Stellar Atmospheres: Radiative Equilibrium 1 Radiative Equilibrium Energy conservation

Stellar Atmospheres: Radiative Equilibrium

1

Radiative Equilibrium

Energy conservation


2


Assumption:

Energy conservation, i.e., no nuclear energy sources Counter-example: radioactive decay of Ni56 Co56 Fe56 in supernova atmospheres

Energy transfer predominantly by radiation

Other possibilities:

Convection e.g., H convection zone in outer solar layer

Heat conduction e.g., solar corona or interior of white dwarfs

Radiative equilibrium means, that we have at each location:

Radiation energy absorbed / sec =

Radiation energy emitted / sec

integrated over allfrequencies andangles


3


Absorption per cm2 and second:

Emission per cm2 and second:

Assumption: isotropic opacities and emissivities

Integration over d then yields

Constraint equation in addition to the radiative transfer equation; fixes temperature stratification T(r)

4 0

)( vIvdvd

4 0

)(vdvd

0)( )()(000

dvSJvvdvJvdv vvv


4

Conservation of fluxAlternative formulation of energy equation

In plane-parallel geometry: 0-th moment of transfer equation

Integration over frequency, exchange integration and differentiation:

vvv SJ

dt

dH

0 0

4eff

0

4eff

0 0

0 because of radiative equilibrium

const for all depths. Alternatively written:4

(1st moment of transfer equat4

v v v

v

v

v

dH dv J S dv

dt

H H dv T

dv T dvdK

Hd

4eff

0

ion)

( ) (definiton of Eddington factor)

4

v vddv T

d

f J


5

Which formulation is good or better?

I Radiative equilibrium: local, integral form of energy equation

II Conservation of flux: non-local (gradient), differential form of radiative equilibrium

I / II numerically better behaviour in small / large depths

Very useful is a linear combination of both formulations:

A,B are coefficients, providing a smooth transition between formulations I and II.

0)(

00

Hdvd

JfdBdvSJA vv

vv


6

Flux conservation in spherically symmetric geometry

0-th moment of transfer equation:

HRLLdvHr

dvJSrdvHrr

JSHrrr

v

vvv

vvv

222

0

2

0

2

0

2

22

16 because 16

1const

0

1


7

Another alternative, if T de-couples from radiation field

Thermal balance of electrons

mllmlml

C

mllmlmm

H

kl

kThvv

lkvlkk

C

kl

lkvlkl

H

j

kThvvjj

C

jvjj

H

CH

hvTqnnQ

hvTqnnQ

dvec

hvJ

v

vJvnQ

dvv

vJvnQ

dvec

hvJTvNnQ

dvJTvNnQ

QQ

,ec

,ec

, 02

3

bf,bf

, 0

bf,bf

02

3

ff,eff

0

ff,eff

)(

)(

21)( 4

1)( 4

2

),( 4

),( 4

0


8

The gray atmosphereSimple but insightful problem to solve the transfer equation together with the constraint equation for radiative equilibrium

Gray atmosphere:

Moments of transfer equation

with

Integration over frequency

Radiative equilibrium ( ) ( ) 0

and because

v vv v v

v v v v

dH dKI J S II H dt

d d

dH dKI

J

J S II Hd d

J S dv J S dv J

S

S

I

1 2 1 2

2

2

of conservation of flux 0

from follows , see below0

dH

d

dKII K c c II c H

d

dc

K

d


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The gray atmosphere

Relations (I) und (II) represent two equations for three quantities S,J,K with pre-chosen H (resp. Teff)

Closure equation: Eddington approximation

Source function is linear in Temperature stratification?

In LTE:

2K 1 3J J 3K IIIS 3H 3c

4

42

4eff

4 4eff 2 2

( ) ( ( ))

insert into : 3 3

with we get: 43

( ) 3 is now determined from boundary condition ( =0)4

S B T T

III T H c

H T

T T c IV c


10

Gray atmosphere: Outer boundary condition

Emergent flux:

3

23 ,

3

2

4

3

3

2

2

1

3

1

2

3)0(

)(12

1)( and

!)(with

)()( 2

3

)(332

1

from with )()(2

1)0(

4eff

4

22

12

0

0

22

0

2

0

22

0

2

HSTT

HccHH

ttEetEnl

ldttEt

dEcdEH

dEcH

IIISdESH

tn

l

from (IV): (from III)


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Avoiding Eddington approximation

Ansatz:

Insert into Schwarzschild equation:

Approximate solution for J by iteration (“Lambda iteration“)

))((4

3)(

function Hopf )(

oftion generaliza ))((3)(

4eff

qTJ

q

IIIqHJ

0

1)(2

1)(

Jfor equation integral )(

dEqq

JSJ

)(2

1)(

3

1

3

23)32(3

)32(3

32)1()2(

)1(

EEHHJJ

HJ

(*) integral equation for q, see below

i.e., start with Eddington approximation

(was result for linear S)


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At the surface

At inner boundary

Basic problem of Lambda Iteration: Good in outer layers, but does not work at large optical depths, because exponential integral function approaches zero exponentially.

