10/13/2003

7. Non-LTE – basic concepts

LTE vs NLTE

occupation numbers

rate equation

transition probabilities: collisional and radiative

examples: hot stars, A supergiants

Sprin

g 20

12 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(-h1

/kT)3. Boltzmann: ni

/ n1

= gi

/ g1

exp(-h1i

/kT)

However:

volume elements not closed systems, interactions by photons

LTE non-valid if absorption of photons disrupts equilibrium

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

Sprin

g 20

12LTE vs

NLTE

NLTE if

rate of photon absorptions >> rate of electron collisions

I

(T) »

T,

> 1 »

ne

T1/2

LTE

valid:

low temperatures & high densities

non-valid:

high temperatures & low densities

Sprin

g 20

12LTE vs

NLTE in hot stars

Kudritzki 1978

Sprin

g 20

12NLTE1. 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

niXj 6=i

Pij =Xj 6=i

njPji

rate out = rate in

rate equations

Pi,j

transition probabilities

i

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

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

Xj 6=i

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

lines ionization lines recombination

Transition probabilities

radiative

collisional

Rij = Bij∞R0

ϕij(ν) Jν dν

Rji = Aji+Bji∞R0

ϕij(ν) Jν dν

absorption

emission

Cij = ne∞R0

σcoll(v)vf(v)dv

Cji = gi/gjeEji/kT Cij

RATE EQUATIONS

niXj 6=i

Pij =Xj6=i

njPji

Occupation numbers

can prove that if Cij À Rij or J

B

(T): ni ni (LTE)

We obtain a system of linear equations for ni :

A ·

⎛⎜⎜⎝n1n2...np

⎞⎟⎟⎠ =X

combine with equation of transfer: μdIν(ω)

dr=−κνIν(ω) + ²ν

∞Z0

ϕij(ν)

Z4π

Iν(ω)dω

4πdνWhere matrix A contains

terms:

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

Sprin

g 20

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

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

Transition probabilities: collisions

probability of collision between atom/ion (cross section ) and colliding particles in time dt: ~

v dt

rate of collisions = flux of colliding particles relative to atom/ion (ncoll v) ×

cross section .

for excitations:

and similarly for de-excitation

Cij = ncoll

∞Z0

σcollij (v)vf(v)dv

v

coll from complex quantum mechanical calculations

collisioncylinder:all particles inthat volumecollide withtarget atom

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 the collisional transition probabilities for excitation and de-excitation are related by

Cij =gj

gie−Eij/kTCji

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

Transition probabilities: collisions

Approximations for Cji

line transition i j

ji = collision strengthfor forbidden transitions: ji '

1

for allowed transitions: ji = 1.6 ×

106 (gi / gj ) fij ij (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’

Transition probabilities: collisions

for ionizations Cik = ne1.55× 1013T1/2e

g0iσphotik (νik)

e−Ei /kT

Ei/kT

photoionization cross section at ionization edge= 0.1 for Z=1

= 0.2 for Z=2

= 0.3 for Z=3

Transition probabilities: radiative processes

Line transitions

absorption i j (i < j)

probability for absorption

integrating over x and d d

= 2

d

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

Transition probabilities: radiative processes

Bound-free

photo-ionization i k

photo-recombination k i

Rik =

∞Zνik

σik(ν) c nphot(ν)dν

×

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

LTE vs NLTE in hot stars

Kudritzki 1978

LTE vs NLTE in hot stars

difference between NLTE and LTE H

equivalent width as a function of log g for Teff = 45,000 K

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

Kudritzki 1978

LTE vs NLTE: departure coefficients - hydrogen

bi =nNLTEi

nLTEi

Orionis (B8 Ia)

Teff = 12,000 K

log g = 1.75 (Przybilla 2003)

Rom

e 200

5N I

N II

Nitrogen atomic modelsPrzybilla, Butler, Kudritzki2003

LTE vs NLTE: departure coefficients - nitrogen

LTE vs NLTE: line fits – nitrogen lines

log ni /niLTE

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

FeII

LTE vs NLTE: line fits – hydrogen lines in IR

bi =nNLTEi

nLTEi

Brackett lines

α

Her Davies, Kudritzki, Figer, 2010 MNRAS, 407,1203

J-band spectroscopy of red supergiants

Cosmic abundance probes out to 70 Mpc

distance

Red supergiants, NLTE model atom for TiI

Bergemann, Kudritzki

et al., 2012

Red supergiants, IR lines and connected transitions for TiI

Bergemann, Kudritzki

et al., 2012

TiI

TiII

Bergemann, Kudritzki

et al., 2012

Red supergiants, IR lines departure coefficient for TiI

Bergemann, Kudritzki

et al., 2012

Red supergiants, IR lines fine structure levels for TiI

Bergemann, Kudritzki

et al., 2012

Red supergiants, IR lines for TiI: NLTE vs. LTEBergemann, Kudritzki

et al., 2012

LTE

NLTE

Red supergiants, IR lines NLTE abundance corrections for TiI

D(NLTE-LTE) = logN(Ti)/N(H)NLTE – logN(Ti)/N(H)LTE

Bergemann, Kudritzki

et al., 2012

AWAP 05/19/05

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

AWAP 05/19/05

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