Solution of the radiative transfer equation in NLTE stellar atmospheres

Solution of the radiative transfer equation in NLTE stellar atmospheres Jiří Kubát kubat@sunstel.asu.cas.cz Astronomický ústav AV ČR Ondřejov Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 1

Outline 1. Basic equations for stellar atmospheres 2. Radiative transfer equation 3. Equations of statistical equilibrium 4. Discretization 5. Formal solution of RTE 6. Lambda iteration 7. Complete linearization 8. Accelerated lambda iteration Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 2

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) radiatively driven stellar wind Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 3

Basic equations assume static atmosphere ( v = 0) stationary atmosphere ( / t = 0) 1-dimensional atmosphere given temperature structure T ( r) given density structure ρ( r) equations to be solved (numerically): radiative transfer equation equations of statistical equilibrium Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 4

Basic equations assume static atmosphere ( v = 0) stationary atmosphere ( / t = 0) 1-dimensional atmosphere given temperature structure T ( r) given density structure ρ( r) equations to be solved (numerically): radiative transfer equation equations of statistical equilibrium Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 4

Basic equations assume static atmosphere ( v = 0) stationary atmosphere ( / t = 0) 1-dimensional atmosphere given temperature structure T ( r) given density structure ρ( r) equations to be solved (numerically): radiative transfer equation equations of statistical equilibrium account for full angle and frequency dependence (I (µ, ) I µ ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 4

Radiative transfer equation µ di µ(z) dz = η (z) χ (z)i µ (z) = χ [S (z) I µ (z)] η (z) emissivity χ (z) opacity S (z) = η (z) χ (z) source function Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 5

Radiative transfer equation µ di µ(z) dz = η (z) χ (z)i µ (z) = χ [S (z) I µ (z)] introduce dτ = χ dz optical depth J = 1 1 I 2 1 µ dµ mean intensity K = 1 1 2 1 µ2 I µ dµ f = K /J variable Eddington factor (Auer & Mihalas 1970) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 5

Radiative transfer equation µ di µ(z) dz = η (z) χ (z)i µ (z) = χ [S (z) I µ (z)] introduce dτ = χ dz optical depth J = 1 1 I 2 1 µ dµ mean intensity K = 1 1 2 1 µ2 I µ dµ f = K /J variable Eddington factor (Auer & Mihalas 1970) 2nd order equation d 2 [f (z)j (z)] dτ 2 = J (z) S (z) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 5

Radiative transfer equation 2nd order equation d 2 [f (z)j (z)] dτ 2 = J (z) S (z) + boundary conditions d [f (z)j (z)] = g (z) H dτ upper d [f (z)j (z)] = H + + g (z) dτ lower g = R 1 0 µ (I + µ I µ) dµ J Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 5

Radiative transfer equation 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 ] Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 6

Radiative transfer equation 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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 6

Equations of statistical equilibrium in LTE: n i = f (n e, T ) Saha-Boltzmann distribution Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 7

Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,... NL l i {n l [R li + C li ] n i (R il + C il )} = 0 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 7

Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,... NL l i {n l [R li + C li ] n i (R il + C il )} = 0 collisional rates n i C il = n i n e q il (T ), Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 7

Equations of statistical equilibrium outside LTE: n i = f (n e, T, J ) statistical equilibrium for i = 1,... NL l i radiative rates {n l [R li + C li ] n i (R il + C il )} = 0 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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 7

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 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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 7

Final radiative transfer equation d 2 [f (z)j (z)] dτ 2 = J (z) S (z, J ) S = η χ f = K J K = χ = f χ (, n e, T, n i ) Z 1 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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 8

Two-level atom d 2 [f (z)j (z)] dτ 2 = J (z) S (z, J ) the source function may be written as (Mihalas 1978) S = (1 ε) ϕ J d + εb ε = ε 1 + ε ε = C 21 ( 1 e h/kt ) A 21 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 9

Discretization J continuous function of z, depth (z d) equidistant in log m or log τ some 4 5 depth points per decade frequency ( n) not equidistant need to resolve lines continuum edges angle (µ m) 3 directions all quantities expressed as J dn Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 10

Discretized radiative transfer equation for a frequency point n and angle m, depth points d = 1,..., ND a d f d 1 J d 1 + (b d f d + 1) J d + c d f d+1 J d+1 = S d (b 1 f 1 + g 1 + 1) J 1 + c 1 f 2 J 2 = H + S 1 a ND f ND 1 J ND 1 + (b ND f ND + g ND + 1) J ND = H + + S ND» 1 1 a d = τ 2 d + τ 12 d+ τ 12 d 12» 1 1 c d = τ 2 d + τ 12 d+ τ 12 d+ 12 b d = a d c d. Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 11

Discretized frequency integration for lines, ensure that J() d φ() d = NF n=1 NF n=1 w n J n w n φ n = 1 if not, it is obligatory to renormalize w n for continuum edges place 2 frequency points (on both sides) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 12

Discretized angle integration I(µ) dµ NF m=1 w m I m 3 points sufficient if Gaussian quadrature is used also necessary to check normalization Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 13

Discretized equations of statistical equilibrium for all d = 1,..., ND, i = 1,..., NL {(n l ) d [(R li ) d + (C li ) d ] (n i ) d [(R il ) d + (C il ) d ]} = 0 l i (n i ) d (R il ) d = (n i ) d 4π NF n=1 (n l ) d (R li ) d = (n l ) d g i g j 4π w n (α il ) n h n NF n=1 J dn w n (α il ) n h n ( 2h 3 n c 2 + J dn ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 14

