Download - (c) Denis Alan Leahy, 1976
c-1
A STEADY STATE SELF-CONSISTENT MODEL
FOR PULSAR MAGNETOSPHERES
by
DENIS ALAN LEAHY
B.A.Sc., U n i v e r s i t y of Waterloo, 1975
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS tROR THE DEGREE OF
MASTER OF SCIENCE
i n
THE FACULTY OR GRADUATE STUDIES
Department of Physics
We accept t h i s t h e s i s as conforming
to the r e q u i r e d standard
THE UNIVERSITY OF BRITISH COLUMBIA
September,1976
(c) Denis Alan Leahy, 1976
In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements for
an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree that
the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.
I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s
f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head of my Department or
by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t copying or pub l i ca t ion
of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my
w r i t t e n p e r m i s s i o n .
Depa r tment
The U n i v e r s i t y o f B r i t i s h Co l umb i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5
Date O C T . -6 , r37<b
ABSTRACT
A steady s t a t e s e l f - c o n s i s t e n t model f o r a p u l s a r magnetosphere
i s developed. I t i s shown that the c e n t r a l neutron s t a r o f a p u l s a r
should possess a magnetosphere. In the f i r s t approximation, the
i n e r t i a of the magnetospheric p a r t i c l e s i s n e g l e c t e d . Steady s t a t e
c o r o t a t i n g models are developed t o c a l c u l a t e the s t r u c t u r e of the
magnetosphere f o r the axisymmetric case and the case of the a r b i
t r a r i l y o r i e n t e d d i p o l e . Two r e s u l t s are t h a t charge d e n s i t y i s
p r o p o r t i o n a l t o the z component of the magnetic f i e l d and t h a t
the z component of the magnetic f i e l d vanishes at the l i g h t c y l i n d e r .
The l i g h t c y l i n d e r i s where the c o r o t a t i o n v e l o c i t y reaches the
speed of l i g h t . The p u l s a r s p i n a x i s i s a l i g n e d w i t h the z a x i s .
I l l u s t r a t i o n s o f the f i e l d s are presented f o r the cases of magnetic
d i p o l e a x i s p a r a l l e l and p e r p e n d i c u l a r t o the s p i n a x i s .
Next these models are a l t e r e d t o take i n t o account the non
zero mass of the p a r t i c l e s i n the magnetosphere. An e x t r a e l e c t r i c
f i e l d i s r e q u i r e d t o h o l d the p a r t i c l e s i n c o r o t a t i o n . Charge
se p a r a t i o n i s assumed. The f o l l o w i n g r e s u l t s are found: 1) F i e l d
l i n e s which p r e v i o u s l y were h o r i z o n t a l i n s i d e the l i g h t c y l i n d e r ,
now have a cusp i n them where they were h o r i z o n t a l . T his cusp
inc r e a s e s i n s i z e as i t s l o c a t i o n approaches the l i g h t c y l i n d e r .
2) F i e l d l i n e s no longer are h o r i z o n t a l at the l i g h t c y l i n d e r but
e r t i c a l , and d i v i d e i n t o two groups those w i t h p o s i t i v e B^
( c a r r y i n g negative charge) and those w i t h negative ( c a r r y i n g
p o s i t i v e charge).
We next cease t o r e q u i r e t h a t the p a r t i c l e s be f i x e d i n the
c o r o t a t i n g frame. F i r s t the s i n g l e p a r t i c l e motion i s c a l c u l a t e d
f o r a r b i t r a r y f i e l d s assuming small v e l o c i t i e s i n the r o t a t i n g
frame. We f i n d that the motion can be separated i n t o a slow d r i f t
along streamlines(which very n e a r l y f o l l o w magnetic f i e l d l i n e s )
and a s p i r a l i n g about these s t r e a m l i n e s . The energy i s conserved,
and can be separated i n t o a l o n g i t u d i n a l and a t r a n s v e r s e energy
a s s o c i a t e d w i t h the two types of motion. The t r a n s v e r s e energy
d i v i d e d by the frequency of s p i r a l i n g i s an a d i a b a t i c i n v a r i a n t .
For the axisymmetric case, a model i s developed from which the
f i e l d s , charge d e n s i t y , and v e l o c i t i e s can be computed. With
a r e s t r i c t i o n on the boundary c o n d i t i o n s , an a n a l y t i c a l model i s
o u t l i n e d f o r the case of a r b i t r a r y magnetic m u l t i p o l e s .
TABLE OF CONTENTS i i i .
page
L i s t of Tables i v .
L i s t of Figures v.
Acknowledgement v i .
I n t r o d u c t i o n 1.
Magnetosphere Ex i s t e n c e 3.
The Force Free Assumption 6.
The Axisymmetric ForceFree Magnetosphere 7.
Covariant Formulation 12.
The Force Free Magnetic P o t e n t i a l 14.
R e l a x a t i o n of the Force Free Assumption 23.
R e l a x a t i o n of C o r o t a t i o n 28.
S i n g l e P a r t i c l e Motion 29.
C o l l e c t i v e Motion 31.
Axisymmetric Case 34.
The General Case 37.
D i s c u s s i o n 42.
B i b l i o g r a p h y . 44.
Appendix 1 46.
Appendix 2 49.
Appendix 3 51.
Appendix 4 56.
LIST OF TABLES
page *
lO.Table I. Numerical E v a l u a t i o n of V ( x , y ) , Where
V(x,y)=(u/Tft (n/c)V(x,y)
* 18.Table 2. Numerical E v a l u a t i o n of x (x,y), Where
2 * <XC x>y) = 0J/7r) (n/<0 X Cx,y) coscb
V . LIST OF FIGURES
page •
10. F i g . 1. Magnetic F i e l d L i n e s , Axisymmetric Force Free Case
11. F i g . 2. E l e c t r i c F i e l d L i n e s , Axisymmetric Force Free Case
11.Fig. 3. Charge Density, Axisymmetric Force Free Case
18. F i g . 4. Lines of Constant Magnetic P o t e n t i a l , x , Axisymmetric .
_ Force Free Case
19. F i g . 5. Lines o f Constant Magnetic P o t e n t i a l ^ , m=l Force Free Case
20. F i g . 6. E l e c t r i c F i e l d L i n e s , m=l Force Free Case
21. F i g . 7. Magnetic F i e l d L i n e s , m=l Force Free Case
22. F i g . 8. Charge D e n s i t y , m=l Force Free Case
27.Fig. 9. Magnetic F i e l d Lines For The Axisymmetric C o r o t a t i o n Case
With I n e r t i a
33.Fig. 10. Streaming and C o r o t a t i n g 'Dead' Zones: Axisymmetric Case
v i . ACKNOWLEDGEMENT
I am indebted t o Dr. M. H. L. Pryce, who has supervised my
t h e s i s work. In a d d i t i o n to p r o v i d i n g the t o p i c of t h i s t h e s i s ,
he has guided my research to enable me to produce t h i s work. I
would l i k e to thank Dr. Pryce f o r h i s h e l p , f o r many u s e f u l
d i s c u s s i o n s , and e s p e c i a l l y '"-''.for appendix 4 , which i s based
on h i s work.
1. INTRODUCTION
Pulsars are astronomical objects which r e g u l a r l y give put
r a d i o pulses w i t h a p e r i o d , which v a r i e s from p u l s a r t o p u l s a r ,
of about a second. The pulses have v a r i a b l e amplitude but a p r e c i s e
p e r i o d . Over 100 p u l s a r s are now catalogued, w i t h p e r i o d s ranging
from 33 m i l l i s e c o n d s to 4 seconds. D. t e r Haar* summarizes the
o b s e r v a t i o n a l r e s u l t s . «
The f i r s t p u l s a r was discovered l a t e i n 1967 by Hewisti et a l
at Cambridge. They used a new r a d i o telescope b u i l t to study
s c i n t i l l a t i o n s , c a u s e d by plasma clouds i n the s o l a r wind,of r a d i o
sources of small angular size.>-Since.-the--time -scale of the s c i n
t i l l a t i o n s i s a f r a c t i o n of a second, the instrument was i d e a l
f o r r e c o r d i n g p u l s a r s i g n a l s .
Some ideas about p u l s a r s are f i r m l y e s t a b l i s h e d . Because o f
the short p e r i o d and i t s slow increase w i t h the passage of time
due to l o s s of energy, a r o t a t i n g neutron s t a r i s most c e r t a i n l y
the c e n t r a l o b j e c t . I t s rotational energy i s coupled to i t s sur
roundings by a huge magnetic f i e l d , having a value at the surface of
12
Bs - 10 Gauss. The reason f o r the p u l s a t i o n i s the asymmetry
introduced by non-alignment of the magnetic d i p o l e and s p i n axes.
A coherent emission process i s necessary f o r the intense r a d i o
emission. However, the emission mechanism and the s t r u c t u r e o f
the magnetic f i e l d s and of matter surrounding the neutron s t a r
i s p o o r l y understood. Many d i f f e r e n t t h e o r e t i c a l models have
been proposed but no c o n c l u s i v e model has yet emerged.
In the f o l l o w i n g work, a p a r t i c u l a r , s e l f - c o n s i s t e n t , steady-
s t a t e model f o r the p u l s a r magnetosphere i s presented. The emission
mechanism i s not considered here.
3. MAGNETOSPHERE EXISTENCE
.The neutron s t a r i s a r o t a t i n g , conducting body and thus w i l l
have an i n t e r i o r e l e c t r i c f i e l d :
(1) FT=-V7C x if
where V<=PJ-<$ i n c y l i n d r i c a l p o l a r c o o r d i n a t e s u n i t v e c t o r s )
centred on the neutron s t a r , w i t h the z-axis along the r o t a t i o n a x i s •
Q i s ' the angular frequency of r o t a t i o n . For an e x t e r i o r
d i p o l e magnetic f i e l d , an i n t e r i o r f i e l d i s 2" w i t h B q a
constant. Thus from (1) one ob t a i n s :
(2) E*=-(arB_/c) £
Th i s can be obtained from the i n t e r i o r e l e c t r o s t a t i c p o t e n t i a l
(3) -$ =ftr 2B o/2c + $ Q
$o a constant, by E=-V$ . For a t y p i c a l p u l s a r B q i s of the order 12
of 10 gauss, the r a d i u s R i s approximately 7 to 10 k i l o m e t e r s and ft i s of the order of 20 sec *. This gives a p o t e n t i a l d i f f e r e n c e
2 17
between poles and equator of A$ =fi R Bq/2C- 10 v o l t s . More
preciselyA<J> = 3.1K10 B 1 2 R 6 /P v o l t s , where B 1 2 = B q /10 gauss,
R^=R/10^cm and P-2v/Q i s the p e r i o d of r o t a t i o n .
E a r l y p u l s a r models assumed a vacuum surrounding the magnetic
neutron s t a r . The e x t e r i o r f i e l d i n t h i s case is' given by s o l v i n g 2
Laplace's equation: V $=0, and f i t t i n g t h i s s o l u t i o n to the boundary
c o n d i t i o n s a t the neutron s t a r s u r f a c e . The t a n g e n t i a l component of
E i s continuous across the su r f a c e . In s p h e r i c a l p o l a r coordinates i t i s , from (2) :
-+ (4) = -(ffcB /c) s i n 6 cos 9 9 ' tan v o . One obtains f o r the e x t e r i o r e l e c t r o s t a t i c p o t e n t i a l (5) _(B_m 5/3cr 3) P 2(cos9)+ (C QR+W 3B Q/3c) 1/r
where ^ s t n e second Legendre polynomial and C Q i s a constant.
T h i s r e s u l t s i n a s u r f a c e charge on the neutron s t a r ^ due-to the
d i s c o n t i n u i t y i n the normal component of the e l e c t r i c f i e l d , o f :
(6) a =-(B « R/4uc)(5cos 26 -3)/2 +(C 0R +^R 3B q/3C)/4^R 2
The second term vanishes i f the net charge on the neutron s t a r i s .
zero( Q= C QR+QR B o/3c ). The r e s u l t i n g e l e c t r i c field(Q=0) i s :
(7) £=-v$ = - ( B o ^ 5 / c r 4 ) P (cos6) r -(B o«t^/2cr 4) sin26 6
The magnitude of the surface component p a r a l l e l to the magnetic
f i e l d B i s 10
(8) R/c- 6x10 B 1 2 R 6 / P volts/cm
To understand what e f f e c t t h i s enormous e l e c t r i c f i e l d would
have on the surface m a t e r i a l of the neutron s t a r , one must examine
other f a c t o r s . The g r a v i t a t i o n a l f i e l d on a neutron s t a r i s l a r g e .