Exact solution of (*) for Hopf function, e.g., by Laplace transformation (Kourganoff, Basic Methods in Transfer Problems)

Analytical approximation (Unsöld, Sternatmosphären, p. 138)

358.034

1

3

1

3

23

2

1)0( , 1)0( , 0

)2(

32

HHJ

EE

2 3

(2)

, ( ) 0 , ( ) 0

23

3

E E

J H

972.11167.06940.0)( eq

exact: q(0)=0.577….


13

Gray atmosphere: Interpretation of resultsTemperature gradient

The higher the effective temperature, the steeper the temperature gradient.

The larger the opacity, the steeper the (geometric) temperature gradient.

Flux of gray atmosphere

d

dT

dt

dT

Td

dT

Td

dTTT

d

d

4eff

4eff

34

~

4

34

2 2

0

1/ 4

eff eff

LTE: ( ( ))

1 1( ) ( ( )) ( ) ( ( )) ( )

2 2

with , 3 4( ( )) ( ) ( )

and

v v

v v v

v

S B T

H B T E t dt B T E t dt

hv kT T T q p hv kT p

H d H dv H

3 3 3

2 3 3

4eff

43eff 2 2

4 3 2eff 0

1 2 4

2

4

( ) ( )4 4( ) /

exp( ( )) 1 exp( ( )) 1v

v

hv k k

hc h v

T

H kT E t E tdv kH H H dt dt

H d T h h c p p


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Gray atmosphere: Interpretation of resultsLimb darkening of total radiation

i.e., intensity at limb of stellar disk smaller than at center by 40%, good agreement with solar observations

Empirical determination of temperature stratification

Observations at different wavelengths yield different T-structures, hence, the opacity must be a function of wavelength

4 4eff

3 2I( 0, ) S( ) B(T( )) T ( ) T

4 3

I(0, ) 2 / 3 2 3(1 cos )

I(0,1) 1 2 / 3 5 2

TTBSSI ))(()()( ),0( measure


15

The Rosseland opacityGray approximation (=const) very coarse, ist there a good mean value ? What choice to make for a mean value?

For each of these 3 equations one can find a mean , with which the equations for the gray case are equal to the frequency-integrated non-gray equations. Because we demand flux conservation, the 1st moment equation is decisive for our choice: Rosseland mean of opacity

transfer equation

0-th moment

1st moment

non-gray gray

)( ISdz

dI ))(( vvv ISv

dz

dI

))(( vvv JSv

dz

dH

vv Hv

dz

dK)(

0)( JSdz

dH

Hdz

dK


16

The Rosseland opacity

Definition of Rosseland mean of opacity

3

0

340

0

00

4)(

1

1

4 and with

)(1

1

: LTE and 3/1ion approximatEddington with )(

1

1

1

)(

1const

T

dvdTdB

v

dz

dTTT

dz

d

dz

dB

dz

dT

dT

dB

dz

dB

dzdB

dvdzdB

v

BJJK

dzdK

dvdzdK

v

dz

dKdv

dz

dK

vdvH

v

R

vv

v

R

v

R

R

vv


17

The Rosseland opacity

The Rosseland mean is a weighted mean

of opacity with weight function

Particularly, strong weight is given to those frequencies, where the radiation flux is large. The corresponding optical depth is called Rosseland depth

For the gray approximation with is very good,

i.e.

R1

)(

1

v dT

dBv

R1Ross

z

RRoss zdzz0

)()(

))((4

3)( 4

eff4

RossRossRoss qTT


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Convection

Compute model atmosphere assuming • Radiative equilibrium (Sect. VI) temperature stratification• Hydrostatic equilibrium pressure stratification

Is this structure stable against convection, i.e. small perturbations?

• Thought experiment

Displace a blob of gas by r upwards, fast enough that no heat exchange with surrounding occurs (i.e., adiabatic), but slow enough that pressure balance with surrounding is retained (i.e. << sound velocity)


19

Inside of blob outside

Stratification becomes unstable, if temperature gradient

rises above critical value.

( ), ( )T r r

ad ad

ad ad

( )

( )

T T T r r

r r

r

( ), ( )T r r

rad rad

rad rad

( )

( )

T T T r r

r r

ad rad

ad rad

ad rad

a

sta

( ) ( ) further buoyancy,

( ) ( ) gas blob falls back,

i.e.

with ideal gas equation p= and pressure balance

ble

stable

unstable

unsta

ble

H

r r r r

r r r r

d d

dr dr

kT

Am

d ad rad rad

ad radunstable

s

=

table

T T

dT dT

dr dr

addT dr


20

Alternative notation

Pressure as independent depth variable:

Schwarzschild criterion

Abbreviated notation

eff eff

eff

eff eff

ad rad

hydrostatic equation: ( )

(ln )

(ln )

(ln ) (ln )

(ln ) (

ide

unstabl

al gas

stabln

e

le )