Formal solution of the RTE solution for given χ and η (given S ) relatively simple d 2 [f (z)j (z)] dτ 2 = J (z) S (z) 2nd order differential equation same numerical scheme as for the Feautrier solution crucial for the total accuracy of the whole problem solution Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 15

Λ iteration radiative transfer equation for J d 2 [f (z)j (z)] dτ 2 = J (z) S (z) 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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 16

Complete linearization to stellar atmospheres introduced by Auer & Mihalas (1969) for the case of NLTE model atmospheres solution of equations: radiative transfer (J ) hydrostatic equilibrium (ρ) radiative equilibrium (T ) statistical equilibrium (n i ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 17

Complete linearization to stellar atmospheres introduced by Auer & Mihalas (1969) for the case of NLTE model atmospheres for a restricted problem NLTE line formation (Auer 1973) solution of equations: radiative transfer (J ) statistical equilibrium (n i ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 17

Complete linearization to stellar atmospheres introduced by Auer & Mihalas (1969) for the case of NLTE model atmospheres for a restricted problem NLTE line formation (Auer 1973) solution of equations: radiative transfer (J ) statistical equilibrium (n i ) vector of solution ψ = (J 1,..., J NF, n 1,..., n NL ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 17

Complete linearization vector of solution ψ = (J 1,..., J NF, n 1,..., n NL ), dimension NF + NL formally F ( ) ψ = 0 current estimate ψ 0 correct solution ψ = ψ 0 + δ ψ corrections δ ψ = [ F ψ ( ψ0 ) ] 1 matrix NL + NF for each depth point d F ( ψ0 ) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 18

Complete linearization radiative transfer equation (from Mihalas, 1978, Stellar atmospheres) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer equation J = (J 1,..., J NF ) A d δ J d 1 + B d δ J d + C d δ J d+1 = L d δs nd = NL l=1 S n n l (δn l ) d d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer equation J = (J 1,..., J NF ) A d δ J d 1 + B d δ J d + C d δ J d+1 = L d δs nd = NL l=1 S n n l (δn l ) d d A d δj n,d 1 + B d δj n,d + C d δj n,d+1 + D d δn l,d = L n,d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer equation A d δj n,d 1 + B d δj n,d + C d δj n,d+1 + D d δn l,d = L n,d statistical equilibrium A n = b, n = (n 1,..., n NL ) [ ] A b n n n + A d δ n d + [ A J n,d n b J n,d ] δj n,d = b d A d n d E d δn l,d + F d δj n,d = K l,d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer equation A d δj n,d 1 + B d δj n,d + C d δj n,d+1 + D d δn l,d = L n,d statistical equilibrium E d δn l,d + F d δj n,d = K l,d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer equation A d δj n,d 1 + B d δj n,d + C d δj n,d+1 + D d δn l,d = L n,d statistical equilibrium E d δn l,d + F d δj n,d = K l,d n l J n = δn l,d = NL r=1 A 1 lr NF n=1 [ n l J n b r J n δ(j n ) d d NL s=1 A rs J n n s ] Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 19

Complete linearization radiative transfer + statistical equilibrium A d δ J d 1 + B d δ J d + C d δ J d+1 = L d J d = (J 1,..., J NF ), d = 1,..., ND A 1 = 0, C ND = 0 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 20

Complete linearization reformulation using net radiative bracket (Auer & Heasley 1976) t transition index (δz t ) d = (n i ) d (δr il ) d (n l ) d (δr li ) d δn i = t n i Z t δz t δ Z t + R t n i + S t δ n l = L t ( ) n i n l 1 + R t + S t δz t = Z t Z L t R t t t t n i Z t δz t S t t t n l Z t δz t system is solved for (Z t ) d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 21

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution J = Λ S Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution J = Λ S + Λ S Λ S Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution J = Λ S + (Λ Λ ) S Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution iteration scheme J = Λ S + (Λ Λ ) S J (n+1) = Λ S (n+1) + (Λ Λ ) S (n) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution iteration scheme J = Λ S + (Λ Λ ) S J (n+1) = Λ S (n+1) + (Λ Λ ) S (n) }{{} J (n) correction term Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

Accelerated lambda iteration huge matrices in complete linearization laborious and computationally expensive simple Λ-iteration convergence problems way out > Accelerated lambda iteration (ALI) (Cannon 1973) formal solution iteration scheme J = Λ S + (Λ Λ ) S J (n+1) = Λ S (n+1) + (Λ Λ ) S (n) }{{} J (n) correction term compare: J (n+1) = Λ S (n) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 22

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 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 23

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 n i R il = n i 4π n l R li = n l g i g j 4π αil () h αij () h [ Λ S (n+1) ( 2h 3 c 2 + ] + J (n) d [ Λ S (n+1) ] ) + J (n) d Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 23

Accelerated lambda iteration equations of statistical equilibrium l i {n l [R li (n i, n l ) + C li ] n i (R il (n i, n l ) + C il )} = 0 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 10000 Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 24

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 solution by Newton-Raphson method (linearization) preconditioning Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 24

Summary two basic methods of the solution of the RTE+ESE system 1. complete linearization (Kiel) 2. accelerated lambda iteration (DETAIL) Non-LTE Line Formation for Trace Elements in Stellar Atmospheres, Nice 30.07.2007 p. 25