

James Liebert Department of Astronomy and Steward Observatory University of Arizona

F. Wesemael D~partement de Physique Universitfi de Montreal

G. Wegner Department of Physics and Astronomy Dartmouth College

A B S T R A C T

PG 1346+082 is both a photometric and a spectroscopic variable, spanning the B-magnitude range 13.6-17.2. High-speed photometric data reveal rapid flickering in the low-state light curve. The system also shows spectroscopic variations, displaying broad, shallow He I absorption lines at maximum light, and a weak emission feature at He 1 447I~ at minimum light. Hydrogen lines are conspicuous by their absence.

We conclude that PG 1346+082 is an interacting binary white dwarf system. Furthermore, because continuum fits to IUE high-state data suggest temperatures consistent with membership in the DB white dwarf instability strip, we suggest that some of the photometric variations we observe may arise from pulsations.

I N T R O D U C T I O N

We present an overview of the results of an extensive study of the Palomar Green (Green et al. 1986; hereafter PG) survey object PG 1346+082 (Nather 1984). We have studied the object using high-speed photometry, multi-color photometry, spectroscopy, the International Ultraviolet Explorer (IUE) Satellite, and the archival Harvard Meteor Program films. We discuss the clues that this object presented us, the model that they demanded, and the implications of this model. A detailed discussion of this object can be found in Wood et al. (1987).

t Alfred P. Sloan Research Fellow

348

TI-IE C L U E S

Our observations revealed that PG 1346+082 is both a photometric and spectroscopic variable with a wide range of observed properties. The B magnitude spans the range 13.6-17.2, and the system was as bright or brighter than a V ~ 14.1 comparison star on roughly 74% of the archival Harvard Meteor Program films that we examined. We observe rapid, large-amplitude (~10%) photometric flickering in the low-state, and a saw-toothed light curve with a dominant 1490 s periodicity in the high-state (Fig. 1).

W* II, I

, - L , . ,

"..: j:../,/..f " L :, f . f :~'...:;:':';'.,./. . .,~;',../: ..

,o'~o ~.o'~, ,o'oo ,o'oo ~ogo ~2oo ,ogo ,,~'oo ? u , i I i l c o n d l l

Figure 1. A summary of portions of the light curves of the system on the nights (top to bottom) 1984 Apr 4, Apt 82 May 4, and May 5 (UT}, demonstrating the variable character of the light curves of PG 1346+082. The light curves have all been normalized about zero, and then constant values were added to the upper three to offset them for display.

We find only lines of He I in the optical spectra: the low-state spectra display a weak emission feature at 4471A, whereas the high-state spectra are characterized by broad (N100A), shallow (~10%) absorption lines. We find no evidence for hydrogen in a n y of our spectra; in addition, we find no evidence for He II features, nor any evidence that the system is a strong X-ray source (J. Osborne and N. White, private communication). Lastly, we fit theoretical helium atmosphere continua (Wesemael 1981; Koester 1980) to our I U E data, and found characteristic temperatures of ~24,000 K for the high state, and --.18,000 K for the low state (Fig. 2).

THE MODEL

The observations outlined above are sufficient to narrow the number of plausible models to

one, as follows. • The rapid photometric flickering and the large-amplitude, quasi-periodic variability are both

indicators of mass transfer in a close binary system. Furthermore, the manner of spectroscopic variability (broad absorption lines at maximum, emission lines at minimum) indicates the presence

of an accretion disk.

349

-25.5

, ,> - 2 a o

-26 .5

~LFEs . . . . . . Oo o o ° O ~ a

" ~ ~'~'XXXxxx ° ° o - o o o o~b'bo.

lIDS

I

I

2

• • •

LWR. LWP 1 SWP

I t ' t ..... AM CVn

, I I

4 5 6 7

I / Z ( ~ m "I )

Figure 2. rUE flux distribution. The top panel shows the combined optical and ultraviolet flux distributions. The open circles represent the 1983 data, and the solid circles represent the 1984 data. The crosses correspond to a 1984 Jan 9 Kitt Peak IIDS intermediate-state spectrum. Also shown are fits to the continuum, Fits a and c correspond respectively to the 22,000 K and 18,000 K models of Koester (1981)~ and fits b and d correspond respectively to the 25,000 K and 18,000 K models of Wesemae! (1980). Note that fits a and b have been offset from each other slightly to avoid confusion. The bottom p~nel shows schematically the flux from AM CVn~ from Greenetein and Ok• (1982), for comparison.

• T h e spec t ra l fea tures we observe are ~ 1 0 0 ,~ wide, which could indica te pressure b roaden ing in the a t m o s p h e r e of a single degenera te object , b u t are more p laus ib ly expla ined as b o t h pressure and Doppler b r o a d e n i n g in an opt ica l ly- th ick accre t ion disk su r round ing a compac t mass accretor .

• We de tec t o n l y fea tures of He I in our spec t ra . T h e only objects observed in n a t u r e which show pure-hellum spec t r a are the he l i um-a tmosphe re whi t e dwarfs. We therefore identify the mass- losing s t a r in th i s sys tem as a he l ium a t m o s p h e r e whi t e dwarf s tar .

• T h e accre t ing objec t c a n n o t be a n e u t r o n s t a r or a black hole for th ree reasons: the char- ac ter i s t ic t e m p e r a t u r e of P G 1346+082 is 18,000 K-25 ,000 K, no He II fea tures are de tec ted in t he opt ica l s p e c t r u m , a n d the s y s t e m is no t observed to be a s t rong X- ray source.

O n t he basis of t he above , we conclude t h a t P G 1346+082 is a n i n t e r ac t i ng b i n a r y whi te dwar f ( I B W D ) sys tem.

I M P L I C A T I O N S O F T H E M O D E L

T h e o rb i t a l s epa ra t i on of an I B W D m u s t be small (so t h a t the compac t secondary can fill its

Roche lobe) , and so the o rb i t a l pe r iod mus t be shor t . Because t he 1490 s per iod ic i ty is near ly always found, it is possible t h a t i t is in fact the o rb i ta l per iod, a l t hough we have no t yet been able to prove it. If 1490 s is in fact the o rb i ta l per iod, and if we fu r the r assume t h a t the secondary fills i ts Roche lobe and t h a t the mass of the p r i m a r y is ~ 1 M®, t h e n we can derive an orbi ta l

s epa ra t i on of a,,-0.3 RE) , and a mass of the secondary of roughly 0.03 ME). In F igure 1, we find the c h a r a c t e r of the V~15 .4 l ight curve to be r emarkab ly s imi lar to t h a t

of the pu l sa t ing whi t e dwarfs ; t he power s p e c t r u m of th is r un shows a b a n d of e n h a n c e d power

s p a n n i n g 200-400 s, and the UBV colors on th i s n igh t were ident ical ( to w i t h i n t he errors) of

those of the pu l sa t ing he l ium-a tmosphe re wh i t e d w a r f GD 358. In addi t ion , t he der ived IUE h i g h - s t a t e t e m p e r a t u r e , 24,000 K, is cons i s t en t w i t h m e m b e r s h i p in t he ins tab i l i ty regime for

he l ium a tmosphe re s (cf. L ieber t st al. 1986}. T he lack of Ion~- term coherence in these var ia t ions

3 5 0

could be an indication that some, at least, arise from pulsations. In fact, because every known star in the DA and DB white dwarf instability strips pulsate, and because the temperatures and gravities indicated are very much like a white dwarf stellar envelope, we see no reason why there should not be pulsations present.

The data allow us to propose an unambiguous model for PG 1346+082: an IBWD model, similar to the models proposed for AM CVn (Falkner~ Flannery, and Warner 1972) and G61-29 (Nather, Robinson, and Stover 1981). In this context, we find two exciting prospects: (i) If indeed the temperature of the system crosses into or passes through the DB instability strip, then we may be afforded the opportunity to map out the position and extent of the DB instability strip in temperature--Le., determine its blue and red edges. (ii) The =outbursts ~ of the system may be similar in physical origin to the dwarf novae outbursts. Because the thermodynamics of a pure-helium accretion disk should be easier to model accurately than those of a solar-composition system, PG 1346+082 may help solve the puzzle of dwarf novae outbursts.

Clearly, PG 1346+082 is an object which merits further study. We hope that the observations we have presented will stimulate further observations of this system and also stimulate theoretical investigations of the dynamics of the accretion process which include the simplifying (and physical) assumption of a pure-helium gas.

Acknowledgments - - We thank J. Osborne and N. White for communicating their EXOSAT results, and also the personnel of the Harvard Plate Stacks for their assistance. This work was sup- ported in part by the National Science Foundation under grants AST 82-18624 through Steward Observatory, AST 81-08691 and AST 83-16496 through the University of Texas and McDonald Observatory, AST 83-19475 through Dartmouth College, in part by NASA (IUE) grant 5-38 and NAG 5-287, in part by the Foundation for Research Development of the CSIR of South Africa, and in part by the Natural Sciences and Engineering Research Council of Canada.

I ~ E F E R E N C E S

Faulkner, J., Flannery, B., and Warner, B. 1972, Ap. J. Lett., 175, L79. Green, R. F., Schmidt, M., and Liebert J. 1986, Ap. J. Suppl., 61, 305. Greenstein, J. L. and Oke, J. B. 1982, Ap. J., 258, 209. Koester, D. 1980, Astron. Astrophys. Suppl., 39, 401. Liebert, J., Wesemael, F., Hansen, C. J., Fontaine, G., Shipman, H. L., Sion, E. M., Winget, D. E.,

and Green, R. F. 1986, Ap. J., 309, 230. Nather, R. E. 1984, in Proceedings of the NATO Advanced Study Institute on Cataclysmic Variable

Stars, ed. P. Eggleton and J.E. Pringle (Dordrecht:Reidel) p. 349. Nather, R. E., Robinson, E. L.~ and Stover, R. J. 1981, Ap. J., 244, 269. Wesemael, F. 1981, Ap. J. Suppl., 45, 177. Wood, M. A., Winget, D. E. , Nather, R. E., Hessman, F. V., Liebert, J., Kurtz, D. W., Wesemael,

F., Wegner, G. 1987, Ap. J., in press.

351

A S E A R C H F O R H O T P U L S A T O R S S I M I L A R T O P G 1150-035 A N D T H E C E N T R A L S T A R OF K 1-16

Howard E. Bond Space Telescope Science In~tltute

Albert D. Grauer Unlver•tt/ of Arkansas, Little Rock

James Liebert and Thomas Fleming Universitll of Arizona Richard F. Green

Kitt Peak National Observatorlt

I. The PG llSO-035-type Pulsators

The variations of PG 1150-03S (GW Vir) were discovered by McGraw et oL (1079). This

object is the prototype of a new class of pulsating stars located in an instability strip at the

left-hand edge of the HR diagram. PG 1159-035 and the spectroscopically similar objects

PG 1707÷427 and PG2131+066 (Bond et aL 1984) display complex non-radial modes with

periodicities of order I0 minutes. Grauer and Bond (1984) recently discovered that the central

star of the planetary nebula Kohoutek 1-16 also exhibits similar pulsation properties, with

dominant periodicities of 25-28 minutes.

These four objects display the followlng characteristics:

• High effective temperatures (~I0 s K) and moderately high surface gravities (log g _~ 6-

8)

• He If, C IV, and O VI absorption lines in the optical spectra, often reversed with

emission cores

• No hydrogen lines clearly detected

Starrfield cta/. (1984) have attributed the pulsational instability to partial ionization of

carbon and/or oxygen.

2. The Photometric Survey

We have carried out a search for additional hot pulsators, with the aim of delineating the

temperature~ luminosity, and compositional boundaries of this new hmtability strip.

A variety of telescopes at KPNO, CTIO, and the University of Arizona Observatory was

used, and most of the data were taken with the two-star photometer described by Grauer and

Bond (1981).

High-speed photometric observations were obtained for 14 candidates, typically on 2-3

different nights for each object. The candidates were the following:

1. Pianctarl/ nuclei with helium-rich atmospheric compositions: the central stars of

AbeU 30, Abell 78, IC 1747, and NGC 246.

352

2. DO white dwarfs: KPD0005+5106, PG0108+I01, P G 0 1 0 9 + I l l , PG0237+116,

BE UMa, and H 1504+65.

3. $d~ PG0216+032.

4. Objects with PGI159-tvpe spectra: PG 1151-029, PG 1424+535, and PG 1520+525.

None of the stars showed evidence for periodic variability on time scales of 60 sec up to as

long as 1-2 hours; typical amplitude limits were a few millimagnitudes.

3. Conclusions

The four known pulss:tors lie at 80,000K < Teff < 160,000K and 6 < logg < 8, or about

1.5 < log L/L® < 3, in the HR diagram. Our non-pulsatlng DO white dwarfs may be outside

the temperature range of the instability strip, being slightly cooler than the GW Vir pulsators

(or slightly hotter in the ease of H 1504+65). The non-pulsating planetary nuclei have high

temperatures and show helium and carbon features in their spectra, but are more luminous

than the pulsators, since they have decidedly lower surface gravities.

More puzzling are our null results for the three PG objects 1151-029, 1424+535, and

1520+525, which have spectra closely resembling those of the GW Vir pulsators (Wesemael et al.

1985). Possibly the pulsational instability may be critically sensitive to the CNO abundances

(which have not yet been determined in these objects) or to temperature or surface-gravity

differences too small to be apparent in the spectra.

A more detailed version of this paper has been submitted for publication in The

Astrophysical Journal.

H.E.B. and A.D.G. were visiting astronomers at Kitt Peak National Observatory, and

A.D.G. was visiting astronomer at Cerro Tololo Inter-American Observatory, both of which are

operated by the Association of Universities for Research in Astronomy, Inc., under contract

with the National Science Foundation. We acknowledge support from the National Science

Foundation through grants AST 82-11905 and 84-13647 (A.D.G.) and 85-14778 (J.L.), and

from the U.S. Air Force Office of Scientific Research through grant 82-0192 (A.D.G.; Principal Investigator: A.U. Landolt).

R E F E R E N C E S

Bond, H.E., Graner, A.D., Green, R.F., and Liebert, J. 1984, Ap. J., 27{}, 751.

Grauer, A.D., and Bond, H.E. 1981, Pub. A.S.P., 93,388.

• 1984, Ap. J., 2TT, 211.

McGraw, J.T., Starrfleld, S., Liebert, J., and Green, R.F. 1979, in IAU Coll. No. 53, White

Dwarfs and Variable Degenerate Stars, eds. H.M. Van Horn and V. Weidemann (Rochester:

Univ. of Rochester Press), p. 377.

Starrfield, S , Cox, A.N., Kidman, R.B., and Pesnell, W.D. 1984, Ap. J., 381, 800.

Wesemael, F., Green, R.F., and Liebert, J. 1985, Ap. J. Suppl., 58, 379.

353

WHAT ARE HIGH L MODES, IF ANYTHING?

J. E. Hestand

Department of Astronomy, University of Texas

Austin, TX 78712

Wr Present Address: Department of Physics, Arizona State University, Tempe, AZ

85712

Abstract

Theoretical analyses of the pulsational instabi l i ty of compact objects

indicate that pulsation modes of high spherical harmonic index,l, are expected to

be unstable. The amplitudes of the observed, presumably low l,modes with I>3 wi l l

bring them below the observable threshold for ground based photometry. We point

out, however, that observations with a space telescope should reveal the presence

of modes of l 4-5 simply because the absence of sc int i l la t ion noise does not l imit

the amplitude sensit ivi ty. We also point out that rotational spl i t t ing has been

identified in a large number of the co~pact pulsators; in principle, this generates

a set of (21+1) modes for each k and I, and we show that the beating together of

these modes may have observable consequences. We examine the prospects for

observing the beating of the pulsation modes with high spherical harmonic index,l.

We show that in some cases this beating may result in something which would look

essentially like a flare. We discuss the prospects for observing this effect, and

the very real possibil ity that i t has already been observed.

White-dwarf variables separate into four distinct classes of pulsators

which span nearly the fu l l range of the white-dwarf cooling sequence in the H-R

diagram. These stars are all multi-periodic pulsating variable stars with periods

typically in the range from 100 to 1000 seconds. The periods of these modes are

long compared to radial pulsation timescales (<~10s); as a result they have been

interpreted as nonradial g-modes. Partial ionization of hydrogen, helium, or

carbon and oxygen in the envelope was determined as the driving mechanism behind

the pulsations of all of these objects. For the envelope regions to be able to

drive the instabi l i ty , they had to respond thermally on the timescale of the mode's

pulsation period. Since the thermal timescale increases with depth in a star, the

354

partial ionization zone must descend to deeper layers, as the star cools, unt i l i t

penetrates an area where the thermal timescale allows nonradial g-modes. These

arguments suggest that just above and just below the observed blue edge, high-I

modes, for which the pulsation period is of order the thermal time-scale, should be

unstable. The question became whether or not these high-I modes could produce

observable l igh t variations. Dziembowski(1977) points out that geometrical

cancellation causes a dramatic reduction in the apparent amplitude of single modes

of high l , making them essentially unobservable using the usual photometric

techniques. However, the rotational sp l i t t ing of the 21+1 modes would produce

frequencies which d i f fe r by integral multiples of the rotation frequency. This

means that these sp l i t frequencies would beat with a period which is equal to the

rotation period of the star. Would these beats be able to produce an observable

amplitude? Exact integer or "magic number" period ratios, such as those observed

in PG1351+489 and GD154, have no natural explanation from the low order g-mode

spectrum. This implies that beating of closely spaced, rotat ional ly (or m-) sp l i t

high-1 modes could be responsible for these integer ratios. Are these real ly

integer ratios between beat frequencies and are the beat frequencies the pulses we

observe?

In answering these questions, we essentially took an observer's approach.

Given a single high-I mode, rotat~onaliy sp l i t , how wou]d these pulsations appear

to someone observing the star? I created synthetic l igh t curves by adding together

sine waves with uniformly spaced, m-split frequencies. Harmonics, the f i r s t and

second, were included to achieve the non-sinusoidal pulse shape characteristic of

observed pulses. I assumed a pulsation period of 50 seconds, a rotation period of

500 seconds, and varied the value of 1. We decided to use sc in t i l l a t i on noise to

drown al l amplitudes in these synthetic curves, except the high amplitude beats,

and then take an FFT to see i f the only remaining features were the beat

frequencies. We thus hoped to produce an FFT resembling actual observations. I

plotted the l igh t curve of a real, constant star and took an FFT of the data. From

this FFT I determined a value of 5.0E-04 as the average height of the noise level.

