1

7. Non-LTE – basic concepts

LTE vs NLTE

occupation numbers

rate equation

transition probabilities: collisional and radiative

examples: hot stars, A supergiants

2

Equilibrium: LTE vs NLTE

LTEeach volume element separately in thermodynamic equilibrium at temperature T(r)

1. f(v) dv = Maxwellian with T = T(r)2. Saha: (np ne)/n1 ∝ T3/2 exp(-hν1/kT)

3. Boltzmann: ni / n1 = gi / g1 exp(-hν1i/kT)

However: volume elements not closed systems, interactions by photons

LTE non-valid if absorption of photons disrupts equilibrium

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

4

Equilibrium: LTE vs NLTE

non equilibrium if

rate of photon absorptions >> rate of electron collisions∼ Iν (T) ∼ Tα, α≥ 1 ∼ ne T1/2

LTE

valid: low temperatures high densities

non-valid: high temperatures low densities

5

Calculation of occupation numbers

NLTE1. f(v) dv remains Maxwellian

2. Boltzmann – Saha replaced by dni / dt = 0 (statistical equilibrium)

for a given level i the rate of transitions out = rate of transitions in

rate out = rate in

niXj6= i

Pij =Xj6=i

njPji

niXj 6=i(Rij + C ij) + ni(Rik + Cik) =

Xj 6=i

nj(Rji+ Cji) + np(Rki+ Cki) RATEEQUATIONS

recombinationlinesionizationlines

Rij = Bij∞R0

ϕij(ν) Jν dν

Rji = Aji+Bji∞R0

ϕij(ν) Jν dν

Transition probabilities

radiative

collisional

absorption

emission

Cij = ne∞R0

σcoll(v)vf(v)dv

Cji = gi/gjeEji/kT Cij

6

Occupation numbers

can prove that if Cij À Rij or Jν Bν (T): ni ni (LTE)

We obtain a system of linear equations for ni:

A ·

⎛⎜⎜⎝n1n2...np

⎞⎟⎟⎠ =X∞Z0

ϕij(ν)

Z4π

Iν(ω)dω

4πdνwhere A contains terms:

µdIν(ω)

dr=−κνIν(ω) + ²νcombine with equation of transfer:

κν =

NXi=1

NXj=i+1

σlineij (ν)

µni − gi

gjnj

¶+

NXi=1

σik(ν)³ni− n∗i e−hν/kT

´+nenpσkk(ν, T )

³1− e−hν/kT

´+neσe

²ν = ...

non-linear system of integro-differential equations

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 sections20...1000 levels per ion – 3-5 ionization stages per species – ∼ 30 species

fast algorithm to calculate radiative transfer required

8

Transition probabilities: collisions

vcollisioncylinder:all particles inthat volumecollide withtarget atom

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 (ncoll v) × cross section σ.

for excitations:

and similarly for de-excitation

Cij = ncoll

∞Z0

σcollij (v)vf(v)dvσcoll from complex quantum mechanical calculations

9

Transition probabilities: collisions

in a hot plasma free electrons dominate: ncoll = ne

vth = < v > = (2kT/m)1/2 is largest for electrons (vth, e ' 43 vth, p)

Cij = ne

∞Z0

σcollij (v)vf(v)dv = ne qij

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

Cij =gj

gie−Eij/kTCji

10

Transition probabilities: collisions

for bound-free transitions:

Cki = Cik negi

g+1

1

2

µh2

2πmkT

¶3/2eEi/kTcollisional recombination (very

inefficient 3-particle process) =

collisional ionization (important at high T)

In LTE: µnjni

¶∗=gjgie−Eij/kT Boltzmann

µn1

n+1

¶∗= neφ1(T) Saha

Cij =

µnjni

¶∗Cji Cki =

µnin+1

¶∗Cik

11

Transition probabilities: collisions

Approximations for Cji

line transition i j

Ωji = collision strengthfor forbidden transitions: Ωji ' 1

for allowed transitions: Ωji = 1.6 × 106 (gi / gj) fij λij Γ(Eij/kT)

Cji = ne8.631× 10−6

T1/2e

Ωjigjs−1 Ωji ∼ σji

max (g’, 0.276 exp(Eij/kT) E1(Eij/kT)

g’ = 0.7 for nl -> nl’ =0.3 for nl n’l’

12

Transition probabilities: collisions

Cik = ne1.55× 1013T1/2e

g0iσphotik (νik)

e−Ei /kT

Ei/kTfor ionizations

= 0.1 for Z=1

= 0.2 for Z=2

= 0.3 for Z=3

photoionization cross section at ionization edge

13

Transition probabilities: radiative processes

Line transitions

absorption i j (i < j)

probability for absorption

integrating over x and dω (dω = 2π dµ)

emission (i > j)Rji = Aji+BjiJji

dWij = BijIxdω

4πϕ(x)dx x=

ν − ν0∆νD

Rij = BijJij Jij(r) =1

2

∞Z−∞

1Z−1I(x,µ, r)ϕ(x)dµdx

14

Transition probabilities: radiative processes

Bound-free

photo-ionization i k

photo-recombination k i

Rik =

∞Zνik

σik(ν) c nphot(ν)dν

σ × v for photons

uν = hν nphot (ν) =4π

cJν energy density

Rik = 4π

∞Zνik

σik(ν)

hνJνdν

Rki =

µnin+1

¶∗Rki

Rki = 4π

∞Zνik

σikhν(2hν3

c2+ Jν) e

−hν/kT dν

15

LTE vs NLTE in hot stars

Kudritzki 1978

16

LTE vs NLTE in hot stars

Kudritzki 1978

NLTE and LTE emperaturestratifications for two different Helium abundances at Teff = 45,000 K, log g = 5

difference between NLTE and LTE Hγequivalent width as a function of log g for Teff = 45,000 K

17

LTE vs NLTE: departure coefficients - hydrogen

bi =nNLTEi

nLTEi

β Orionis (B8 Ia)

Teff = 12,000 K

log g = 1.75 (Przybilla 2003)

18

Rom

e 200

5N I

N II

Nitrogen atomic modelsPrzybilla, Butler, Kudritzki2003

19

LTE vs NLTE: departure coefficients - nitrogen

20

LTE vs NLTE: line fits – nitrogen lines

21

log ni/niLTE

Przybilla, Butler, Kudritzki,Becker, A&A, 2005

FeII

22

LTE vs NLTE: line fits – hydrogen lines in IR

bi =nNLTEi

nLTEi

Brackett lines

AWAP 05/19/05 23

complex atomic models for O-stars (Pauldrach et al., 2001)

AWAP 05/19/05 24

NLTENLTE AtomicAtomic ModelsModels in modern model atmosphere codesin modern model atmosphere codeslines, collisions, ionization, recombination

Essential for occupation numbers, line blocking, line forceAccurate atomic models have been includedAccurate atomic models have been included

26 elements149 ionization stages5,000 levels ( + 100,000 )20,000diel. rec. transitions4 106 b-b line transitions

Auger-ionizationrecently improved models are based onrecently improved models are based on SuperstructureSuperstructure

Eisner et al., 1974, CPC 8,270

AWAP 05/19/05 25

AWAP 05/19/05 26

consistent treatment of expanding atmospheres along withspectrum synthesis techniquesspectrum synthesis techniques allow the determination of

stellar parameters, wind parameters, andstellar parameters, wind parameters, and abundancesabundancesPauldrach, 2003, Reviews in Modern Astronomy, Vol. 16