7. non-lte – basic concepts - institute for astronomy · lte vs nlte in hot stars kudritzki 1978...
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7. Non-LTE – basic concepts
LTE vs NLTE
occupation numbers
rate equation
transition probabilities: collisional and radiative
examples: hot stars, A supergiants
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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’
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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
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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
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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
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LTE vs NLTE in hot stars
Kudritzki 1978
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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
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LTE vs NLTE: departure coefficients - hydrogen
bi =nNLTEi
nLTEi
β Orionis (B8 Ia)
Teff = 12,000 K
log g = 1.75 (Przybilla 2003)
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Rom
e 200
5N I
N II
Nitrogen atomic modelsPrzybilla, Butler, Kudritzki2003
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LTE vs NLTE: departure coefficients - nitrogen
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LTE vs NLTE: line fits – nitrogen lines
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log ni/niLTE
Przybilla, Butler, Kudritzki,Becker, A&A, 2005
FeII
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LTE vs NLTE: line fits – hydrogen lines in IR
bi =nNLTEi
nLTEi
Brackett lines
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complex atomic models for O-stars (Pauldrach et al., 2001)
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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
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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