The- a t t r a c t i v e f o r c e at the surface i s :
(9) F g= 1 . 6 x l 0 " 1 3 M/R 62 dynes
f o r an e l e c t r o n mass(M i s the neutron s t a r mass i n s o l a r mass
u n i t s ) and correspondingly greater f o r i o n s ( about 10^ times,
assuming n u c l e i near Fe*'*'). The r a t i o of electromagnetic to g r a v i
t a t i o n a l f o r c e i s :
(10) eE/F = 7 . 5 X 1 0 1 1 B 1 0R. 3/PM g 12. 6
4 Thus g r a v i t a t i o n a l b i n d i n g i s i n s i g n i f i c a n t . I t has been shown
56 12 that Fe n u c l e i i n a magnetic f i e l d of 2*10 Gauss w i l l form an
a n i s o t r o p i c , very t i g h t l y - b o u n d l a t t i c e w i t h a b i n d i n g energy of
14 kev per i o n and e =750 eV per e l e c t r o n . L a t t i c e spacing i s
of the order of 10 cm. To remove ions r e q u i r e s a f i e l d o f
(11) E * e j / Z e l = 5 x l O 1 1 v o l t s / c m
The nuclear charge,Z, of F e i > b i s 2 6 ,and 1 i s the l a t t i c e spacing.
Compare t h i s t o the maximum e l e c t r i c f i e l d a v a i l a b l e , c a l c u l a t e d
f o r vacuum i n equation (3). Probably only the f a s t e s t p u l s a r
(the Crab p u l s a r , w i t h P=.033 sec) w i l l be able to p u l l ions from
i t s s u r f a ce. E l e c t r o n s are removed more e a s i l y than ions but the
d e t a i l s have not yet been c a l c u l a t e d . In any case, i t i s assumed
tha t a source of charged p a r t i c l e s f o r the magnetosphere e x i s t s
and serves to reduce the huge value of i n (8) .A p o s s i b l e source
i s the r e g i o n oivtside the magnetosphere. More l i k e l y , p a r t i c l e s
trapped i n the magnetosphere during the formation of the p u l s a r i n
a supernova e x p l o s i o n , so t h a t a huge E«S never develops.
THE FORCE FREE ASSUMPTION 6.
Another ' f o r c e ' enters the problem. Due to the enormous
r o t a t i n g magnetic f i e l d and the p r o p e n s i t y of charged p a r t i c l e s
to f o l l o w f i e l d l i n e s , an i n e r t i a l f o r c e due to the a c c e l e r a t i o n
of the p a r t i c l e s a r i s e s . For a p a r t i c l e i n c o r o t a t i o n w i t h the
c e n t r a l neutron s t a r at d i s t a n c e from the a x i s r , the energy i s
v.mc , wi t h :
(12) Y ( j ) = ( l - n 2 r 2 / c 2 ) - J s
T h i s becomes s i n g u l a r at the ' l i g h t c y l i n d e r ' where the c o r o t a t i o n
v e l o c i t y i l r i s the speed o f l i g h t c. The magnetic energy d e n s i t y i s 2 2 2 much grea t e r than J^THC : .-•> B /8ir >>Y^mc , except f o r very c l o s e
2 2 to r. =c/H . For an e l e c t r o n these are equal f o r B /8ffmc =
5 2 6 2x10 B, , where B, i s B i n u n i t s of 10 gauss. Thus fir i s very D O
n e a r l y c at t h i s p o i n t . Estimates f o r B at the l i g h t c y l i n d e r ( f r o m
energy l o s s by a r a d i a t i o n d i p o l e ) vary from lO^gauss f o r the
Crab p u l s a r downwards.
The f o r c e f r e e assumption i s :
(13) E+v/c XB =0
i . e . that the electromagnetic f o r c e s on a p a r t i c l e v a n i s h . We
neglect non-electromagnetic f o r c e s and assume zero mass p a r t i c l e s .
T h i s i s a good approximation everywhere i n the magnetosphere (
which i s here considered as being l i m i t e d to the r e g i o n between
the neutron s t a r and the l i g h t c y l i n d e r ) except very near the
l i g h t c y l i n d e r .
t
7. THE AXISYMMETRIC FORCE FREE MAGNETOSPHERE
The b a s i c equations are t h e f o r c e f r e e assumption (13), and Maxwell's equations:
(14) V- t=0 Vx £=0 VX&=4IT/C J V»l:=4irp
Note t h a t because o n l y steady s t a t e (3/3t - -fi3/3<{> i n the s t a t i o n a r y
. r e f e r e n c e frame) i s being c o n s i d e r e d , i n combination w i t h a x i a l
symmetry (3/3<j>=0) , Maxwell's equations take the form ( 1 4 ) , i . e .
3/3t=0. In a d d i t i o n we assume o n l y t o r o i d a l p a r t i c l e v e l o c i t i e s
and t h a t the v e l o c i t t a o f a l l p a r t i c l e s at one p o i n t are the same
(15) v=w(r,z)r <j>
The c u r r e n t i s g i v e n by
(16) j=pv .
C y l i n d r i c a l c o o r d i n a t e s are used. Since VxE=0,one w r i t e s E=-W o r :
(17) E r=-8V/3r E z=-3V/3z
V«B=0 gi v e s 3 ( r B z ) / 3 z +3 ( r B r ) / 3 r =D . Thus one can d e f i n e i> by:
(18) B =l/r 3*/3z Bz
= ~ 1 / r ^ / 9 r
The f l u x through a c i r c l e p e r p e n d i c u l a r t o , and cen t r e d on
the z-axis i s t*it = B z 2irr dr = -2ir f^dii /3r d r =-2n^(R,z)-\|>(0,z
The f l u x through the s i d e s o f a c y l i n d e r about the z- a x i s from
z=0 t o z=Z i s f'0;• B 2irr dz = 2n^)(r,Z)-if) ( r , 0 ) ) . Thus the stream f u n c t i o n
ty i s d i r e c t l y r e l a t e d t o the magnetic f l u x .
Using(17)§(18)i .equation(13)has'components -3V/3r -(cor/c}|l/r^i|//3r)=0
and -3V/3z 4>r/c)(-|/r)(3'|»/3z) =0 . Therefore one has:
(19) . 3V/3r = (-co/c) 3*/3r 3V/3z = (-to/c) 3^/3z
8.
From (19) one gets -3 2V/3r3z =3/3z(w/c 3ijj/3r) =3/3r(w/c 3*/3z)
or ato/az 3ij./3r - 3co/3r3if)/3z =0 . But t h i s i s j u s t the Jacobian
J(u)>tjO;r,z) = 0 J so that w i s a f u n c t i o n of-ty . Now, one has:
(20) vi|> 't = 3^/3r (1/r) 8i(;/3z + 3tj,/3z (-1/r) 3i^/3r =0
so t h a t i|> and thus toa re constants along f i e l d l i n e s . S i n c e the f i e l d l i n e s
o r i g i n a t e iri the neutron s t a r which i s - r i g i d l y r o t a t i n g at angular
frequency ft, we must have to=ft everywhere. I.e. the magnetosphere
r i g i d l y corotates with the neutron s t a r .
From (19) we have:
(21) V=(-ft/c) *
From the inhomogeneous Maxwell's equations (14) one gets:
(22) Vx $ = (3B r /8 z -3Bz/6 r)= - <£/ft)(l/r)( 32V/ 3z 2 + 3/3r((l/r)3 V/3r))
= (4TT/C) j = (4 ir/c) pQrdJ
(23) v-E -A^ (-1/r) 3/3r(r3V/3r) -3 2V/3z 2 = 4irp
E l i m i n a t i n g p gives a d i f f e r e n t i a l equation f o r V:
(24) ( l - Q 2 r 2 / c 2 ) ( 3 2 V ^ r 2 + 3 2 V / 9 z 2 ) - g / r ) ( l + f ? r 2 / c 2 ) &V/3r =0
The s o l u t i o n of t h i s equation i s obtained • i n appendix 1. The r e s u l t i s :
(25) V(r,z) = ( o / c ) V u ) _ " x ( l - f t 2 r 2 / c 2 ) / X ( l ) e i a r f e / c dx
wherep i s the magnetic d i p o l e moment of.the neutron s t a r and
r*/c ) i s given i n appendix 1. The boundary conditions
assumed were a d i p o l e magnetic f i e l d at. the o r i g i n , and" f i n i t e
f i e l d s at the l i g h t c y l i n d e r .
For each value of ( r , z ) , the s e r i e s s o l u t i o n f o r x.was'evaluated
using a hand c a l c u l a t o r and p e n c i l , to 3 f i g u r e s o f accuracy, f o r
5 values of a. The i n t e g r a t i o n i n (25) was then c a r r i e d out using
Simpson's Rule f o r numerical i n t e g r a t i o n . The c a l c u l a t i o n r e s u l t s
a r e presented i n t a b l e 1. Magnetic f i e l d l i n e s are p a r a l l e l to l i n e s
o f constant V and e l e c t r i c f i e l d l i n e s p e r p e n d i c u l a r t o l i n e s o f constant
V. F i g u r e s 1,2 and 3 i l l u s t r a t e the magnetic and e l e c t r i c f i e l d s
•and the charge d e n s i t y . As x>1 i . e . as one approaches the l i g h t
c y l i n d e r ^ v ^ l - f l r /c- approaches zero so t h a t X(v) approaches a constant
p l u s terms of second degree and higher i n v (see appendix 1 ) . Thus:
B z=-(l/r)3i{)/9r =-(1/r)(dv/dr)3^/3v approaches zero. I.e. f i e l d
l i n e s approach the l i g h t c y l i n d e r h o r i z o n t a l l y . A l s o from the f a c t
t h a t X(v) i n v o l v e s o n l y even powers of a, one sees t h a t B = l / r 3v^ z
i s zero at z=0. T h i s r e s u l t s i n a n u l l p o i n t i n the magnetic f i e l d '
at z=0,r=c/Q (see f i g u r e 1 ) .
The charge d e n s i t y can be computed from (23)^using (24)to e l i m i n a t e
the second d e r i v a t i v e o f V, Using the d e f i n i t i o n o f ikf.18), the r e s u l t i s :
(26) 4 up — (-2 jD/c) Y2 B z
Thus p j the charge d e n s i t y ^ i s d i r e c t l y p r o p o r t i o n a l t o the z component
of the magnetic f i e l d .
10.
TABLE I NUMERICAL EVALUATION OF V*(x,y) ,WHERE
V(x,y)=M/« V (:
?\ 0 0.25 0.5 O.75 1.0
0 singular 6.0 h.3' 3-2 2.9
0.25 0 6.0 2.9 2.5 1.8
0.5 0 - 0 . 2 0.8 1.2 1-7
1.0 o- 0.5 0.1 -0.1 . -0.1
FIGURE 1 MAGNETIC FIELD LINES ; AXISYMMETRIC
FORCE FREE CASE
y
1.0—4
0.75«*
0.25
f 9 ' i r-" 0 0.25 0.5 0.75
FIGURE 2: ELECTRIC FIELD LI-NES, AXISYIMBTRIC FORCE FREB CASS y •
I.
•2*, x I 1.0
0.75-
0.5—
0.25—
0—
/
locus of zero charges^
(B z =0)
- A •
^++ + +
/
! 0
8 I • f .0.25 . 0.5 O.75
FIGURE 3: CHARGE DENSITY, MISYt-METRIC FCRCE FREE CASE
fi 1.0
12.
COVARIANT FORMULATION
So f a r we have considered only the axisymmetric s i t u a t i o n .
In t h a t case a l l f i e l d q u a n t i t i e s are time independent. For the
more general case of non-aligned s p i n and magnetic d i p o l e axes,
or the existence of higher order m u l t i p o l e s , time independence
i s i n v a l i d a t e d , except i n the re f e r e n c e frame c o r o t a t i n g w i t h the
neutron s t a r . However t h i s frame i s n o n - i n e r t i a l and even becomes
s i n g u l a r at the l i g h t c y l i n d e r . In order t o express the equations
i n t h e i r c o r r e c t form we use the c o v a r i a n t formalism of general
r e l a t i v i t y .
'We work i n the coordinate system r o t a t i n g at the angular fx
v e l o c i t y of the p u l s a r u s i n g c y l i n d r i c a l coordinates x = ( t , z , r , <f>) .
