Temperature Correction Schemes

Transcript

1 Temperature Correction Schemes Motiation Up to now: Radiation transfer in a gien atmospheric structure No coupling between radiation field and temperature included Including radiatie equilibrium into solution of radiatie transfer Complete Linearization for model atmospheres (next chapter) Separate solution ia temperature correction + Quite simple implementation + Application within an iteration scheme allows completely linear system next chapter No direct coupling Moderate conergence properties 2

2 Temperature correction basic scheme. start approximation for. formal solution 2. correction 3. conergence? T() T () =ΛS( T) T() T() + T() Seeral possibilities for step 2 based on radiatie equilibrium or flux conseration Generalization to non-lte not straightforward With additional equations towards full model atmospheres: ydrostatic equilibrium Statistical equilibrium 3 LTE Strict LTE Scattering S( ) = ( T( )) S ( ) = ( β ) ( T( )) + β ( ) e e = Simple correction from radiatie equilibrium: = ( ) (,) (,) ( T(),) d ( [ T( )]) (,) (,) T() + d= T = T d = T= T( ) T = ( ) d d = ( ) d d T = = T= T( ) = = 2

3 LTE Strict LTE Scattering S( ) = ( T( )) S ( ) = ( β ) ( T( )) + β ( ) e e Simple correction from radiatie equilibrium: = = ( ) (,) (,) ( T(),) d ( [ ( T( )]) (,) (,) T ) + d = T = T d = T= T( ) T = ( ) d d = = T= T( ) 5 LTE Problem: T ( ) d d = = T= T( ) = independent of the temperature T Gray opacity ( independent of frequency): = ( ) () d ( ) ( ) = ( ) = d.moment equation dt = deiation from constant flux proides temperature correction 6 3

4 Unsöld-Lucy correction Unsöld (955) for gray LTE atmospheres, generalized by Lucy (96) for non-gray LTE atmospheres d -th moment: dt ( ) = d Ld = = d = d d = dt,,, = = dk st moment: dt = dk Ld =, = d = now new quantities,, K fulfilling radiatie equilibrium (local) and flux conseration (non local) radiatie equilibrium: flux conseration: d = = dk σ = = T π 7 Unsöld-Lucy correction Now corrections to obtain new quantities: X = X X d K = integrate K = K() + = K = K d = f d = f, () = () d = h () d = h () f() () K = + = f h = d () () = d f = + + fh f = 3 σt d d () () = = + f T + + π fh f = π () () f T = σt fh f = 8

5 Unsöld-Lucy correction π f() () T = d σt fh f = Radiatie equilibrium part good at small optical depths but poor at large optical depths Flux conseration part good at large optical depths but poor at small optical depths d Unsöld-Lucy scheme typically requires damping Still problems with strong resonance lines, i.e. radiatie equilibrium term is dominated by few optically thick frequencies 9 Generalization for scattering Unsöld-Lucy correction d -th moment: = = dt ) d Ld = tot ( S ) ( ( βe βe ) = ( β ) d, = ( β ) d, = dt M e e = = All the rest is the same Difficulties for scattering dominated regions: weak coupling between radiation field and temperature β e d 5

6 Unsöld-Lucy correction Generalization to non-lte (Werner & Dreizler 998, Dreizler 23) d -th moment: = ( ) = % S γ dt d Ld = = % d, = ( γ ) d, = dt M = = All the rest is the same % should contain only terms which couple directly to the temperature, i.e. bf and ff transitions Depth dependent damping (need to play with parameters c): / π / / ( ) () 3( ) = 3 + f c e T ce c2( e ) + σt fh f = Stellar Atmospheres This was the contents of our lecture: Radiation field Radiation transfer Emission and absorption Radiatie equilibrium ydrostatic equilibrium Stellar atmosphere models 2 6

7 Stellar Atmospheres This was the contents of our lecture: Radiation field Radiation transfer Emission and absorption Radiatie equilibrium ydrostatic equilibrium Stellar atmosphere models The End 3 Stellar Atmospheres This was the contents of our lecture: Radiation field Radiation transfer Emission and absorption Radiatie equilibrium ydrostatic equilibrium Stellar atmosphere models The End Thank you for listening! 7

8 Arett-Krook method In case that flux conseration and radiatie equilibrium is not fulfilled, Unsöld-Lucy can only change the temperature Change of other quantities, e.g. opacity, is not accounted for Arett & Krook (963) strict LTE assumed, generalization straightforward Current quantities: di µ = ( ( )) with some kind of mean opacity ( I ) T { = χ Does not fulfill flux conseration and radiatie equilibrium New quantities: di µ = I ( T( )) with mean opacity { ( ) = χ 5 Arett-Krook method Linear Taylor expansion of the new quantities from old ones: = + = + = + = + = + = T T T I I I ( ) d σ πt dχ d χ = χ + χ = χ + = + = + T dt Radiatie transfer equation: di µ = χ( I ( T( )) ) di di µ + µ = ( χ + χ)( I + I ) di χ I + µ = + χ + χ I + I di χ I + µ = χ + χ + χ I + I di µ χ ( I ) χ + χ I ( ) ( )( ) ( ) ( ) = + ( ) 6 8

