what dynamic changes in the sun drive the evolution of oscillation frequencies through the activity...

What Dynamic Changes in the Sun Drive the Evolution of Oscillation Frequencies through the Activity Cycle?

Philip R. GoodePhilip R. Goode

Big Bear Solar ObservatoryBig Bear Solar Observatory

New Jersey Institute of TechnologyNew Jersey Institute of Technology

11 March 200211 March 2002Big Bear Solar ObservatoryBig Bear Solar Observatory

The Sun’s Irradiance Variations are Small, but Hard to Explain


The Solar Cycle Changes in the Sun

Naively, luminosity varies like irradiance and the Sun Naively, luminosity varies like irradiance and the Sun is a blackbody. Then, allowed size of the changes in is a blackbody. Then, allowed size of the changes in the Sun’s outputthe Sun’s output

-In truth, we have cast the Sun’s changing -In truth, we have cast the Sun’s changing

magnetic field, thermal structure and turbulent magnetic field, thermal structure and turbulent velocity as temperature changes. Further, is there a velocity as temperature changes. Further, is there a role for a changing solar radius?role for a changing solar radius?

Principal Collaborator:Principal Collaborator: -Wojtek Dziembowski, University of Warsaw-Wojtek Dziembowski, University of Warsaw

4 2L T R

L T R


Variations in the Solar Radius

Brown & Christensen-Dalsgaard (1998): 1981-1987 Brown & Christensen-Dalsgaard (1998): 1981-1987 annual average Rannual average Rss each year the same within each year the same within measurement errors (measurement errors (±±37 km)37 km)

--comparable to those of Wittman (1997) for --comparable to those of Wittman (1997) for same time intervalsame time interval

--much smaller than annual changes reported --much smaller than annual changes reported by Ulrich & Bertello (1995) -- sun grows with by Ulrich & Bertello (1995) -- sun grows with growing activity -- and Laclare et al. (1996) – sun growing activity -- and Laclare et al. (1996) – sun shrinks with growing activity -- for the same interval shrinks with growing activity -- for the same interval ((200 Km from maximum to minimum)200 Km from maximum to minimum)


Variations in the Solar Radius from Space

Emilio et al. (2000) Emilio et al. (2000) MDI limb MDI limb observations: observations: annual radius annual radius increase with increase with increasing activity increasing activity

of of 5.95.9±±0.7 0.7 Km/yearKm/year


Normal Modes of the Sun( )

0n 1n

, 2n l n l


Helioseismic Radius Using MDI f-modes

ll22ll((ll+1)GM+1)GMss/R/Rll

33 : Asymptotically surface : Asymptotically surface

waves, but f-modes see different effective waves, but f-modes see different effective gravities – depending on gravities – depending on ll

ll / / ll=-3/2 =-3/2 RRss/R/Rss : : means model value means model value

minus true valueminus true value

From this, Schou et al. (1997) determined From this, Schou et al. (1997) determined RRss= =

695.68±0.03 Mm695.68±0.03 Mm


The Surface Radius of the Sun

Auwers (1891): RAuwers (1891): Rss=D=Dss/2=695.99±0.04 Mm/2=695.99±0.04 Mm

--standard value for 100 years--standard value for 100 years

Schou et al. (1997): RSchou et al. (1997): Rss= 695.68±0.03 Mm= 695.68±0.03 Mm

--MDI f-modes--MDI f-modes Brown & Christensen-Dalsgaard (1998): Brown & Christensen-Dalsgaard (1998):

RRss=D=Dss/2=695.508±0.026 Mm/2=695.508±0.026 Mm

--HAO Solar Diameter Monitor--HAO Solar Diameter Monitor


Solar Cycle Variations in the f-mode Radius

Must look more carefully at this form of the Must look more carefully at this form of the equation because there are problems with using it to equation because there are problems with using it to determine the considerably smaller changes the (f-determine the considerably smaller changes the (f-mode) radius from year to year:mode) radius from year to year:

Even though Even though RRllRRs s , , cannot expect cannot expect RRllRRs s . The . The problem is we must contemplate changes beyond a problem is we must contemplate changes beyond a simple radius change because near surface effects simple radius change because near surface effects also contribute to frequency changes also contribute to frequency changes — turbulence — turbulence (Brown 1984) and/or magnetic fields (Evans & (Brown 1984) and/or magnetic fields (Evans & Roberts 1990). Roberts 1990).

minminmin

3, w.r.t. Min.