I mult ipl ied my synthetic curves by this factor to hide them in the noise and then

added them to the constant star l igh t curve. The result was graphed to see i f any

observable beat pulses would rise up out of the noise in the l ight curve or i ts

FFT.

For intermediate I values, 4<I<I0, the width of the beat pulses in the

synthetic l igh t curve appeared too narrow to agree with observed pulse shapes. I

found an expression for the pulse width in terms of the rotation period, pulsation

Period, and I . I calculated the width for a large range of l ' s and found that the

numerical values were indeed too smal] to agree with observed values even when ]

was in the low ranges. Since my calculations showed that the pulse width decreased

355

as I increased, the poss ib i l i ty of these beats explaining the exact integer period

ratios was already ruled out. In the combined l igh t curve, the beats did not reach

above the sc in t i l l a t i on noise and the beat frequency could not be distinguised fn

the FFT. I t seemed that for these l values, the beats were not what was being

observed in the l igh t curves. On further consideration, I determined that not

seeing the beat in the FFT was a logical result . The answer lay in the nature of

the FFT i t s e l f . An FFT te l l s you only what frequencies were used to create a l igh t

curve and nothing else. A beat is merely a phenomenon resulting from the way wave

functions behave when more than one mode is being driven. A beat is, therefore,

not an input frequency. This means that there is no way to produce a pulsation

l ight curve such that the only signi f icant feature you ident i fy in i ts FFT is the

beat frequency. Another point to consider in this argument is that the FFT te l l s

you al l the frequencies which were input into the l ight curve. Even i f a pulse's

amplitude is squelched below the noise level, i ts frequency can s t i l l be found by

the FFT. You cannot fool a Fourier transform.

For high l values, with 15<I<100, the amplitudes increased and the pulse

width decreased in the synthetic l igh t curve. The beat pulses are observable above

he sc in t i l l a t ion noise and the FFT tapers upwards towards small frequencies in

agreement with observational data. This tapering is due not only to degeneracies

caused by the harmonics but also by the frequencies folding around zero when the

absolute value is taken of negative frequencies in calculating the m-split

frequency values. Although the amplitudes for these high I values are large enough

to be seen, their infrequent spacing makes them d i f f i c u l t to ident i fy . Since these

pulses occur once every ro ta t ion period, i f t h i s period is on the order of days, i t

could make identifying these as def in i te ly repeating beats very d i f f i c u l t . Their

narrow spiked pulse shapes also make i t d i f f i c u l t to prove they are not due to

instrumental error effects.

Very high l values, with I>100, are deemed unlikely to be observable

because of energy equipartit ion which would damp down the osci l lat ions. The energy

being distributed equally over al l modes would keep the individual amplitudes from

building above the noise level. An example of this kind of behavior is the

osci l lat ions of the sun.

One of the conclusions drawn from these results is that to ident i fy the

intermediate l beats the sc in t i l l a t i on noise must be lessened. The only way to

accomplish this is by means of the Space Telescope, as suggested by Don Kurtz in

his research on the l imi ts of ground based photometry [Kurtz,1981]. Another point

is that extended observations of objects just above the observed blue edge could be

the key to identifying high-I mode beating. Run times equal to many rotation

periods would be necessary for th is . The Global Photometry Network proposed by

Nather and Winget would be ideal.

A further consideration is that perhaps hlgh-I modes have already been

356

observed, In the past, observations have occurred where abrupt, large jumps In

amplitude were attributed to f lares, equipment error, or unknown origin. Two such

examples are observations of DBV G44-32, by Nather et al in 1970, and observations

of AM CVn, by Bhattacharyya et al in 1985. Cases such as these should be

re-examined with longer data runs to investigate the possibi l i ty of high-l mode

beating. Perhaps phenomena such as flare stars and neutron stars should be

considered strong candidates for this model. The l ight curve of a neutron star

closely resembles the narrow, large pulses of the l ight curves where l has a large

value, I t seems that further observations are warranted.

References

Br ickhi l l , A. J. 1975, M.N,R.A.S. 170, 405.

Cox, John P. 1984, P.A.S.P. 96, 577.

Dziembowski, W. 1977, Acta Ast. 27, I .

Kurtz, D, W, 1986, Preprint. Winget, D. E., Van Horn, H. M., Tassoul, M,, Hansen, C, J. , Fontaine, G,, and

Carroll, B. W. 1982, Ap. J. 252, L65.

Winget, D. E., and Fontaine, G. 1982, in Pulsations in Classical and Cataclysmic

Variable Stars, ed. J. P. Cox and C. J. Hansen (Boulder Joint Inst i tute

for Laboratory Astrophysics), 46.

Winget, D. E., Van Horn, H. M., Tassoul, M., Hansen, C. J., and Fontaine, G.

1983, Ap. J. 268, L33,

Winget, D. E. 1986, Preprint.

357

Pulsations of White Dwarf Stars With Thick Hydrogen or Helium Surface Layers

Arthur N. Cox Los Alamos National Laboratory Los Alamos, New Mexico 87545

Sumner G. Starrfield Arizona State University Tempe, Arizona 85287

Russell B. Kidman Los Alamos National Li~boratory Los Alamos, New Mexico 87545

W. Dean Pesnell University of Colorado Boulder, Colorado 80309

1. Introduction

In order to see if there could be agreement between results of stellar evolution theory and those of nonradial pulsation theory, calculations of white dwarf models have been made for hydrogen surface masses of 10 -4 M®. Earlier results by Winget et al. (1982) indicated that surface masses greater than 10 - s M® would not allow nonradial pulsations, even though all the driving and damping is in surface layers only 10 -12 of the mass thick. We show that the surface mass of hydrogen in the pulsating white dwarfs (ZZ Ceti variables) can be any value as long as it is thick enough to contain the surface convection zone.

Evolution calculations that produce white dwarfs from the asymptotic giant branch have for a long time found that there seems always to be a small residual hydrogen surface layer. Fujimoto (1977,1982) first pointed out that it was very difficult to remove all the hydrogen from these red stars, because as soon as the hydrogen is almost gone, any reasonable wind would be very weak for the then small radius, high gravity star. This was also the result also of the early calculations of Sch&nberner (1979,1981), and subsequent ones by Iben and his many collaborators such as Iben and McDonald (1985,1986).

This residual hydrogen has been a problem for those stars that clearly have no surface hydrogen at all, such as the GW Vir variables and the DB white dwarfs. This situation has been elucidated by Iben (1984) who shows that the final abstraction of all the hydrogen is due to a helium shell ttash during some stage after the star has left the asymptotic giant branch. Then the increased luminosity creates a surface convection zone that transports the hydrogen to deep, hot levels where it is all burned. This helium flash also makes the star return again to the red giant region for another wind mass loss

358

episode to blow away the remaining hydrogen with again a possible planetary nebula shell illuminated.

The results of our white dwarf pulsation calculations are mostly based on a Lagrangian method for getting eigensolutions for low order g modes. The method is described by Pesnell (1987).

Figures 1 and 2 show the growth rates for pulsation modes at the blue edge of the instability strip for, respectively, 10 -4 and 10 - s M o layers of hydrogen at the surface. One can see that in both cases the growth rates generally increase with period (with g mode order) up to the longest unstable period, and they range from 10 -8 to 10 -1 per period. This independence of the pulsation upon hydrogen layer thickness agrees with the observed fact that apparently all the white dwarfs that are in the instability strip are pulsating in one or more nonradial modes. If only the thin layer ones were unstable, then a thick hydrogen shell white dwarf would be stable. There would then be some nonpulsating white dwarfs in the ZZ Ceti instability strip.

2. Pulsation Mechanisms

In the process of making these calculations it became apparent that the pulsation driving was frequently at the base of the deep convection zone at temperatures far above those where the hydrogen ionization ~¢ and "7 effects operate. We found that the frozen-in convection assumed in this work acts to periodically block the convection luminosity, and the driving was not at all due to the ~c and ~, effects. Adaptation of the convection by a crude model shows that convection blocking in some way is necessary to make the star pulsate. Figures 3 and 4 show this convection blocking driving at the effective temperature of l l ,000K for our 0.6 M• models. In all cases we have needed to assume a rather large convection efficiency to produce white dwarf pulsations at all, as others have also noted.

Figure 3 plots versus zone number and surface mass fraction 5 different structure variables. The FRFT is the fraction of the total luminosity that is being carried by radiation. It is less than 10 - a in the middle of the convection zone. The mass zoning is also given to indicate that the Lagrangian mass shells range in mass between 10 is and 102o grams in the convection zone and just below where all the driving is present. The Fs - ] dip shown at the surface between zones 500 and 600 is due to the hydrogen ionization, and this low F3 makes the gas very compressible so that at the contraction stage of the pulsation the density is increased considerably. The two logarithmic derivatives of the opacity with, respectively, the density and temperature are key variables in determining if there is radiation blocking during the contraction stages. The density derivative is always positive tending toward increased opacity at this time, but often the temperature derivative is negative allowing for a net decrease in the opacity, and radiation leaking at maximum

compression.

Figure 4 gives the pulsation driving and damping for the g7 to gl0 modes for 1=2 for this model. At zone 490 where the driving is a maximum, the radiation blocking

359

mechanisms are not effective because the gamma has already returned to the completely ionized value, and the opacity derivative with respect to temperature is very negative. The pulsation driving is clearly operating at the bo t tom of the convection zone, because of the convection blocking effect, and not because of radiation blocking.

Figures 5 and 6, however, show that just 500K hotter at l l ,500K, very near the pulsation instability strip blue edge, the usual radiation blocking (~ and ~ effect) mechanisms are operating at the much cooler zones near zones 522. These temperatures are near 31,000K instead of the 94,000K for the previous model.

3. Other Results

In addition to these results we also have the following details:

1. Modal selection is caused by the interaction of the eigenvector shape with the composition and the convection luminosity gradients.

2. The theoretical blue edge of the pulsational instability strip is between ll ,500K and 12,000K. If the observations really give blue edge temperatures up to 13,000K, we do not know any mechanism for such hot star pulsations. The Greenstein (1982) blue edge, that we agree with, adopts the Hayes-Latham temperature scale.

3. The longest periods (up to over 1000 seconds) are predicted to occur in the middle of the instability strip as observations indicate. They are not so strongly driven at the blue and red edges, and there only the shorter, more unstable periods are predicted.

4. Radial pulsation modes are still definitely predicted. They must be stabilized by time-dependent convection in a way not yet theoretically known.

5. Helium surface white dwarfs pulsate at 24,000K to 27,000K surface effective temperature. It is possible that a thin hydrogen layer can cover the helium (DBV) star to have an apparent DA white dwarf pulsate in the DB star instability strip.

360

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361

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0.8

0.| - " ~

0.I

0.2

OA

,-O.a'

-0.4

-,,O.I

W H I T [ O W M I f 11.SOOK L / I . I P w 3 . 5

I , i - t l

Z m

4

2

1

¢

;L

WI~II~ OWAIIIr ~SOOK L/HPa2.5 t=2 i 1 i i

Y.I-I I &l, , . l i 4..I,,41 4.1-1

Zm

Fig. 5 St ructure var iab les for the I I , 500K model.

Fig. 6 Work per zones to cause pu lsa t ion versus zone number and surface msss f r s c t i o n for four g modes driven by r ad ia t ion blocking.

REFERENCES

Fujimoto, M.Y. 1977, Pub. Ast. Soc Japan 29, 331. Fujimoto M.¥. 1982, Ap. J. 257, 752. Greenstein, J. L. 1982, Ap. J. 258, 661. Iben, I. 1984, Ap. J. 277, 333. Iben, I. and McDonald, J. 1985, Ap. d. 296, 540. Iben, I. and McDonald, J. 1986, Ap. J. 301, 164. Pesnell, W.D. 1987, Ap. J. submitted. SchSnberner, D. 1979, Astron. Astrophys. 79, 108. SchSnberner, D. 1981, Astron. Astrophys. 103, 119. Winget, D.E., Van Horn, H.M., Tassoul, M., Hansen, C.J., Fontaine, G. and

Carroll, B.W. 1982, Ap. J. Left. 252 L65.

362

ENSAMPLING WHITE DWARF g-MODES

W. Dean Pesnell Joint Institute for Laboratory Astrophysics, University of Colorado and

National Bureau of Standards, Boulder, Colorado 80309-0440

ABSTRACT

We examine the pulsation spectrum of zero t4mperature models as a first ap-

proximation to white dwarf stars. The g-mode spectrum of such objects is found by

USing a constant adiabatic exponent and we have found that the core is an important

region for the determination of the eigenvalue. A comparison to an evolutionary

model is given.

I. INTRODUCTION

Pulsations of white dwarf star models have come a long way since the calcula-

tions of Sauvenier-Goffin (1949), Schatzman (1961), and Harper and Rose (1970).

These early calculations were based on the perturbations of completely degenerate

(or zero-temperature) objects described by Chandrasekhar (1967). Here the equi-

librium model assumes that the pressure and density are related by a parametric

equation of state, reducing the equations governing hydrostatic equilibrium to a

polytropic like expression. Such models do not have any g-mode spectrum as they

are neutrally stratified.

More realistic models of white dwarf stars can show pulsations of all three

classifications, p-, f-, and g-modes (see e.g. the review of Winget and Fontaine

[19~2]). While investigating evolutionary models of 0.6 M~ white dwarfs, we found

the g-mode spectrum quite different from that of earlier calculations using the same

model. The difference is due to the treatment of the square of the Brunt-V~is~l~

~requency (N2). The particular model being investigated had a varying helium con-

tent throughout. When the chemical composition is included by using a numeric de-

rivative for N ~, the eigenvalues found using the Eulerian version of the pulsation

Code agreed with the Lagrangian version. To verify this behavior, a simple model of

this phenomena has been devised.

In a completely degenerate model the adiabatic exponent is assumed to vary with

position in a known fashion that maintains N 2 = 0. After trying various equations

of state, the simplest scheme was to use the fully degenerate equation of state for

the initial model but to set r I equal to 5/3 instead of the usual formula. This

gave an N 2 with a spatial variation resembling that of the evolutionary model. Near

the surface, realistic models have a convection zone that has an almost neutral

363

stratification; this is reflected in the simple model by having F I approach 5/3 at

the surface. A broad flat region in the interior is reproduced, and N 2 decreases to

zero at the center due to the effects of the variation of the local gravity. The g-

mode spectrum of this simple model has a weight function similar to the realistic

model, allowing us to conclude that g-modes of white dwarf stars probe much deeper

than previously was thought.

II. INITIAL MODEL AND PULSATION ANALYSIS

The equations that govern completely degenerate configurations are well known

and can be found in Chandrasekhar (1967, Chap. XI). Although it is possible to cast

the equations of linear, adiabatic stellar pulsations into a form specific to these

models (Sauvenier-Goffln 1949), this was not done here. Therefore, the polytropic

function, ~, was used to construct a stellar model with a radius and mass appro-

for the value of I/y~ chosen. The run of the pressure, density, gravity, and prlate

mass with radius was found in a form compatible with the dual-centered model builder

described by Pesnell (1983).

For calculating the adiabatlc~ nonradial pulsational properties~ we use the

Lagranglan formalism presented by Pesne11 (1986a, Paper I). The boundary conditions

used in this investigation are discussed in Paper I and in Pesnell (IgS6b). To

compare with another calculation, the periods were verified using an Eulerian code

written by C. J. Hansen (Kawaler, Hansen, and Winger 1985).

III. SUMMARY OF PULSATION RESULTS

The primary purpose of this paper is to show that a simple model exists for g-

modes in models of white dwarf stars. In Fig. I, N 2 from an evolutionary model,

kindly provided by D. E. Winget, is shown. This model has a mass of 0.6 M@ and an

effective temperature of 13,969 K. This curve corresponds to using the numeric

derlatlve of the pressure and density in N 2. If we calculate the gl mode for this

model, we find a period of 59.00 aec.

Next, in Fig. 2, N 2 for the simplified model, i/y~ - 0.4 and r I = 5/3, is

shown. The primary features are the long flat region from x ~ 0.3 to x ~ 0.9 in

both models. This should be compared with the behavior in a non-degenerate star

(e.g. Fig. 4 of Scuflaire 1974). It is this similarity of N 2 that leads to this

model. The gl mode for this model is 47.57 sec, but must be multiplied by (~e)I/2

(~1.4) to compare with the realistic model. The scaled period is 67.27 sec, ap-

proximately 14% too high.

We note that various authors have stated that white dwarf g-mode pulsations are

limited to the envelope (see, e.g., Koester 1976). The only method possible to de-

scribe where an eigenvalue is determined is to consult the weight function of the

364

4 3

2

!

0 ~ - I

-.2

- 5 0

M = 0 . 6 0 M ® , T e f f = 1 3 , 9 6 9 K

1 I I I I I I I O.t 0.2 0.3 0.4 0.5 0.6 0.7 O.8 o.9 #.0

X

I / y o ~' = 0 . 4 0 , 9o = 3 . 5 2 4 5

5

6

7

g l ~ ~ I I I i I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X

Fig, i. The variation of the logarithm of the square of the Brunt-VHisHla fre- quency with normalized radius for an evolutionary model of a hydrogen white dwarf with a mass of 0.6 M@ and an ef- fective temperature of 13,969 K.

Fig. 2. The variation of the logarithm of the square of the Brunt-V~isHIH frequency with nor- malized radius for completely de- generate object with I/y~ = 0.40.

oscillation (Chandrasekhar 1964; Schwank 1976). A more recent example is evolution-

ary period changes of hot pre-whlte dwarf models (Kawaler, Hansen, and Winget 1985).

In Figs. 3 and 4, we present the total weight functions for the realistic model

With the numeric derivative for N 2 (Fig. 3) and the completely degenerate model with

r I = 5/3 (Fig. 4). Each weight function is plotted as a fraction of the total weight

so that the area under a curve is unity or I/R, depending on whether the simple model

or evolutionary model is displayed. It is obvious from these pictures that this mode

is determined in the central regions of the star when plotted against radius.

M = 0 . 6 0 M ~ 3 , T e f f = I 3 , 9 6 9 K ~ 1 / y o 2 = 0 . 4 0 , ~ 0 = 3 . 5 2 4 5

o ,4 "~ o.oor

¢0,2 E = I 0

~ s st

~t 6

I-.

5 Q.

tO 0 ~ o 0.1 0.2 o.Z, 0.4 0.5 0.6 0.7 0.8 0.9 I.o

X

Fig. 3. The weight function of the gl-mode for the evolutionary model, plotted against the normalized radius.

