5. Atomic radiation processes
Einstein coefficients for absorption and emission oscillator strength line profiles: damping profile, collisional broadening, Doppler broadening continuous absorption and scattering
1 2 A.A. LineLine transitionstransitions
Einstein coefficients
probability that a photon in frequency interval in the solid angle range is absorbed by an atom in the energy level El with a resulting transition El Eu per second:
abs dω dw (ν, ω,l,u)=Blu Iν(ω) ϕ(ν)dν 4π
atomic property ~ no. of incident probability for probability for photons absorption of with photon with
probability for absorption transition l u profile
Blu : Einstein coefficient for absorption
3 Einstein coefficients
similarly for stimulated emission
st dω dw (ν,ω,l,u)=Bul Iν (ω) ϕ(ν)dν 4π
Bul : Einstein coefficient for stimulated emission
and for spontaneous emission
sp dω dw (ν, ω,l,u)=Aul ϕ(ν)dν 4π
Aul : Einstein coefficient for spontaneous emission
4 st dω abs dω dw (ν,ω,l,u)=Bul Iν (ω) ϕ(ν)dν 4π dw (ν, ω,l,u)=Blu Iν(ω) ϕ(ν)dν 4π
Einstein coefficients, absorption and emission coefficients
Number of absorptions & stimulated emissions dF dV = dF ds Iν in dV per second: abs st ds nl dw dV, nu dw dV
absorbed energy in dV per second:
abs abs st stimulated emission counted dE = nl hν dw dV nu hν dw dV ν − as negative absorption and also (using definition of intensity):
abs L dEν = κν Iν ds dω dν dF Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening hν κL = ϕ(ν)[n B n B ] ν 4π l lu − u ul
L hν for the spontaneously emitted energy: ²ν = nu Aul ϕ(ν) 4π 5 L hν L hν κ = ϕ(ν)[n B n B ] ²ν = nu Aul ϕ(ν) ν 4π l lu − u ul 4π
Relations between Einstein coefficients
Einstein coefficients are atomic properties do not depend on thermodynamic state of matter L ²ν We can assume TE: Sν = L = Bν (T) κν n A n A B (T)= u ul = u ul ν nu nlBlu nuBul nl Blu Bul − − nl
nu gu hν From the Boltzmann formula: = e− kT nl gl
2 nu gu hν 2ν for hν/kT << 1: = 1 ,Bν (T)= kT n g − kT c2 l l µ ¶
2 2ν gu hν Aul kT = 1 2 gu hν c gl − kT Blu 1 Bul µ ¶ − gl − kT ¡ ¢ glBlu = guBul for T ∞ 0 6 2 2ν gu hν Aul kT = 1 2 gu hν glBlu = guBul c gl − kT Blu 1 Bul µ ¶ − gl − kT ¡ ¢ Relations between Einstein coefficients
2 2ν gu hν Aul kT kT = 1 c2 g − kT B hν l µ ¶ lu
3 2 hν Aul gu hν 2 = 1 c Blu gl − kT for T ∞ µ ¶ 3 Note: Einstein coefficients atomic 2 hν gl quantities. That means any Aul = 2 Blu c gu relationship that holds in a special thermodynamic situation (such as T 2h ν3 A = B very large) must be generally valid. ul c2 ul
hν g κL = ϕ(ν) B n l n ν 4π lu l − g u · u ¸ only one Einstein 3 L hν gl 2hν coefficient needed ²ν = ϕ(ν) nu 2 Blu 4π gu c 7 Oscillator strength
Quantum mechanics The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation. Eigenvalue problem using using wave function: p2 Hat om ψl >= El ψl >Hat om = + Vnucleus + Vshell | | 2m Consider a time-dependent perturbation such as an external iωt electromagnetic field (light wave) E(t) = E0 e . The potential of the time dependent perturbation on the atom is: N V (t)=e E ri = E d d: dipol operator · · i=1 X
[Hat om + V (t)] ψl >= El ψl > | | 2 with transition probability < ψl d ψu > 8 ∼ | | | | Oscillator strength
hν πe2 The result is Blu = flu 4π mec
flu: oscillator strength (dimensionless) classical result from electrodynamics = 0.02654 cm2/s
Classical electrodynamics electron quasi-elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion (damped harmonic oscillator):
2 e x¨ + γx˙ + ω0 x = E ma = damping force + restoring force + EM force me the electron oscillates preferentially at resonance damping (incoming radiation ν = ν0) constant resonant The damping is caused, because the de- and 2 2 2 2 2 (natural) accelerated electron radiates 2 ω0 e 8π e ν0 frequency γ = 3 = 3 3 mec 3 me c 9 Classical cross section and oscillator strength
Calculating the power absorbed by the oscillator, the integrated “classical” absorption coefficient and cross section, and the absorption line profile are found:
2 n : number σ cl: classical L,cl πe cl l tot integrated over the line profile density of cross section κν dν = nl = nl σtot m c absorbers (cm2/s) Z e
