Radiative transfer equation in spherically symmetric NLTE model stellar atmospheres
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1 Radiative transfer equation in spherically symmetric NLTE model stellar atmospheres Jiří Kubát Astronomický ústav AV ČR Ondřejov Zářivě (magneto)hydrodynamický seminář Ondřejov p.
2 Outline 1. Basic equations for stellar atmospheres 2. Radiative transfer equation in spherical symmetry 3. Formal solution 4. Equations of statistical equilibrium 5. Λ-iteration and accelerated Λ-iteration 6. Model atmosphere problem 7. Linearization 8. Applications 9. Conclusions Zářivě (magneto)hydrodynamický seminář Ondřejov p.
3 Role of radiation in stellar atmospheres source of information about star and stellar atmosphere influence on matter in stellar atmosphere non-local (long distance) interaction (photon mean free path particle mean free path) change of the population numbers (non-equilibrium values) Zářivě (magneto)hydrodynamický seminář Ondřejov p.
4 Basic equations assume static atmosphere ( v = 0) stationary atmosphere ( / t = 0) 1-dimensional spherically symmetric atmosphere equations to be solved (numerically): radiative transfer equation equations of statistical equilibrium equation of radiative equilibrium equation of hydrostatic equilibrium radial depth equation Zářivě (magneto)hydrodynamický seminář Ondřejov p.
5 Basic equations assume static atmosphere ( v = 0) stationary atmosphere ( / t = 0) 1-dimensional spherically symmetric atmosphere equations to be solved (numerically): radiative transfer equation equations of statistical equilibrium equation of radiative equilibrium equation of hydrostatic equilibrium radial depth equation account for full angle and frequency dependence (I (µ, ) I µ ) Zářivě (magneto)hydrodynamický seminář Ondřejov p.
6 Radiative transfer equation µ I µ(r) r + 1 µ2 r I µ (r) µ = χ (r)i µ (r) + η (r) η (z) emissivity χ (z) opacity θ µ = cos θ r Zářivě (magneto)hydrodynamický seminář Ondřejov p.
7 Radiative transfer equation µ I µ(r) r + 1 µ2 r I µ (r) µ = χ (r)i µ (r) + η (r) = χ (r) [I µ (r) S (r)] η (z) emissivity χ (z) opacity S (z) = η (z) χ (z) source function Zářivě (magneto)hydrodynamický seminář Ondřejov p.
8 Radiative transfer equation with scattering µ I µ(z) r + 1 µ2 r I µ (r) µ = [κ (r) + σ (r)]i µ (r) + η (r) + σ (r)j (r) Zářivě (magneto)hydrodynamický seminář Ondřejov p.
9 Radiative transfer equation with scattering µ I µ(z) r + 1 µ2 r I µ (r) µ = [κ (r) + σ (r)]i µ (r) + η (r) + σ (r) 0 I µ (r)dµ Zářivě (magneto)hydrodynamický seminář Ondřejov p.
10 Moment radiative transfer equations ( ) 1 d r 2 H r 2 dr = χ J + η + σ J. where J = 1 2 we introduce f = K J 1 dq q dr 1 3 f = r dk dr + 3K J = χ H, r 1 1 I µ dµ, H = µi µ dµ, K = 1 2 variable Eddington factor the sphericity function 1 1 µ2 I µ dµ. d (f q J ) dx = r 2 H Zářivě (magneto)hydrodynamický seminář Ondřejov p.
11 Moment radiative transfer equations d 2 (f q J ) dx 2 where dx = q χ /r 2 dr boundary conditions = r4 q ( J η ) + σ J, χ where g = d (f q J ) = r 2 ( g J H ) dx d (f q J ) = r 2 ( H + ) g J dx ( ) 1 0 µj µ dµ / (J ) at the surface at depth Zářivě (magneto)hydrodynamický seminář Ondřejov p.
12 Formal solution of the RTE solution for given χ and η (given S ) relatively simple crucial for the total accuracy of the whole problem solution Zářivě (magneto)hydrodynamický seminář Ondřejov p.