~0,05 I v O.04F b- z o.o~ 0

0.01

r:, o

~ o.ol o

I I I I L l I I I O. I 0.2 0.3 0,4 0.5 06 0.7 08 0.9 I.O

X

Fig. 4. The weight function of the gl-mode for a completely degenerate object with I/y~ = 0.40, plotted against normalized radius.

365

The implications for the ZZ Ceti stars are quite clear. First, the order of a

mode that fits the observed period will be larger than in earlier calculations. Sec-

ond, the effects of the core's evolution will govern the pulsation evolution -- not

the envelope. This means that crystallization of the core can change the period

structure. Third, in a nonadlabatlc calculation the growth rates for a single t-

value will have a much smaller variation with increasing period (see Pesnell 1986c).

Fourth, resonances with the composition gradient regions near the surface of the star

will be less effective in "filtering" the pulsations. Resonances are not eliminated,

but they should be interpreted in terms of a lobe of the weight function fitting into

a region of constant composition. Lastly, the blue edge will be hotter than the pre-

dictions of Winget (1981) as the transition zon~ for a shorter period is closer to

the surface.

IV. CONCLUSIONS

A simple model showing that g-modes in degenerate stars are not necessarily

confined to the non-degenerate layers is presented. The calculated period and weight

function can be favorably compared to an evolutionary model. Various implications of

such g-modes are listed.

Support for this project has been provided, in part, by National Science

Foundation grant AST85-15489 through the University of Colorado. The numeric portios

of this work was carried out on the JILA VAX 8600.

V. REFERENCES

Chandrasekhar, S. 1964, Astrophys. J. 139, 664. Chandrasekhar, S. 1967, An Introduction t o the Study of Stellar Structure (New York:

Dover Publ.). Cox, J. P. and Giuli, R. T. 1968, Principles of Stellar Structure (New York: Gordon

and Breach). Harper, R. V. R. and Rose, W. K. 1970, Astrophys. J. 162, 963. Kawaler, S. D., Hansen, C. J., and Winger, D. E. 1985, Astrophys. J. 295, 547. Koester, D. 1976, Astron. Astrophys. 52, 415. Pesnell, W. D. 1983, Ph.D. Dissertation, Univ. of Florida. Pesnell, W. D. 1986a, Astrophys. J., in prep. (Paper I). Pesnell, W. D. 1986b, in preparation. Pesnell, W. D. 1986c, Astrophys. J., in press. Sauvenier-Coffin, E. 1949, Ann. d'Astrophys. 12, 39. Schatzman, E. 1961, Ann. d'Astrophys. 24, 237. Schwank, D. C. 1976, Astrophys. Space Sci. 43, 459. Scuflalre, R. 1974, Astron. Astrophys. 36, 107. Winger, D. E. 1981, Thesis, Rochester University. Winger, D. E. and Fontaine, C. 1982, in Pulsations in Classical and Cataclysmic

Variable Stars, papers presented at a conference held in Boulder, CO June I-4, 1982, eds. J. P. Cox and C. H. Hansen (Boulder: JILA), p. 78.

366

UNIFORM PERIOD SPACINGS IN WHITE DWARF MODELS

Steven D, Kawaler

Center for Solar and Space Research,Yale University

New Haven, CT USA

Introduction

Asymptotic analysis of the equations of nonradial adiabatic oscillation shows that there is a uniform

period spacing for g-modes with the same degree (l) and consecutive values of the radial wave

number (n) (Tassoul 1980). If modes with the same I but different n are presem in a pulsating star,

then comparison of the period spacings in appropriate stellar models with period differences between

the observed pulsation periods can provide mode identifications, and thereby constrain other physical

properties of the star.

Since the pulsating white dwarfs are g-mode pulsators, those with rich power spectra may have period

spacings that are integer mt~fiples of some uniform period interval. Indeed, we recently demonstrated

that PG1159-035, a DOV star, shows very strong evidence for such spacings (Kawaler 1987). The

periods of PC1159 show spacings that correspond very closely to theoretical models for modes with

1--.1 and/or I =3. Since the period spacing of g-modes in DOV stars is sensitive to total stellar mass

alone, this property of the period spectrum of PG1159 aiso strictly constrains its mass to be 0.60"20.02

M e. New photometric observations of the other DOV stars are being undertaken (Hill, private

communication), and progress continues in unravelling the complex lightcurve of the pulsating nucleus

of the planetary nebula K1-16 (Grauer, Bond, and Green 1987). Hence we will soon be able to extend

this technique to other DOV stars, and improve our understanding of the modes of pulsation and

physical properties of this class of pulsators.

Of course, most compact pulsators are cooler than the DOV stars: the DBV and ZZ Ceil, or DAV,

stars. In this paper, we calculate the period spacings for some representative DBV and DAV white

dwarf models. We then briefly examine the applicability of the analysis of period spacings to the DBV

and DAV stars.

Evaluation ofg -Mode Period Spacings

Tassoul (1980) shows that the periods of high radial overtone g-modes with the same degree I obey the

asymptotic relation:

367

l-In -r ino "~ rio [ / ( l + 1)]-1/2(n - no) (1)

where l'I n is the period for a mode of order n, and no is a reference mode. The "characteristic period

spacing" r i o is determined by the following integral:

i-io(l-i) = 292 [Jb (N/r) dr] -1 (2)

where N is the Bmnt-V~iisal~i frequency and a and b are the irmer and outer boundaries of the

propagation region for a g-mode with a period of r i seconds. (For a thorough derivation of the above

expressions and a discussion of their use, see Tassoul [ 1980].) The asymptotic period spacing, I']o(~),

is obtained by integrating over the whole equilibrium model. We evaluated 17[ 0 using equation (2) for

periods of 100 s and longer in models of DOV, DBV and ZZ Ceil stars.

Results for Models of White Dwarfs

Models of evolving DO white dwarfs CKawaler 1986) show that I'I o is insensitive to luminosity and

composition at L>10L,; it is most sensitive to total stellar mass. This is because the periods are formed

within the degenerate core of all models over most of this luminosity range CKawaler et al. 1985).

Thus, the periods and period spacings are determined in a degenerate region of uniform chemical

composition, where the stratification depends only on total stellar mass.

In cooler models, the region of period formation lies closer to the surface, and therefore the periods are

more sensitive to the composition and stratification in the nondegenerate outer layers. Therefore, in this

study, we use compositionally stratified models of white dwarfs. The models of DBV and 71 Ceil

stars are similar to those of Iben and Tutukov (1984); the DBV models have also been discussed in

Kawaler et al. (1986).

In Table 1, we show the values of ]'I o for various 0.60 M o models. In the DOV models the period

spacing is within 2% of the asymptotic value for the observed range of periods (300-800 seconds).

The remarkable uniformity of the period spacings in PG1159 is therefore consistent with the model; in

fact, it is this property that helps make the uniform spacing detectable at a statistically significant level.

In the DBV and DAV models, however, the period spacing decreases by up to 10% over the same

period range. This more rapid decrease of I I o with increasing ]'I results from the behavior of the

Brunt-Vttis~ilti frequency with depth. One condition for propagation of g-modes is that the pulsation

368

frequency be less than N. N is large in the outer layers, where the composition transition layers lie, and

falls to zero in the center. So, as the pulsation frequency decreases (1"I increases), the mode samples

deeper into the core. With more of the star contributing to 1-I o at longer periods, the value of the

integral in equation (2) is large, and therefore I I o is small, for large 11. The difference between 1-I o for

modes with short periods and I'I o for modes with long periods directly reflects the value of N as a

function of depth.

The value of I'I o in the DAV models is 20% smaller than that of the DBV models. This results from

the difference in the composition of the outer layers. As shown by (}said and Hansen (1973), the periods of g-modes are inversely proportional to the square-root of the specific heat. Since the specific

heat of the ions varies inversely with the mean atomic weight, the g-mode periods are proportional to

the square-root of the mean atomic weight in the region of period formation. Thus the "light"

hydrogen-rich outer layers of the DAV models give a smaller value of I'I o than the "heavy" helium-rich

I~BVs. Hence, the period spacings provide probes of the envelope composition and structure of the

COol white dwarfs.

Table I: N o (seconds) for White Dwarf Models

1-lo(100s) Ho(200s) rlo(400s) l'Io(800s) Iio(,,,)

DOV Model L--100L e 31.20 29.73 29.15 29.15 29.15

DBV Model Te=30,000K 60.19 55.05 52.79 51.66 50.87

Te--25,000K 70.39 62.15 59.32 58.00 56.61

Te--20,000K 84.46 72.32 68.58 66.72 64.79

DAV Model Te--10,400K 42.97 42.39 40.96 40.26 40.05

Te= 9,650K 44.18 43.82 42.21 41.47 41.22

Conclusions and Prospects

Attempts to identify uniform period intervals in pulsating white dwarfs have already provided mode

identifications and a mass determination for the DOV star PG1159-035. The frequency dependence of

the period spacings of DBV and DAV stars complicates the identification of such spacings in these

369

stars. While some white dwarfs have several periods (i.e. the DBV star GD358 [Winget et al. 1982],

and the DAV stars GD66 [Fontaine et al. 1985] and VY Hor [=BPM 31594, O'Donoghue 1986]), the

relatively small number of periods makes it difficult to identify statistically significant uniform period

intervals. However, the preliminary results presented here indicate that with careful examination of

realistic models, such spacings may be obtainable from current and forthcoming data, and will provide

a vaiuable probe of the composition and envelope structure of these stars.

This work was sponsored by N.A.S.A. Grant NAGW778 to Yale University.

References

Iben, I. Jr, and Tutukov, A. V. 1984, Astrophys. J., 282, 615. Fontaine, G., Wesemael, F., Bergeron, P., Lacombe, P., Lamontagne, R., and

Saumon, D. 1985, Astrophys. J., 294, 339. Grauer, A., Bond, H., and Green, R. 1987, these proceedings. Kawaler, S. D., Hansen, C. J., and Winget, D.E. 1985, Astrophys. J., 295, 547. Kawaler, S. D., Winget, D. E., Iben, I. Jr., and Hansen, C. J. 1986,Astrophys. J., 302, 530. KawaIer, S. D. 1986, Ph.D. thesis, University of Texas at Austin. Kawaler, S. D. 1987, LA.U. Symposium #123: Advances in Helio- and Asteroseismology, ed.

J. Christensen-Dalsgaard (Dordrecht: Reidel). O'Donoghue, D. 1986, in Seismology of the Sun and the Distant Stars, ed. D. Gough (Dordrecht:

Reidel ), p. 467. Osald, Y., and Hansen, C. J. 1973, Astrol~hys. J., 185, 277. TassouI, M. 1980, Astrophys. J. Suppl., 43, 469. Winget, D. E., Kepler, S. O., Robinson, E. L., Nather, R. E., and O'Donoghue, D. 1985,

Astrophys. J., 292, 606. Winget, D. E., Robinson, E. L., Nather, R. E., and Fontaine, G. 1982, Astrophys. J, (Letters), 262,

Ll l .

370

THEORETICAL EXPRESSION FOR THE RATES OF CHANGE OF NON-RADIAL PULSATION

PERIODS IN RAPIDLY EVOLVING STARS

P. Smeyers and P. Bruggen

Astronomlsch Instituut

Katholleke Unlversitelt Leuven) Belgium

Consider a spherically symmetric star that is rapidly evolving on the Helmholtz-

Kelvin time scale. Assume that the star is in instantaneous hydrostatic equilibrium

but generally in thermal imbalance. Suppose that the star is subjected to a small

non-radial pulsation. Both the evolution and the pulsation are considered to be

reversible processes.

We use spherical coordinates and write the equations that govern the star's

eVolution and pulsation in a Lagrangian description. We characterize the mass

elements by means of three parameters: the mass m contained in the sphere with

radius equal to the distance of the mass element to the center, and the angular

COordinates 0 and ~. T h e equations that govern the star's evolution and those that

gOVern the s t a r ' s p u l s a t i o n i n the l i n e a r a p p r o x i m a t i o n can be decoup led .

In view of the subsequent treatment, we introduce dimensionless quantities. In

Particular, the dimensionless time t::) radial distance r:: to the center, and mass

distribution ~ are defined as

t: ~ t r % m ( 3 ~ ) l / 2 ' r;" = ~ , m = ~ .

Any non-radlal pulsation mode of a rapidly evolving star

~ a v e e q u a t i o n

(i)

satisfies the vector

r~2(Sqi)]

+ (US i" + UNs,ij)6ql = 0 . (2)

In t h i s e q u a t i o n , g t j a re the components o f the m e t r i c t e n s o r , 6q 1 s tand f o r 6 r , 60,

~ , and the second and t h i r d terms o f the l e f t - h a n d member c o r r e s p o n d t o the

a d i a b a t i c and n o n - a d i a b a t i c t e rms , r e s p e c t i v e l y .

371

In solving the eigenvalue problem that governs periodic non-radial pulsations of

a rapidly evolving star, we must deal with the small cumulative effects of the

star's evolution and the non-adlabatic phenomena, both of which act on the much

longer Helmholtz-Kelvin time scale. The problem can be treated adequately by making

use of the method of the two time variables. In this connection, the reader is

referred to our previous investigation devoted to purely radial pulsations of

rapidly evolving stars (Smeyers and Bruggen, 1984).

An essential step in the method is the introduction of a slow time variable

and a fast time variable t +. The slow time variable is defined as

t = C t :c , (3)

where C is the ratio of the dynamic time scale to the Helmholtz-Kelvin time scale

and is assumed to be a small quantity. The fast time variable is introduced here in

the general way

dt + = B(~) dt :: • (4)

The function ~(~) will be specified below.

We first consider t h e eigenvalue problem of the isentropic non-radial

oscillations of an evolving star a t time variable ~. In this approximation, the

eigenfrequency ~0(~) and the Lagrangian displacement 6qi(~| ~, 8, ~) of a mode k

satisfy the vector wave equation

~,0 gij(6qi)k,0 + US,ij(6qi)k,0 = 0 . (5)

The integro-differential operator US,Ij is known to be Hermltian, and the set of

eigenfunctlons is complete (Kanlel and Kovetz, 1967; Eisenfeld, 1969; Dyson and

S c h u t z , 1979).

In order to solve wave Eq, (I) for mode n, we express the second time derivative

in terms of derivatives with respect to the time variables t + and ~ and expand the

function p(~) and the eigenfunctlon (~ql) n as

~(~) = mn,0(t) + C=n,l(t) ...

• ~ i

(6qZ)n(t +, t; ~, 8, ~) = Fn,0(t +) (~q)n,0

In using the latter expansion, we assume that a

valid zeroth-order approximation In the whole mass of the star.

(6)

+ C ~Fk, I (t +, t)(~q )k,0 " ' " (7)

the isentropic approximation is

By s u b s t i t u t i n g i n t o Eq. ( I ) , we a r e a b l e t o d e r i v e s e t s o f e q u a t i o n s f o r t h e

various orders of C.

372

It follows that t h e zeroth-order approximation of the solution

function of the fast time variable

Fn,o(t +) An,O + sin t + = cos t + Bn, 0 ,

where An, 0 and Bn, 0 are two undetermined integration constants.

is an harmonic

(s)

Next, we require the first=order approximation of the solution to remain bounded

in the fast time variable for large values of this variable. The requirement leads

to equations that relate the derivatives d~n,O/d~ and ~(~qi)n,0/~. In order to

eliminate the latter partial derivatives, we differentiate Eq. (5) applied to the mode n with respect to t. By integrating over the whole mass and making use of the

Hermitlcity property of the operator OS,ij, we derive the following integral

expression for function of t=

the rate of change of the zeroth-order pulsation period Pn,O as a

1 dPn,o 1 L x'rW1 t2 f Ogl.j . . . . (6q3)n,O (6qI)n, 0 dm:~" Pn,0 d~ 2 n,O ~ ', ~n,0 Tn,0 M ~t

+ 1 3US~ ij " dm;Q 7 ; (~qJ)n,0 ~ ,(6qZ)n,O

M

where • i

Tn, 0 ~ I giJ(~qJ)n, 0 (~q)n,O M

d~ dm ~: = sin 0 dO d~ ~ •

, (9)

din:: , (!0)

(11)

A bar on a quantity denotes the complex conjugate.

The integral Expression (9) can also be derived from the property that the mean

values of the kinetic energy and the potential energy taken over a pulsation period

in the fast time variable remain equal as a function of the slow time variable in an

evolving star.

From integral Expression (9), it follows that the rate of change of the zeroth-

order approximation of the period as a function of the slow time variable is

determined by the isentroplc approximation of the pulsation mode and by the rate of

change of a number of physical quantities relative to the evolving star, a l l taken

at time variable t considered.

We perform the i n t e g r a t i o n s wi th r e s p e c t to the angular v a r i a b l e s O and ~ in

Express ion (9 ) for s p h e r o i d a l modes

373

(6r)n,0 = a(r) Y~(O, ~) ,

(~O)n, 0 = b(r) b0 '

I ~Y~(0, ~) (6~) = b(r)

n,0 sin 2 0 ~

(12 )

with

If we neglect the terms relative to the rate of change of V~', we find

~r x b 2 l dPn,0 l I{2 u) 2 I(£ + I) r::

Pn,0 d~ 2 2 n,0 ~ n,O Nn,0 0 J

(r:;2a) ~x _~2 ~r :~ I 2 c 2 1 ~_2o+ __)

r ........... ~m

+4 2 ~ ~_~+&~'_2") ~:)d~ (7~? r ' - ~? r '"

I' = [ a 2 + £(£ + 1 ) r : , ' 2 b 2 ] d ~ ,

Nn'o 0

- - 8(r~a) - £(£ + I) b

~m

1 ~:," =

.,2 ~r:: r~

r u

m g = - ~ - .

r

(13)

(14)

(15)

( 1 6 )

Eventually, we hope to apply integral Expression (9) to models used by Kawaler

et el. (1985) in their study of the evolution of pulsation properties of hot

pre-white dwarf stars.

Acknowledgement

The authors like to thank Drs. S.D. Kawaler, C.J. Hansen, and D.E. Winger for

providing them with models for their further investigation.

References

Dyson) J., and Schutz) B.F.~ 1979, Proc. R. Soc. London Ser. A 368, 389 Eisenfeld, J.~ 1969, J. math. Anal. AppI. 26, 357 Kaniel, S.) and Kovetz) A., 1967, Physics Fluids I0, 1186 Kawaler, S.D., Hansen, C.J., end Winger, D.E.: 1985, Astrophys. J. 295, 547 Smeyers, P., Bruggen) P.z 1984, Astron. Astrophys. 141, 297

Erratum Astron. Astrophys. 144, 516

374

Secular Instabilities of Rotating Neutron Stars

Robert A. Managan Department of Astronomy University of Toronto

Toronto, Ontario. Canada M5S IA7

Introduction

A secular instability is distinguished by the fact that it has a growth time scale that is long compared to the dynamical time scale. They appear in the stability analysis when dissipation is included. In rotating stars the secular instabilities

that are of interest are caused by viscosity and gravitational radiation reaction

(GRR).