1 γ/4π ϕ(ν)dν = [Lorentz (damping) line profile] π (ν ν )2 +(γ/4π)2 − 0 oscillator strength flu is quantum mechanical correction to classical result (effective number of classical oscillators, ≈ 1 for strong resonance lines) L hν From κ = ϕ ( ν ) n l B lu (neglecting stimulated emission) ν 4π 10 Oscillator strength
hν πe2 absorption cross section; ϕ(ν) Blu = ϕ(ν) flu = σlu(ν) dimension is cm2 4π mec
1 mec flu = hν Blu 4π πe2
Oscillator strength (f-value) is different for each atomic transition Values are determined empirically in the laboratory or by elaborate numerical atomic physics calculations Semi-analytical calculations possible in simplest cases, e.g. hydrogen
5 3 2 g 1 1 − f = 3/2 5 3 2 2 : Gaunt factor lu 3 π l u l − u g ¡ ¢ Hα: f=0.6407 Hβ: f=0.1193 Hγ: f=0.0447 11 LineLine profilesprofiles
line profiles contain information on physical conditions of the gas and chemical abundances analysis of line profiles requires knowledge of distribution of opacity with frequency several mechanisms lead to line broadening (no infinitely sharp lines)
- natural damping: finite lifetime of atomic levels
- collisional (pressure) broadening: impact vs quasi-static approximation - Doppler broadening: convolution of velocity distribution with atomic profiles
12 1. Natural damping profile
finite lifetime of atomic levels line width
NATURAL LINE BROADENING OR RADIATION DAMPING
-8 t = 1 / Aul (¼ 10 s in H atom 2 1): finite lifetime with respect to spontaneous emission
Δ E t ≥ h/2π uncertainty principle
Δν1/2 = Γ / 2π 2 Δλ1/2 = Δν1/2λ /c line broadening -4 e.g. Ly α: Δλ1/2 = 1.2 10 A -4 1 Γ/4π Hα: Δλ1/2 = 4.6 10 A
ϕ(ν)= 2 2 π (ν ν0) +(Γ/4π) − Lorentzian profile 13
Natural damping profile
resonance line excited line
natural line broadening is important for strong lines (resonance lines) at low densities (no additional broadening mechanisms) e.g. Ly α in interstellar medium but also in stellar atmospheres
14 2. Collisional broadening
radiating atoms are perturbed by the electromagnetic field of neighbour atoms, ions, electrons, molecules
energy levels are temporarily modified through the Stark - effect: perturbation is a function of separation absorber-perturber
energy levels affected line shifts, asymmetries & broadening
ΔE(t) = h Δν = C/rn(t) r: distance to perturbing atom
a) impact approximation: radiating atoms are perturbed by passing particles at distance r(t). Duration of collision << lifetime in level lifetime shortened line broader in all cases a Lorentzian profile is obtained (but with larger total Γ than only natural damping) b) quasi-static approximation: applied when duration of collisions >> life time in level consider stationary distribution of perturbers 15 Collisional broadening n = 2 linear Stark effect ΔE ~ F for levels with degenerate angular momentum (e.g. HI, HeII) field strength F ~ 1/r2 ΔE ~ 1/r2 important for H I lines, in particular in hot stars (high number density of free electrons and ions). However , for ion collisional broadening the quasi-static broadening is also important for strong lines (see below) Γe ~ ne
n = 3 resonance broadening atom A perturbed by atom A’ of same species important in cool stars, e.g. Balmer lines in the Sun
Γ ~ n ΔE ~ 1/r3 e e
16 Collisional broadening
n = 4 quadratic Stark effect ΔE ~ F2 field strength F ~ 1/r2
ΔE ~ 1/r4 (no dipole moment) important for metal ions broadened by e- in hot stars Lorentz profile with Γ ~ n e e
n = 6 van der Waals broadening atom A perturbed by atom B important in cool stars, e.g. Na perturbed by H in the Sun Γe ~ ne
17 Quasi-static approximation
tperturbation >> τ = 1/Aul
perturbation practically constant during emission or absorption atom radiates in a statistically fluctuating field produced by ‘quasi-static’ perturbers, e.g. slow-moving ions
given a distribution of perturbers field at location of absorbing or emitting atom statistical frequency of particle distribution
probability of fields of different strength (each producing an energy shift ΔE = h Δν) field strength distribution function line broadening
Linear Stark effect of H lines can be approximated to 0st order in this way 18
Quasi-static approximation for hydrogen line broadening
Line broadening profile function determined by probability function for electric field caused by all other particles as seen by the radiating atom.