The no n - r o t a t i n g coordinates are: c a r t e s i a n x =(f,x,y,z) ; and
c y l i n d r i c a l x - ( t , z , r , <f^). Thus <j)= <H-ftt • The met r i c i s then
=c 2 (dt) 2 -(dx) 2 -(dy) 2 -
5 % e d x
(dz)2
a, 3 dx
=c 2 (dt) 2 -(d z) 2 - (dr) 2 - (rd*. ) 2
= (c 2-n 2 r 2 ) (dt ) 2 - (dz) 2 -(dr) 2 (rd<j>)2- 'flrclMt
Thus o n e has: c 2 - f t V 0 0 -ft r 2
(27) g = ag
0
0
-1
0
0
-1
0
0
-ftr2 0 0 - r 2
and i t s inverse:
(28) _a3
g
o o -n/c'
- 1 0 0
1/c
0
0 0 - 1 0
-pVc 2 0 0 fi2/c2-l/r2=-l/^2r2) J
The Einstein summation convention has been used.
Greek indices range from 0 to 3, Latin indices from 1 to.3.
The f i e l d variables F^ v are expressed i n terms of the f i e l d
E and B i n the stationary frame, in appendix 2. The covariant **a
current density has the components, in an i n e r t i a l frame, J:=
(p,^) and satisfies the equation of continuity (conservation of
charge):
(29)
The comma(semicolon) denotes ordinary(covariant) differentiation
Maxwell's equations are then:
(30) F ^ J ^ T T / C ) ^
J a ; =0 a
e a y v X F ,=0 uv X (31)
(30) can also be written: (32) (WcX-gljV =((-g)V),v
2 2
where g=determinant g = -c r for the metric (27). aB
14. THE FORCE FREE MAGNETIC POTENTIAL 1 3
We now assume c o r o t a t i o n so that the current i n . t h e r o t a t i n g ' f r a m e
i s zero: J =0. This assumption has some j u s t i f i c a t i o n : 1) i t
holds i n the axisymmetric case, of which t h i s i s a g e n e r a l i z a t i o n t
2) A h i g h l y conducting magnetosphere w i l l have the magnetic f l u x
f r o z e n i n t o i t , thus w i l l be dragged by the f i e l d . The f i e l d i s so
lar g e that i t w i l l be locked i n t o c o r o t a t i o n .
Since;we have time independence i n the r o t a t i n g frame (the
steady s t a t e assumption), (32) reads:
(33) ( r F k l ) 5 1 =0 k l klm
Since F i s an antisymmetric t e n s o r , w r i t e rF = E A m- Then (33)becomes A ,,-A,, =0 so th a t A i s d e r i v a b l e from a p o t e n t i a l : A =X,m Thus we have: m 1 1 m • r m . 111
(34) F k l = ^ l m ( l / r ) X , m
X i s c a l l e d the magnetic p o t e n t i a l .
Now we use the f o r c e f r e e assumption, i . e . the LOrentz f o r c e a
f on each p a r t i c l e vanishes: f =^/cj F v au v=0 where u i s the p a r t i c l e ' s
v e l o c i t y . As pu i s J t h i s g i v e s : v v 6
. ctv (35) F J v =0
We use t h e metric (27) to r e l a t e J to J v : J ^ r g v ^ 0 , so th a t we have: v °
(36) J _ = ( c 2 - f t 2 r 2 ) J 0 J x=J 2=0 J 3=-Or 2J°
One gets from (35): ( F a 0 ( c 2 - ^ 2 r 2 ) -P0^ r 2)J°=0 or s i n c e J°= p(in
the r e s t frame) • i s not i d e n t i c a l l y zero, we have: r-zi-i caOro 2,, 2 2 2 ^ c
a 3 u 2 , 2-. 2_«3
(37) F u = pr / ( c -QT ) J F =\ftr /c JY^F
The c o v a r i a n t form of (35): F j V J V =0 g i v e s :
(38) F ^ O
Thus the homogeneous Maxwell's equations are just:
<39> • F 1 2 ' 3 + F 2 3 ' l + F 3 r 2= 0
ct 3 We know F in terms of x from (34) and (37). To get the
desired equation for X from (39) use:
W V g i « g j s F a B
The results are:
(41) f1 2
= f 1 2
o 2^10 2^13 -ff.2 2, 2» 2 ... 2r13 2 2r13 F = ^ r F +r F =l(fi r /c )y, +l)r F = r y. F. „ 2„20 2„23 r„2 2.2-, 2 2^23 2 2r23 F 2 3=^r F + r. F =[(fi r /c JY^ +l)r F = r F
Thus (39) becomes ( ( / r ) x , 3 ) , 3 + ( n ^ X ) 1), 1+(r Y J X . ^ . J ^ or:
(42) ' S^/a? 2 +r 2Y ( ) )2(9 2x/3z 2 +8 2
X/3r 2) +y 2 (2y 2 - l ) 3 X /3 r =0
The solution to this equation is obtained in appendix 3 for
a dipole magnetic f i e l d ,with axis oriented at angle 9 Q with respect
to the rotation axis(z-axis) , at the neutron star. The second
boundary condition was the requirement of f i n i t e fields at the light
cylinder. No attempt was made in matching the solution within the
magnetosphere to an outgoing wave solution in the region outside
the magnetosphere. This is a p e s s i b i l i t y for further study. The
magnetic potential obtained for the stated boundary conditions i s :
(43) X (z,r,9)<„R o(v)e i a>* * ( e * + e " * O ^ e ^ d x
= - ^ ( P / c ) 2 { c o s V " > / ^ b ) ) % /a V ^ e ^ d x
.' + sin ecosfr/"Jj/(~c )\zk J9. ^ v n + 2 e i x y d a }
where R Q(v) and R^(v) are given e x p l i c i t l y in appendix 3: by equations
(23)5(31) and (23)5(36) respectively, where v=l-fi 2 r 2 / c 2 , y=« z/c •
Vi is the magnetic dipole moment of the neutron star.
16.
From F U V , i n terms of E andB i n the s t a t i o n a r y frame from
appendix 2, we have: •
(44) E ^ r / c ^ 2 3x/ 9r r/c) y^ 2 3 X/ & =0
(45) B = Y^ 23x / a z B =Y^ 23x/3r B 4/rJ3x/3*
(44)gives E"*Vx=0 : l i n e s of constant X are e l e c t r i c f i e l d l i n e s .
A lso i n the r,z plane^B i s p r o p o r t i o n a l to V x . The charge d e n s i t y
p can be c a l c u l a t e d u s i n g : d / d r ( r 2 y ^ 2 ) = 2ry^ to get:
(46) 4up=V-E= (-l/r]3/3r{(f2r 2/c)Y ( j )23 X/3z)+3/3z((f2r/c) Y ( ) )
23x/3r)
= (-2fi/c) Y43 X/3z .= -(2Q/c)y^hz
Thus the r e l a t i o n betweenp and B found f o r the axisymmetric case(26)
i s s t i l l v a l i d i n the more general case.
In f i g u r e 4 are shown l i n e s of constant X f o r the a x i
symmetric case . Note t h a t these are orthogonal to l i n e s of constant
if), the stream f u n c t i o n , as given i n f i g u r e 2.
For the m=l case, a d i p o l e o r i e n t e d at 90° w i t h respect to the
r o t a t i o n a x i s , see f i g u r e 5. The e l e c t r i c and magnetic f i e l d s and
charge d e n s i t y i n the plane c o n t a i n i n g both;:rotation and d i p o l e axes,
are shown i n f i g u r e s 6,7 and 8. The e l e c t r i c f i e l d i s b a s i c a l l y
a quadrupole f i e l d . :.As one approaches the l i g h t c y l i n d e r ,v approaches 0.
Thus from (43) we have:
(47) X+ ( - y / ^ ( n / c ) 2 v 2 c o s < i . r j ^ i a y / ( Z cj) da
Using (40) and r3/3r=2(v-l) 3/3v : one o b t a i n s :
(48) E+ -fr/Mc)5 4 ( v - l ) c o s 9 / 0 ° i e : L a y / ( i : c )) da
-(u/TTXoyc)3(r-v)\ c o s 9 / 1 | a 9 i a y / ( I c j da
Also from (44)5(45) ,as v-*0, one has B ->--E and B -*E . E and B ' v ' z .. r r z r z
. t
are zero at the l i j h t cylinder. Thus at the light cylinder where
Ezchanges sign (see figure 6), there i s a null in the electric and
magnetic fi e l d s . The null follows two circles parallel to the
equatorial plane, since the dependence i s contained entirely in
the coscb term of (47) .
Some estimates of the order of magnitude of the various quantities
can now be given. The magnetic dipole moment of the neutron star i s
roughly BS
R ~ 10 esu for Bj= 10 gauss, R=10 cm. From (44J $ (45)
we have , for regions intermediate between the neutron star and
the light cylinder: E= (QAO^lO^su* 10 5volts/cm forft=30 sec'}
Then we have B= E= 10 Gauss, and p= (ft/c)B- 10 esu- 10 electronic
charges per cm3.
TABLE I I : NUMERICAL EVALUATION OF 'X(x,y), WHERE
X ( x , y ) ^ A ) W c ) 2 X ( x , y ) cos*
c 0 0.25 0.5 0.75 1.0
0 s i n g u l a r -50 -11 -1.8 0
jt/8 0 -11 -5.2 -1.2 0
«/h 0 6.0 -0.1 -0.2 p"
it/2 0 25 k.o 0.6 0
I I 1 I •• I 0 0.25 0.5 0.75 1-0
FIGURE 5 : LINES OF CONSTANT X. FORCE FREE m=l CASE
FIGURE 6: ELECTRIC FIELD LINES, FORCE FREE m=l CASE
22.
l'.O
1.0
+ + + • +
+ + + + + 0 1.0
+
-1.0
FIGURE 8: CHARGE DSIISITY, FORCE FREE m=l CASE
23.
RELAXATION OF THE FORCE FREE ASSUMPTION
Up u n t i l now a l l e f f e c t s of a f i n i t e p a r t i c l e mass have been
ignored because of the presence of huge electromagnetic f i e l d s .
To maintain c o r o t a t i o n net electromagnetic f o r c e on the p a r t i c l e s
i s necessary. This means the charge to mass r a t i o of a l l p a r t i c l e s
i n any a r b i t r a r i l y small volume must be the same. Thus we assume
charge s e p a r a t i o n and the i d e n t i t y of a l l p a r t i c l e s at a given
p o i n t i n the magnetosphere.
In r o t a t i n g c y l i n d r i c a l coordinates the Lagrangian f o r s i n g l e
p a r t i c l e motion i s :
(49) L=-mc2/y . - (q/c) (A^+A^+A3<p +A Q)
where *=d/dt- m and q are the mass and charge of the p a r t i c l e . We d e f i n e
(50) y = c ( c 2 - r i r 2 - z 2 - f 2 - 2ftr 2cp- r 2<p 2 )~^ 2
Since L does not e x p l i c i t l y depend on time(assuming t h a t the c o v a r i a n t
f o u r p o t e n t i a l A^ i s time independent i n the r o t a t i n g frame), the
energy H i s conserved:
(51) H= (3L/3 x 1 ) x 1 - L = m ( ( c 2 - f t 2 r 2 ) - n r 2 < f )+ti/c) A Q
The Lagrangian equations of motion are:
(52) d/dt (Ymz) = ( q / c ) F l a x a
2 ct * d/dt (ymf) -• ymr(Q+$) =(q/c)F2 ax
d/dt ( Ymr 2(<£+n)). = ( q / c ) F 3 a x a
3 ct *
C o r o t a t i o n has: z=f=<£=0. Using F = 9A /9x - 9A„/9x .(52) reads
0= 3A Q/9z -Ymrft2= (-q/c) 9A Q/9r 0= 3A /3<f>
Thus apart from a t r i v i a l ' c o n s t a n t we have: (53) A q= (-mc2/q) ( c 2 - ^ ^ 2 ) ^ = C-c/q) m c 2 ^ .
This provides the • necessary electromagnetic f i e l d to h o l d
the p a r t i c l e s i n c o r o t a t i o n .
4
24,
The previous force free models involving the stream function
^(axisymmetric case) and the magnetic potential x c a n n o w be modified.