When a rotating star is secularly unstable one of the nonaxisymmetric normal modes grows on a time scale set by the strength of the dissipation mechanism. When both viscosity and GRR are present they compete against each other. This happens be- cause viscosity drives the star toward a nonaxisymmetric uniformly rotating equi- librium while GRR drives the star toward a nonaxisymmetric equilibrium with inter- nal motions but a static quadrupole moment. Under the influence of both types of dissipation the rotating star initially becomes nonaxieymmetric but as energy is lost the star eventually becomes axlsymmetric again, this time with a low enough

angular momentum to be secularly stable.

Lindblom (1986) has derived a method to estimate for a neutron star the critical

rotation period where it becomes secularly unstable to GRR as moderated by viscos- ity, This critical rotation period is the upper limit on the rotation rate of the neutron star. Any attempt to make the star spin more results in a nonaxisymmetric perturbation growing and quickly dissipating (on an astrophysical time scale) the excess energy and angular momentum, The estimate is derived by using the effect of rotation on the secularly unstable mode in rotating polytropic sequences and normalizing this behavior to the frequency and damping time of the mode as calcu-

lated from relativistic neutron star models.

Method

The modes that go unstable have been best studied for the Maclaurin spheroids. These analytic studies (Comins 1979, Lindblom 1986) show that the time scale for

the unstable mode, which is proportional to a6(~t+m~), to damp or grow can be writ-

ten as

. , - . . , . (n ) = ,o , . , . , ( r~) - - + ~ . ( 1 ) .,..,,.,,., , ,-G..,. , L~,,.,(n)o,-,,(o)J

In this formula ~ is the angular velocity of the star, m is the azimuthal wavenum- ber of the mode, ru,m and rGRR,m are the damping times for the mode in the spher- ical limit under the influence of viscosity and GRR respectively, ~m(~) is the

375

frequency of the mode, and ~m(~) and qm(~) are dimensionless quantities that con-

rain the angular velocity dependence of the time scale (they are equal to 1.0 for = 0 and do not vary significantly as n increases).

When the time scale changes sign from positive to negative the mode becomes unsta" ble. It is interesting to note that when viscosity is neglected (r~ = O) that

the time scale changes sign when the frequency of the mode does. Figure 1 show that this means that the modes with higher m values become unstable first. This result was first pointed out by Friedman and Schutz (1978). Equation (I) shows

that when viscosity is included the critical angular velocity (the angular ve- locity where the time scale changes sign) increases. Because the increase in the

critical angular velocity is larger for larger values of m there is some value of for which the critical angular velocity reaches a minimum value.

Setting the time scale to zero and rearranging terms results in this convenient expression for the critical angular velocities:

m \ r~,m / J

where ~ m ( a ) ~ [ a m ( n ) + m a ] / a m ( O ) . The f u n c t i o n a m i s u s e d b e c a u s e t h i s d i m e n s i o n - less quantity should not be sensitive to whether it is determined by using rotating

neutron star models (not possible at this time) or by using rotating polytropes (see Fig. 2).

To use equation (2) ffm(0) and rGRR,m are calculated using a fully relativistic spherical neutron star model, ~m and ~m are determined using the pulsation fre-

quencies of a sequence of rotating polytropes, and ru, m is taken from the studies of Maclaurin spheroids. This approach uses all the information we have from rel- ativistic neutron stars (Lindblom 1986) and models the effect of rotation with

polytropic sequences. The frequencies of the modes for the rotating polytropes are determined by a variational principle (Managan 1986). A range of minimum ro-

tation periods can be obtained by varying the polytropic sequence used to determine

am and qm and also by using spherical neutron star models with different masses and equations of stats.

It is also possible to estimate the growth time directly. The first term in the

Taylor expansion of the time scale formula can be used to set an upper limit on the time scales:

rGRR,mOm(O) \ r~,m / (3)

This approximation implies time scales on the order of several years for the modes when (~- ~m)/am is about 10X.

Results

i. To determine the critical rotation periods for rotating neutron stars vis-

cosities in the range I < v < i00 cm 2 s -I are used, as estimated by Fried-

man (1983). The values of G(0) and TGRR,m are taken from Lindblom (1986).

The values for ~m(n) and qm(~) are taken from Managan (1986). The value of

376

0.6

0.4

0.2

0

--0.2 -

--0.4 0

F i g u r e 1.

I

m=2

m=4

m=5

t t , I O.Ofl 0.16

The frequencies of the f- modes with l =m for the ~ = 1.5 poly- tropic sequence are plotted on the vertical axis in units of (4~Gpe) I/~ vs. the angular velocity in the same ~tnits (Pc is the central density).

1 . 4

a

1 . 0

0 . 0

I 0 . 2 ...... ' 0 0 . 2

tq 0.4

Figure ~. The dimensionless ratio ~m vs. the angular velocity in units

of (4~G~D z/2 f o r n = 0 (solid lines)'5nZ n = 1 .0 (short-dashed lines), a n d = . (long.dashed lines). The curves have been offset from each other by 0.2 for clarity and am is actually equal to i.O for n =0 for all values of n~ (Po is the average density of the spherical member of the sequence with the same equation of state and mass).

2.

3.

4.

ru~ ~ = ( 2 m + l ) ( m - - 1 ) v / R ~ , where Re i s the r a d i u s o f the s p h e r i c a l member of the sequence.

The modes with m = 4 turn out to have the lowest nm (or highest critical

period) for most cases.

The critical periods vary from 0.8 ms to i.8 ms. Most of this variation is

associated with the different equations of state and masses of the neutron star models. This can be seen in the fact that the Keplerian periods for

rotating neutron stars calculated with these equations of state vary from

0.40 ms to 1.84ms (Friedman ct~. 1988). Varying Just the polytropic sequence used can change the result by about i0~.

The millisecond pulsar is inconsistent wi~h the stiffest proposed equations

of state if its mass is about 1.4M® and its viscosity is in the range given

above. The rotating neutron star models of Friedman etal. (1986) also show

that rotating sequences constructed using the stiffest equations of state with

masses about 1.4fvf® never rotate as fast as the millisecond pulsar before breaking up due to centrifugal forces. The secular stability limit allows

377

tighter limits to be placed on the mass of the neutron star if these equations of state are used.

5. A pulsar with a period less than 0.5 ms is not consistent with any of the

proposed equations of state. If such a pulsar were found then either the equations of state are in question or using rotating polytropes to model the effect of rotation in neutron stars is a much worse approximation than is thought,

i.

2.

3.

Future Directions

The detailed effect of viscosity on the modes for nonuniform density models is not known. This problem has been solved~exactly only for the modes of

incompressible stars. Presumably the solution for compressible fluids is not significantly different from the solution for incompressible fluids.

A fully relativistic calculation of the frequencies of the modes of rotating

neutron star models is needed, Undoubtedly the angular velocity dependence of the mode frequencies will be somewhat different in the relativistic cal- culation than it is in the Newtonian calculation used here. Unless the rel- ativistic effects are large the changes in the results given here should not be large.

Some neutron star models have solid crusts. What effect does the presence

of a crust have on the modes? It is probably small, but the combined effect with rotation is unknown.

References

Comine, N. 1979, M.N.R.A.S., 189,233. Friedman, J. L. 1983, PAys. Rev. L~tt, 51, 11. Friedman, J. L., Ipser, J. R., and Parker, L. 1986, Ap. J.. in press. Friedman, J, L., and Schutz, B. F, 1978, Ap. J., 223,281. Lindblom, L. 1986 Ap. J.. 303,146, Managsn, R, A, 1986 Ap. J., 309,000.

378

Nonl inea r Pu l sa t ions of L u m i n o u s He S ta r s

Charles R. Proffitt I'2 and Arthur N. Cox 1

1Los Alamos National Laboratory Los Alamos~ New Mexico 87545

2Department of Astronomy University of W~hington

Seattle~ Washington 98105

A b s t r a c t

Radial pulsations in models of R Cor Bor stars and BD+1°4381 have been studied with a

nonlinear hydrodynamic pulsation code. Comparisons are made with previous calculations and

with observed light and velocity curves.

I. I n t r o d u c t i o n

The R Cot Bor stars are characterized by atmospheres with extremely low hydrogen abun-

dances, enhanced carbon, high luminosities (104L®) and effective temperatures of about 7000 K.

At irregular intervals they show abrupt declines in brightness for several months that are at-

tributed to the formation of a dust shell (Feast 1975). Many pulsate with small amplitudes and

periods of 40 to 120 days (Saio 1986).

BD+1°438I is also a hydrogen deficient star (Drilling 1979), with an effective temperature

of 9500 =t: 400 K (Drilling et al 1984), which was discovered by Jeffery and Malaney (1985) to

pulsate with a 22 day period and an amplitude AV = 0.06 magnitudes.

Linear non-adiabatic studies of luminous He stars have been done by Saio, Wheeler, and

Cox (1984), Cox et al. (lg80), Wood (1976), and Trimble (1972) among others. They find

that the pulsations are very nonadiabatic, with the thermal timescates for such stars becoming

comparable to the dynamic timescales. There is no longer a one to one corespondance between

the adiabatic and non-adiabatic modes, new "strange" modes appear and modal identification

379

becomes difficult. Hydrodynamic models of luminous He stars have been studied by Trimble

(1972), Wood (1976), King et al. (1980) and Saio and Wheeler (1982).

H. Models

We have calculated linear and hydrodynamic models for R Cor Bor stars, using the same

parameters as Saio and Wheeler used for their models 7 and 8. We compare our results to theirs

in table 1. The results of the two calculations are very similar, except that our linear growth

rates are consistently about twice those obtained by Saio and Wheeler, and we get nonlinear

amplitudes that are slightly larger. In addition to the modes discussed by Saio and Wheeler

we also list the closest adjacent modes that we found in our linear analysis. Because of the

complexity of the modal structure for these stars we do not attempt to identify the adiabatic

counterparts of these modes.

In addition to using a composition of 90% helium and 10% carbon by mass, we also com-

puted linear models using a mixture with 98% helium and 2% metals in solar proportions. These

mixtures are designated "hegcl" and "cxhdsn2a" on the Los Alamos opacity tables. The best

abundance analysis of R Cor Bor suggests that carbon is about 1.2~0 of the mass of the at-

mosphere, with nitrogen and oxygen in approximately solar abundances (Cottrell and Lambert

1982). So the appropriate opacity to use is probably intermediate between these two mixtures.

The models with lower metal abundance tend to have smaller linear growth rates, but numerical

difficulties have so far prevented calculation of nonlinear models with these opacities. Since the

models with 10% carbon show nonlinear amplitudes significantly larger than actual R Cot Bor

stars, nonlinear models with appropriate compositions should be calculated.

Our nonlinear models show a double periodicity, with alternate cycles having larger ampli-

tudes. We attribute this to the presence, with a small amplitude, of a higher order mode. Our

linear models show such a mode, with a period close to two thirds of the period of the dominant

mode. We do not know if this mode is locked into a resonance with the dominant mode or if

it will damp out after a large number of periods. While the light and velocity curves of R Cot

Bor stars are observed to vary from cycle to cycle, there has been no suggestion of this kind of

double periodicity being observed in high luminosity He stars.

SchSnberner (1979) calculated models of He shell burning stars with a CO core and a thin

He envelope in the range of 0.7 to 1.0 M®. Stars with the same total mass but with a smaller

core mass could have lower luminosities than these models, but it would be difficult for a star

380

of the same mass to have a higher luminosity. If we assume a M-L relation for BD÷1°4381

consistent with SchSberner's models and a Teff of 9500 K, we find a pulsationally unstable mode

with the observed period of 22 days if we use a mass of 0.95M O (See table 2). For lower masses

the longest pulsationally unstable mode that we find has too short a period, unless we assume a

luminosity higher than consistent with SchSberner's models. Our models show very high linear

growth rates and large nonlinear amplitudes (see table 2). As with our R Cor Bor models a lower

metal abundance has a strong effect on the growth rates of the pulsational modes and there is

a 2:3 period ratio between the 22 day mode and a higher order mode that results in a double

periodicity in the light curve. Published light curves of BD÷1°4381 are not detailed enough to

show whether or not such behavior is actually present.

Such a large mass and high luminosity seems inconsistent with BD-t-l°4381's location about

25 ° below the galactic plane. It would require a very large error in the measured effective

temperature to reduce this mass significantly. An effective temperature of about 8000 K is

needed for a 0.TM~ star with the luminosity of SchSnberner's model to have a unstable mode

with a period near 22 days.

Re fe rences

Cotrell, P.L. and Lambert, D.L. 1982, Ap. J., 261, 595. Cox, J.P., King, D.S.,Cox, A.N., Wheeler, J.C., Hansen, C.3., and Hodson, S.W. 1980, Space

Sci Rev., 27, 519. Drilling, J.S. 1979, Ap. d. 228,491. Drilling, J.S., SchSnberner, D., Heber, U., and Lynas-Gray, A.E. 1984, Ap. J., 278, 224. Feast, M.W. 1975, in IAU Syrup. ~67, Variable Stars and Stellar Evolution, eds. Sherwood,

V.E. and Plaut, L. (Dordrecht: Reidel), p. 293. Jeffery, C.S. and Malaney, R.A. 1985, M.N.R.A.S. 213, 61p. King, D.S., Wheeler, J.C., Cox, J.P., Cox, A.N., and Hodson, S.W. 1980, in Nonradial and

Nonlinear Stellar Pulsation, eds. H.A. Hill and W.A. Dziembowski (Berlin: Springer-Verlag), p. 161.

Saio, H. 1986 preprint. Saio, H. and Wheeler, J.C. 1985, Ap. J. 295, 38. Saio, H., Wheeler, J.C., and Cox, J.P. 1984, Ap. J. 281, 318. SchSnberner, D. 1979, Astr. Ap. 79, 108. Trimble, V. 1972, M.N.R.A.S. 156, 411. Wood, P.R. 1976, M.N.R.A.S. 174, 531.

381

Table 1

0.9M® 17000L® 7100 K

PL (days) y Comp

This Paper 72.2 -2.8 He9C1 " 32.6 +1.9 " " 21.0 -0.002 ~'

Saio and Wheeler 32.9 +1.0 He9CI This Paper 75.8 -2 .3 Cxhdsn2a

" 37,0 --0.57 "

20.0 -0.60 s

This Paper Saio and Wheeler

PNL (days) Av(km/see) AMbo! Comp

37 55, 1.4 He9C1 37 46 1.2 He9C1

0.TM® 15000L o 7000 K

PL (days) ~ Comp

This Paper 64.1 -1.5 He9C1 " 35,8 +2.9 " " 23.5 --1.1 "

Saio and Wheeler 36.2 +1.4 He9C1 This Paper 64.2 - 1 . i Cxhdsn2a

" 38.7 +2.1 " " 22.9 --0,I "

This Paper Saio and Wheeler

PNL (days) Av(km/sec) A Mbo, Comp

41 46 1.2 He9C1 42 40 LO He9C1

Table 2

0.95M®

/%

39.1 21.7 14.3 42.0 18.9 14.6

33000L® 9500 K

r/ eomp

-0.63 He9C1 +0.51

+0.05 " -0.53 Cxhdsn2a +0.92

+2 .1

PNL = 21.4 days AMbol = 0,6 AU = 65 km/sec

Observed values for BD+1°4381 are P = 22 days, and AV = 0.06.

382

PULSATIONS OF CATACLYSMIC VARIABLES

Brian Warner Department of Phgslcs and Astronomy, Dartmouth College, New Hampshire,

U.S.A., and Department of Astronomy, University of Cape Town, 7700 Rondebosch,

South Africa

I. ~N~TRODUCTION

The rapid oscillatory phenomena observed in the light curves of

cataclysmic variables can be broadly classified into the highly

coherent, typified by DQ Herculls which was the first to be discovered

(Walker 1954); the quite coherent, characteristic of some of the

oscillations seen in dwarf novae during outburst, discovered by Warner

& Robinson (1972) and referred to nowadays as DNO; the quasi-periodic

oscillations, also seen in dwarf novae during outburst, first

described by Robinson & Nather (1979) and referred to as QPO; a~]d the

stochastic flickering, present in all cataclysmic variables and

variously ascribed to the bright spot, accretion disc, or dlsc-star

boundary layer.

AS the writer has recently published a review of the

observational characteristics of the rapid oscillations in cataclysmic

variables (Warner 1986a), we will offer only an outline of their

properties here, but give more coverage of the various suggestions

that have been made in explanation of the phenomena.

2. DO HERCULIS

The 71s oscillations in DQ Her are potentially the most useful

for an initial understanding of at least one type of cataclysmic

variable pulsation. As well as amplitude and phase variations through

eclipse, the oscillations show variations around orbit (Patterson,

Robinson & Nather 1978). The variations through eclipse are

convincingly modelled by a radiating beam rotating at what is

presumably the white dwarf rotation period and illuminating the

surface of the accretion dlsc (Petterson 1980). The variations around

orbit are qualitatively understood in terms of the beam alternatively

illuminating and being obscured by a raised rim on the accretion disc,

possibly associated with the heated region at and down stream of the

bright spot (Chester 1979, O'Donoghue 1985). There is, however, still

no satisfactory quantitative agreement between model and observations:

384

in particular, not only must the phase and amplitude of the ?Is

oscillations be modelled around orbit, the harmonlc content of the

oscillations must also be reproduced (O'Donoghue 1985). Until this

very detailed information available for DQ Her is convincingly

interpreted we feel it premature to claim any certain understanding of

the greater variety but less constrained observations of the DNO.

3. ~ DWARF NOVA OSCILLATIONS

The observed low amplitude optical oscillations in dwarf novae

during outburst, and in some nova-llke variables, are listed in

Table i of Warner (1986a). The 40s ~ in V2051 Oph has since been

reclassified as a Q~ (Warner & O'DOnoghue 1986), which results in the

DNO being confined to the range 7.5-39s, with the range at maximum

light (i.e. at minimum observed period) being even further restricted

to 7.5-32s with a preference for 24±4s. Why there should be such a

strong concentration of periods in this latter range is not

understood, but it indicates a uniformity in the underlying mechanism

that may result from a narrow range of white dwarf masses. The only

stars to lle significantly outside the 24t4s range are SS Cyg (7.5s),

RU Peg (ll.6s), EM Cyg (14.6s) and Z Cam (16.0s), all of which have

orbital periods longer (Porb > 6h) than most dwarf novae. However, AH

Her with Porb = 5.93h has not been observed to oscillate with a period

less than 24s.

