stellar atmospheres: solution strategies 1 solution strategies

Stellar Atmospheres: Solution Strategies

1

Solution Strategies


2

All equations

Radiation Transport Iv(z), Jv(z), Hv(z), Kv(z)

Energy Balance T(z)

Hydrostatic Equilibrium ne(z)

Saha-Boltzmann / Statistical Equilibrium nijk(z)

Huge system with coupling over depth (RT) and frequency (SE)

Complete Linearisation (Auer Mihalas 1969)

Separate in sub-problems


3

RT: Short characteristic method

Olson & Kunasz, 1987, JQSRT 38, 325

Solution along rays passing through whole plane-parallel slab

max

maxmax

0

( , , ) ( , , ) exp ( )exp

( , , ) (0, , ) exp ( ) exp

Solution on a discrete depth grid , 1, with boundi

dI v I v S

dI v I v S

i ND

1

max

ary conditions:

( , ) (0, , )

( , ) ( , , )ND

I v I v

I v I v


4

Short characteristic method

Rewrite with previous depth point as boundary condition for the next interval:

1

1 1

11

( , , ) ( , , ) exp ( , , )

( , , ) ( , , ) exp ( , , )

with

using a linear interpolation for the spatial variation of

the intergrals can be evaluated

i i i i

i i i i

i ii

i

I v I v I S v

I v I v I S v

S

I

1 1

as

i i i i i i iI S S S


5


Out-going rays:

1 1

1

/ / / /

/ / / /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 11

1

i i

i i

i ii

ii i

a a b ai a

a b b ai b

d dI S v S S

x g x x a b

ew e e e e

w e e e e

1ee


6


In-going rays:

11

11

/ / / /

/ /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 1

1

i i

i i

i ii

ii i

b a b ai a

b b bi b

d dI S v S S

x g x x a b

ew e e e e e

w e e e

/ / 11a e

e


7


Also possible: Parabolic instead of linear interpolation

Problem: Scattering

Requires iteration

1

1

1 , ( )

2e e e e e en J I d


8

Solution as boundary-value problemFeautrier scheme

Radiation transfer equation along a ray:

Two differential equations for inbound and outbound rays

Definitions by Feautrier (1964):

( )( ) ( )

pp:

sp:

vv v

dII S

ddt

dd

d dZ

symmetric, intensity-like

antisymmetric, flux-l

1

2

k1

i e2

u I I

v I I


9

Feautrier scheme

Addition and subtraction of both DEQs:

One DEQ of second order instead of two DEQ of first order

2

2

( )( ) ( )

( )( )

(1)

(2

( )( )

)

( )

v

v

dvu S

ddu

vd

d uu S

d


10

Feautrier scheme

Boundary conditions (pp-case)

Outer boundary ... with irradiation

Inner boundary

Schuster boundary-value problem

0

(2

(

)

0) 0 ( 0) ( 0)

( ) ( 0)

I u v

duu

d

max max

max

max

max max max

max

( ) ( ) ( )

(( )

) )2 (

I I u v I

duI u

d

0

00

( 0)

( )( 0)

I I u v I

duu I

d


11

Finite differencesApproximation of the derivatives by finite differences:

1 2 1 2

2

2

1 2 1

1 1

1 1

discretization on a scale

at interfirst derivative

second

mediate points:

1

2( ) ( )

and

(

derivative

)

:

)

(

i i

i

i i i

i i i i

i i i i

d uu S

d

u u u udu du

d d

dudd du

d d

1 2 1 2

1 2 1 2

1 12

1 12

1 1

( )

( )12

i i

i

i i

i i i i

i i i i

i i

dud

u u u u

du

d


12

Finite differences

Approximation of the derivatives by finite differences:

2

2

1 1

1 1

1 1

1 1

1

1 1 1

1

1 1 1

discretisation on a scale

, 2 112

, 2 1

1

2

1

2

1

i i i i

i i i ii i

i i

i i i i i i i

i i i i i

i i i i i

i

d uu S

du u u u

u S i ND

Au B u C u S i ND

A

C

B

i iA C


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Linear system of equations

Linear system for ui

Use Gauss-Jordan elimination for solution

11 1 1 1

2 2 2 2 2

1 1 1

2

1*

0

0

=

ND ND

ND

ND ND

ND ND ND ND

B C u W

A B C u W

C u

S

S

S

S

W

A B u W


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Upper diagonal matrix

1st step:

1 11

2 22

1 11

1 11 1 1 1 1 1

1 1

1 1

1

1 0

0

1

1

2

ND NDND

ND ND

i i i i i i i i i

u WC

u WC

u WC

u W

i C B C W B W

i ND C B AC C W B AC

1i i iW AW


15

Back-substitution

2nd step:

Solution fulfils differential equation as well as both boundary conditions

Remark: for later generalization the matrix elements are treated as matrices (non-commutative)

11 1ND ND

i i i i

i ND u W

i ND u W C u


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Complete Linearization

Auer & Mihalas 1969

Newton-Raphson method in n

Solution according to Feautrier scheme

Unknown variables:

Equations:

System of the form:

1 , 1 , , , ,Ti

i i ND

i

Ji ND

n

, 1, , , , 1, , ( ) 0 NF transfer equations

( ) 0 NL equations for SE

i k i k i k i k i k i k i k i

i i i

A J B J C J S n

P J n b

, ( ) 0 , 1if NF NL


17


Start approximation:

Now looking for a correction so that

Taylor series:

Linear system of equations for ND(NF+NL) unknowns

Converges towards correct solution

Many coefficients vanish

0, ( ) 0if

0, ( ) 0 , if i

0

0, ,

, ,0, , ,

1 1 1, ,

0 ( ) ( )

( )

i i

ND NF NLi i

i i k i li k li k i l

f f

f ff J n

J n

, , , i k i lJ n


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Complete Linearization - structureOnly neighbouring depth points (2nd order transfer equation

with tri-diagonal depth structure and diagonal statistical equations):

Results in tri-diagonal block scheme (like Feautrier), , 1 1( ) ( , , )i i i i if f

1 1

, 1 ,

1

,

0 0

0

0 0

0 0

0

0

0

0 0

i i i i i i i

i k i i k i

i i

i k

A B C L

A J B J

n n

C

1

0,

1

( )i

i

i

J

f

n


19

Complete Linearization - structure

Transfer equations: coupling of Ji-1,k , Ji,k , and Ji+1,k at the same frequency point:

Upper left quadrants of Ai, Bi, Ci describe 2nd derivative

Source function is local:

Upper right quadrants of Ai, Ci vanish

Statistical equations are local

Lower right and lower left quadrants of Ai, Ci vanish

2

2

d J

d


20

Complete Linearization - structureMatrix Bi:

,,

,

,, ,

1 ,

0

0

i ki k

i l

i

NLi l m

i m i l lm i k

SB

n

B

Pn P

J

1 ... NF 1 ... NL1

... NF

1 ... N

L


21


Alternative (recommended by Mihalas): solve SE first and linearize afterwards:

Newton-Raphson method:• Converges towards correct solution• Limited convergence radius• In principle quadratic convergence, however, not achieved

because variable Eddington factors and -scale are fixed during iteration step

• CPU~ND (NF+NL)3 simple model atoms only– Rybicki scheme is no relief since statistical equilibrium not as

simple as scattering integral

1( ) 0 ( )i i i i i iP J n b n P J b


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Energy Balance

Including radiative equilibrium into solution of radiative transfer Complete Linearization for model atmospheres

Separate solution via temperature correction Quite simple implementation Application within an iteration scheme allows completely linear

system next chapter No direct coupling Moderate convergence properties


23

Temperature correction – basic scheme

0. start approximation for

1. formal solution

2. correction

3. convergence?

Several possibilities for step 2 based on radiative equilibrium or flux conservation

Generalization to non-LTE not straightforward

With additional equations towards full model atmospheres:• Hydrostatic equilibrium• Statistical equilibrium

0( ) ( )T T ( )v v vJ S T

( ) ( ) ( )T T T


24

LTE

Strict LTE

Scattering

Simple correction from radiative equilibrium:

( ) ( ( ))v vS B T ( ) (1 ) ( ( )) ( )v e v e vS B T J

0

0

( )0

( )0 0

( , ) ( , ) ( ( ), ) 0

( , ) ( , ) ) 0

0

( )(

v v

v

v vTv

vv v

T Tv

vv v

T Tv v

v J v B T v dv

v J v B T dv

BJ B dv

T

BJ B dv dv

T

T

T

T


25

LTEProblem:

Gray opacity ( independent of frequency):

deviation from constant flux provides temperature correction

( )0 0

independent of the temperature 0

vv v

T Tv v

v v

BT J B dv dv

T

J B T

0

0.Moment equation

( ) ( )

( ) 0

( )

v v

v

v J B dv J B

J B B

J B B

dHB

dt


26

Unsöld-Lucy correction

Unsöld (1955) for gray LTE atmospheres, generalized by Lucy (1964) for non-gray LTE atmospheres

0 0

0

0-th moment

1st m

:

, , ,

:

,

now new quantities , , fulfill

omen

ing ra

t

diat

vv v v

JB v v J v v B

B v v

vv v

HH v v

B v

dHJ B

dt

dHdv J B B B dv J J dv d dt

d

dKH

dt

dKdv H H H dv

d

J H K

4eff

radiative equi

ive equilibrium (local) and

flux conservation (non local)