For the former case write:
(54) A = ((-c/q)mc 2/ Y,, 0,0,^.1 y <p
so that B is given by (18) and E by (17) when m=0 (see appendix 2 Ci R
for the relation between F and E § B). Using g from (28) one exp
gets:
(55) F 0 1= ( - l / c 2 ) F 0 1 + ^ / c 2 ) F 3 1 <Q / c 2 ) 9 ^ / o Z
F 0 2=(-l/c 2)F 0 2 +Cn/c^ 3 2 = ( . c / q ^ j i 2r / c 2 ) Y ^ / c 2 ) 8 * / 3 r
03 12 F =0 F = F 1 2=0
F 2 3 = ( f i / c 2 ) F 2 0 + C l / r 2 Y ( j )2 ) F 2 3 = C-mcfi/q^ 2r/c 2) Y ^ Y ^ r 2 ) 3,J,/3r
F 3 1= ( o y c 2 ) F 0 ± + ( l / r 2
Y ^ 3 1 =(l/r 2Y #
2)3*/3z
Thus Maxwell's equations:
v (56) (4T!r/c)jy=(r F y v ) , .
with J^=p and J*=0 (corotation) read:
(57) y =0: (4Trr/c)p=3/3z((ox/c2) 3i|)/3z) + 3/3r(fnr/c2) 3IJJ/3r-<c/q)fmft2r2/c2) Y )
p=3: 0=3/3z((l/ Y^ 2r) 31i)/3z) + 3/3r((mcoy q)(fi 2r2/c 2) Y^/ Y^ 2r) 3^/3r)
These give, respectively, the equations for \JJ and p when inertia
is included. When the mass m is zero, these reduce to (23) and
(24), using (21).. To find the solution to (57) write:
(58) ty =tyh + ty^
where ty^ is the solution to the homogeneous equation(m=0) as given
by (21)and (25). Since the inhomogeneous term is a function of r 2 2 2 2t alone, we have: d/dr((l/ Y r) d ty^/dr +(mc9/q^fl r /c^Y^O . Setting
the constant of integration to 0,we get; 4JjP={-mcfyqXfi2r2/c2)rY,3
25.
=(-mcfi/q)r( YJ-Y^). With d/dr( y^)=ny^n+2ii2?:/c2 we o b t a i n :
(59) V(-m cVq)(Y^l/Y^) For the magnetic p o t e n t i a l , F1-1 are known from (34) i n terms
of X but F ^ are not. Wc c a l c u l a t e F ^ as f o l l o w s :
F 0 1 4 l / c 2 ) F ( ) 1 < n / c 2 ) F 3 1 = 0 ^ / c 2 j ( n r 2 F 0 1+ r 2 F 3 1 ) g i v e s F 0 1 = Y fcr/c2j 3 X / 8 r
S i m i l a r l y F 0 2= - Y ^ c / q ^ m ^ r / c 2 ) Y ^ frr/c2) 3 X/9z) and F° 3=0 are found.
The only Maxwell's equation that.changes i s f o r u=0, i n (56):
(4Trr/c)p=-mn 2(r/qc)Y ( i )3(Y (j )
2-l) - (P/c2) 2 r Y ( ( )4 3 X/3z
The equation f o r x g i v e n by the V=3 equation o f (56) i s unchanged
so t h a t F1-* are unchanged. Thus o n l y P and F^ 1 are m o d i f i e d .
However, r e f e r i n g to appendix 2, one sees that both E and B are •
a l t e r e d .
In both cases, f o r and X , one has the r e s u l t :
(60) . 4Trp(qc/ft)Y^ - m c ^ (Y^ 2 + l ) -2qB z
•'The l e f t hand s i d e of (60) i s never negative (Pq>Q) . Thus we have 2
2qB^ <-mcftY(j) (Y^ +1) < 0 so tha t q and B^ are always of op p o s i t e sign.Thus one
has |B z >(mcft/2| q|) Y^ ( Y ^ + l ) . In t h i s model q may have d i f f e r e n t
v a l u e s i n d i f f e r e n t r e g i o n s . F i e l d l i n e s along which q does not
change s i g n clo not have a change i n the s i g n o f IV . I f
q changes s i g n , B^ must change d i s c o n t i n u o u s l y , r e s u l t i n g i n a
cusp i n the f i e l d l i n e . Since P=0 at the p o i n t o f change o f q , ( 6 0 ) y i e l d s
(61) LZz= cft(m 2/2q 2 - m ^ q ^ ( Y ^ 2+ l )
For a change from e l e c t r o n s t o protons i n the i n t e r m e d i a t e magneto
sphere, one'has : AB^~ cft(m p+m e)/e~ 2* 10~ 3gauss ( f o r ft-20 s e c " 1 ) .
26.
Thus the cusp i n the f i e l d l i n e s i s n e g l i g i b l e u n t i l one gets
c l o s e to the l i g h t c y l i n d e r . From (61) and the estimate of B - - 3 3 3
on page 17 : B^/B^ 10 y /10 . This i s of the order o f u n i t y 6 2 wheny^-10 . At t h i s p o i n t the p a r t i c l e energy,Y^roc > exceeds
the magnetic energy d e n s i t y . C o r o t a t i o n i s no longer v a l i d , so
t h i s model i s no longer v a l i d . N e g l e c t i n g the f a c t t h a t the mode
i s no longer p h y s i c a l f o r such l a r g e values of y ^ j t h e change i n
f i e l d l i n e s . i s i l l u s t r a t e d i n f i g u r e 9 f o r the axisymmetric case.
Compare t h i s : t o f i g u r e 1. Now the f i e l d l i n e s are not h o r i z o n t a l
at the l i g h t c y l i n d e r but v e r t i c a l and thus do not penetrate the
l i g h t c y l i n d e r .
27.
FIGURE 9: MAGNETIC FIELD LITISS FCR MISTOMETRIC CCROTATION
CASE WITH INERTIA
RELAXATION OF COROTATION 28.
I f one no longer assumes c o r o t a t i o n , what p a r t i c l e motions
can occur and how w i l l the f i e l d s be a l t e r e d ? To answer t h i s
question one procedes as f o l l o w s : C a l c u l a t e the s i n g l e p a r t i c l e
motion i n an a r b i t r a r i l y given f i e l d assuming small departures
from c o r o t a t i o n . I.e. p a r t i c l e v e l o c i t i e s i n the r o t a t i n g frame
are assumed to be of order e where e<<l. Next make assumptions
which give the c o l l e c t i v e motion i n terms of the s i n g l e p a r t i c l e
motion and use t h i s to c a l c u l a t e a l t e r e d f i e l d s . The a l t e r e d
f i e l d s can be used to r e c a l c u l a t e the s i n g l e p a r t i c l e motion. '
One repeats t h i s procedure u n t i l the d e s i r e d accuracy of r e s u l t s
i n orders of e i s a t t a i n e d . The model i s then s e l f - c o n s i s t e n t
to that order.
29. SINGLE PARTICLE MOTION
For s i n g l e p a r t i c l e motion we assume the f i e l d s are such that
c o r o t a t i o n i s p o s s i b l e .I.e. we imagine that a l l the p a r t i c l e s
except the t e s t p a r t i c l e i n question are i n c o r o t a t i o n . Then by 2
(53) we have (q/c)A Q= -mc /y^ s i n c e a l l p a r t i c l e s at any p a r t i c u l a r p l a c e i n the magnetosphere are i d e n t i c a l , w i t h mass m and charge q.
We now w r i t e the s i n g l e p a r t i c l e Lagrangian (49):
2 2 (62) L= -mc ly +mc /y -(q/c) (A 1 z+ A 2f+ A^)
= (m/2XY 9(z 2+ * V Y ^ r V ^ q / c ) A ^ q / c j A ^ 2
-((q/c) A^- my^1- ft)<J>+ terms cubic i n the v e l o c i t i e s
Since we are assuming the v e l o c i t i e s are of order e, e<<l, we
keep terms t o second order and w r i t e L . i n the form:
(63) L=%( Ax 2+ By 2+ C z 2 ) - S^x- <J>2y- * z 3 2
where (x,y,z) = (z ,r ,<j>), A=B=mY(j), C=mY^ r , $ 1=^/c)A ] L, $ 2=^/c)A 2, 2
and $3=^/c) A^- my^r Lagrangians of t h i s form are t r e a t e d i n
appendix 4. The main r e s u l t s a r e :
1) The motion c o n s i s t s of a slow d r i f t tangent to s t r e a m l i n e s ,
s p e c i f i e d by dx/X = dy/Y. = dz/Z where:
(64) X= Cq/c)rBz ^ Y ^ Y ^ l )
Y=(q/c\rB r Z ^ q / c ^
and a s p i r a l i n g motion around these s t r e a m l i n e s .
2) The energy E i s conserved.
3) The energy can be separated i n t o a l o n g i t u d i n a l energy 2
a s s o c i a t e d w i t h the d r i f t — % P J l ' , and a t r a n s v e r s e energy assoc-
30.
i a t e d w i t h the s p i r a l i n g — Noo .
4) An a d i a b a t i c i n v a r i a n t e x i s t s f o r the motion, namely M= *
N a a /IM> the t r a n s v e r s e energy d i v i d e d by the angular frequency
of the s p i r a l i n g .
t P, J2jN and a are d e f i n e d by equations (8), (9), (10), and (16)
2 o f appendix 4. E s s e n t i a l l y , %P£ i s the l o n g i t u d i n a l k i n e t i c
* 2 * 2 2 2*2 ^ energy- h^Y^t z H+ r u+ r <£M ) and Naa i s the t r a n s v e r s e k i n e t i c . 2 • 2 2 2 • 2
energy- ^ Y ^ t ZjJ- r A+ r y^ <J>) . The s u b s c r i p t s r e f e r t o motion
p a r a l l e l t o , and p e r p e n d i c u l a r to s t r e a m l i n e s .
•I
COLLECTIVE MOTION 3 1 *
If,instead of allowing just one particle to move,one allows a l l to
move,then the fields will be altered. In the formalism of the
single particle treatment this can be taken into account since
the four potential, except for A q , was- not assumed to be of a
particular form. Thus one on l y needs t o all o w f o r a different A o.
We write:
(65) A q= (-c/q>c 2/Y 9 + A Q
This modifies the Lagrangian L and thus the energy, which becomes:
(66) E = %P*2 + Cq/c) A Q (for M=0)
2 Thus one sees that A is second order in e since both E and 1
o 2
are of order e (see (67) below).
To determine the fields we must make assumptions about how
the currents derive from the single particle motion. Since we
have steady state*all particles which move along a given streamline
must be of the same type. The Coulomb interaction between streaming
particles results in no spiraling motion (M=0) and ensures that
a l l particles along the same streamline have the same energy E.
If M were not zero, collisions would occur between particles on
neighboring streamlines which damp out the transverse motion.
Thus we have convection current only: j=pv or: (67) J =p(l,x,y,z) = p(l, JtX, £Y, 11 )
using (16) of appendix 4(with c=0). X,Y,Z are given by (64). The equation
of continuity (yJ^),^ =0 reads:
(68) y(P*X),x + (yP*Y),y + y (p£Z) , z = 0
32.
From the d e f i n i t i o n (2) o f appendix 4,one has the i d e n t i t y
(corresponding to v-ko) : x y + Z = 0., thus (68) becomes:
(69) X(p£y), x + Y(P£y), y + Z ( P * / ) , Z = 0
Therefore P£y i s constant along the streamlines s p e c i f i e d by
dx/X = dy/Y = dz/Z . We w r i t e :
(70) p£y =(c 2/4Ttq) K
where K i s constant along a s t r e a m l i n e , the v a l u e of which i s
determined by the boundary c o n d i t i o n s . Write p = P Q + P J where
PQ i s zero order ..in e and p ^ i n c l u d e s a l l higher orders. (70)
•gives £ t o f i r s t order i n e as a f u n c t i o n of p o s i t i o n along a
s t r e a m l i n e s i n c e p ^ i s known from the c o r o t a t i o n case as given
by (60). Maxwell's equations(56) w r i t t e n out are:
(71) ( r F ? 1 ) ^ =(4Trr/c) J V = (4Trrp/c)(l, *X, £Y, IZ)
= (4Trrp/c, Cc/q)KX, fc/q) KY, Cc/q) KZ)
Equation (70) r e q u i r e s t h a t K and £ be zero along a s t r e a m l i n e
where p changes s i g n . In steady s t a t e , and w i t h charge s e p a r a t i o n ,
no streaming can occur along any s t r e a m l i n e along which the charge
changes s i g n . From (60) p Q changes s i g n , to a good approx
imation except very near the l i g h t c y l i n d e r , when B z changes s i g n .