2fl

l f

R

R


Rate of Shrinking from f-modes

To account for near-surface contribution to the To account for near-surface contribution to the frequency change we add a second termfrequency change we add a second term

— — IIll 0, as0, as l l , , ll weak function of weak function of ll

min minmin 3

2f fl

l f l l

R

R I


Evolution of f-mode Radius With With ::

dRdRff/dt/dt = -1.51 = -1.51±±0.31 km/y0.31 km/y

Without Without ::

dRdRff/dt/dt = -1.82 = -1.82±±0.64 km/y0.64 km/y

Results imply at a depth of 6-Results imply at a depth of 6-10 Mm, the sun shrank by 10 Mm, the sun shrank by some 5 km during the rising some 5 km during the rising phase of this activity cyclephase of this activity cycle

ddff/dt= 0.180/dt= 0.180±±0.051 0.051 Hz/y, Hz/y,

noisy with some cross-talknoisy with some cross-talk As small as it is, a shrinking As small as it is, a shrinking

sun is not easy to explainsun is not easy to explain


Near-Surface Terms from p-modes

Note that rise of cycle Note that rise of cycle starts early 1997, so fits starts early 1997, so fits for dR/dt start thenfor dR/dt start then

Note Note systematically systematically

rises, but behavior is rises, but behavior is much more muted than much more muted than anisotropic terms anisotropic terms — — this is a clue to the this is a clue to the shrinking of outermost shrinking of outermost layers of the sunlayers of the sun


f-mode Radius Changes -- Entropy and Random R.M.S. Field, Goldreich et al. (1991)

2

0 00

2 2 2 2

Lagrangian change in local radius

( )

Horizonally averaged in presence of random field

where the magnetic pressure and anisotropy are

and 8 8

w

g

g m

h r h rm

m

xr r r r dx

r

P

P P

B B B BP

P

1here = for isotropic field and =-1 for radial field

3


02

T1 0

T

T1

Using thermodynamic relations yields,

1 ( )( )

where =the logarithmic derivative of at constant .

1At relevant depths, gas ideal, so =-1 and =0.6, and

b

r

m

g pr

P S xr dx

P c r

P

S

0.4

The irradiance of an active sun is higher than average. If the

same were true about luminosity, then 0 because CZ

is losing heat. Hence a negative contribution to , as

activity

g

p g

PT

c T P

S

r

rises.


How Big Can the Shrinking Be?

cycle CZ

CZCZ

cyc

Thermodynamic change in the luminosity from

activity minimum to maximum

SL

and the timescale for the convection zone is

.

Combining the two reveals the problem

r

p r

p

TdMt

c TdM

tL

tS L

c L

le 3 75

CZ

10y10 10

10 y

Alternative: throw out mixing length theory?

t


More Acceptable: Variation in the R.M.S. Magnetic Field

For a purely radial random field (For a purely radial random field (=-1), an =-1), an increasing field implies contraction.increasing field implies contraction.

For an isotropic random field (For an isotropic random field ( =1/3), an =1/3), an increasing field implies expansion! A non-increasing field implies expansion! A non-trivial constraint!trivial constraint!


What about the Radius?

Need to have accurate information about the Need to have accurate information about the outer ~4 Mm to combine with the shrinking outer ~4 Mm to combine with the shrinking beneathbeneath

This outermost region is where one expects the This outermost region is where one expects the largest activity induced changeslargest activity induced changes

— — because of rapid decline of gas pressurebecause of rapid decline of gas pressure — — thermal structure most susceptible to field thermal structure most susceptible to field

induced changes in convective energy induced changes in convective energy transport efficiencytransport efficiency


p-mode ’s for the Last 4 Mm

p-mode spectrum is >10x richer than that for p-mode spectrum is >10x richer than that for f-modes f-modes — reminder can’t use p-modes for — reminder can’t use p-modes for radius (radius (RR-1.5-1.5 valid only if changes valid only if changes homologous throughout whole sun)homologous throughout whole sun)