4. 0UASI-~IODIC OSCILLATIONS

The observed 0Po are also listed in Table 1 of Warner (1986a).

In general, the OPo have periods a few times greater than the DNO in

the same object. Thus in Ss Cyg the DNO lle in the range 7.5-9.7s and

the OFO in the range 32-36s. However, one very important exception is

VW Hyl, which probably has the most complex set of oscillations

observed in any of the cataclysmic variables.

VW Hyi shows DNO in the range 20-32s late in the outburst. In

addition, OpO near 88, 250 and 410s have each been observed on more

than one occaslon. The ~250s modulation shows up both as a general

brightness variation near maximum light and as a modulation of the PNO

late in outburst (Robinson & Warner 1984). This long period Qpo

modulation of the DNO produces a power spectrum (Figure I) that

resembles that of a OPO itself. Figure l shows a range of such

effects. In the lowest frame the ~250s modulating QPO shows as a

385

0.0

0.0

-2.0

-2.0

-2.0

-2.0

i

2.0 4.0 8.0 8.0 lO.O

-2.0

- 4 . 0

I 0 -0 2.0

I

4.0 6.0 8.0 lO.O E-2

Figure I. Power spectra of VW Hyl showing the presence of ~ru~sl- periodic oscillations with a range of amplitudes. The vertical scale is logarithm of power; the abscissa is frequency in Hzo From top to bottom the observations were made respect Ively on the nights of 8 January 1973, 6 December 1973, 12 September 1972, 25 December 1972 and 6 January 1978.

386

spike of power near 0.4xl0-2Hz; in the next frame the ~400s

modulating QFO is also visible. The upper three frames show

diminishing amplitude of the periodicity centred on -30s but no

obvious modulating QPO - although one is probably present. Thus the

normally quite coherent DNO at ~30s is spread out in the Fourier

amplitude spectrum by the effect of modulation of its amplitude (and

possibly also its frequency or phase) by a typical QPO. This has not

been seen in any other star but it demonstrates an important

interaction between the 0Ff) and DNO.

More commonly, the DNO decrease in coherence and become QPO as

they decline in apparent amplitude during the later stages of an

outburst. That is, the power in a DNO becomes spread over a wider

range of frequencies (e.g. Cordova et al. 1984).

5. X-~6Y OSCILlaTIONS

The soft X-ray observations observed in SS Cyg and U Gem are

described and analysed by Cordova et al. (1980, 1984) and in VW Hyi by

Heise, Faerels & van der Woerd (1984). In SS Cyg the oscillations are

in the same period range as those seen in the optical but are in

general less coherent. However, there is a wide range of coherence

observed and, as the signal-to-noise is so much greater in the X-ray

emission, probably indicates only that the optical oscillations must

be quite coherent before they can be detected at all. Certainly in

the dwarf nova TY PsA (PS74) where relatively large amplitude optical

oscillations are sometimes seen (O'Donoghue 1985) the coherence

Properties resemble those seen in the SS Cyg and U Gem X-ray

observations: Figure 2.

6. IN~PRETATIONS

B o t h DNO and OPO h a v e p e r i o d s i n t h e r a n g e e x p e c t e d f o r Keplertan orbits around a white dwarf. The low viscosity in gaseous accretion

discs results in near-Keplerlan circular motion in the simplest

models. As a result, several proposed models appeal to disturbances

in the disc for the origin of the oscillatory phenomena. Other models

introduce channeled conduction from a corona and surface oscillations

of the white dwarf.

387

.oo u3 ~-

31[

~-

C~

C3

=

~0,00

TIMF_ (S] ,I0' 2o.00 40.00 60.00 8,0.00 _ ~oo.oo ,2o.oo

k

k

;'0,00 40.00 60.00 80,00 I00.00 TIME ( 5 1 ~lO'

l j40.00

t

t

". :

,'~o.oo 1'4o.oo

(:3

[ 6~. O0

o o

"~LLJ 13f £.9 UJ CD

OO--

'T" n

CD

==

'6~ .oo

F i g u r e 2. Phase of 25.2s oscillations in TY PsA (P874) as a function of time. Each point and associated error bar is a 150s data segment with 50 percent overlap.

6.1 OSCILLATIONS OF ACCRETION DISCS

Because of the weak coupling in the radial direction, vertical

oscillations of an accretion disc are well described by considering

only the motions of individual annull (Kato 1978; Van Horn, Wesemael

& Winget 1980; Cox 1981; Cox & Everson 1982). It is found in this

approximation that the period of oscillation of an annulus is

proportional to, and not very different from, the Keplerian period at

the radius of the annulus. Superposltion of the contributions from

the visually brightest annull produces a spread of power with periods

in the range 10-150s, very similar to what is observed in the QFO. An

excitation mechanism for such pulsations is not yet known. If a

highly selective mechanism were found, then disc oscillations near the

white dwarf might also account for the DNO.

More general treatments, performing perturbation analyses of thin

or thick accretion discs, produce several spectra of oscillations,

analogous to p-mode and g-mode non-radial oscillations in spherical

388

stars (Abramowicz et el. 1984; Blumenthal, Yand & Lin 1984; Carroll

et al. 1985). In particular Carroll et al. (1985) find that there are

three groups of oscillations= (i) 2ZpZI40s, which overlap the DNO and

some of the QFK), (ll) 49~PZ640s, which overlap the QFO, and

(Ill) F~480s, which, if spread over a wide range of periods, would be

difficult to detect observatlonally unless of large amplitude.

There are interesting consequences of disc pulsations for which

more observational data are needed. As there is a range of annull

oscillatinG, shorter periods correspondinG to smaller disc radii and

higher temperatures will be radiated more effectively at shorter

wavelengths. There should therefore be a strong colour-perlod

relationship.

Middleditch & cordova (1982), in what appears to be the only

Study so far of the flux distribution of a QPO, find the ~160s QPO in

X Leo to be red compared to the disc, whereas the ~ 30s DNO in AH Her

and SY Cnc are much bluer than the disc (Hildebrand, Spillar &

Stlenlng 1981a; Middleditch & Cordova 1982). However, Middleditch

(public communication at Los Alamos meeting) has reservations about

the veracity of the X Leo observation.

Of more direct diagnostic value would be observations in a

particular star of the variations of power spectrum as a function of

the wavelenGth of observation. If the QPO are the result of the

superposltlon of many oscillations of disc annull then the peak power

should move to shorter periods at shorter observational wavelengths.

6.2 INHOMOGENEOUS DISCS

One of the earliest models proposed to explain the DNO was that

of orbiting Inhomogeneltles (Bath 1973; Bath, Evans & Papaloizou

1974). In order to account for the short periods, bright spots in the

disc must be orbiting close to the white dwarf. In the original model

it was proposed that eclipse of the spot by the white dwarf would

provide the modulation as seen by the observer. Some modifications of

the model are now regulred: oscillations are seen in SS Cyg and RU

Peg which are thought to have inclinations -30 ° which therefore could

not produce any significant eclipse as seen by the observer. The

further requirement that, from the phase shifts seen through eclipse,

the whole of the disc is involved in the pulsed component suggests

that eclipse as seen from a point on the disc could be involved. This

might also account for why the DNO have P~30s - for inhomogeneities

orbiting with longer periods the radial distance is too great to

generate significant obscuration by the white dwarf.

389

An attractive aspect of the inhomogeneous disc model is its

provision of a tlmescale for the lifetime of the oscillations= an

Inhomogeneity is sheared apart by differential rotation in 50-200

orbits, depending on Its size (Bath, Evans & Papaloizou 1974). such

lifetimes are characteristic of the times between the abrupt period

changes observed in TY PsA (O'Doncghue 1985 and Figure 2). However,

the coherence lifetime required to detect the 0.0002 mag oscillation

in SS Cyg (Patterson, Robinson & Klpllnger 1978) is an order of

magnitude greater than suggested by this model.

6.3 INSTABILITIES AT THE INNER BOUNDARY ~

This mechanism is distinct from the last one because It involves

intrinsic oscillations in luminosity rather than eclipse effects.

Cordova et al. (1980) first suggested instabilities of the accretion

flow at the inner edge of the accretion disc as a possible source of

the DNO observed in the soft X-ray flux of SS Cyg during outburst.

The amplitude of modulation in the X-ray region is typically 30

percent and can reach i00 percent. Clearly the modulation mechanism

is controlling a very large fraction of the accretion flow in the

vicinity of the white dwarf.

Papalolzou & Stanley (1986) find that the boundary between disc

and star can be subject to oscillations driven by viscous

instabilities. However, these have a low amplitude of awx%ulation and

although a feasible mechanism for the optical DNO, cannot be held to

account for the X-ray oscillations. If simultaneous observations shoW

that X-ray and optical DNO are different aspects of the same

oscillation mechanism (e.g. with the optical pulsating component being

the Rayleigh-Jeans tail of the X-ray flux modulation, or the former

arising from reprocessing of the latter) then this mechanism is

excluded.

6.4 .MAGNETICALLy CONTROLLED ACCRETION FROM Thee D~SC

The necessity of finding a mechanism that will account in a

natural way for the observed correlation between D~O period and system

luminosity led Warner (1983) to propose an intermediate polar model

for dwarf novae. Inhomogeneltles at the inner edge of the accretion

disc, as they orbit periodically through that region of the magnetic

field most efficacious in attaching the gas, will modulate mass

transfer onto the white dwarf. This will occur at the beat period

390

Pbeat between the Kepler period PKep at the inner edge of the dlsc

(l.e at the outer edge of the magnetosphere) and the rotation period

Prot of the white dwarf. As the radius of the magnetosphere is given

by

r 2.7 x 101° ~+/7 - 1 / V = ~33 M~i 7 (MI/Mo) cm (I)

(King, Frank & Rltter 1985), the increase in M during outburst wlll

decrease r a and hence reduce Pbeat" Subsequent reduction of M after

maximum luminosity causes Pbeat to increase again.

The model has several attractions : in the optical the

oscillations are caused by a broad beam of X-rays (from the accretion

column on the white dwarf) sweeping around the disc. Unlike the DO

Her model, where the front-back asymmetry of the disc causes the white

dwarf rotation period itself to be seen, in most DN the inclination is

great enough for such asymmetry to be small so that we see no

contribution from Prot" Instead the beam itself Is modulated at

Pbeat" As a possible extension of thls model we could consider the

effect of the beam falling on the inflated bright spot region (or the

secondary star), as in the proposed models of intermediate polars

(Warner 1983). If this happens in a 'noisy' way, because the beam

itself has rapidly varying angular size, then there wlll be an

amplitude modulated component at the beat period between Fro t and

Porb Thls could be identified with the QPO. It is a nice irony in

such a model that the fluctuating mass transfer produces a relatively

coherent [X~K) whereas the hlgh precision beat period between Prot and r

Porb appears as a QPO.

This beat period model has had some success in explaining the QPO

seen in X-ray binaries (e.g. Lamb et al. 1985), but there is a serious

problem with its application to the cataclysmic variables - namely its

inability to generate the phase shifts seen through ecliPses of the

dwarf nova HT Cas and the nova-llke variable tPX UMa.

6. S HAGN~'ICP, f-~Y COH~ O[-r-n% ACCRETION ~OM A CORONA

The ~st recent p ~ s ~ of a mec~nlsm for t h e DNO I s that b y

Klng (1985) which invokes a model in which the hard X-rays observed in

'non-magnetlc' cataclysmic variables originate In a guasl-hydrostatlc

corona surrou~Ing the white dwarf {King, Watson & Heise 1985). It is

proposed that transient magnetic fields in the surface of the white

dwarf, generated by dynamo action resulting from differential rotation

of the surface layers (which Is itself caused by the increased angular

391

momentum transfer during outburst), channel conduction electrons from

the corona to produce spots which radiate their energy in the soft X-

ray region. The variation in period of the DNO arises from slippage

of the surface layers relative to the interior of the white dwarf as

the high angular momentum material is acquired and distributed (a

similar model was considered by Warner (1985) in connection with an

intermediate polar model of SU UMa stars). Incoherence of the DNO

occurs because of the short lifetime of the individual coronal loops

that channel the electrons.

Among the attractlve features of this model is that it explains

why the soft but not the hard X-ray~ are m~mdulated; a similar

situation obtains in the intermediate polar EX Hya (Cordova & Mason

1985) but there a combination of corona and accretion column appears

more appropriate.

From the flux and spectrum of soft X-rays in SS Cyg, Cordova et

al. (1980) find that the emitting region has an area 2

~2×10~S(d/200 pc)2cm . The currently favoured distance to 88 Cyg is

76 pc (warner 1986b) so the region has an area only ~2x10 -4 of that of

the white dwarf surface. This is smaller than the areas of accretion

columns in polars but is not an unreasonable cross section for a flux

tube created in the outer envelope (c.f. the areas of sunspots).

An unsolved mystery is the 14.06s modulation of the soft X-ray

flux emitted by VW Hyl during a superoutburst (Helse, Paerels & van

der Woerd 1984) which was apparently constant for I0 days and could be

safely interpreted as the rotation period of the white dwarf were it

not that in a subsequent superoutburst a similarly constant, but

different, period was observed (reported at the B~m~berg meeting on

cataclysmic variables in 1985). With only two discordant observations

it cannot yet be decided whether the two periods are manifestations in

the X-ray of what is observed in WZ Sge in the optical: namely the

occasional presence of one or other (or sometimes both) coherent

periods.

The absence of very strong soft X-ray flux from OY Car during

superoutburst (Hassall et al. 1986), and the absence of modulation in

what X-ray emission was actually observed, is compatible with King's

model provided the soft X-rays have been lost from our direction

through obscuration by the disc in this high inclination (I~81 °)

system.

392

6.6 OSCIL~TION8 OF THE WH,~TE DWARF

The nightly variations of periods of the DNO exclude the

Possibility that they are p- or g-modes of the white dwarf. However,

the wind of accretlng gas blowing around the surface of the white

dwarf can raise waves which wlll reinforce themselves if their periods

are Pm = Prot/m' where Prot is the rotation period of the surface of

the white dwarf and m Is the azimuthal wave number (PaPalolzou &

Prlngle 1978, 1980). As the excitation of these Rossby type

oscillatlons involves little mass, and the rotation period of that

mass may vary through the interplay of accreted angular momentum and

coupling to the interior, variable periods can arise. If many high

order modes are excited simultaneously then the observed oscillations,

with their rapid amplitude modulation and phase shifts, could be a

result of the interference pattern produced by the modes. It seems

unlikely, however, that the observed large amplitude slnusoldal (but

phase shifting) modulation of the soft X-rays can be produced by such

an interference mechanism: the large amplitude requires excitation of

at most a few modes.

~%CKNOWLEIDGER~J~TS

This review was written while the author was a Visiting Professor

in the DePartment of Physics and Astronomy at Dartmouth College. The

financial assistance and hospitality of that institution are

gratefully acknowledged. The work was also partially funded by the

Foundation for Research Development of the C.S.I.R.

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Soc. 167, 7P.

Blumenthal, G.R., Yang, L.T. and Lin, D.N.C., 1984. AstrophNs. J.

287, 774.

Carroll, B.W., Cabot, W., McDermott, F.N., Savedoff, M.P. and

Van Horn, H.M., 1985. Astrophys. J. 296, 529.

Chester, T.J., 1979. Astroph~s. J. 230, 167.

Cordova, F.A. & Mason, K.O., 1985. Mon. Not. R. astr. Soc. 212, 447.

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Cordova, F.A., Chester, T.J., Tuohy, I.R. & Garmlre, G.P., 1980.

Astroph~s. J. 235, 163.

Cordova, F.A., Chester, T.J., Mason, K.O., Kahn, S.M. & Garmlre, G.P.,

1984. Astrophys. J. 278, 739.

COX, J.P., 1981. Astrophys. J. 247, 1070.

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p. 42.

Hassall, B.J.M., Naylor, T., Bath, G.T., Charles, P.A., Sonneborn, G.,

van der Woerd, H. & van ParadlJs, J., 1986. I.A.U. S~mp. No. 93,

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Hildebrand, R.H., Splllar, E.J. & Stlenlng, R.F., 1981a. Astrophys.

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395

On the 9.25 Minute X ray Oscil lations of V471 Tauri

Edward M. Sion Department of Astronomy and Astrophysics Vil lanova University Villanova, PA 19085 U.S.A.

Abstract

The 9.25 minute x ray osc i l la t ions discovered recently by K. Jansen (1985) in V471 Tauri using ~XQSAT, arise from the DA2 white dwarf (Jensen, et el. 1986) . One poss ib le exp lana t ion o f the o s c i l l a t i o n s is x ray modu la t ion due to a ro ta t i ng , magnet ized, accret ing degenerate (Jensen, e~ al. 1986). The results of model calculat ions of accret ion and d i f fus ion of w ind/ f la re material onto the V471 Teu white dwarf are presented as a test of this interpretat ion. Weak line features cons t ra i n the f i e l d i f they ar ise f rom a po la r acc re t i on area. It is shown that r o t a t i o n a l modulation of x - ray , opaque accret ion poles is probably may not be the correct explanation unless the white dwarf is strongly magnetic, accreting matter at a rate of t 0 - ' e M o y r . -~, bu t has much l a rge r p o l a r a c c r e t i o n a reas than that ob ta ined f rom k i nema t i ca l models o f the acc re t i on f low.

I. Introduction

The eclipsing, spectroscopic binary V471 Tauri CK2V+DA2; P =o.=5) has y i e l d e d a w e a l t h o f important new in format ion in several d is t inc t areas o f s te l la r astrophysics; C1) common envelope evo lu t i on (c f . Vauc la i r 1972, Paczynskt 1976, Webbink 1985), (2) the genesis o f cataclysmic va r iab les Ccf. Paczynski 1976; Verbunt and Zwaan 1981; R i t t e r 1986; (33 the s o l a r - s t e l l a r connect ion, i .e. f la res , impulsive mass eject ion, chromosphere-coronal t ransi t ion region structure, "RS CVn" l ight curve wave d is tor t ion, starspots (cf . Young e t a l . 1983, Oswalt 1979, Guinan and $ion 198t , DeCampti and Baltunas 1979, Guinan et el. t986)1 [4) V471 Tauri as a possible ancient nova (Bruhweiler and Sion 1986a= Hertzog 1986, Pskovski 1979); (5) w ind- f la re accret ion/di f fus ion onto the DA2 white dwarf ($ion and $ ta r r f ie ld 1984; Bruhweiler and $ion 1986b) and most recently, the EXOSAT detect ion of a strong sof t x - r a y f lux from the white dwarf pulsed with a 9.25 minute p e r i o d , an a m p l i t u d e o f 20% and a d o u b l e - peaked pulse p r o f i l e (Jensen et e l . 1986~.