13 Formal solution of the RTE ray method Hummer & Rybicki (1971, MNRAS 152, 1) µ = cos ϑ µ r + 1 µ2 r µ = d ds ± di± µ(r) ds = χ (r)i ± µ(r) + η (r) + σ (r)j (r) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
14 Feautrier solution of the RTE solution along a ray (specific intensities I + and I ) introduce Feautrier variables j µ = 1 2 h µ = 1 2 ( I + µ + I µ) ( I + µ I µ) the transfer equation d 2 j µ dτ 2 s = j µ S where S = η + σ J κ + σ, dτ s = (κ + σ ) ds (µ absorbed to ds) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
15 Feautrier solution of the RTE boundary conditions upper lower dj µ dτ s = j µ I µ dj µ dτ s = I + µ j µ dj µ dτ s = 0 core rays (at depth) tangent rays (in the middle of the ray). Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
16 Formal solution of the RTE with scattering 1. determine j µ by solution along each ray (J given): d 2 j µ dτ 2 s = j µ η + σ J κ + σ 2. f = 1 0 µ2 j µ dµ 1 0 j µ dµ, g = 1 0 µj µ dµ 1 0 j µ dµ, 1 dq q dr = 3 f 1 r 3. determine J by solution: 4. check if δf < ε d 2 (f q J ) dx 2 = r4 q ( J η ) + σ J, χ Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
17 Opacities and emissivities opacity χ = i emissivity l>i [ n i g ] i n l α il () + g l i k ( n i n i e h kt n e n k α kk (, T) ) α ik ()+ ( 1 e h kt ) + n e σ e η = 2h3 c 2 [ i l>i n l g i g l α il () + i n i α ik ()e h kt + k n e n k α kk (, T)e h kt ] Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
18 Opacities and emissivities opacity χ = i emissivity l>i [ n i g ] i n l α il () + g l i k ( n i n i e h kt n e n k α kk (, T) ) α ik ()+ ( 1 e h kt ) + n e σ e η = 2h3 c 2 [ i l>i n l g i g l α il () + i n i α ik ()e h kt + n e n k α kk (, T)e h kt ] k Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
19 Equations of statistical equilibrium in LTE: n i = f (n e, T) Saha-Boltzmann distribution Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
20 Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,...NL {n l [R li + C li ] n i (R il + C il )} = 0 l i Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
21 Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,...NL collisional rates {n l [R li + C li ] n i (R il + C il )} = 0 l i n i C il = n i n e q il (T), Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
22 Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,...NL {n l [R li + C li ] n i (R il + C il )} = 0 l i radiative rates αil () n i R il = n i 4π h J d ( g i αij () 2h 3 n l R li = n l 4π g j h c 2 + J ) d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
23 Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,...NL {n l [R li (J ) + C li ] n i (R il (J ) + C il )} = 0 radiative rates l i αil () n i R il = n i 4π h J d ( g i αij () 2h 3 n l R li = n l 4π g j h c 2 + J ) d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
24 Final radiative transfer equation d 2 (f q J ) dx 2 f = K J = r4 q K = 1 dq q dr = 3 f 1 r χ = f χ (, n e, T, n i ) Z 1 [ J η ] + σ J, χ 1 «µ 2 I µ dµ η = f η (, n e, T, n i ) X {n l [R li (J ) + C li ] n i (R il (J ) + C il )} = 0 l i Z n i R il = n i 4π Z g i n l R li = n l 4π g j αil () h αij () h J d 2h 3 c 2 + J «d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
25 Λ iteration radiative transfer equation for J d 2 (f q J ) dx 2 = r4 q [ J η ] + σ J, χ formally may be written as J = Λ S iteration scheme: J ESE S RTE J ESE S l i J (n+1) { [ ] n (n) l R li (J (n) ) + C li = Λ S (n) n (n) i [ R il (J (n) ) + C il ]} = 0 converges extremely slowly for stellar atmospheres (τ 1) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
26 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
27 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S + Λ S Λ S Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
28 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S + (Λ Λ )S Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
29 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution iteration scheme (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S + (Λ Λ )S J (n+1) = Λ S (n+1) + (Λ Λ )S (n) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
30 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S + (Λ Λ )S iteration scheme J (n+1) = Λ S (n+1) + (Λ Λ )S (n) }{{} J (n) correction term Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
