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Stellar Atmospheres: Solution Strategies
1
Solution Strategies
Stellar Atmospheres: 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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
5
Short characteristic method
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
Stellar Atmospheres: Solution Strategies
6
Short characteristic method
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
Stellar Atmospheres: Solution Strategies
7
Short characteristic method
Also possible: Parabolic instead of linear interpolation
Problem: Scattering
Requires iteration
1
1
1 , ( )
2e e e e e en J I d
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
13
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
Stellar Atmospheres: Solution Strategies
14
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
16
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
Stellar Atmospheres: Solution Strategies
17
Complete Linearization
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
Stellar Atmospheres: Solution Strategies
18
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
21
Complete Linearization
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
Stellar Atmospheres: Solution Strategies
22
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
27
Unsöld-Lucy correction
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
Stellar Atmospheres: Solution Strategies
28
Unsöld-Lucy correction
„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
Stellar Atmospheres: Solution Strategies
29
NLTE Model Atmospheres
Radiation Transport and Sattistical Equilibrium are very closely coupled
Simple separation (Lamda Iteration) does not work
Complete Linearization
Accelerated Lambda Iteration
Stellar Atmospheres: Solution Strategies
30
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
Stellar Atmospheres: Solution Strategies
31
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
Stellar Atmospheres: Solution Strategies
32
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
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
Stellar Atmospheres: Solution Strategies
36
Stellar Atmospheres
This was the contents of our lecture:
Radiation field
Radiation transfer
Emission and absorption
Radiative equilibrium
Hydrostatic equilibrium
Stellar atmosphere models
The End
Stellar Atmospheres: Solution Strategies
37
Stellar Atmospheres
This was the contents of our lecture:
Radiation field
Radiation transfer
Emission and absorption
Radiative equilibrium
Hydrostatic equilibrium
Stellar atmosphere models
The End
Thank you forlistening !