For f-modes: For f-modes: ddff/dt= 0.180/dt= 0.180±±0.051 0.051 Hz/yHz/y

For p-modes: dFor p-modes: dp,0p,0/dt= 0.149/dt= 0.149±±0.008 0.008 Hz/yHz/y

— — A much more precise value to describe the A much more precise value to describe the last 4Mmlast 4Mm


Consider Radial R.M.S. Random

Field Growth (=-1), then T/T as Source

Solid lines and dotted lines Solid lines and dotted lines fit fit , other forms possible, other forms possible

<<BBphph> < 100 G> < 100 G <<BB> < 300 G at 4 Mm> < 300 G at 4 Mm T/T too large at surfaceT/T too large at surface cannot exclude cannot exclude T/T at T/T at 1010-3-3 level in level in

subphotospheric layers.subphotospheric layers.


Isotropic Random R.M.S. Field in Outermost Layers?

=1/3 is precluded in f-mode layer because =1/3 is precluded in f-mode layer because that layer shrinks how can it be present above that layer shrinks how can it be present above when field above wants to be radial? Radial when field above wants to be radial? Radial random field is the most economical to account random field is the most economical to account for frequency changes.for frequency changes.

For For =-1, the splitting kernels (=-1, the splitting kernels (k>0k>0) are much ) are much

larger than those for the isotropic part. This is larger than those for the isotropic part. This is consistent with the anisotropic consistent with the anisotropic ’s (like ’s (like 33) )

being much larger than for isotropic being much larger than for isotropic ’s (’s (00).).


Is the Sun Hotter or Cooler at Activity Maximum?

Required changes in turbulent flows are Required changes in turbulent flows are probably too large to account for frequency probably too large to account for frequency changes changes

Limiting problem to magnetic field and Limiting problem to magnetic field and temperature alone, for aspherical part can use temperature alone, for aspherical part can use condition of mechanical equilibrium to pose condition of mechanical equilibrium to pose problem for field or temperature changeproblem for field or temperature change

Spherical part goes through thermal Spherical part goes through thermal equilibrium condition – much harder to treatequilibrium condition – much harder to treat


Small-scale Aspherical Random R.M.S. Field

Each component, Each component, k,k, gives rise to gives rise to PP2k2k distortion of distortion of

sun’s shape and for sun’s shape and for kk>0 temperature pert.>0 temperature pert.

, , 2

1( ) ( ) cos

2j k jk rk r k H k k k kB B r P M M

,,

0

,,

ln1 ln1 2 ( ) ( )

8 ln ln

ln ln ln1 ( ) ( ) ,

ln ln ln

r kr k

k T

H kH k

T

dT d

T d r d P

d d P

d r d P

MM

MM


For k>0, Eliminate T/T, and

, ,( )B Bk r r k H H k dD M K M K


Is the Sun Hotter at Activity Minimum?2

2rms 2 2

min

min

min

A simple model of inward field increase,

, if 1

(3 2 ), if y 1

where

,

is the depth variable and is depth at temp. min.

and is depth

b

b

b

b

B yB

B y y

D Dy

D D

D D

D

2

beneath which field is constant.

is negative at the minimum of , but function is

rapidly changing, so inconclusive

T


What Next ? Treat condition of thermal equilibrium to Treat condition of thermal equilibrium to

constrain surface averaged temperature change constrain surface averaged temperature change because of the sharper minimum in because of the sharper minimum in 22 for k=0 for k=0

More thorough analysis of MDI and GONG More thorough analysis of MDI and GONG data – more years, etc.data – more years, etc.

Use BBSO Ca II K and Disk Photometer data to Use BBSO Ca II K and Disk Photometer data to constrain the field to link irradiance and constrain the field to link irradiance and luminosityluminosity

Three color photometry from Disk PhotometerThree color photometry from Disk Photometer Use formalism to probe for buried magnetic Use formalism to probe for buried magnetic

fieldfield

what dynamic changes in the sun drive the evolution of oscillation frequencies through the activity...

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