31 Accelerated lambda iteration simple Λ-iteration convergence problems huge matrices in complete linearization laborious and computationally expensive way out > Accelerated lambda iteration (ALI) formal solution (Cannon 1973, JQSRT 14, 627; ApJ 185, 621) J = Λ S + (Λ Λ )S iteration scheme J (n+1) = Λ S (n+1) + (Λ Λ )S (n) }{{} J (n) correction term compare: J (n+1) = Λ S (n) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
32 Accelerated lambda iteration iteration scheme J (n+1) = Λ S (n+1) + (Λ Λ ) S (n) = Λ S (n+1) + J (n) J (n) correction term S RTE J, J ESE+ALI S solution of the radiative transfer equation is transferred to the solution of the statistical equilibrium equations Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
33 Accelerated lambda iteration iteration scheme J (n+1) = Λ S (n+1) + (Λ Λ ) S (n) = Λ S (n+1) + J (n) J (n+1) radiative rates αil () n i R il = n i 4π h g i αij () n l R li = n l 4π g j h [ Λ S (n+1) ( 2h 3 c 2 + ] + J (n) d [ Λ S (n+1) ] ) + J (n) d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 1
34 Accelerated lambda iteration equations of statistical equilibrium {n l [R li (n i, n l ) + C li ] n i (R il (n i, n l ) + C il )} = 0 l i for i = 1,...NL ALI eliminated radiation field from the explicit solution equations of statistical equilibrium are nonlinear in n i, n l savings important, for a typical problem NF Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
35 Accelerated lambda iteration equations of statistical equilibrium {n l [R li (n i, n l ) + C li ] n i (R il (n i, n l ) + C il )} = 0 l i for i = 1,...NL ALI eliminated radiation field from the explicit solution equations of statistical equilibrium are nonlinear in n i, n l savings important, for a typical problem NF solution by Newton-Raphson method (linearization) preconditioning Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
36 Model atmosphere problem additional structural equations hydrostatic equilibrium radiative equilibrium radial depth preconditioning does not help for them, linearization necessary Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
37 Radial optical depth equation modified column mass depth ( ) R 2 dm = ρ r 2 dr m independent variable in model atmosphere calculation Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
38 Hydrostatic equilibrium equation upper boundary condition: dp g dm = GM R 2 4π c 0 1 d [q f J ] q dm d p 1 = GM m 1 R 2 4π c ( r1 ) 2 R 0 χ 1 ρ 1 [ g1 J 1 H ] d g 1 = 1 0 µj 1µ dµ/ 1 0 j 1µ dµ H is the incident flux at the upper boundary Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
39 Hydrostatic equilibrium equation dp g dm = GM R 2 4π c upper boundary condition: 0 1 d [q f (Λ S + J )] q dm d p 1 = GM m 1 R 2 4π c ( r1 ) 2 R 0 χ 1 ρ 1 [ g1 (Λ S 1 + J 1 ) H ] d g 1 = 1 0 µj 1µ dµ/ 1 0 j 1µ dµ H is the incident flux at the upper boundary J was eliminated using ALO: J = Λ S + J Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
40 Equation of radiative equilibrium integral form 0 [κ J η ] d = 0 κ [J S ] d = 0 differential form (from F = 0) L (4πR) 2 = 0 ρ d [q f J ] q χ dm d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
41 Equation of radiative equilibrium integral form 0 [κ (Λ S + J ) η ] d = 0 κ [(Λ 1)S + J ] d = 0 differential form (from F = 0) L (4πR) 2 = 0 ρ d [q f (Λ S + J )] q χ dm d J was eliminated using ALO: J = Λ S + J Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
42 Complete linearization to stellar atmospheres introduced by Auer & Mihalas (1969, ApJ 158, 641) for the case of NLTE model atmospheres solution of equations: radiative transfer (J ) hydrostatic equilibrium (ρ) radiative equilibrium (T ) statistical equilibrium (n i ) radial depth (r) vector of solution ψ = (n e, T, r, n 1,...,n NL, J 1,...,J NF ) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
43 Complete linearization to stellar atmospheres introduced by Auer & Mihalas (1969, ApJ 158, 641) for the case of NLTE model atmospheres solution of equations: radiative transfer (J ) included with a help of ALO hydrostatic equilibrium (ρ) radiative equilibrium (T ) statistical equilibrium (n i ) radial depth (r) vector of solution ψ = (n e, T, r, n 1,...,n NL, J 1,...,J NF ) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