H

H

H H

dp g dr g dr

kTdp

Am g p

Am AmdT dT T d Tg g

k dp p k d p

d T d T

d

d

Am p

k T

p d

r

dr

p

ad rada

ad d

d rad

ra

(ln ) (ln );

(ln ) (l

sta

n

b

)

le

d T d T

d p d p


21

The adiabatic gradient

Internal energy of a one-atomic gas excluding effects of ionisation and excitation

But if energy can be absorbed by ionization:

Specific heat at constant pressure

V V

0 (no heat exchange)

(1st law of therm

intern 0 (al

ody

energy

namic

*)

s)

dQ

dQ pdV

c d

dE

dE c dT T pdV

V

3 3

2 2E NkT c Nk

V

3

2c Nk

p V Vt t

p V

( )

p cons p cons

c c

Q dE dV d NkT p Nkc p c p c p

T dT dT dT

Nk

p


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p V

p VVp V

V

p

p V

V

V

p

V

p

V

Ideal gas:

(**)

/

from(*) with (**) 0

0

0

(ln ) (l

pV

n )

defi

pV NkT Vdp pdV dT dT

Vdp pdVdT

c c

V pc cc pdV

c cc

c cdV

p V c

cdp dV

dp d

p V c

cd V d p

V

d

Nk c

p d

V

c

c

V

p

V

(ln ) 1nition: :

(ln ) c d V

c d p


23


ad

ad

rad

(ln )needed:

(ln )

/

ln ln ln ln( )

(ln ) (ln )1

(ln ) (ln )

(ln ) 1 1

1 st

1(ln )

b e

1

a l

d T

d p

T pV Nk

T p V Nk

d T d V

d p d p

d T

d p

Schwarzschild criterion


24


• 1-atomic gas

• with ionization• Most important example: Hydrogen (Unsöld p.228)

V p V

ad

3 2 5 2

5 3 2 5 0.4

c Nk c c Nk Nk

ad1 0 convection starts effect

2Ion

ad 22Ion

2

2 5 2

5 5 2

( ) ( ) ( )with ionization degree

2 2

x x E kT

x x E kT

f T f T f Tx

N N N


25


2Ion

ad 22Ion

2

2 5 2

5 5 2

( ) ( ) ( )

2 2

x x E kT

x x E kT

f T f T f Tx

N N N


26

Example: Grey approximation

4 4eff

4eff

1

3 2( ) 4 3

3 24ln ln ln4 3

2ln 34

hydrostatic equation: Ansatz: ( here a mass absorption coefficient)

(ln ) 124 3

1 integrate

1

b

b bb

T T

T T

d

d

dp gAp

dg g g

p pA b A

d T

Ap

dp

d

d

1

1

rad

rad

1

( 1)

1 1

( 1)24 3

becomes large, if opacity strongly increases with depth (i.e

(ln ) 1

(

. exponent b large).

The absolute value of is

1)

l

not

l

essenti

n

n

b b

b

g g

p p

d T

A

d

d p

d b

d p

p

p

d p

d

d

A

b

rad ad

a

l but

large

the change o

(> ): c

f with depth (g

onvection starts,

radient

-Ef ekt

)

f


27

Hydrogen convection zone in the Sun

-effect and -effect act together

Going from the surface into the interior: At T~6000K ionization of hydrogen begins

ad decreases and increases, because a) more and more

electrons are available to form H and b) the excitation of H is responsible for increased bound-free opacity

In the Sun: outer layers of atmosphere radiative

inner layers of atmosphere convective

In F stars: large parts of atmosphere convective

In O,B stars: Hydrogen completely ionized, atmosphere radiative; He I and He II ionization zones, but energy transport by convection inefficient

Video


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Transport of energy by convectionConsistent hydrodynamical simulations very costly;

Ad hoc theory: mixing length theory (Vitense 1953)

Model: gas blobs rise and fall along distance l (mixing length). After moving by distance l they dissolve and the surrounding gas absorbs their energy.

Gas blobs move without friction, only accelerated by buoyancy;

detailed presentation in Mihalas‘ textbook (p. 187-190)

( ) = pressure scale height

mixing length parameter

=0.5 2

l H r H


29

Transport of energy by convection

Again, for details see Mihalas (p. 187-190)

For a given temperature structure

4rad eff

conv

conv

ad rad

( )

compute ( )

flux conservation including convective flux

new temperature stratification ( )

wit

(

h

)

F r T F r

F r

T r

iterate


30

Summary: Radiative Equilibrium


31

Radiative Equilibrium:

Schwarzschildt Criterion:

Temperature of a gray Atmosphere

0)(

00

Hdvd

JfdBdvSJA vv

vv

4 4eff

3 2

4 3T T

ad rad(ln ) (ln )

(

unst

stable ln ) (

bl

ln

e

)

ad T d T

d p d p

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Stellar Atmospheres: Radiative Equilibrium 1 Radiative Equilibrium Energy conservation