9 Arett-Krook method st moment: di µ = χ ( I ) + χ + χ ( I ) µ dµ L dk dχ = χ : + χ + χ = χ + + χ = dχ + = d χ L d dχ σ + d = T χ π d, linear DEQ of first order x σt dχ = M ( x) dx, M ( x) exp dy d M( ) = π ( x) χ 7 Arett-Krook method Outer boundary: () = h () Ld σ σt σt () = T () = () () = d π π () π () σt () () d= h d π () σt () () = π () h dk = : f ( σt () ( ) = π () fh σt ) = const = f () = π () h () 8 9

10 -th moment: Arett-Krook method di µ = χ ( I ) + χ + χ ( I ) dµ L d = + + χ χ χ ( ) ( ) d d dχ = χ T χ ( ) d dt + + L d d d dχ d χ d T χ d + ( ) d dt + = = T = χ d dt d () d d d d d f h dχ d σt ( ) χ π () 9 Radiatie equilibrium and Complete Linearization (LTE) Simultaneous solution of RT and RE radiation transfer: 2 d (, ) (, ) 2 (, T()) = () r r r r fik ( k, T) = k = (, k, L, i, k, L, ND, k ) T = ( T, L, Ti, L, TND ) r r r r r r r r r r k, T fik( k, T ) correction δk, δt fik( k + δk, T + δt) = r r Taylor expansion: fik ( k + δk, Ti + δt) r r fik fik fik fik = = fik ( k, T ) + δi k + δik + δi+ k + δti i k ik i+ k i r r r Tkδk + UkδT = Kk Tk : tri-diagonal with usual Aik, ik, Cik ( k, Ti) Uk : diagonal ( Uk) = ii i r r r K = f (, T ) ( ) k ik k i 2

11 Radiatie equilibrium and Complete Linearization (LTE) Simultaneous solution of RT and RE radiatie equilibrium: = ( ) (, ) (, ) (, T) d = i i i ( L L ) ( ) f,,,,, T = w (, T) = ( ) inf, + i, ik, inf, i k ik k i k = r r Taylor expansion: f, T T ( ) ( ) NF ( + δ + δ ) inf, + k k i k=, NF r r NF f NF inf, + f r r r inf, + inf, + k δ ik δ i kδ k δ k = ik i k= = = f (, T ) + + T W + D T = L W : diagonal W = w k k ii k NF D: diagonal D = w r NF L = w i k = ii k = k k ( ik ( k, Ti )) ( k, Ti) i 2 Radiatie equilibrium and Complete Linearization (LTE) Together: Rybicki scheme: r r T U δ K r r Tk Uk δ k K k = r r TNF U NF δ NF K NF W W W D r r δt L k NF RE takes the part of the scattering integral Instead of sole for temperature corrections Non-linear iteration During the iteration: new opacities, Eddington factors 22

12 Radiatie equilibrium and Complete Linearization (NLTE) Direct generalization at least problematic due to weak coupling of NLTE source function to the temperature Take into account the change of the population numbers Add RE to the Complete Linearization scheme (Auer & Mihalas 969) Radiatie equilibrium in NLTE: (, )( (, ) S (, )) d= = Linearization: i i i ( L L ) ( η ) f,,, n,, n, T = w (, ) (, ) = inf, + NL+ i, inf, i, inl, i k k i ik k i k = k η k wk r r ik finf, + NL+ ( ψ + δψ) = nil nil NF NL r f i, NF + NL+ fi, NF+ NL+ fi, NF + NL+ = finf, + NL+ ( ψ ) + δik+ δnil+ δti k= ik l= nil i NF 23 Radiatie equilibrium and Complete Linearization (NLTE) Together with RT and SE: f ( ψ + δψ) = α = L NF + NL i, α ND NF NL fi, α fi, α f i, α = fi, α ( ψ ) + δi, k + δni, l + δti i= k= ik, l= nil, i r r r r Aδψ + δψ Cδψ = L i i i i i i+ i O O O Aik, r δ i r ik, δ i C, r ik δ i+ O O O + r δn r = fi, α ( ψ ) i δn i r δ n i+ δt i δt i δt i 2 2

13 Radiatie equilibrium and Complete Linearization (NLTE) Matrix i:... NF... NL T O M Sik, S ik, ik, L L nil, i O M = M M i NL ( Pi) ( P) lm, i lm, nim, ( Pi ) L L L L T ll, i m= ik, i M M finf + NL+ finf+ NL+ finf+ NL+ ik, nil, i... NF... NL T 25 3