44 Summary of equations equations of statistical equilibrium X {n l [R li (n i, n l ) + C li ] n i (R il (n i, n l ) + C il )} = 0 i = 1,... NL l i n i R il = n i 4π R α il h [Λ g S + J ]d, n l R li = n i l 4π R α ij g j h 2h 3 c 2 + [Λ S + J ] d equation of hydrostatic equilibrium dp g dm = GM R 2 4π c equation of radiative equilibrium Z 0 1 d[q f (Λ S + J )] q dm d Z 0 κ [(Λ 1) S + J ] d = 0. or H 0 = Z 0 ρ d[q f (Λ S + J )] q χ dm d. radial depth equation R 2 «dm = ρ r 2 dr Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
45 Solution by linearization vector of solution ψ = (n e, T, r, n 1,...,n NL ), dimension 3 + NL formally F ( ) ψ = 0 current estimate ψ 0 correct solution ψ = ψ 0 + δψ corrections δ ψ = [ F ψ ( ψ0 ) ] 1 F ( ψ0 ) matrix NL + 3 for each depth point d Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
46 Accelerated linearization if we express the linearized b-factors as δb i = b i n e δn e + b i T δt + b i r δr from equations of statistical equilibrium (cf. Auer 1973, ApJ 180, 469; Anderson 1987, NRT, 163), [ b i x = A ij + A im b j b m B i b j ] 1 ( Ajm x b m B j x ) (x stands for n e, T, r) further reduction of the number of explicitly linearized variables Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
47 Accelerated linearization solution of equations: hydrostatic equilibrium (ρ) radiative equilibrium (T ) radial depth (r) statistical equilibrium (n i ) vector of solution ψ = (n e, T, r, n 1,...,n NL ) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
48 Accelerated linearization solution of equations: hydrostatic equilibrium (ρ) radiative equilibrium (T ) radial depth (r) statistical equilibrium (n i ) included with a help of implicit linearization of b-factors vector of solution ψ = (n e, T, r, n 1,...,n NL ) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 2
49 Solution by linearization vector of solution ψ = (n e, T, r), dimension: 3 significant savings of computing time Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
50 Application of static spherically symmetric model atmospheres generalization of plane-parallel atmospheres in extended atmospheres large radiaton pressure O and B stars strong stellar wind static approximation unusable Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
51 Application of static spherically symmetric model atmospheres generalization of plane-parallel atmospheres in extended atmospheres large radiaton pressure O and B stars strong stellar wind static approximation unusable Do static extended atmospheres exist? Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
52 Application of static spherically symmetric model atmospheres generalization of plane-parallel atmospheres in extended atmospheres large radiaton pressure O and B stars strong stellar wind static approximation unusable Do static extended atmospheres exist? they existed! Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
53 Application of static spherically symmetric model atmospheres generalization of plane-parallel atmospheres in extended atmospheres large radiaton pressure O and B stars strong stellar wind static approximation unusable Do static extended atmospheres exist? they existed! first stars in the Universe consisted only of primordial hydrogen and helium Krtička & Kubát (2006, A&A 446, 1039) showed that such stars can not have stellar winds Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
54 Application to moving atmospheres static radiative transfer equation may be used for continuum radiative transfer in moving stellar atmospheres, as it was done by Krtička & Kubát (2004, A&A 417, 1003) static spherically symmetric model atmospheres can be used as a lower boundary condition in calculations of stellar winds (core-halo approximation) Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
55 Final remarks 1. static spherically symmetric model atmospheres are the first step towards more complicated structures 2. for stellar wind calculations they provide the lower boundary condition 3. for axially symmetric problems they give the zeroth order approximation 4. for the first stars in the Universe they provide full description Zářivě (magneto)hydrodynamický seminář Ondřejov p. 3
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