Thus the p u l s a r must have a c o r o t a t i n g 'dead' zone, where no streaming
occurs, which very n e a r l y c o i n c i d e s w i t h the r e g i o n c o n t a i n i n g
f i e l d l i n e s which go h o r i z o n t a l i n s i d e the l i g h t c y l i n d e r . This
s i t u a t i o n i s i l l u s t r a t e d f o r the axisymmetric case i n f i g u r e 10.
33.
FIGURE 10. Streaming and C o r o t a t i n g 'Dead' Zones
Axisymmetric Case
LEGEND -
- C o r o t a t i o n Region
1 Streaming Region
v
0 1 Note: the s i g n of charge i s i n d i c a t e d by + and - above.
34. AX I: SYMMETRIC CASE
Write (72): A^= (fc-c/ql mc2/y^, 0, 0,* ) + ( A Q, A j , A 2 , Ap
so t h a t the zero order p a r t i s i d e n t i c a l t o (54) . The • terms
higher order i n e are contained i n the second set o f b r a c k e t s .
For a x i a l symmetry the p a r t i a l d e r i v a t i v e of any q u a n t i t y , w i t h r e s p e c t
t o 9 , i s zero. Thus A 1 and A 2 always appear i n the
combination 8A ] L/3r - 3A 2/3z = F j 2 = . Now from (64) we have:
(73) X= Q-q/c) (3A 3/3r +3i|//3r ) + nSlry ( Y ^ + l ) "
Y=(q/c)(3A 3/3z +3i|>/3z) Z= Cq/c) B^
F a 3=3A a/3x 3 -3A e/3x a i s c a l c u l a t e d from (72) and F a 3 c a n be found
u s i n g g from (28). The r e s u l t i s :
(74) F 0 1 = ( - l / c 2 ) 3 A 0 / 3 z + (Q/c 2) (3<J//3 z + 3A 3/3z) . 02 ? 9 9 9 F = ( - l / c ) 3 A Q / 3 r - (c/q^mQ r/ c .) y +(«/c ) (3^/3r + 3A 3/3r)
F 2 3 = (-mcA/q^r/c^Y - ( ^ / c 2 j 3 A Q / 3 r 4/Y^ V ^ ^ r +3A 3/3r)
F 3 1=(fi/c 2)3A 0/3z + ( l / Y ( J )2 r 2 ) ( 3 ^ / 3 z + 3A 3/3z)
The zero order p a r t s o f (74) are i d e n t i c a l t o (55), as necessary
f o r c o n s i s t e n c y .
Maxwell's equations equations (56) w i t h JV g i v e n by (67) a r e :
(75) y = 0: (4irr/c> (pQ+p j)=3/3 z (C-r/c 2) 3A Q/3 z^r/c2) (3^/3 z+9A 3/3 z))
+9 /3 r ((-r/c 2) 9 A Q/3 r - t / q X m f 2 2 r 2 / c 2 ) Y^ - ^ r / c ^ / S r+3 A 3/3r ) )
(76) v=l: (4irr/t) (pQ+P^ftX = 9/9r (rB^)
(77) p=2: (4irr/c) (P Q + P J R Y = -3/3z (rB^)
(78) u =3: (4rrr/c) (p Q+p ^ = 3 /3 z(C2r/c 2) 3A Q/3 Z-^/Y^ 2r) (3^/3 z+3 Aj/3 z) )
+3 /3 i ( ( c m n / q ) ^ 2 r 2 / c 2 J Y^ -$r'/c 2) 9 A Q/9 r+fr/y^ 2 i ) ( 3 * / 3 r+9 Aj/3 z) )
35.
The zero order p a r t s o f (75) and (78) are the same as (57) and
g i v e ^and p Q . "
Using (70) and (73), (76) and (77) have f i r s t order p a r t s :
( c / q U 3/3r ( ( - q / c ) * + n t f r ^ ) = 3/3 r (rB^)
(c/q)K 3/3r ((q/c) = -3/3 z (r B^)
.We d e f i n e :
(79 ) = ty-(c™n/q)r2Y^
so t h a t these equations become:
(80) K 3/3x X t|>' = -3/3X 1 (rB^) w i t h ( x 1 ,x 2) = ( r , z ) . .
Thus we have the f o l l o w i n g r e l a t i o n among K,^1, and B^:
(81 j K = -d(rB^Vd*«
Note t h a t K,^', andrB^ are a l l constant along s t r e a m l i n e s ( a c t u a l l y
the s u r f a c e s dx/X = dy/Y f o r t h i s case o f a x i a l symmetry). From
(81) i t i s seen t h a t i n r e g i o n s where streaming does not occur
(K=0), rB^ i s constant .everywhere.^/
A Q i s known from the energy equation (66) t o second order s i n c e
A i s known to f i r s t order i n e f r o m (70). From (78) we'can c a l c u l a t
t o second order:
(82) l l / v f o 3 2 / 3 z 2 A 3 + 3 / 3 r ( ( l / Y 4 )2 r ) 3 A 3 / 9 r ) = K B ( { ) ^ / c 2 ) V 2 A 0
2 where V i s the L a p l a c i a n operator i n c y l i n d r i c a l c o o r d i n a t e s .
(75 ) g i v e s t o second o r d e r :
36.
(83) (4Trr/c)p1=«:/c2)V2(nA3-A{))
In summary:The zero order q u a n t i t i e s ^and are given by
(57). The f i r s t order q u a n t i t i e s K,£,.and are given by boundary
c o n d i t i o n s , (70), and (81) r e s p e c t i v e l y . . The second order
q u a n t i t i e s A n,A_, and p 1 are given by (66),(82), and (83).
THE GENERAL CASE 37.
The magnetic p o t e n t i a l x cannot be used i n the case where
streaming c u r r e n t s occur s i n c e the d e f i n i t i o n (34), when i n s e r t e d
i n Maxwell's equations (56), i m p l i e s zero current J 1 .
However t h i s problem can be handled i n the f o l l o w i n g manner.
We assume a z,<\> dependence of the form e 1 U , where u=kz+mc|> (m an
i n t e g e r , k a r e a l number). From Maxwell's equations(71) we have:
(84) K ( r F 1 J ) = (c/q)K(K X+ K , Y + K Z)
Thi s i s i d e n t i c a l l y zero, from (69). We have,from the assumption
concerning the z,<j> dependence : 3/3z=k3/3u and 3/3<j>=m3/3u so t h a t
(84) becomes:
(85) 0=K, i(rF 1 ; i) ,j=3K/3u 3/3r ( k r F 1 2 - m r F 2 3 )
-3K/3r 3/3u ( k r F 1 2 - m r F 2 3 )
= J(K, k r F 1 2 - m r F 2 3 5 r,u ) 1 2 - 2 3
I.e.. the Jacobian of K and krF -mrF w i t h respect t o the v a r i a b l e s
r a n d u vanishes. Thus the Jacobian of the general stream f u n c t i o n
A a l s o vanishes, where A i s defined by:
(86) KkA= n i r F 2 3 - k r F 1 2
Since K i s constant along streamlines dx/X = dy/Y = dz/Z , A i s
a l s o constant along streamlir.es.
With a z,9 dependence of e 1 U the s o l u t i o n s w i l l be h e l i c a l
waves. To get a p h y s i c a l l y acceptable s o l u t i o n , s o l u t i o n s of d i f f e r e n t
k and ni must be superposable through F o u r i e r a n a l y s i s . In t h i s way the
t o t a l s o l u t i o n can f i t the boundary c o n d i t i o n s . Thus Maxwell's equations
(71) must be l i n e a r . This i s t u r n r e q u i r e s K to be constant or at
38.
l e a s t piecewise constant i n d i f f e r e n t regions of the p u l s a r mag-
netosphere.e.g. K=0 f o r the c o r o t a t i n g dead zone and K = K q (a constant)
o u t s i d e the c o r o t a t i n g dead zone. We r e s t r i c t K to be piecewise
constant f o r the c a l c u l a t i o n o f A. In the f o l l o w i n g the lower
case l e t t e r s r e f e r to Fourier components of the f i e l d s :
(87) F a e(z,r,<(,) = z c o s ^ c j , ) / 0 0 C-(k) f a 3 ( r ) e l k z d k
Note that (86) i s no longer v a l i d but i n s t e a d we have:
(88) K k X = m r f 2 3 - k r f 1 2
ct 8 ot 8 Now f and f . ., us i n g A Afrom (65) and g from (28),, a r e : Ctp u (89) f 0 1 = i k i 0 f 0 2 = d a Q / d r -tfncft 2r/q} y ^ ( k ) f
o z
= i ^ 0
(90) £ =b f 2 3 = r b z f 3 1 = r b r
r u l _ / ; i , /„2N - ./ n /_2 >^ ,-03 .-- , 2 2 (91) f U H ^ / O v ( f t / c ) f 3 1 •• f = " i m i 0 / c 2 r ^
f too 4/c 2;. 2 3 f 0 2<-l/c 2;d£ 0/dr -(mft 2r/cq)Y d )5(k) -$/c2)fr
(92) f l 2 = f1 2
f 3 1 4 f i k / c 2 ) a 0 ^ / r 2 Y ( { )2 ) f 3 1
f 2 3= ( . f j / c 2 ; d a 0 / d r - ^ f t 3 r / c q ) Y ( { ) 6 ( k ) 4/r2y*)f23
where 6(k) i s the D i r a c d e l t a f u n c t i o n . W r i t i n g out the components
o f Maxwell's equations (71) ,using (64) , we get:
(93) u=l: d / d r ( r f 1 2 ) + imrf 1 3=K(f 2 3<cmft/q)d/dr(r 2Y ( j )) «(k)) 21 23 u=2 i k r f +imrf = K f 3 ^
u=3 i k r f 3 1 + d / d r ( r f 2 3 ) = K f J 2
From the y=2 equation one has:
(94) f 3 1 = i k X
Combining.the u=l and u =3 equ a t i o n s , one o b t a i n s :
K(mf 1 2+kf 2 3)+Kk(cmfyq >)d/dr(r 2Y (j 1) <$(k)=d/dr ( k r f 1 2 + m r f 3 2 ) = -Kkd X/dr . «• ,
From (88) and (91) one f i n d s :
Kk^=(m/Y^ 2r)f 2 3-<mrft/c 2)di 0/dr-<mma 3r 2/cq)Y ( j )5(k) - k r f 1 2
39.
Using the previous two r e l a t i o n s , one o b t a i n s :
(95) f =(-Kk 2Mmk/Y 62r)dA/dr-(mknr/c 2)da /dr^cm^mk/q)Y, 6 (k) )
(96) f 2 3-(Kkm X - k 2 r d X / d r ^ n 2 f t r / c 2 ) da 0 /dr-^irfVq) (m 2(Y ( j ) + l / Y ( j ) )
- k 2 r 2 Y ( j ) ( Y ( j )2 + l ) ) 6 ( k ) ) H k 2 r + m 2 / ( Y ^ r ) )
Thus (94), (95) , and (96) g i v e f „ i n terms o f Xand a^.
To o b t a i n the d i f f e r e n t i a l equation f o r X, take the u = 3 component
i n (93) and s u b s t i t u t e f o r f , f , and f 1 0 . We d e f i n e :
(97) t, =k2r\2+m2
The d i f f e r e n t i a l equation f o r X i s :
'(98) k 2 d/di((r/S) dX/dr)+k 2((Kr/C) Y ^ i U k / C ) Y^*+K)-1/Y^ 2 r ) '
=-d/dr ( ( fJr/c 2 Xk 2r 2 Y 4 )2/C)d£ 0/dr) -{Cmkr/Cj vfar/c2) d a Q / d r
. - ^ r / c 2 ) S 0 + ^ / q ) 6 ( k ) ( d / d < ( m 2 / ^ ) ( Y ^ - l / Y ^ V / O Y ^ f Y ^ + l ) )
J$2x/c2)y^(y2+l)^Y^T/K)y*)
For the f o r c e f r e e case(aQ=0,m=0) the r i g h t hand s i d e o f (98)
vanishes to g i v e the e c u a t i o n f o r the f o r c e f r e e stream f u n c t i o n .
