7. Non-LTE basic concepts

Transcription

1 7. Non-LTE basic concepts LTE vs NLTE occupation numbers rate equation transition probabilities: collisional and radiative examples: hot stars, A supergiants 1

2 Equilibrium: LTE vs NLTE LTE each volume element separately in thermodynamic equilibrium at temperature T(r) 1. f(v) dv = Maxwellian with T = T(r) 2. Saha: (n p n e )/n 1 T 3/2 exp(-hν 1 /kt) 3. Boltzmann: n i / n 1 = g i / g 1 exp(-hν 1i /kt) However: volume elements not closed systems, interactions by photons LTE non-valid if absorption of photons disrupts equilibrium 2

3 Equilibrium: LTE vs NLTE Processes: radiative photoionization, photoexcitation establish equilibrium if radiation field is Planckian and isotropic valid in innermost atmosphere however, if radiation field is non-planckian these processes drive occupation numbers away from equilibrium, if they dominate collisional collisions between electrons and ions (atoms) establish equilibrium if velocity field is Maxwellian valid in stellar atmosphere Detailed balance: the rate of each process is balanced by inverse process 3

4 Equilibrium: LTE vs NLTE non equilibrium if rate of photon absorptions >> rate of electron collisions I ν (T) T α, α 1 n e T 1/2 LTE valid: low temperatures high densities non-valid: high temperatures low densities 4

5 Calculation of occupation numbers NLTE 1. f(v) dv remains Maxwellian 2. Boltzmann Saha replaced by dn i / dt = 0 (statistical equilibrium) for a given level i the rate of transitions out = rate of transitions in rate out = rate in X n i P ij = X n j P ji j6=i j6=i X X n i (R ij + C ij )+n i (R ik + C ik )= n j (R ji + C ji )+n p (R ki + C ki ) j6=i j6=i RATE EQUATIONS lines ionization lines recombination Transition probabilities radiative R ij = B ij R 0 R ji = A ji + B ji R ϕ ij (ν) J ν dν 0 ϕ ij (ν) J ν dν absorption emission collisional C ij = n e R 0 σ coll (v)vf(v)dv C ji = g i /g j e E ji/kt C ij 5

6 Occupation numbers can prove that if C ij À R ij or J ν B ν (T): n i n i (LTE) We obtain a system of linear equations for n i : n 1 A n 2... = X where A contains terms: n p Z Z ϕ ij (ν) 0 4π I ν (ω) dω 4π dν κ ν = combine with equation of transfer: NX NX i=1 j=i+1 µ σ line ij (ν) n i g i n j g j NX + σ ik (ν) i=1 µ di ν(ω) dr = κ ν I ν (ω) +² ν ³n i n i e hν/kt +n e n p σ kk (ν,t) ³1 e hν/kt +n e σ e ² ν =... non-linear system of integro-differential equations 6

7 Occupation numbers Iteration required: radiative processes depend on radiation field radiation field depends on opacities opacities depend on occupation numbers requires database of atomic quantities: energy levels, transitions, cross sections levels per ion 3-5 ionization stages per species 30 species fast algorithm to calculate radiative transfer required 7

8 Transition probabilities: collisions probability of collision between atom/ion (cross section σ) and colliding particles in time dt σ v dt rate of collisions = flux of colliding particles relative to atom/ion (n coll v) cross section σ. for excitations: C ij = n coll Z 0 σij coll (v)vf(v)dv v σ coll from complex quantum mechanical calculations collision cylinder: all particles in that volume collide with target atom and similarly for de-excitation 8

9 Transition probabilities: collisions in a hot plasma free electrons dominate: n coll = n e v th = < v > = (2kT/m) 1/2 is largest for electrons (v th, e ' 43 v th, p ) C ij = n e Z 0 σ coll ij (v)vf(v)dv = n e q ij f(v) dv is Maxwellian in stellar atmospheres established by fast belastic e-e collisions. One can show that under these circumstances there the collisional transition probabilities for excitation and de-excitation are related by C ij = g j g i e E ij/kt C ji 9