The 9.25 minute sof t el. 1986): (1) ro ta t i ona l ro tat ional modulat ion of an g-mode pu lsa t ions of the required accretion rate from 12.5 eV po la r hot spots. osci l lat ions, and because the degenerate dwarf instab i l i ty of this temperature have not

x - ray osci l lat ions suggest at least three possible explanations (Jensen et modulat ion of an accre t ion hot spot on a magnetic whi te dwarf l (2) x - r a y opaque pole due to accreted helium and metals or l C3] non-radial DA2 s tar . The f i r s t p o s s i b i l i t y is considered un l ike ly because the the K2V star would have to be I~1 > 1011~.e Moy r . - , , in order to yield The th i rd p o s s i b i l i t y is un l i ke l y because o f the ampl i tudes of the DA star ( T . = 3 2 , 0 0 0 - 38,OOOK) does not l i e w i th in any known str ip. However, non-radia l pulsat ional driving mechanisms in DA stars as yet been explored.

396

Poss ib i l i t y (2) appears at this time to be the most a t t r ac t i ve fo r several reasons. First , re la t ive ly l i t t l e accreted matter would be needed to provide the required x - ray opaci ty (cf. Kahn et ah 1983). Moreover, any w ind / f la re mass loss from the chromospherical ly very act ive KV star would very l ikely lead to some accretion onto the DA star in this very c l o s e s y s t e m . T h e poss ib i l i t y of w ind / f l a re accre t ion in this system and other such post-common envelope detached binaries has been discussed elsewhere (Sion and Starr f ie td 1984, Stauf fer 1986~. The signature of any accre t ion would be the presence o f metal ions and/or helium at the E inste in- redsh i f ted rest frame o f the whi te dwarf photosphere provided that one can rule out lev i ta t ion due to radiat ive fo rces in a hot DA atmosphere (Vauc la i r , Vauc la i r and Greenstein 1979; Morvan, Vauc la i r and Vauclair 1986). Recent u l t rav io le t spectra of V471 Tauri obtained with the International Ultraviolet Explorer (IUE) have revealed possible weak narrow absorption associated with the white dwarf, both at high reso lu t ion end low reso lu t ion . These are summarized in Table 1. These very tentat ive ident i f icat ions, together with the accurately known physical parameters of the white dwarf, and the s t rong l i k e l i h o o d that some acc re t i on does occur in the system, a l l encourage a quan t i ta t i ve exp lorat ion o f acc re t ion-d i f fus ion on a magnetic white dwarf as suggested by the EXOSAT pulses.

In part icular, two questions emerge in connection with x - ray opacity due to accreted gas. First what ion abundances are expected at the DA2 photosphere for a range of possible mass accret ion ra tes f rom the K2V star? Second, what accre t ion rates are in fac t required to supply the ion abundances implied by the line strengths in Table 1? In order to answer these questions, and test the va l i d i t y o f poss ib i l i t y (2) , a c c r e t i o n - d i f f u s i o n ca lcu lat ions were carried out as described in s e c t i o n 2. The c o m p u t a t i o n a l r e s u l t s and i m p l i c a t i o n s e re d i s c u s s e d in s e c t i o n 3.

2. Accret ion-Dif fusion Computations

A 0.8 M 0 white dwarf model (Iogg = 8.205) was constructed using the Paczynski code with the Los Alamos equat ion o f state and opac i t ies CPesnell 1986). The parameters of the model at the sur face and at op t i ca l depth T = 1 are l i s ted in Table 2. For th is model, we consider the dif fusion of trace metals in hydrogen.

Fo l lowing the nota t ion of Iben and Macdonald (1985) , the binary d i f fus ion coe f f i c ien ts for screened Coulomb potentials are related to the resistance coeff ic ients I<~j by

Dij = kT n,n~/(ni+nj)K,j ( I )

where Kjj is evaluated from a f i t to the results of Fontaine and Michaud (1979), the f i t is given by

K,]/K%} = 1.6249 In (1 + 0.187679 x 1.=,j) inCl+X,j=)

where Ko=j is the resistance coeff ic ient for Coulomb col l is ion cross sections (Chapman and Cowling 1970):

K%i = [ 0-94~ ~t= e4Z=Z=/(kT)~'e ] n~ j |P.(l+Xlj =)

with the reduced atomic mass

.JJ = Aj Aj MH/(AI+Aj)

Xa,j = 16 kaTa~a/ZaaZ2Je4 ,

and ~ being the la rger o f e i the r the Debye length or the mean ion ic separa t ion , both functions of the total ion density. The dif fusion veloci ty of an ion is given by

v= = D=j.m (g - g .=~ k T

397

where g is the downward a c c e l e r a f i o n due to g rav i ta t i ona l d i f fus ion~and thermal )and grad is the r a d i a t i v e a c c e l e r a t i o n here taken to be the maximum rad ia t i ve acce le ra t ion , for unsaturated lines due to trace di f fus ing ions in background hydrogen (Vauclair et el. 1979].

I f acc re t ion of solar compos i t ion gas f=om the K2V star replenishes ions at the same rate as they d i f fuse, we can wr i te

A?v,=(ZIH) * I~I[ZlH) = M(ZIH)~) (2]

where A is the area accreted onto, M is the accre t ion rate, (Z /H) is the ion /hydrogen abundance ra t i o , and v d is the d i f f us ion v e l o c i t y . . T h e f rac t iona l area, f , requi red to f i t the EXOSAT pulse p ro f i l e is ~15% of the to ta l whi te dwarf surface. Thus i f accreted ions are confined to the poles, making them x - ray dark, the observed line strengths, should scale as f - 1 [Shipman 1 9 8 6 )

3. Results and Discussions

I f the features in Table 1 are real and ar ise at or near the photosphere, the ion abundances corresponding to the u l t r a v i o l e t metal l ines can be determined by using the theoret ica l line pro f i le g r ids o f Henry, Shipman and Wesemeel ( 1 9 8 5 ) . I f the l ines ar ise f rom mater ia l over the ent i re surface then no scal ing of the measured equivalent width is needed. In the interest of brev i ty only CIV and Si IV are considered here. The result ing d i f fus ion coef f ic ien ts are D=j ( C I V ) = 2 . 0 x 10 e and D~j [Si lV) = 5.97 x 10. 6 . For both ions, we have log g r= l ' ~ "~ < l o g g. The d i f f u s i o n ve loc i t ies are v,=(CIV) = 1.2 x 10~cm/s and vd(Si lV] = 3.5 x 10Scm/s.

I f one assumes g rav i ta t i ona l capture by the whi te dwarf of wind ou t f low from the K2V star at the solar wind rate (c f . Sion and S ta r r f i e l d 1984) then the parameters of the V471 Tauri system yield an accret ion rate Ivl ~ 1 0 - 1 6 M o y r . - l . Us ing t h i s I~1 in e q u a t i o n (2 ) and s o l v i n g f o r the abundance rat ios (CIV/H) and [S i lV /H) we obtain 10 -a.== and 10 -7 , r e s p e c t i v e l y . These abundances are su f f i c i en t to y ie ld observable features in the far u l t rav io le t in the case of acc re t ion - d i f fus ion equi l ib r ium• I f the Si lV fea ture in Table 1 is real , and Si lV is d i s t r i bu ted over the ent ire surface area, then Si /H = 10 -== and equation (2) y ie lds I~1 = 8 x 10 -1== M e y r . - I in o rder to replenish SilV ions at the same rate as they d i f fuse• However, i f Si lV is conf ined to a f rac t iona l area of 15% f o r b o t h po les (Jansen 1 9 8 6 ) , then a s i l i c o n abundance g r e a t e r than s o l a r ( ( S i / H ) ~ = 3 x 10-Oi ls implied, i.e. (S i /H) = 5 x 10-==)]. An u n r e a s o n a b l y l a r g e a c c r e t i o n r .a te, M = 9 x 10 -~= MGyr . -~ , wou ld be requ i red in o rder to rep len ish the d i f fus ing Si lV ions. The problem is even more severe fo r the CIV in Table 1, i f i t is conf ined to the same f rac t i ona l area. These resu l t s suggest some in te res t i ng conc lus ions : (1) i f r a d i a t i v e f o r ces are unimportant (e.g. log g m== = 8.091 fo r Si lV) , the implied large abundances o f CIV and Si lV can not result from a steady s ta te equ i l ib r ium between accre t ion and d i f fus ion. Such large abundances would not be inconsistent, however, wi th episodic a c c r e t i o n ; [ 2 ) i f the p h o t o s p h e r i c me ta l f e a t u r e s , and c o r r e s p o n d i n g abundances ar ise in a po lar acc re t ion cap def ined by a magnet ica l ly contro l led accret ion f low, then the l ine fea tures const ra in the magnetic f ie ld st rength by imposing a lower l im i t to the f rac t iona l area ( i .e. the scal ing f ac to r ) such that the impl ied abundances are kept "reasonable" (several times so lar or less). I f one adopts the k inemat ica l model of Oavidson and Ost r iker (1973) , a f ract ional area of 15% implies a magnetic f ie ld of only ~ I G for an accret ion rate, M 1 0 - t = M ® y r - / y r -1 A f i e l d tha t weak is unable to channel the f l o w and an incons is tency ar ises because the magnetospher i¢ rad ius becomes less than the s te l la r radius. Note, however, that the k inemat ica l model provides a lower l imi t to the polar cap area.

398

Therefore, a strong f ie ld may indeed be present but have a much larger polar cap than that obtained from the kinematical formula. If the DA2 star has a 104G f i e l d and the same acc re t i on rate (~10-1SM~yr . - t ) , a lower limit cap area of 5.7 x 10 ~4 cm = is obtained. The cap area would have to be much la rger than th is value (15 -20% of the to ta l sur face area~ in order to avoid absurdly high abundances o f accreted metals implied by line features. Polar cap areas that large, a s s o c i a t e d wi th s t r o n g l y magnet ic wh i t e dwar fs , may be observed in the DO Her systems.

Final ly, as an al ternat ive to the rotat ion of x - ray opaque magnetic poles as the explanation of the g.25 minute osc i l l a t ions , the poss ib i l i t y of non- rad ia l osc i l la t ions should be fu l l y explored. Since the 9.25 minute per iod ic i ty is in the range of periods for non-radia l g-mode osci l lat ions of white dwarfs, non-rad ia l pulsational models of DA white dwarfs, possibly including composit ional ly s t r a t i f i e d a tmospheres and even nuclear she l l burning, should be exp lo red in th is p rev ious l y unexamined regime of DA white dwarf surface temperatures.

D iscuss ions w i th Drs. K. Jensen, M. L iv io , H. Shipman and S. S t a r r f i e l d are g r a t e f u l l y acknowledged. This work was supported by NSF grant AST85-17125, and in part by NASA grant NAG5-343, both to Vil lanova University.

References

Bruhweiler, F., and Sign, E. M. 1986a, Ap.J- Lett. 304, 626. Bruhweiler, F., and Sign, E. M. 1986, in preparation. Chapman, S. and Cowling,T. G. 1970. The Mathematical Theory of Non-

Uniform Gases (3rd ed.) Cambridge: Cambridge University Press). Oavidson, K., and Ostriker, J. P- 1973, Ap. J. 179, 585. De Campli, and Baliunas, S. 1979, Ap.J. 23,0, 815. Fontaina, G., and Michaud, G. 1979, Ap. J. 231, 826. Guil~n, E. F., and Sign, E. M. 1981, Butt. Am. Astron. Soc. 13, 8 t7 . Guinan, E., Wacker, S., Baliunas, S., Loesser, t., and Raymond, J. 1986,

Proceeding of the IUE Symposium "New Insights in Astrophysics CESA SP - 263), in press.

Henry, R. B. C., Shipman, H., and Wesemael, F. 1985, Ap. J. Suppl. 57, 154. Hertzog, K. P. 1966, The Observatory, 106, 38. Iben, I., and Macdonald, J- 1985, Ap. J. 296, 540. Jensen, K. 1985, IAU Circ. No. 4102. Jensen, K., Swank, J., Petre, R., Guinan, E., Sign, E. M., and Shipman,

H. L. 1986, Ap. J. (;Letters), in press. Jansen, K. 1986, private communication. Kahn, S., Wesemael, F., Liebert, J., Raymond, J. Steiner, J., and Shipman,

H. L. 1984, Ap. J. 279, 255. Oswalt, T. D. 1979, Publ. Astron. Soc. Pac. 91, 222. Paczynski, B. 1976, Acta Astr. 21, 417. Pesnell, D. 1986, private communication. Pskovski, Yu P. 1979, Soy. Astr. Lett. 5, 209. Ritter, H. 1986, Astr. Ap., in press. Shipman, H. 1966, private communication- Sion,E. M., and Starrf ield, S. G. 1984, Ap. J. 26.~., 760. Stauffer, J. 1986, preprint. Vauclair, G. 1972, Astr. Ap. 17, 437. Vauclair,G.0 Vauclair, S., and Greenstein, J. L. 1979, Astr.Ap. 80, 79. Verbunt, F., and Zwaan, C. 1981, Astr. Ap. 100, 27. Webbink, R. 1985, Invited Talk, Kitt Peak Workshop on Stellar Remnants. Young, A., Klimke, A., Africano, J. L., Ouigtey, R., Radick, R. R. and

Van Buren, D. 1983, Ap.J. 267, 655.

399

T a b l e 1

P o s s i b l e A b s o r p t i o n F e a t u r e s I d e n t i f i e d a t o r n e a r t h e DA2 P h o t o s p h e r e

I o n ~ = = C A ) E. W. ( m ~ )

H e l l C ~ ) 1 6 4 0 . 4 7 4 140 C I V C ~ ) 1 5 4 8 . 2 0 2

1 5 5 0 . 7 7 4 " 1 8 0 S i l l i C X ) 1 2 9 8 . 0 " 1 2 0 S i l l ( X X ) 1 2 6 0 . 4 2 1 - 1 9 0 S i l V C X Z ) 1 3 9 3 . 7 5 5 ~ 5 0 0

1 4 0 2 . 7 6 9

Cx ) H i g h r e s o l u t i o n ( o r b i t a l v e l o c i t y - c o m p e n s a t e d ) IUE S p e c t r u m C B r u h w e i t e r a n d S i o n 1 9 8 6 b ) .

( x ~ ) A v e r a g e d l o w r e s o l u t i o n IUE s p e c t r a C G u i n a n e t 8 1 . 1 9 8 6 ) .

T ~ b l e 2

M o d e t P a r a m e t e r s V471 T a u

Mw= = 0 . 8 M O , Te = 3 2 0 0 0 K , Log g = 8 . 2 0 5

S u r f a c e Cr = O) R = 8 . 1 3 6 6 x 10ecm P = 2 . 0 8 2 x 10 - e g cm - =

To = 2 5 , 2 3 9 K P = 8 . 7 3 7 5 x 104 d y n e s cm - = K = 3 4 . 9 8 c m = / g

A t r = 1 6 . 1 3 5 5 x 10Scm

3 . 8 6 5 x 1 0 - T g c m - =

3 9 9 2 3 K 2 . 5 3 5 2 x 10 e d y n e s cm - a 2 7 5 . 8 5 c m = / g

400

HYDRODYNAMIC STUDIES OF OXYGEN, NEON, AND MAGNESIUM NOVAE

Sumner Starrfield +

Theoretical Division, Los Alamos National Laboratory, and

Joint Institute for Laboratory Astrophysics

University of Colorado, Boulder CO, 80309

W. M. Sparks

Applied Theoretical Physics Division

Los Alamos National Laboratory

Los Alamos, NM 87545

J. W. Truran

Department of Astronomy, University of Illinois

Urbana, Iii, 61801

Supported in part by NSF Grants AST83-14788 and AST85-16173 to Arizona State

University, by NSF Grant AST83-14415 to the University of Illinois, by NASA grant

NAGS-481 to Arizona State University, and by the DOE. +

Permanent Address: Department of Physics and Astronomy, Arizona State

University, Tempe, AZ, 85287

ABSTRACT

In this paper we present the results of recent theoretical studies that have

examined the properties of nova outbursts on ONeMg white dwarfs. These outbursts

are much more violent and occur much more frequently then outbursts on CO white

dwarfs. Hydrodynamic simulations of both kinds of outbursts are in excellent

agreement with the observations.

1 Introduction

In this paper we assume that the model for a cataclysmic variable also holds

for the nova: a close binary system with one member a white dwarf and the other

member a star that fills its Roche lobe. Because it fills its lobe, any tendency

for it to grow in size because of evolutionary processes or for the lobe to shrink

because of angular momentum losses will cause a flow of gas through the inner

401

Lagranglan point into the lobe of the white dwarf. The size of the white dwarf is

small compared to the size of its lobe and the hlgh angular momentum of the

transferred material causes it to spiral into a disc surrounding the white dwarf.

Some viscous process~ as yet unknown, acts to transfer material inward and angular

momentum outward so that a fraction of the material lost by the secondary

ultimately ends up on the white dwarf. The accreted layer grows in thickness until

the bottom reaches thermonuclear burning temperatures. The further evolution of the

white dwarf now depends upon its mass and luminosity, the rate of mass accretion,

and the chemical composition of the reacting layer. Given the proper conditions, a

thermonuclear runaway (hereafter: TNR) occurs, dr~vlng the temperatures in the

aecreted envelope to values exceeding 108K. At this time the positron decay

nuclei become abundant which strongly affects the further evolution of the

outburst. Theoretical calculations demonstrate that this evolution releases enough

energy to eject material with expansion velocities that agree with observed values

and that the predicted light curve produced by the expanding material can agree

quite closely with the observations. The hydrostatic and hydrodynamic studies of

accretion onto white dwarfs have identified those conditions which will result in a

TNR. In order for a fast nova to occur, it is necessary to accrete at a rate M <

10 -8 M yr -I onto a white dwarf with Mwd > I.i M and a luminosity Lwd 10 -2 L . o o o

In addition, it is also necessary to enhance the CN0 nuclei in order to provide

enough energy at the peak of the outburst to eject a shell at sufficient velocities

to agree with the observations. Published reviews of the classical nova outburst

[1-9] summarize the work up to 1985.