Write ^ - ^ o + ^ i where i s zero order i n the streaming v e l o c i t i e s
and ^ i n c l u d e s a l l other o r d e r s . The zero order p a r t o f (98) i s :
(99) d/dr((k 2r/ c ) dX Q/dr ) - $ < 2 /Y ( | )2r)X 0 =
cmfi Y ( { )46(k ) (5c 2r 2/qrC 2J (iH% ( l ^ 2 ) ^ 2 ^ ( 1 - Y ^ 2 ) )
For the axisymmetric case(m=0) (99) becomes:
(100) d/di((l/ rY ( j ) V x ( )/dr)--k 2X 0 /Y ( j )2r^cmn/qr)2Y ( { )(l - Y ( { )
2)6(k)
The F o u r i e r transform of t h i s i s not the same as the equation
. f o r i|> (57), but i s the same as the equation f o r i> ' = (-cm^/q) r Y^+S'
(from ( 7 9 ) ) . . We have: d / d i { ( l / r 2 Y ^ 2 ) d/dr ((-cm^/q) r 2 Y ^ y ) = 3 3
•: (-m /cq) rY^ (Y 9 - 1 ) . When t h i s i s added t o the r i g h t hand s i d e
of (57) one gets the equation f o r 4": (101) V ^ C l/rY^ 2) W'/tW/yfo 9V/9z2<cm^/q) 2Y<f| (1-Y^ 2)
40.
T h i s i s j u s t the F o u r i e r transform of (100). This r e s u l t i s c o n s i s
t e n t s i n c e i t i s and not ij>which i s constant along s t r e a m l i n e s .
The f i r s t order p a r t of (98) i s :
(102) d / d r f t k ^ / O d A j d r ^ - O ^ / Y ^ r ) A 1 =
(-2KkirY^ 2/0((crafi/q) <5(k) Y <^ k^^/cV
This gives X=AQ+ X l U P t o s e c o n c ^ order i n the streaming v e l o c i t i e s ,
The u=0 equation o f (71) can be w r i t t e n , u s i n g (91), as:
(4irr/c) (p0+p 1)=(c/c 2r)£ ( f(Y 4 )2/c 2)d/dr(r d^ / d r ) - ( m a 2 r / c q ) y^ 3 ( Y ^ + l ) 6 (k)
2 2 2 2 - _ •*§x /c Kf 1 2 - ^ Q / c ) y ( j ) f 2 3 where p Q and p j are the F o u r i e r components o f p Q.andpj. The zero order p a r t of t h i s equation i s :
(103) 4T r rp 0/c=(-ft/c) T ( f )2r((mcft/q)Y ( ( )( Y^ 2+l)6(k) + 2b z) •
The F o u r i e r transform of t h i s i s i d e n t i c a l t o (60) ,as necessary
f o r consistency. The remainder i s :
(104) 4 1 r r p 1/c=(l/c 2;((?a 0/r ) - Y ( j )2d/dr(r d i Q / d r ) Hft/e2) Y ^ r 2 ^
T h i s has f i r s t order and second order order p a r t s , i n c o n t r a s t
to the axisymmetric case. This i s because b^ now has a zero order
p a r t , a^ i s s t i l l second order ,as given by the F o u r i e r transform
of (66).
The f i e l d components given by (94), (95),(96) have zero order
p a r t s :
(105) h w = f u 0 .4mk/c)(dX0/dr ^ m f y q^Y ^ C k ) )
r b r 0 = f 3 1 0 = i k X 0 r b z 0 = f 2 3 0 ^ -k^^OY^dXQ /dr^mJVq^Y^raCk) . ( m ^ - l / Y ^ )
2 2 2 -k r Y^(Y^ +1))
41.
The f i r s t order p a r t s o f the F o u r i e r components of the f i e l d s
are given by:
(106) b + 1= - K k ^ Y ^ / C - t m k / ^ d A j / d r
r b r l = i k X x
r b z l = ia£ry2XQ/^ - ( k ^ V / ^ d X ^ d r
One can c a l c u l a t e the magnetic f l u x through a v e r t i c a l s l i c e ,
width rA<j>, o f a c y l i n d e r w i t h a x i s c o i n c i d i n g w i t h the s p i n a x i s
( z - a x i s ) , extending from z=0 t o Z = Z Q . - : . T h i s i s : z z
(107) rA<j>/00 B rdz =A<|,/0
0 F 3 1 d z =A(f»(A(z0,r,<j»)-A(0,r,<|)))
-a simple r e s u l t . I f one i n t e g r a t e s over <j>, t h i s w i l l v a n i s h except
f o r the m=0 case, f o r which one obtains the same r e s u l t as f o r the
stream f u n c t i o n if>: 2iTjfi ( z Q , r ) - A (0,r))
From the F o u r i e r transform (87) and the formulae f o r f „ i n Ctp
terms of Xand a^ i t i s p o s s i b l e t o c a l c u l a t e a l l the f i e l d s . To
c a l c u l a t e X^ f o r a s p e c i f i c set of boundary c o n d i t i o n s there
are two approaches: i ) C a l c u l a t e the s e r i e s s o l u t i o n to (100), and
compute s e r i e s f o r ^°ZQ>^TQ>^^Q• Match these to the F o u r i e r transforms
of the f i e l d s at the neutron s t a r t o get C-(k) . i i ) Use the r e s u l t
(107) and c a l c u l a t e the c y l i n d r i c a l s l i c e f l u x near the neutron
s t a r ( e . g . f o r an obli q u e d i p o l e f i e l d ) . Then d i f f e r e n t i a t e w i t h
respect to <f> t o get A(z,r,<f>) at the boundary, Inverse F o u r i e r t r a n s
form to get X at the boundary. Match t h i s to the s e r i e s s o l u t i o n
t o (100) to determine C^(k). The f i r s t procedure i s s i m i l a r t o
tha t used i n s o l v i n g e x p l i c i t l y the axisymmetric f o r c e f r e e c o r o t a t i o n
case f o r a d i p o l e f i e l d at the neutron, s t a r , i n appendix 1.
42. D I S C U S S I O N
A f t e r arguing f o r the existence o f a magnetosphere we s t a r t e d
h i t h a simple but r e s t r i c t i v e model u t i l i z i n g the stream f u n c t i o n
IJJ. T h i s was generalized t o f i t any boundary f i e l d at the neutron
s t a r v i a the magnetic p o t e n t i a l x Next the f o r c e f r e e
assumption was r e l a x e d , ivjVich had the e f f e c t of a l t e r i n g the f i e l d s
to hold the p a r t i c l e s in c o r o t a t i o n . F i n a l l y the p a r t i c l e s were
allowed to move. The type o f motion th a t occurs was c a l c u l a t e d 'as1.was
i t s r e s u l t i n g e f f e c t on the f i e l d s .
The two major assumptions are: 1) the model i s steady
state(time independence i n the r o t a t i n g frame); 2) p a r t i c l e motions
are small departures from c o r o t a t i o n . .The f i r s t assumption seems .
q u i t e reasonable. However some p u l s a r models, most notably those
i n v o l v i n g some type of r e l a x a t i o n o s c i l l a t i o n to explain the emissions,
such as the spark gap model of Ruderman and Sutherland 4, abandon assumption
1) e n t i r e l y . In any case, keeping 1) i s reasonable i n order to
c a l c u l a t e the conditions under which non-steady processes occur.
The second assumption i s on l e s s s o l i d f o o t i n g . I t i s q u i t e reason
able well within the l i g h t c y l i n d e r , but as one approaches regions where
i s very large, the p h y s i c a l s i t u a t i o n i s ' u n c l e a r . Many other
processes become important. Others have proposed a shock f r o n t
near the l i g h t c y l i n d e r . Associated with 2 ) i s the assumption
of charge separation, which follows n a t u r a l l y from the steady
9 s t a t e c o r o t a t i o n case. Again some authors assume countering views.
43.
The l a s t models presented here, a l l o w slow p a r t i c l e
motions. The c o n t i n u i t y equation gives r i s e to the parameter K
which i s constant along a s t r e a m l i n e . Except f o r the axisym
metric case; K i s f u r t h e r r e s t r i c t e d t o be piecewise constant.
K must be known before one can proceed w i t h the c a l c u l a t i o n o f a
model. It i s determined by the c o n d i t i o n s at the neutron s t a r sur
f a c e . Since one knows the p o s t i o n of the streamlines from the
zero order c a l c u l a t i o n , K i s - then known everywhere i n s i d e the
l i g h t c y l i n d e r . From (64),(67) and(70) we have:
KCB^cmn/q^CY^.+l), B r, B ^ ^ P / O (z,*,*)
The zero order charge d e n s i t y and f i e l d s are known. The physi c s
of the emission process enters the problem i n determining the
i n i t i a l v e l o c i t i e s and thus K. If the neutron s t a r has a uniform
surface and a pure d i p o l e f i e l d , the surface p r o p e r t i e s and thus
K should be a function only of the angle from the d i p o l e a x i s .
P r e c i s e l y what t h i s f u n c t i o n i s depends on a knowledge of the phy s i c
of the s u r f a c e , which i s outside the scope of t h i s i n v e s t i g a t i o n .
The r e s t r i c t i o n f o ttatise a, piecewise constant K f o r the case of
the general stream f u n c t i o n A ,may or may not be an unreasonable
approximation to the a c t u a l p h y s i c a l s i t u a t i o n .
44.
BIBLIOGRAPHY
1. t e r Haar D, Pulsars,'Phys .Rep .3_, no . 2,57-126, (1972)
2. Ruderman M, P u l s a r s : S t r u c t u r e S Dynamics,A.R.A.A., 427-476(1972)
3. G o l d r e i c h P § J u l i a n PulsarEiectrodynamics,Ap.J,157_,869-880(1969
4. Ruderman M § Sutherland P,Theory of P u l s a r s : P o l a r Gaps, Sparks
and Coherent Microwave Radiation,Ap.J.,196,51-72,(1975)
5. Arfken G, Mathematical Methods f o r P h y s i c i s t s , Academic Press
1970 (2nd e d i t i o n )
6. G o l d s t e i n H, C l a s s i c a l Mechanics, Addison Wesley 1965(7th p r i n t i n g )
7. P r o t t e r M § Morrey C, Modern Mathematical A n a l y s i s , Addison
Wesley 1970
8. Ardavan H, Magnetosphere Shock D i s c o n t i n u i t i e s i n Pu l s a r s . I . '
A n a l y s i s of the I n e r t i a l E f f e c t s at the Lig h t Cylinder,Ap.J,203,226-232(1976)
9. Henriksen R § Rayburn D, Hot Pu l s a r Magnetospheres, M.N.R.A.S.,
166, 409-424,(1974)
10. Henriksen R § Norton J , Obiique R o t a t i n g P u l s a r Magnetospheres
With Wave Zones, Ap.J,201,719-728,(1975) •
11.. Mestel L, P u l s a r Magnetosphere, Nat .Phys.Sci. ,233,149-152, (1971)
12. M i c h e l F, Ro t a t i n g Magnetosphere: A Simple R e l a t i v i . s t i c
Model, Ap.J,180, 207-225,(1973)
13. Cohen J § Rosenblum A, Pu l s a r Magnetosphere, Astr.SSpace S c i . ,
6,130-136,(1972)
45.
14. Mestel L, Force-Free P u l s a r Magnetospheres, Astr.§Space S c i . ,
24,289-298,(1973)
15. Okamoto I, Force-Free P u l s a r Magnetosphere I,M.N.R.A.S.,167,
457-474, (1974)
16. Scharlemann E 5 Wagoner R, A l i g n e d R o t a t i n g Magnetospheres I
General A n a l y s i s , Ap.J,182, 951-960,(1973)
APPENDIX 1
( l - n 2 r 2 / C 2 ) 0 2V / 8 r 2 + 9
2 V / 9 z2 ) - ( 1 / r ) ( l + n 2 r 2 / c 2 ) 9 V / 9 r = 0
(1)
Since t h i s equation i s separable i n r and
V(r,z)=X(x) Y(y)
we write:
where x=9x/c, y=ftz/c. Separation of va r i a b l e s gives:
(2) (l/Y)d 2Y/dy 2= -a= (1/X)((1/x)(1+x 2)/(1-x 2) dX/dx -d 2X/dx 2)
(3) v= e i o r y •
Writing d/dx = ', one has
(4) X " -(1/x) ( l + x 2 ) / ( l - x 2 ) . X' - a 2 X= 0
•(4) has regular singular points at x=0,l, and" ..
The boundary conditions we use are: 1) The magnetic f i e l d
at the neutron star i s a dipole f i e l d aligned with the spin axis.
2) The f i e l d s at the l i g h t c y l i n d e r are f i n i t e .