: librium

flux conservation

0

: 4

J

B

H H

B B

dHJ B

d

dKH T

d


27


Now corrections to obtain new quantities:

0

0 0 0 0

0

integrate (0)

, (0) (0) (0)

(0)

(0) (0)

(0) (0

) 1

HH

B

J

B

v v v v v v

H

B

B

J

B

X X X

K K Hd

K K dv f J dv fJ H H dv h J dv hJ

f H

d H

d

K Hd fh

f HB

fh

d KH

d

d H

fB

J

Jd

0

3

0

30

4 (0) (0) 1

(0) (0) 1

4

J

B

H

B

J H

B B

J H

B B

Hd

T dH dH

d d

f HB T Hd

fh f

f HT HdB

fh fJ

T


28


„Radiative equilibrium“ part good at small optical depths but poor at large optical depths

„Flux conservation“ part good at large optical depths but poor at small optical depths

Unsöld-Lucy scheme typically requires damping

Still problems with strong resonance lines, i.e. radiative equilibrium term is dominated by few optically thick frequencies

30

(0) (0) 1

4J J H

B B B

f HT J B Hd

T fh f

J B

0dH

d


29

NLTE Model Atmospheres

Radiation Transport and Sattistical Equilibrium are very closely coupled

Simple separation (Lamda Iteration) does not work


Accelerated Lambda Iteration


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Lambda Iteration

Split RT and SE+RE:

• Good: SE is linear (if a separate T-correction scheme is used)

• Bad: SE contain old values of n,T (in rate matrix A)

Disadvantage: not converging, this is a Lambda iteration!

0

( , )

( , )

( , , ) ( , , ) 0

new old

new

v v

J S n T

A J T n b

v n T J S v n T dv

RT formal solution

SE

RE


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Accelerated Lambda Iteration (ALI)

Again: split RT and SE+RE but now use ALI

• Good: SE contains new quantities n, T• Bad: Non-Linear equations linearization (but without RT)

Basic advantage over Lambda Iteration: ALI converges!

* *

0

( , ) ( , ) ( , )

( , )

( , , ) ( , , ) 0

new old old old new new new old old old

newnew new

new new new new newv v

J S n T S n T S n T

A J T n b

v n T J S v n T dv

RT

SE

RE


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Example: ALI working on Thomson scattering problem

amplification factor

* *

* *

*

*

:=formal solution on

solve for

1

1

1 1

new

new newe e

new new newe e

e e

old old

old old old

new old

FS

FS

FS

e

old new newe e

new

e

S B J

S S

B J B

S

B J

B J

J S

J

J

J J J J

J

B

1* *

1*

1 subtract on both sides

1

FS old olde e

new old FS olde

J J J

J J J J

Interpretation: iteration is driven by difference (JFS-Jold) but: this differenceis amplified, hence, iteration is accelerated.Example: e=0.99; at large optical depth * almost 1 → strong amplifaction

source function with scattering, problem: J unknown→iterate


33

What is a good Λ*?The choice of Λ* is in principle irrelevant but in practice it decides about the success/failure of the iteration scheme.First (useful) Λ* (Werner & Husfeld 1985):

A few other, more elaborate suggestions until Olson & Kunasz (1987): Best Λ* is the diagonal of the Λ-matrix(Λ-matrix is the numerical representation of the integral operator Λ)

We therefore need an efficient method to calculate the elements of the Λ-matrix (are essentially functions of ).Could compute directly elements representing the Λ-integral operator, but too expensive (E1 functions). Instead: use solution method for transfer equation in differential (not integral) form: short characteristics method

* ( ), ' ( ')

0

vv v

SS


34

Towards a linear scheme

Λ* acts on S, which makes the equations non-linear in the occupation numbers

• Idea of Rybicki & Hummer (1992): use J=ΔJ+Ψ*ηnew instead• Modify the rate equations slightly:

*

0 0

*3

20

*3

*2

0

4 4 ( )

24

24 ( )

ij ijij i i v

ijiji j j v

j

i

i

jji

j

R n n J dv J dvhv hv

n hvR n n J dv

n hv c

n hvJ dv

n

n n

nhv

nc


35

Stellar Atmospheres

This was the contents of our lecture:

Radiation field

Radiation transfer

Emission and absorption

Energy balance and Radiative equilibrium

Hydrostatic equilibrium

Solution Strategies for Stellar atmosphere models


36

Stellar Atmospheres


Radiation field

Radiation transfer


Radiative equilibrium


Stellar atmosphere models

The End


37

Stellar Atmospheres


Radiation field

Radiation transfer


Radiative equilibrium


Stellar atmosphere models

The End

Thank you forlistening !

stellar atmospheres: solution strategies 1 solution strategies

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