The entire character of the outburst: light curve, ejection veloelties, and

speed class depends upon the amount of CNONeMg nuclei initially present in the

envelope. In addition, the fact that a fast nova outburst demands enhanced CNO

abundances was one of the first and clearest predictions of the TNR theory of the

nova outburst. Optical observations by Willlams and Gallagher and their

collaborators concluded that not only are nova shells enhanced in CN0 nuclei but

that there is a correlation (with a few exceptions) between degree of enhancement

and nova speed class [3,10-12]. A summary of the observed abundances for novae can

be found in Truran and Livlo [32].

Studies of recent novae have led to some very interesting results. A recent

outburst was that of the recurrent nova U Sco [15,16] which at maximum showed

strong H and HeII 4686 but at minimum showed only lines of helium. The optical

data imply that He/H in the eJeeta was ~ 2, while the UV data imply nearly normal

CNO abundances. U Seo was an extremely fast nova declining by more than eight

magnitudes in one month and its ejection velocities may have exceeded 104 km/sec.

Of great importance to our understanding of classical novae have been the recent

studies using the International Ultraviolet Explorer Satellite. These include Nova

Cygnl 1978 which showed enhanced CNO [18], in agreement with the theoretical

402

calculations of Starrfield, Sparks, and Truran [19]; the studies of V603 Aql [17]

and U Sco [16]; Nova Corona Austrlna 1981 [20, 28, 29], and Nova Aquila 1982

[21,29,30]. The interpretation of Nova Corona Austrlna, Nova Aql 1982 and Nova Vul

1984 #2 is that they all ejected material from an oxygen, neon, magnesium white

dwarf that had been processed through a hot hydrogen burning region by the nova

outburst [28, 29]. The most likely scenario suggests that the white dwarf had a

main sequence mass of 8-12M ° and must now have a mass of ~I.IM ° to have survived

nondegenerate carbon burning . Enhanced neon was also reported in VIb00 Cygnl

[131.

2 Hydrodynamic Calculations

The most detailed calculations of the TNR theory for the nova outburst are

found in a series of papers by the authors [14,19,22-25,27,28]. In our most recent

studies [27,28], we have evolved TNR's on massive white dwarfs (1.38M ° and 1.25Mo)

in successful attempts both to produce outbursts which resemble those of recurrent

novae such as U Sco and also outbursts which resemble those that occur on ONeMg

white dwarfs. We used a spherical accretion code to secrete solar composition

material at a variety of rates onto white dwarfs with various luminosities. Our

results produced sequences that took less than 40 years to reach the peak of the

outburst and then ejected a small amount of material by radiation pressure. This is

in good agreement with the observations.

The evolu,tlonary studies done with the envelope consisting of half solar

material plus half carbon and oxygen or half solar material plus half carbon

produced very different results. Accretion onto luminous white dwarfs produced an

outburst, but no mass was lost and a major fraction of the outburst luminosity was

radiated in the EUV. Because carbon is so highly reactive, the runaway occurred

before the envelope had secreted sufficient material to become degenerate and only

a weak outburst occurred. At low white dwarf luminosities, an outburst occurred

and a major fraction of the envelope was ejected. The evolutionary sequences done

with half solar and half oxygen were very violent and a very large fraction of the

accreted envelope was ejected [29]. This composition is not so far-fetched as it

seems since both theoretical and observational analyses of PGI159-035 (a pulsating

variable star) suggest that it is very rich in oxygen near the surface [26,31]. We

identify this calculation with the recently discovered outbursts occurring on 0NeMg

white dwarfs.

3 Summary and Discussion

In this paper we have presented both theoretical and observational evidence

that leads to the inescapable conclusion that the classical nova outburst is the

403

direct result of a TNR in the accreted hydrogen rich envelope of a white dwarf.

The most important evidence in favor of this theory has been the predictions and

confirmation both of enhanced CNO nuclei in the ejecta and of a constant luminosity

phase in the outburst. The recent studies of novae shells and the UV observations

of novae in outburst demonstrate that such a correlation exists with two notable

exceptions: DQ Her and Nova Vul 1984 #2. DQ Her was a slow nova with the largest

amount of carbon in the ejecta of any well studied nova. In addition, analysis of

its spectrum near maximum indicated non-solar 12C/13C and 14N/15N isotopic ratios -

the strongest evidence for the operation of a TNR in the nova outburst. The

existence of this object underscores the wide variety of initial conditions that

are possible in a pre-nova object.

One final point, yet to be answered, about th~ nova phenomena is the source of

the enhanced nuclei in the accreted envelope. It does not seem likely that these

nuclei are produced in the secondary, and numerical studies of shear instabilities

have not produced a nova outburst. It may be possible that the enhancement is the

result of combined hydrogen-helium runaways in the accreted envelopes but the

defining conditions for such runaways have yet to be identified.

This paper has greatly benefitted from discussions with J. Gallagher, and R.

Williams. We would also llke to thank Drs. A. N. Cox, E. Sion, G. Shaviv, and H.

Van Horn for valuable discussions. Support from the Association of Western

Universities and the Joint Institute for Laboratory Astrophysics for sabbatical

leave fellowships is gratefully acknowledged. S. Starrfleld would also llke to

thank Dr's. G. Bell, S. Colgate, A. Cox, M. Henderson, and J. Norman for the

hospitality of the Los Alamos National Laboratory.

References

i. Gallagher, J.S., and Starrfield, S., Ann. Rev. Astr. Astro.,16, 171 (1978). 2. Starrfield, S.G., Sparks, W.M., and Truran, J.W., in "Structure and

Evolution of Close Binary Systems", Ed. P. Eggleton, S. Mitton, and J. Whelan (Reldel, Dordrecht, 1976) p. 155.

3. Truran, J.W., in Essays in Nuclear Astrophysics, Ed., C.A. Barnes, D.D.Clayton, and D. Schramm (Cambridge, Cambridge University Press, 1982).

4. Starrfleld, S., in "The Classical Nova", ed. N. Evans, and M. Bode, (New York: Wiley), in press, 1987.

5. Bode, M. F., and Evans, A. N., "The Classical Nova", (New York: Wiley), in press, 1987.

6. Starrfleld, S., in "Radiation Hydrodynamics", ed. D. Mihalas and K.-H. Winkler, (Dordreeht: Reidel), in press, 1986.

7. Starrfleld, S., in "The Scientific Accomplishments of the IUE", ed. Y. Kondo, (Dordrecht:Reldel), in press, 1987.

8. Starrfleld, S., in "New Insights in Astrophysics", ed. E. Rolfe, (ESA, SP265), in press, 1986.

9. Bode, M. F., "RS OPH (1985) and the Recurrent Nova Phenomenon", (Utrecht: VNU Science Press), in press, 1986.

I0. Williams R.E., Woolf, N.J., Hege, E.K., Moore, R.L., and Koprlva, D.A.,Ap. J., 224, 171 (1978).

II. Williams, R.E., and Gallagher, J.S., Ap. J., 228, 482(1979). 12. Gallagher, J.S., Hege, E.K., Kopriva, D.A., Williams, R.E., and Butcher,

404

H.R., Ap. J., 237, 55 (1980). 13. Ferland, G.J., and Shields, G.A., Ap. J., 226, 172 (1978). 14. Starrfield, S., Truran, J.W°, and Sparks, W.M., Ap. J.,226, 186(1978). 15. Barlow, et al., M.N.R.A.S., 195, 61 (1981). 16. Williams, R.E., Sparks, W.M., Gallagher, J°S., Ney, E.P.,

Starrfield,S.G., and Truran, J.W., Ap. J., 251, 221 (1981). 17. Ferland, G.J., Lambert, D.L., McCall, M.L., Shields, G.A., and Slovak,

M.H., Ap. J., 260, 794 (1982). 18. Stickland, D.J., Penn, C.J., Saaton, M.J., SniJders, M.A.J., and Storey,

P.J., M.N.R.A.S., 197, 197 (1981). 19. Starrfield, S., Sparks, W.M., and Truran, J.W., Ap. J. Supp.,28, 247

(1974). 20. Sparks, W.M., Starrfield, S., Williams, R.E., Truran,J.W., and Ney, E.P.,

in Advances in Ultraviolet Astronomy, ed. Y. Kondo,J.M. Mead, and R.D. Chapman, (NASA Publication 2238, 1982) 478.

21. Snljders, M.A.J., Seaton, M.J., and Blades, J.C., in Advances in Ultraviolet Astronomy, ed. Y. Kondo, J.M. Mead, and R.D. Chapman (NASA Publication 2238, 1982) 625.

22. Starrfield S.,Truran, J.W., Sparks, W.M., and Kutter, G.S., Ap. J., 176, 169 (1972).

23. Starrfleld, S., Sparks, W.M., and Truran, J.W., Ap. J., 192, 647(1974). 24. Sparks, W.M., Starrfield, S., and Truran, J.W., Ap. J., 220,1063 (1978). 25. Starrfleld, S., Kenyon, S.K., Sparks, W.M., and Truran, J.W.,Ap. J., 258,

683 (1982). 26. Starrfield, S., Cox, A.N., Hodson, S.W., and Pesnell, W.D. Ap. J. Lett.,

268, L27 (1983). 27. Starrfield, S., Sparks, W.M., and Truran,J. W., Ap. J., 291, 136 (1985). 28. Starrfield, S., Sparks, W.M., and Truran,J. W., Ap. J., Left., 303, L5

(1986). 29. Williams, R.E., Ney,E.P., Sparks, W.M., ~tarr f ie ld , S., and Truran, J.W.,

M.N.R.A.S., 212, 753, (1985). 30. SniJders, M.A.J., Batt, T.J., Seaton,M.J., Blades, J.C., and Morton, D.C.,

M. N. R. A. S., 211, 7p, (1984). 31. Sion, E.M., Liebert, J. , and Starr f ie ld, S., Ap. J . , 292, 471(1985). 32. Truran, J. W., and Liv io, M., Ap. J. , in press, (1986).

405

C A P A B I L I T I E S O F T H E H U B B L E S P A C E T E L E S C O P E

F O R V A R I A B L E - S T A R R E S E A R C H

Howard E. Bond Space Telescope Science Institute

1. I n t r o d u c t i o n

A new era in observational optical and ultraviolet astronomy will begin with the launch

of the Hubble Space Telescope (HST). In this paper, I will discuss the capabilities of the HST

and its instruments, with particular emphasis on its expected impact in the area of pulsating

variable stars. For more general overviews of the scientific capabilities of the HST, the reader

is referred to Longair and Warner (1979) and Hall (1982).

Detailed information on the telescope and its instrumentation, and instructions for

submitting observing proposals, are provided in the Call.for Propoaal~ and Instrument Handbook8 that were issued by the Space Telescope Science Institute (STScI) in October, 1985. Updates

to this information are published as necessary in the STSeI Newsletter. These publications are

available upon request from the General Observer Support Branch at STScI.

It should be noted that the following discussion reflects the status of the HST project as of

the date it was written (September, 1986). Project modifications and enhancements are possible

between now and launch.

2. H S T O v e r v i e w

HST is a 2.4-m [94-inch) Ritchey-Chreti~n reflector that will be launched by NASA's

Space Shuttle system. Operating above the Earth 's atmosphere, HST will have the major

capabilities of providing hlgh-resolution images and ultraviolet spectra of faint astronomical

objects. Part icipants in this workshop will also be interested in HST's ability to obtain high-

speed photometric observations free of atmospheric effects.

The telescope optics are expected to achieve diffraction-limited performance in the visual

region; the design specification for image quality is that , at 6328/~, 70~ of the energy of an on-

axis stellar image be contained within a radius of 0t!10. The guidance system is designed to keep

pointing j i t ter below q-0~007 rms. The optical coatings give a wavelength coverage of 1150 ~. to

1 mm~ although the initial complement of instruments will cover only the 1150-11,000 ~ range.

Delayed by the Challeneer tragedy, the launch of HST is now expected in late 1988. Its

nominal altitude will be 500kin, giving an orbital period of 95 rain. Several factors, including

the short orbital period with at tendant frequent Earth occultations, the relatively slow slew rate

(6 ° per minute), and the time required to acquire guide stars before each observation, will limit

408

on-target observing efficiency early in the HST mission to .-.20%. As experience is gained, this

figure may ult imately rise to ~35%.

S. Te lescope S t a t u s

At this writing, the HST is located at the Lockheed facility in Sunnyvale, California.

Its initial thermal-vacuum test was completed in July, 1986. Optical throughput tests showed

nominal results. Tests of the scientific instruments (Sis) indicated that, in general, the Sis

should operate to the specifications published in the STScI Instrument Handbooks in 1985.

Indeed, in the case of the High Resolution Spectrograph (HITS), the ultraviolet performance is

now expected to exceed that predicted in the HRS Instrument Handbook.

The thermal-vacuum tests did reveal several problem areas, including an unexpectedly high

power consumption related to insufficient spacecraft insulation, and problems with outgassed

water from the graphite-epoxy material used in construction of the telescope assembly. Solutions

or workarounds for these problems will be devised well before launch. A final thermal-vacuum

test will be conducted before launch in I988.

Once in orbit, the HST will normally be operated from a console at the STScI in Baltimore.

Commands have already been sent successfully from the STScI to the HST in Sunnyvale, and

actual data generated by the Sis have been returned to Baltimore.

4. T h e G e n e r a l - O b s e r v e r P r o g r a m

The STScI is now accepting proposals for the first 12 months of the General Observer (GO)

program (months 6-18 following launch). The proposal deadline has not yet been announced,

but is expected to be in the autumn of 1987. NASA has awarded a fraction of the observing

time to Guaranteed Time Observers (GTOs), who are the scientists involved in the development

of the Sis. However, GOs will be awarded 62% of the observing time during the first scheduling

cycle, and this fraction will increase in steps to 100% after 2.5 years in orbit.

Given the observing efficiency described above, about 1700 hr of exposure time will be

available during the first cycle, of which about 1000 hr will be assigned to GOs. It is apparent

that the observing time will be very heavily oversubscribed.

5. Scient i f ic I n s t r u m e n t s

HST will carry five Sis: the Wide Fie ld/Planetary Camera (WFPC}, the Faint Object

Camera (FOC), the Faint Object Spectrograph (FOS), the High Resolution Spectrograph (HRS),

and the High Speed Photometer (HSP}. Moreover, the Fine Guidance Sensors (FGS) will be

able to carry out astrometric measurements in addition to their function of spacecraft guidance.

The accompanying table briefly summarizes the main capabilities of the Sis, and mentions

a few possible applications in the area of pulsating stars. Detailed documentation of the SI

capabilities is available from STScI as discussed above.

B. A p p i c a t l o n s to V a r i a b l e - S t a r R e s e a r c h

In this section I will discuss examples of possible applications of the HST capabilities to the

study of pulsating variable stars. The examples are intended merely to illustrate a few areas in

409

SI

C a p a b i l i t i e s of the Scient i f ic I n s t r u m e n t s

Applications to Capabili ty Variable Stars

FGS Stellar positions Distance scale to =k0(1003, m v ~_ 18 RR Lyr, Cepheid ~rtrig

HSP High-speed photometry Radial and non-radial free of atmospheric WD pulsations effects Stellar oscillations

WFPC, Diffraction-limited Extragalactic RR Lyr FOC imagery to r n v ~- 28 and Cepheid variables

FOS, Low- and high-resolution Chemical abundances HRS UV & visible in hot pulsators

spectroscopy Time-resolved spectra

which HST has the potential for major contributions to our knowledge; the actual investigations

that the telescope will conduct will be limited only by the ingenuity of the successful proposers.

6.1 D i s t a n c e s o f R r t L y r a e a n d C e p h e i d V a r i a b l e s a n d H0

Pulsating variables of the Cepheid and RR Lyrae types are of fundamental importance as

calibrators of the extragalactic distance scale, and ultimately of the Hubble constant, H0.

Excellent discussions of HST observing scenarios designed to exploit pulsating variables for

these purposes have been given in the 1985 reports of the STScI Stellar and Galaxies Working

Groups (available from STScI) and by Aaronson and Mould (1986).

The HST route to H0 might begin with astrometrlc measurements designed to establish

the absolute luminosities of the RR Lyrae and Cepheid variables. This information is of course

of basic importance in understanding the evolutionary status of these objects, in addition to its

application to the distance-scale problem; these observations should also be supplemented with

ultraviolet spectra along the lines described by Evans and BShm-Vitense at this conference in

order to obtain Cepheid masses and luminosities.

The GTO program that is already in place calls for at tempts to measure direct

trigonometric parallaxes with the FGS for RR Lyr itself and the 1.9-day Cepheid SU Cas.

The positional accuracy that can be achieved by the FGS will not be known until after HST has

been operated successfully in orbit, but it is possible that repeated measurements may reduce

the parallax errors to as low as about :k01.10005.

Figure 1 shows loci of constant percentage parallax error as functions of apparent and

absolute magnitudes for an assumed FGS accuracy of 4-0.5 milliarcseconds. Several objects that

are on the GTO parallax program are plotted. Obviously, measurements of usefully accurate

distances of nearby RR Lyr and Cepheid variables will be at or possibly beyond the limits of the

FGS capability. (Incidentally, the very nearest Cepheids, such as a UMi, are too bright to be

measured by the FGS; on the other hand, the FGS will be capable of measuring positions and

parallaxes for stars to m r -~ 18, considerably fainter than the H i p p a r ~ o 8 mission will be able to

reach.)

410

FGS PARALLAX ACCURACY

m

b.o

¢)

<

5

I 0

15

o ~ O H D 140Z@~ r Cas (Cep, l.9~i.

/ V , v",, I , , , , I , , I , , , , l- I ! ! !

15 10 5 0 - 5

Absolu te Magn i tude -10

Figure 1 Shows apparent/absolute magnitude combinations for which

the indicated percentage error in ~rtrlg is achieved by the FGS, assuming an error of ~0(10005 in the parallaxes.

Approximate positions of several GTO parallax targets are indicated.

Although Figure 1 shows that the distances of RR Lyr and Cepheid variables may be too

great for direct measurement, it also shows that the main-sequence members of the Hyades

cluster, as well as typical field subdwarfs like HD 140283, are easily within range of very accurate FGS parallaxes. Thus it appears quite likely that the HST will be able to establish,

through main-sequence fitting, the distance scale for open and globular clusters, and thus

indirectly calibrate the pulsating variables of Populations I and II (assuming that ground-based

observations will permit removal of the metalticity dependence of the main-sequence luminosity).