From(21)of the main text we have V(x,z) = -(n/c)i|j(x,z)
has the same behaviour as x,z-+0,as the stream function f o r a
dipole f i e l d . In c y l i n d r i c a l coordinates the dip o l e f i e l d i s :
C5). B r = 3 y r z / ( r 2+ z 2 ) 3 / 2 ^ = y ( 2 z 2 - r 2 ) / ( r 2
+ z 2 ) 5 / 2
can be obtained from the stream function:
^dipole= V / C r 2+ z 2 ) 3 / 2
= . ( f i M / c ) x 2 / ( x 2 + y 2 ) 3 / 2
Thus we have:
dipole as x,y-K). We
47.
We need the Fourier transform:
F ( ( 1 i / 2 ) } s x / ( x 2+ y 2 ) 3 / 2 ) = -d/dx ( K Q (<*)) = a K ^ C K )
Here KQ, K 1 are the f i r s t and second Hankel functions. Thus we have:
(j/c) 2vpc (2 7 r )^ F ( x / ( x 2 + y 2 ) 3 / 2 ) = ( f/c) 22 ux aKj ( CK)
We have: Kj+Cl/cx) + (c*/2) log(c#:/2)as x-*0. Thus one obtains:
C o / c ) 2 ( u ^ ( l + ( o 2 x 2 / 2 ) l o g ( a x / 2 ) +• ••) + (£/C)VTT as x*0.
The series f o r the two independent solutions to (4) were
computed:
(7) X(x)=A(cOX 0(x) +B( a)X 1(x)
At the o r i g i n these series.have the values:
X0(0)=0, X1(0)=1
From (7) we then have:
08) B= ( f/c) 2 y/ TT
A i s l e f t undetermined. Both solutions diverge as one approaches
the l i g h t c y l i n d e r , so the requirement of f i n i t e f i e l d s at x=l
cannot be applied i n any simple manner.
We consider the behaviour of X(x) near the l i g h t c y l i n d e r using
the v a r i a b l e v: .
(9) v=l-x 2
We have the r e l a t i o n s :
(JO) dv=-2xdx l+x2=2-v d/dx = -2x d/dv
d 2/dx 2 = -2d/dv + 4(l-v) d 2 / d v 2
With these (4) becomes (with d/dv='):
(11) (v-v 2)X +(l-v)X -(a /2) 2vX =0
We writeX(v) =E a v n + ^ .(il") then becomes:
( 1 2 ) • 1 v n + B - 1 ( ( n + 3 ) 2 a n - ( n ^ - l ) 2 a n _ 1 - ( a / 2 ) 2 a n _ 2 ) =0
48.
The i n J i c i a l equation(n=0) i s 3^=0. Since i t has a double root
the second s o l u t i o n i s (9X/38) _ = X ( v ) l o g ( v ) + r e g u l a r s e r i e s , 3-u
where X(v) i s the s o l u t i o n f o r 6=0. T h i s second s o l u t i o n i s
divergent at the l i g h t cylinder(v->0) . Thus we only need to consider
the. •Mie s o l u t i o n , X ( v ) . The r e c u r s i o n r e l a t i o n f o r the c o e f f i c i e n t s
a^, from (16), i s :
(13) a n = ( ( n - l ) 2 a n _ r ( a / 2 ) 2 a n _ 2 ) / n 2
Taking a^=l we have:
V ( r , z) = A: (a )X (v) e i a y d a =/" (A(a ) X Q (x) + (ft/c) 2 (y /TT ) X j (x) ) e i a y d a
By F o u r i e r transforming t h i s r e l a t i o n , we o b t a i n :
(14) C(a)X(v)= A ( a ) X 0 ( x ) + (ft/c) 2(M / i T)X 1(x)
f o r a l l a and x. For x=0 (v=l),(18) g i v e s :
(15) C(a) = (n/c) 2(y /TT ) 1/X(1)
Thus • the e x p l i c i t s o l u t i o n f o r V ( r , z ) i s :
(16) V ( r , z ) = (n/c ) 2 ( y/7r)/^(X(l-Q 2r 2/c 2)/X(l)) ei a f i z / c da
Note t h a t we could have set x to zero i n the equation f o l l o w i n g
(13), then performed the F o u r i e r transform to get (15) d i r e c t l y ,
without any refe r e n c e to the s o l u t i o n s Xn(.x) and X 1 (x) .
49.
APPENDIX 2 The Physical Meaning of F yv
In non-rotating cartesian coordinates xa=(t,x,y,z) ,
the f i e l d quantities , for Gaussian units, are:
0 E /c X
E /c y
E /c z
(1) ? y v = -E /c X
0 B z
-B y
-E /c y
-B z
• 0 i
B X
-E /c z
B y
-B X
0
In transforming to cylindrical(stationary) coordinates:
x^(t,z,r,9 0) , $ a B = £ a / # ^ 6/35c V F y V is used
have We
«. -0 A0 t=x =x
-3 A l z=x =x
-1 A2 ,A3. KI . ,A3, x=x =x. cos(x ) y=x sm(x )
and
x =(x +x ) 2
9r/ 9^=x/r=cos 9,
A3 ^ x = tan (x /x )
0 3r/ Sy=y/r=sin9 0 99 0/Sx=9/9x(tan" 1(y/x)) = -y/(x2+y2)= - s i n 9 Q /r
9 < f ,0 / 9 y = COSI^Q /r :
Thus one has:
(2) 5 x a / 3 x y =
1
0
0
0
0
0
COS 9n
0
0
s i n 9 n
0
1
0
-sin9 Q/r -cOS9 Q/r 0
Using (2) one can now calculate 1^ in cylindrical coordinat
from (1). The result i s : . es
(3)
0 E /c z E r/c V c
-E /c z 0 - B r / r
-E r/c 0 B /r z B r/r -B /r z 0 .
50.
where:
E =E cos<J> +E sind> r x y B =B cosd> +B s i n * r x y r
The change to r o t a t i n g c y l i n d r i c a l coordinates i s simply ac
complished by u s i n g : d>=d>0-ftt t o o b t a i n the d e r i v a t i v e s :
E =-E sind) +E cosi> <p x r y Y
B( j )
= - Bxsin<{. +EycosiJ>
1 0 0 0
14) 3 x ^ 3 ^ = 0 1- 0 0
0 . 0 1 0
-n 0 0 1
Using (3) and (4) we c a l c u l a t e F y v : F ^ V = ( 3 x y / 3 r ) ( 3 x V / 3 ^ ) r
0 E /c z E /c r y c r
(5) F a p= -E /c z 0 \ UE /c-B / r z r -E r/c " B* 0 ftE /c+B / r r' z'
- ~Vcr B -QE — r - z r c -B -HE -z — r r c 0
Note that E and B are q u a n t i t i e s as measured i n the s t a t i o n a r y
frame. Using g ^ from(27) 0 f the main t e x t , one gets:
0 -cE +ftrB z r -cE -ftrB r z -crE,
cE -ftrB z r 0 N -rB„ A.
cE +ftrB r z 0 rB z
crE^ L *
rB r
-rB z
0
51 APPENDIX 3 S e r i e s S o l u t i o n of:
3 V 3 9 2 + r 2Y ^ 2 ( 3 2 x / 9 r 2
+ 3 2 x / ? z 2 ) + Y ( J )2(2Y ( j )
2-l) 9x/9r = 0
The 9 and z dependences caf* be separated by w r i t i n g x a s :
(1) X (z,r,9) = e i a y e i m * R ( x )
•with x=fir/c, y=Qz/c. m i s an i n t e g e r , anda i s . a r e a l number.
For R , w i t h d/dx= 1 , one obtains the equation:
(2) x 2 ( l - x 2 ) R " + x ( l + x 2 ) R , - C m 2 ( l - x 2 ) + a 2 x 2 ) ( l - x 2 ) R = 0 2 2 2 where 2y^ -l=(l=x ) / ( l - x ) has been used.
Before s o l v i n g (2) f o r R(x),.w e c o n s i d e r t h e problem of matching
t o an o b l i q u e magnetic d i p o l e f i e l d at the o r i g i n . As z,r-K) 'X.
approaches To c a l c u l a t e X ^ ^ p o l e w e must know what F
i s - i n the r o t a t i n g c o o r d i n a t e s . From appendix 2. we have:
F 1 ^ ( l / r ) X , 3 = B F " = C l / r ) x , 1 = f i E r / c + B z / r
F 3 1 = ( l / r ) X , 2 = - f i E z / c + B r / r
As r+0 we have:
(3) X ) 1 - B z X , 2 - B r x, 3=rB^
The f i e l d o f a d i p o l e , i n coordinates z ^ r ' , ^ ' w i t h z' a l i g n e d
w i t h the magnetic a x i s , i s given by -the equations:
B z I = p ( 2 z . 2 - r . 2 ) / ( z . 2+ r . 2 ) 5 / 2 = 9 X d . p o l e / 3 z '
B r . = 3 u r ' z ' / C z - 2+ r ' 2 ) 5 / 2 = 3 X d i p o l e / 3 r '
These .are s a t i s f i e d by the magnetic p o t e n t i a l :
( 4 ) X d i p o l e = - y z , / ( r ' 2 + z , 2 ) 3 / 2
• The diagram next page, i l l u s t r a t e s the r e l a t i o n between
52.
One has the r e l a t i o n s :
c a r t e s i a n coordinates (x,y,z) and
( x ^ y ' j Z ' ) , and c y l i n d r i c a l coord
i n a t e s (z,r,<j>) and ( z ' , r ' , 9 ' } - T h e -
primed coordinates are r e l a t e d
t o the unprimed coordinates by
a r o t a t i o n of 6 Q about the y a x i s .
z' = z cos6 Q+r cos<j)Sine o
,2 ,2 2 2 r ' +z' =r +z 2 2 3/2
C 5 ) x d i p o l e = _ y ( : z c O s e 0 + r c o s < f ' s : L n 9 0 ) / ( : r + z J
From the s e p a r a t i o n of v a r i a b l e s " "(1) ,we have
(6) x(z,r , 9 ) = J e ^ / j c n r / c ) e i a Q z / c da
We i n v e r t t h i s to get: (7) R(x) = ( l / 4 ^ 2 ) / ^d^fjy e " 1 0 " ^ 0 ^ x(cy/n,cx/n,$)
As x.approaches zero, one has X + X d i p o l e . -Thus, from(5) , the
i n t e g r a l i n (7) i s non-zero only f o r m=0,l,-l.
m=0: R Q (x) + (y/2Tr) (R/c) 2cos0 o£dy e~iay(-y/ ( x 2+ y 2 ) 3 / 2 )
Here we use i n t e g r a t i o n by p a r t s The i n t e g r a l i s . t h e n :
/y i a ydci/CxV)%> = iaAxV)-* e'lay dy
This i s j u s t ia(2*)h times the F o u r i e r transform of (x 2+y 2)
which i s (2/7r^K Q(ax) . Thus we have:
(8) R 0(x)-^CM/TT)Cfi/c) 2cos0 0 i a K 0 ( a x ) " l as x->-0
•-(v/ir) (ft/c) icxlog(ax) ) where K (ax)+-log(ax) as x-K) has been used.
53.
Now, the i n t e g r a l ' i n a>is j u s t TT . The i n t e g r a l i n y i s found
d i r e c t l y from appendix 1 tc; be 2 ^ (ax). As x+0 , we have
K 1( ax)->(l/ax) + (ax/2)log(ax/2) Thus one demands that
(9) R ± 1 ( X H - ( v / 2 T T ) ( f t / c ) 2 s i n e 0 ( l / x + (ax 2/2) log (ax/2) ) as x-*0
The so l u t i o n s R(x) were found as s e r i e s about x=0 (two f o r 2 2
m =0, two f o r m =1). However, as i n appendix 1, the boundary
condition at the neutron star determines the c o e f f i c i e n t of only
one s o l u t i o n of each p a i r .
We s h a l l now consider the behaviour near t h e " l i g h t c y l i n d e r 2
using the v a r i a b l e v=l-x (see appendix 1,equations (9 )5(10) ) .