10 Transition probabilities: collisions for bound-free transitions: collisional recombination (very inefficient 3-particle process) = C ki = C ik n e g i g µ h 2 2πmkT 3/2 e E i/kt collisional ionization (important at high T) In LTE: µ nj n i = g j g i e E ij/kt Boltzmann µ n1 = n e φ 1 (T) Saha n + 1 C ij = µ nj n i C ji C ki = µ ni n + 1 C ik 10

11 Transition probabilities: collisions Approximations for C ji line transition i j C ji = n e T 1/2 e Ω ji g j s 1 Ω ji σ ji Ω ji = collision strength for forbidden transitions: Ω ji ' 1 for allowed transitions: Ω ji = (g i / g j ) f ij λ ij Γ(E ij /kt) max (g, exp(e ij /kt) E 1 (E ij /kt) g = 0.7 for nl -> nl =0.3 for nl n l 11

12 Transition probabilities: collisions for ionizations C ik = n e T 1/2 e g 0 i σphot ik (ν ik ) e E i /kt E i /kt = 0.1 for Z=1 = 0.2 for Z=2 = 0.3 for Z=3 photoionization cross section at ionization edge 12

13 Transition probabilities: radiative processes Line transitions absorption i j (i < j) probability for absorption dw ij = B ij I x dω 4π ϕ(x)dx x = ν ν 0 ν D integrating over x and dω (dω = 2π dµ) R ij = B ijjij Jij (r) = 1 2 Z Z 1 1 I(x,µ, r)ϕ(x)dµdx emission (i > j) R ji = A ji + B ji Jji 13

14 Transition probabilities: radiative processes Bound-free photo-ionization i k R ik = Z σ ik (ν) cn phot (ν)dν ν ik σ v for photons u ν = hν n phot (ν) = 4π c J ν energy density photo-recombination k i R ik =4π Z ν ik σ ik (ν) J ν dν hν R ki =4π R ki = Z ν ik µ ni n + 1 σ ik hν (2hν3 c 2 Rki + J ν ) e hν/kt dν 14

15 LTE vs NLTE in hot stars Kudritzki

16 LTE vs NLTE in hot stars Kudritzki 1978 difference between NLTE and LTE Hγ equivalent width as a function of log g for T eff = 45,000 K NLTE and LTE emperature stratifications for two different Helium abundances at T eff = 45,000 K, log g = 5 16

17 LTE vs NLTE: departure coefficients - hydrogen b i = nnlte i n LTE i β Orionis (B8 Ia) T eff = 12,000 K log g = 1.75 (Przybilla 2003) 17

18 Rome 2005 N I N II Nitrogen atomic models Przybilla, Butler, Kudritzki

19 LTE vs NLTE: departure coefficients - nitrogen 19

20 LTE vs NLTE: line fits nitrogen lines 20

21 log {n i /n i LTE } FeII Przybilla, Butler, Kudritzki, Becker, A&A,

22 LTE vs NLTE: line fits hydrogen lines in IR b i = nnlte i n LTE i Brackett lines 22

23 complex atomic models for O-stars (Pauldrach et al., 2001) AWAP 05/19/05 23

24 NLTE Atomic Models in modern model atmosphere codes lines, collisions, ionization, recombination Essential for occupation numbers, line blocking, line force Accurate atomic models have been included 26 elements 149 ionization stages 5,000 levels ( + 100,000 ) 20,000diel. rec. transitions b-b line transitions Auger-ionization recently improved models are based on Superstructure Eisner et al., 1974, CPC 8,270 AWAP 05/19/05 24

25 AWAP 05/19/05 25

26 Pauldrach, 2003, Reviews in Modern Astronomy, Vol. 16 AWAP 05/19/05 26 consistent treatment of expanding atmospheres along with spectrum synthesis techniques allow the determination of stellar parameters, wind parameters, and abundances