(Figure 1 also shows the approximate positions of two nearby cataclysmic variables that

have been selected for parallax measurements by the GTOs, It can be seen that nature has

again conspired to place important objects at or near the limits of the FGS.)

In the next step toward the extragalactic distance scale, the WFPC could be used to refine

further the zero point of the Cepheid P-L relation via main-sequence fitting to determine the

distances of the Magellanic Clouds and other Local Group galaxies. A Population II check on

the Local Group distance scale could be provided with WFPC observations of RR Lyr variables

in the halos of M31 and M33.

In galaxies beyond the Local Group, one could use WFPC observations of Cepheids.to

calibrate the Fisher-Tully IR luminosity/H I line width relation, and the brightest red and blue

supergiants. The WFPC could then be used to detect the brightest supergiants in galaxies out

411

to m - M ~- 33, as well as the brighter Cepheids in the Virgo cluster (where a 20-day Cepheid

would have rn v ~- 26).

The goal of this ambitious program, according to Aaronson and Mould (1986), should

be to determine H0 to an accuracy of =t=10°~! Of course, considerable theoretical work and

ground-based astronomy, in addition to the HST observations, will be required to achieve this

goal. The goal may yet remain elusive because of such effects as interstellar reddening and

chemical-abundance dependences of RR Lyr and Cepheid luminosities, as well as evolutionary

considerations discussed elsewhere in these proceedings.

6.2 Osc i l l a t ions o f S o l a r - T y p e S t a r s

The Sun exhibits periodic radial-velocity and integrated-light oscillations (e.g., at periods

near 5 rain), and it should be possible to extract considerable information about the structure

and composition of the solar interior from observations of this phenomenon. Naturally, one

would like to extend these observations to other nearby solar-type stars because of the wealth

of new information that would be provided (Ulrich 1986).

Detection of radial-velocity oscillations is probably an exclusively ground-based project.

However, it is likely that integrated-light variations of solar-type stars can be detected only

from space, if at all, because of the severe photometric problems introduced by the Earth 's

atmosphere. For the Sun, the largest fractional white-light amplitudes are several times 10 -6

(see, for example, Woodard and Hudson 1983).

Detection of such stellar white-light variations would require HSP observations with a

signal-to-noise ratio of about 108, or about 1012 detected photons. At a 10MHz counting rate,

~30 hr of integration would be required.

A particularly favorable target for such an observation is a Cen A, not only because it is

the nearest solar-type star, but because of its known mass and the fact that it lies in one of HST's

continuous-viewing zones (where continuous coverage is possible without Earth occultation).

It remains to be seen, of course, whether even the HSP can achieve a S/N of 106. The

limiting factor may be the stabil i ty of the FGS pointing-control system over intervals of ~5 rain

(D. Soderblom, private communication).

6.3 P u l s a t i n g W h i t e Dwarfs

It is now known that a t least three pulsational-instability strips exist for white dwarfs.

In addition to the ZZ Ceti-type DA pulsators~ there are hotter DB-type (Winget et aL 1982)

and the recently discovered extremely hot GW Vir (PG 1159-035)-type pulsators (McGraw

et al. 1979; Bond et al. 1984). I believe that HST can make several important advances in our

understanding of these objects.

6.3.1 The Z ~ Cet l and D B Pulaators

The ultraviolet capabili ty of the HST makes for very favorable circumstances for

observations of the DA and DB pulsators. By working in the UV, closer to the flux maximum

for these objects, one expects to observe larger pulsation amplitudes.

Ultraviolet light and color curves would provide a test of the hypothesis that the variations

are due to non-radlal g-modes, for which the brightness changes are due solely to temperature

412

changes at constant radius.

Several of us recently made at at tempt at such a test of the g-mode hypothesis for the

ZZ Ceti variable G29-38, using the IUE short-wavelength spectrograph (Holm et al. 1985).

Because of the prohibitively long readout time, we used simultaneous ground-based photometry

to direct the placement of the stellar spectrum in real time to two different locations on the

detector, corresponding to the =bright" and "faint" phases of the variation. We were able to

show that the temperature difference between these two phases was (290 + 150) K, with a radius

change of ( -3 .3 4- 3.3)%. This is consistent with the g-modes, but clearly HST will be able to

make a much more stringent test.

The GTO program for HST already includes a 10~hr run on the ZZ Ceti pulsator

Gl17-B15A with the HSP in its prism mode (which permits simultaneous photometry at 1350

and 2500.~k), for the purpose of testing the g-mode hypothesis. A test should also be possible

using time-resolved spectra obtained with the FOS, along the lines of our IUE investigation

described above.

Since HSP data will be free of high-frequency atmospheric scintillation, they can also be

used to test for the presence of radial white-dwarf oscillations (which are expected to have

periods of order 1 sec). Such oscillations, although predicted theoretically, have never been

observed for white dwarfs from the ground (Robinson 1984).

6.8.~ G W Vir-t~tpe Pulsators

Two pulsation mechanisms have been proposed for these extremely hot objects. Starrfield

et aL (1984) have suggested partial ionization of carbon and/or oxygen (~¢- or ~/-mechanism), and

have shown that substantial enhancements of the abundances of these elements near the surface

layers are required. On the other hand, Kawaler et al. (1986) have argued that the pulsational

instability could arise in a variable helium-burning shell (c-mechanlsm), so that carbon/oxygen

enhancements would not be required. Efforts to understand these pulsators have been frustrated

by the imprecision of the effective-temperature and chemical-abundance scales at high Teff.

The optical spectra of the GW Vir pulsators do show features of C IV and O VI (Sion ct aL

1985), but high-quality ultraviolet spectra would certainly be of value. The faintness of the

objects means that IUE spectra are rather noisy (at low resolution) or cannot be obtained at all

(at high resolution). The available low-dispersion short-wavelength IUE spectra are fascinating

in showing much stronger metallic lines (C IV, N V, O IV-V, and possibly Si IV) than do other

hot white dwarfs (Wesemael et al. 1985). However, the quality of the spectra is low, and it is

clear that HST observations with the FOS~ with their much higher S/N, would provide much

additional information about the presence of weaker metallic lines and about temperature-

sensitive ratios like 0 V/O VI.

T. Conc lu s ion

I have mentioned just three areas in which HST observations could lead to new advances in

the study of pulsating variable stars; I am sure all of you can think of many other applications.

However, I must emphasize again that the amount of HST observing time that will be available

will be extremely limited. The areas of astronomy that will receive observing time will be

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determined by the scientific merit of the proposals that are submitted. I urge you all to make

a serious effort to prepare outstanding proposals.

I am grateful to Marc Rafat and George Hartig for discussions of the HST status, and

David Soderblom for a discussion of the detectability of stellar oscillations.

R E F E R E N C E S

Aaronson, M., and Mould, J. 1986, Ap. J., 403, 1.

Bond, H.E, Grauer, A.D., Green, R.F., and Liebert, J.W. 1984, Ap. J., 279, 751.

Hall, D.N.B., ed. 1982, The Space Telescope Observatory, NASA CP-2244.

Holm, A.V., Panek, R.J., Schiller, F.H., Bond, H.E., Kemper, E., and Grauer, A.D. 1985, Ap. J.,

280, 774.

Kawaler, S.D., Winget, D.E, Hansen, C.J., and Iben, I. 1986, Ap. J. (Lettera), 406, L41.

Longair, M.S., and Warner, J.W., eds. 1979, Scientific Research with the Space Telescope, IAU Colloq. No. 54, NASA CP-2111.

McGraw, J.T., Starrfield, S., Liebert, J., and Green, R.F. 1979, in IAU Colloq. No. 53, White Dwarfs and Variable Degenerate Stars, eds. H.M. Van Horn and V. Weidemann (l~ochester: Univ. of Rochester Press), p. 377.

Robinson, E.L. 1984, A. J., 89, 1732.

Sion, E.M., Liebert, J., and Starrfield, S.G. 1985, Ap. J., 292, 471.

Starrfield, S., Cox, A.N., Kidman, R.B., and Pesnell, W.D. 1984, Ap. J., 281, 800.

Ulrich, R.K. 1986, Ap. J. (Letters), 406, L37.

Wesemael, F., Green, R.F., and Liebert, J. 1985, Ap. J. Suppl., 58~ 379.

Winger, D.E., Robinson, E.L., Nather, R.E., and Fontaine, G. 1982, Ap. J. (Lettera), 262, Ll l .

Woodard, M., and Hudson, H.S. 1983, Nature, 405,589.

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A N A S T E R O S E I S M O L O G Y E X P L O R E R

Thnothy M. Brown High Alt i tude Observatory/Nat ional Center for Atmospheric Research*

P.O. Box 3000 Boulder, CO 80307

Arthur N. Cox Los Alamos National Laboratory

Los Aiamos, NM 87545

A b s t r a c t . In response to a NASA opportunity, a proposal has been made to s tudy the con- cept of an Asteroseismology Explorer (ASE). The goal of the ASE would be to measure solar-like oscillations on many (perhaps hundreds) of stars during a 1-year mission, including many members of open clusters. We describe this proposal 's observational goals, a straw- man technical approach, and likely scientific rewards.

Background

Solar p-mode oscillations have become a fruitful source of information about solar structure (Christensen-Dalsgaard et al. 1985). Motivated by the solar example, many work- ers have made recent a t tempts to detect solar-like oscillations on other stare. It seems likely that these a t tempts will succeed in the near future, but it is obvious tha t the amounts of large-telescope time required to make even a partial survey of nearby solar-like stars will be prohibitive. In view of this si tuation and st lmulated by a NASA opportuni ty, a small group (including the authors, H.S. Hudson, J.W. Harvey, R.W. Noyes, J. Christensen-Dalsgaard, P. Demarque and J.T. McGraw) has considered whether such a survey might be conducted from space, and what its scientific rewards would be. The result of our efforts was a propo- sal to s tudy the concept of an Asteroselsmology Explorer (ASE). We describe here the current s tate of thinking about the ASE, part ly to inform the community of its possibilities, and par t ly to solicit ideas for improvement.

Compared to familiar stellar pulsations, the solar p-modes are distinguished by their short periods (5 m), their large radial order (typically 20, for large spatial scales) and their

small ampli tudes (10 cm/s in velocity, or 3X10 -6 in relative continuum intensity). Though many inferences about the solar interior rely on observation of strongly nonradial

*The National Center for Atmospheric Research is eponsored by the National Science Foundation

415

oscillations, there is much to learn from oscillations with angular degree l between zero and three. This range o f I may be detected using light integrated over the stellar disk, and hence is accessible even on distant stars. A part icularly striking example of this abil i ty (and the paradigm for the ASE) is the time series of solar irradiance obtained by the ACRIM instrument on the SMM satellite (Willson 1979). This device is a radiometer, intended to

measure changes in the solar constant with absolute precision of about 10 -3 , and with ran-

dom noise for a single 128 s integration of about 3>(10 -5. In spite of this large noise level, power spectra of t ime series taken with the ACRIM show the individual solar p-modes clearly for l ~-- 0-2, and give one of the best current estimates of their frequencies (Woo-

dard and Hudson 1983).

Wha t can one learn from these modes? The frequency spectrum for low-/p-modes con- sists of pairs of modes with l ~ 0,2 and l --~ 1,3. The modes in each pair have nearly identical frequencies, and the pairs are separated bY~uo/2, where u 0 is a parameter deter-

mined principally by the s tar 's mean density. For the Sun, u 0 is about 130 ~uHz; u 0 decreases as the stellar radius increases. Conditions in the stellar core determine the split- ting between modes within each pair. The split t ing between modes with l ~-- 0 and l ~-- 2 is typical ly 8-12 /~Hz in cool dwarfs; it is expected to decrease as the star evolves (Christensen-Dalsgaard 1984, Ulrich 1986). Observing a value for this spli t t ing would require 2-5 days of observation, and would allow an estimate of the s tar 's evolutionary state. In addit ion to s tructural information, one can hope to observe the rotational split t ing of modes with the same radial order and angular degree, but different azimuthal order. This spli t t ing depends on the angular rotation speed of the star, averaged over most of its inte- rior. Combined with photometric or spectroscopic estimates of the surface rotat ion rate, one might thus learn whether stellar envelopes spin down first, leaving a rapidly rotat ing core. Observing the amplitudes of p-mode oscillations for a range of stellar masses, ages, and activi ty levels would certainly cast light on the mechanisms responsible for exciting the modes. Finally, we anticipate tha t a broad survey of many stars may reveal relationships that would not appear in a detailed study of any single one, especially if many of the stars studied are cluster members, and therefore approximately coeval and of similar initial com-

position.

Based on these considerations, the principal goal of the ASE is to observe a large number (perhaps a few hundred) of roughly solar-like stars, with enough precision and tem- poral coverage to detect and classify their p-mode oscillations. A large fraction of these stars should lie in clusters, or should be members of visual binary systems, both to maximize the number of target stars in each field of view, and for the astrophysical reasons just men- tioned. However, because of the limited age range of nearby clusters, it will certainly be necessary to spend much (perhaps most) of the observing time on selected field stars.

II. T e c h n i c a l A p p r o a c h

Mainly for reasons of technical feasibility, we chose to detect oscillations by observing their associated photometric variations. The anticipated mode relative intensity amplitudes

are a few times 10 -8, (i.e., micromagnitudes), and these must be detected within the typical

mode lifetime of 106 s. It is very important to note that one need not do absolute

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photometry at the micromagnitude level. Rather, one can (and probably must!) do pho- tometry that is relative in both space and time: one requires stability only over a small fl'ac- tion of the field of view, for times somewhat longer than the oscillation period. Since we wish to observe solar-like stars in nearby clusters, we must be able to reach mlcromagnitude precision for stars as faint as about m v ----- 10. Finally, in order to make simultaneous

observations on as many stars within a cluster as possible, we would like a field of view at

least 2 ° in diameter.

These observational requirements prescribe many of the system parameters. To obtain

nficromagnitude precision in 108 s, one must detect at least 106 photon/s. To do this for a 10th magnitude star using a broadband telescope with reasonable transmission requires an aperture of about 1 m. Covering a 2 ° field of view with the largest "available" CCD detec- tor (the Tektronix 2048 x 2048) requires a focal length of about 1 m. Finally, to obtain good photometry on bright objects one must avoid saturating any of the CCD pixels, which

implies that the images must be large (to cover many pixe]s), and the detector must be read

out at the highest feasible rate.

These requirements lead to a design incorporating a 1 m aperture, f/1.2 Schmidt tele- scope, with a field flattener and a single 2048 x 2048 pixel CCD detector at the prime focus. This arrangement covers a field 2.56 ° square, with a resolution of 4.5 arcsec/pixel. By defocusing the stellar images to about 50 arcsec diameter, binning multiple plxels on-chip,

and reading the detector at 1 r i t z , one can construct a system that will detect 106 photons/s from stars with m v = 10.1, and begins to saturate at my = 6.0. With exposures

just 100 s in length, such a system could detect mil]hnagnitude changes on stars with m~ ~-- 15. Maki,lg a detector system to these specifications will be a challenge, particularly

because of the large dynamic range encountered and the high readout rates required. Simu- lations based on our experience wigh current CCD systems suggest that we can meet this

challenge, however.

The baseline orbit for this telescope is one with 57 ° inclination and 500 km altitude. Most nearby galactic clusters would fall into the continuous viewing zone of this orbit at one time or another, with periods of 75% visibility lasting for 15 to 20 days. The once-per-orbit data gaps that appear during times of partial target visibility would be obnoxious, but toler- able. The 74-day precession period of the orbit would allow several opportunities to observe each target field during a 1-year mission. The mode of operation would be to observe each chosen field continuously for ten days or more, averaging images over roughly 60 s intervals before transmitt ing them to the ground. We believe that all of the information in each image should be preserved in the transmission process. Since the images involved are pre- cisely as unchanging as the heavens, efficient data compression techniques are possible; we

estimate that a downlink rate of 4X104 blts/s should suffice. Once the data reach the ground, it will be possible (and desirable) to construct a photometric time series for each identifiable object in the field of view. For most fields, the limiting magnitude for effective

photometry will be determined by confusion with faint background stars, rather than instru-

mental or photon noise.

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III. Discussion

The system just described would be able to detect solar-like oscillations on GV stars as far away as the Pleiades; half a dozen other clusters and several hundred late-type dwarf field stars lie within that distance -- enough for an informative first survey. It will be par- ticuiarly productive to study stars with well-establlshed physical parameters, ages, and activity indices. From this point of view the Hyadcs chtster is a very important target, as are the ~ Perseus group and the Pleiades. Obtaining adequate ancillary information about possible field star targets is likely to require a substantial ground-based effort. In addition to these candidates for asteroseismology, one should expect to see many of the more familiar pulsating stars: 5 Scuti variables, Cephleds, rapidly oscillating Ap stars, flare stars and oscil- lating white dwarfs. Detection of low-amplitude variables of the more traditional types would assist in identifying the edges of instability s~rlps and understanding pulsation mechanisms. One could also use thc data set from the ASE to address problems unrelated to oscillations. For example, power spectra of the photometric changes due to stellar con- vection are likely to be observable (ACRIM observes them on the Sun). The way in which these spectra depend on stellar mass, age, and metaliicity could improve our knowledge of the convection process in general. Low mass stellar companions, even those of planetary size, would make detectable light variations if they were to transit a stellar disk; whether or not such events are seen, one can use the observations to set statistical limits on the number

of such companions.

What important questions remain to be answered about the ASE, i.e., what issues should be addressed by a NASA-funded study? The most vital is to verify that one can do mlcromagnitude time-series photometry using current CCD technology; this will require careful laboratory work. It will also be necessary to peL'form a number of tradeoff studies to determine the best choice of parameters for the instrument, the optimum observing

sequence, and suitable means for reducing and accessing the large amounts of data that will be produced. To make these tradeoffs it will be necessary to know exactly what are the scientific priorities, what observations are needed to meet these goal, and how these require- ments affect the instrument design and mission plan. We hope that other stellar astrono-

mers will help us to answer these questions.

References

Christensen-Dalsgaard, J. 1084, Space Research Prospeels in Stellar Activity and Variability, A. Mangeney and F. Praderie (eds.), Observatoil'e de Paris, p. 11.

Christensen-Da]sgaard, J., Gough, D.O. and Toomre, J. 1985, Science 229~ 923.

Ulrich, R.K. 1986, Astrophys. J. 306, L37. Willson, R.C. 1979, Applied Optics 18, 179. Woodard, M. and Hudson, H.S. 1983, Nature 305, 589.

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