Writing d/dv=* ,(2) becomes: (10) v(l-v) 2R+(v-l)R-((m/2) 2+(a/2) 2(l-v))vR=0
0 0 n+3 Writing R(v)=£a v .(18) becomes: rv n f
(11) f v n + 6 ~ 1 ( ( a / 2 ) 2 a T i _ + ((n +3-2)(n +3-3)-(a/2) 2-(m/2) 2)a r» 1 1 - 0
-(n+S-1) (2n+2B-5)a j+Cn+6) (n+3-2)a n)=0
The i n d i c i a l equation i s B(8-2)=0 , so that 8=0,2.. For 8=0 a l = 0 ' a 2 , a 3 e t c ' d i v e r 8 e - Thus we set a Q=3b so that (9R/8B)^
i s a l s o a s o l u t i o n . But we have: co n
(.12) ( 3V3B ) B = 0 = : R m ( v ) log(v) + E b n v "
Here R. (v) i s the solution.for.3=2, as fol l o w s : m
(13) R (v)=fa v n + 2
m ' 5 nm
With a n m = ( ( n + l ) ( 2 n - l ) a n l + ( ( a / 2 ) 2+ ( m / 2 ) 2 - n ( n - l ) ) a n 2
- ( a / 2 ) 2 a n 3 )*/n.(n + 2))
n-2
54,
Both s e r i e s are f i n i t e as v goes to zero. From appendix 2, we have:
F =E /c r Using t h i s and (37) from the main t e x t , we have:
E r=- (f2r 2/e 2) Y (j )
2F 23=--(«r/c) Y ^ x / 3z = - (Or/c) Y ^ i a W c ) x
Since we assume th a t E^ i s w e l l behaved as ftr/c approaches u n i t y ,
X must approach zero as one approaches the l i g h t c y l i n d e r ( v = 0 ) .
Thus the s o l u t i o n (12) i s r e j e c t e d . We are l e f t w i t h , f o r m=0:
(14) R 0 ( v ) = a 0 0 ( v 2 + ( 2 / 3 ) v 3 + ( l / 8 ) ( ( a / 2 ) 2 + 4 ) v 4 + • • • )
and, f o r m=±l:
(15) R 1 ( v ) = a 0 1 ( v 2+ ( 2 / 3 ) v 3
+ ( l / 8 ) ( ( a / 2 ) 2+ ( 1 7 / 4 ) ) v 4 + - - - )
da
The magnetic p o t e n t i a l x i s given by:
(16) X ( v , y ) = / ; R 0 ( v ) e ^ d a + ( e i * + e - i * ) £ R ^ e 1 ^
w i t h RQ and R^ from (13) as given i n ( 1 4 ) 5 ( 1 5 ) . A l s o we have
X"*Xj • i - as x,y ->0 A d i p o l e " , • •, • i_ - i a u .. , - i a u +i<j> , . . . . , , By m u l t i p l y i n g by e , -Uand e e .. a n d - i n t e g r a t i n g over y and <f>
one ge t s , f o r x->-0(v->-l) :
(17) R Q(v+l)+ - ( y / T r ) ( f 2 / c ) 2 c o s 6 o i a l o g ( x )
(18>:Rj(v*l>»- - ( y A ) ( R / c ) 2 s i n 9 o (1/x)
R e c a l l x d/dx=2(v-l)d/dv ((10) i n appendix 1) so th a t (17) i s :
(19) 2(v-l)d/dv R 0 ( v + 1 > - (y /TT) (fl/c) 2cos6 i a
Write the q u a n t i t y on the l e f t hand s i d e as the s e r i e s :
a n r Z b v" with h ~2(n+l)a </ann -2(n+2)a /ann a are the 00$ n n v ' n-1 00 v 1 n 00 n c o e f f i c i e n t s of RQ(V) as given by ( 1 3 ) . Then as v * l , we have:
(20) a 0 ( )|b n= - ( y / 7 r ) ( f i / c ) 2 c o s 6 Q i a
Thi s determines the c o e f f i c i e n t of R r t ( v ) , s i n c e b are known. 0 1 ' n
55.
To determine the c o e f f i c i e n t of Rj(v) from (18), we solve the
d i f f e r e n t i a l equation f o r W(v). We d e f i n e :
(21) W(v)=xR 1(v) = ( l - v ) i s R ( v )
W(v) w i l l be f i n i t e as x-K). With R 1(v) = ( l - v ) 2W(v), we have:
Rj =(1-V)_5SW + JS(1-V)" 3 / 2 W
RJ =(l-v) _ J 2W + ( l - v ) " 3 / 2 W +(3/4) ( l - v ) " 5 / 2 W
Thus (10) (With m=l) becomes,upon m u l t i p l y i n g by (1-v) 2:
(22) v(l-v)W-(l-v)W-(V(a 2/4)v)W=0 0 0 n+8 W r i t i n g W=ar.1Ec v , c n = l , (22) becomes: 01 » n 0
(23) v n + 6 " 1 ( - ( a 2 / 4 ) c N _ 2 - ( J S + ( n + e - ^ ( n + 6 - 3 ) ) c n _ 1 + ( n + 6 ) ( n + 8 - 2 ) c n ) = 0
The i n d i c i a l equation i s 6(B-2)=0. The 8=0 s o l u t i o n i s r e j e c t e d
s i n c e as v-*-0, R (and thus W) must; approach zero.' We are l e f t w i t h :
(24) W(v)=a 0 1Zc nv n w i t h c n = ( ( ( n + l ) ( n - l . ) + % ) c n _ 1 + ( a V 4 ) c n _ 2 )
T((n+2)n) and CQ=1
As x->0, from (18) and (21) we get:
( 2 5 ) a 0 l K =-^/ i r)Cn/c) 2sin8 0
Here a m i s the c o e f f i c i e n t of R 1 ( v ) .
Now x>as given i n (16) , i s completely determined :
S i . APPENDIX 4
»
We consider the Lagrangian:
(1) L=^(Ax 2+By 2+Cz 2) + 01x+$2y+<I>3z
Here A,B,C/!^ >*2»*3 a r e f u n c t i o n s o f x,y,z but not e x p l i c i t l y
o f t or the v e l o c i t i e s . We .define:
(2) X=8*3/Dy -9$ 2/8z and c y c l i c a l l y f o r Y and Z.
Then the Lagrange equations o f motion a r e : (3) d/dt(Ax)=Zy-Yz + i i ( A , x x 2 + B , x y 2
+ C , x z 2 ) d/dt(By)=Xz-Z* + l 2 ( A , y x 2
+ B , y y 2+ C , y z 2 )
d/dt(Cz)=Yx-Xy +3i(A, zx 2+B, zy 2+C, zz 2)
w i t h A,x=8A/8x e t c . The Hamiltonian energy f u n c t i o n i s constant:
(4) E=MAx 2+By 2+Cz 2)
s i n c e L i s not e x p l i c i t l y a f u n c t i o n o f time.
The equations (3) have x=y=z=0 as a p o s s i b l e s o l u t i o n , f o r
a r b i t r a r y x,y,z. We are l o o k i n g f o r sm a l l d e v i a t i o n s from these
p a r t i c u l a r motions. I f the q u a n t i t i e s A,B,C,X,Y and Z were
constant, then the motion would be a s u p e r p o s i t i o n of
s o l u t i o n s of normal mode type, each o f which has a time dependence
o f the form e~ l t 0 t. The c h a r a c t e r i s t i c f r e q uencies are given by
itoA Z -Y
(5) determinant -Z iuB X ' = 0
Y -X iwC
or (6) w(ABCu)2-AX2- 2 2 BY -CZ )=0
The root w=0 corresponds to uniform motion w i t h the v e l o c i t i e s
(x,y,z) • p r o p o r t i o n a l t o (X,Y,Z) . The other r o o t s correspond to
tr a n s v e r s e e l l i p t i c a l motion.
57,
to= ( (AX 2+BY 2+CZ 2)/AEC) J 5
One sees that A,B,C,X,Y and Z, which are functions of •:
p o s i t i o n , w i l l vary to f i r s t order i n the v e l o c i t i e s . They vary
slowly, due to the uniform motion, and rapi.dly.with frequency
w , due to the o s c i l l a t i n g motion. We d e f i n e :
(7)
Thus the roo t s o f (6) are 0,± w. With t h i s value f o r u, we
wr i t e the matrix equation f o r the amplitudes o f the transvei-se motion
K Z -Y (8) -Z . iwB X
Y -X iwC
=0
T h i s has n o n - t r i v i a l s o l u t i o n s (£,n>c) which are determined up to
a n - a r b i t r a r y complex m u l t i p l y i n g f a c t o r .
We s h a l l use the v e c t o r s : (X,Y,Z), (£ ,n ,0 and (£*,ri*^ , as a
ba s i s f o r d e s c r i b i n g the general motion. We introduce the d e f i n i t i o n s
(9) AX 2+BY 2+CZ 2=P •k *k *
(10) AU +Bnri =N
(11) 7 u dt = X
M u l t i p l y i n g (8) on the l e f t by (X,Y,Z) and (£,n,r.), we have:
(12) AX5+BYn+CZC=0
(13) 2 2 2 A£ +Bn +CC =0
From (8) , we can express any one of£,n,r, i n terms of another:
C=-A((XZ-itoBY)/(BY 2+CZ 2)) 5 and c y c l i c r e l a t i o n s
Thus we can express N, from (10), as a f u n c t i o n of only one of
£,n>C and one complex conjugate quantity £,n,r.. Inverting these
r e l a t i o n s g i v e s :
St.
(14) «*=(1/A -X2/P)N/2 and cyclic relations
(15) £*n=(-iwCZ-XY)N/2P and cyclic relations
We now express the velocity vectors as a superposition of
the instantaneous normal modes. These have a real amplitude SL describi
the longitudinal component and a complex amplitude amplitude o*
representing the e l l i p t i c a l component, as follows:
(16) x=£X+o5e~ l x+o*£*e l x " . —iv * * iv y=£Y+ane A+a n e A
z=£Z+a!;e"lx+a*?*e:LX
The factor e~ Aincorporates the rapid time dependence of the traiisvers
e l l i p t i c a l motion.
These equations can be inverted to give £ and ain terms of the •
velocity components. The results are:
(17) ?l= AXx+BYy+CZz
(18) Nae-1*=Ac;*x+Bn*y+(:c*z
Using (16), the energy (4) is given by:
(19) . E=%P£2+Noa*
This shows that the particle energy i s unambiguously expressible
as the sum of a longitudinal partf^PJi') and a transverse part(Noo ).
We can obtain the equation of motion for £ anda by different
iating (17) and (IP.), using the equations of motion (3),and then
using (16) to express the results in terms of % and o. The results
are expressions for I and 6 which contain slowly varying terms,
plus oscillating terms proportional to e ~ l x , e " 2 l x . In considering
59.
the " a d i a b a t i c " aspects of the motion the o s c i l l a t i n g terms can
be ne g l e c t e d , to second order i n the v e l o c i t i e s . When one i n t e g r a t e s
over many c y c l e s the f i r s t order p a r t s of the o s c i l l a t i n g terms
v a n i s h , whereas the f i r s t order p a r t s o f the s l o w l y v a r y i n g terms
do not. The s l o w l y va'rying terms o n l y c o n t a i n £,n, orC as a product * * *
w i t h one of £ ,n , or X, . (14)5(15) are used to e l i m i n a t e these terms.
The r e s u l t of the c a l c u l a t i o n f o r & i s w r i t t e n : (20) PJ--Js.i 2(X3/8x +Y3/3y +Z3/3z)P
*
-Noa (X3/3x +Y3/3y +Z3/3z)logu) + o s c i l l a t i n g terms
The r a t e of change of any f i e l d q u a n t i t y , a l o n g the path o f a
p a r t i c l e , i s gi v e n by a p p l y i n g the operator:
d/dt = x3/3x + y3/3y + z3/3z
The p a r t i c l e ' s v e l o c i t i e s , x,y,z, are given e x p l i c i t l y by (16).
When we average over s e v e r a l c y c l e s , t h i s operator becomes, t o
f i r s t order:
*(X3/3x + Y3/3y + Z3/3z )
Thus from (20), we can w r i t e , f o r a p a r t i c l e moving along a stream
l i n e : P£d£ +%l2dP +No*od(logu>) =0 or
(21) d(%P£ 2) + Nac*d(logu>) = 0
From (19) and the constancy o f E, t h i s y i e i d s : * * *
d(No o)=Na od(logu) o r d(log(No o/u))=0 , i . e . *
(22) Na a/w =M M i s a constant.
T h i s r e s u l t i s a g e n e r a l i z a t i o n of a r e s u l t f a m i l i a r i n many asp e c t s ,
of p a r t i c l e motions i n a magnetic f i e l d . Namely , the magnetic
moment i s an a d i a b a t i c i n v a r i a n t . Combining (19) and (20) g i v e s :
(23) P42=2(E-Mw)
in the a d i a b a t i c l i m i t . . •