W(F) dF: probability that field seen by radiating atom is F
dF ϕ(∆ν) dν = W (F ) dν dν
For calculating W(F)dF we use as a first step the nearest-neighbor approximation: main effect from nearest particle
19 Quasi-static approximation – nearest neighbor approximation assumption: main effect from nearest particle (F ~ 1/r2) we need to calculate the probability that nearest neighbor is in the distance range (r,r+dr) = probability that none is at distance < r and one is in (r,r+dr) Integral equation for W(r) differentiating differential equation
relative probability for particle in shell probability for no particle in (0,r) (r,r+dr) N: total uniform particle density
Differential equation
Normalized solution
20 Linear Stark effect:
Quasi-static approximation
mean interparticle distance: normal field strength:
F r 2 note: at high particle density large F define: β = = 0 0 F0 { r } stronger broadening
from W(r) dr W(β) dβ:
Stark broadened line profile in the wings, not Δλ-2 as for natural or impact broadening
21 Quasi-static approximation – advanced theory complete treatment of an ensemble of particles: Holtsmark theory + interaction among perturbers (Debye shielding of the potential at distances >
Debye length)
Holtsmark (1919), 2β ∞ y3/2 W (β)= e− ysin(βy)dy Chandrasekhar (1943, Phys. Rev. 15, 1) π Z0 Ecker (1957, Zeitschrift f. Physik, 148, 593 & 149,245)
4/3 2βδ ∞ δg(y) 2/3 W (β)= e− ysin(δ βy)dy π 0 number of particles Z inside Debye sphere 2 3/2 ∞ 1 5/2 g(y)= y (1 z− sinz)z− dz 3 − Zy 1/2 T Debye length, field of ion vanishes D =4.8 cm beyond D ne 4π δ = D3N number of particles inside Debye sphere Mihalas, 78 3 22 3. Doppler broadening
radiating atoms have thermal velocity Maxwellian distribution:
3/2 m m (v2+v2+v2) P (vx,vy ,vz ) dvxdvy dvz = e− 2kT x y z dvxdvy dvz 2πkT ³ ´ Doppler effect: atom with velocity v emitting at frequency , observed at frequency : v cos θ ν0 = ν ν − c
23 Doppler broadening
Define the velocity component along the line of sight: The Maxwellian distribution for this component is:
1 ξ 2 − ξ0 P (ξ) dξ = 1/2 e dξ π ξ0 ¡ ¢ 1/2 ξ0 =(2kT /m) thermal velocity
ξ if we observe at v, an atom with velocity ν0 = ν ν − c component ξ absorbs at ν′ in its frame if v/c <<1 line center
ξ ξ ξ ∆ν = ν0 ν = ν = [(ν ν0)+ν0 ] ν0 − − c − − c ≈− c 24 Doppler broadening
line profile for v = 0 profile for v ≠ 0
R R ξ ϕ (ν ν )= ϕ (ν0 ν )=ϕ(ν ν ν ) − 0 ⇒ − 0 − 0 − 0 c
New line profile: convolution
∞ ξ ϕnew(ν ν )= ϕ(ν ν ν ) P (ξ) dξ − 0 − 0 − 0 c Z −∞
profile function in rest velocity distribution frame function 25 Doppler broadening: sharp line approximation
∞ 2 new 1 ξ0 ξ ξ dξ − ξ 0 ϕ (ν ν0)= 1/2 ϕ(ν ν0 ν0 ) e − π − − c ξ0 ξ0 Z ¡ ¢ −∞
: Doppler width of the line
1/2 ξ0 =(2kT /m) thermal velocity
Approximation 1: assume a sharp line – half width of profile function <<
ϕ(ν ν0 ) δ(ν ν0 ) delta function − ≈ −
26 Doppler broadening: sharp line approximation
∞ 2 new 1 ξ ξ dξ − ξ0 ϕ (ν ν0 )= 1/2 δ(ν ν0 ∆νD ) e − π − − ξ0 ξ0 Z ¡ ¢ −∞
1 δ(ax)= δ(x) a
∞ 2 new 1 ν ν0 ξ ξ dξ − ξ0 ϕ (ν ν0)= 1/2 δ − e − π ∆νD − ξ0 ∆νD ξ0 Z µ ¶ ¡ ¢ −∞
ν ν 2 new 1 − 0 Gaussian profile – − ∆νD ϕ (ν ν0)= 1/2 e valid in the line core − π ∆νD ¡ ¢ 27 Doppler broadening: Voigt function
Approximation 2: assume a Lorentzian profile – half width of profile function >
1 Γ/4π ϕ(ν)= π (ν ν )2 +(Γ/4π)2 − 0 ∞ y 2 new 1 a e− Γ ϕ (ν ν0 )= 1/2 2 dy a = − π ∆νD π ∆ν 2 4π∆νD Z ∆ν y + a −∞ D − ³ ´
1 ν ν ϕnew(ν ν )= H a, − 0 − 0 π1/2 ∆ν ∆ν D µ D ¶
Voigt function (Lorentzian * Gaussian): calculated numerically 28
Voigt function: core Gaussian, wings Lorentzian
∞ normalization: H(a, v) dv = √π Z −∞
usually α <<1
max at v=0: H(α,v=0) ≈ 1-α
~ Gaussian ~ Lorentzian
Unsoeld, 68 29 Doppler broadening: Voigt function
Approximate representation of Voigt function:
ν ν 2 ν ν − 0 H a, − 0 e− ∆νD line core: Doppler broadening ∆νD ≈ ³ ´ ¡ ¢ a line wings: damping profile ν ν 2 π1/2 − 0 ≈ ∆νD only visible for strong lines ¡ ¢
General case: for any intrinsic profile function (Lorentz, or Holtsmark, etc.) – the observed profile is obtained from numerical convolution with the different broadening functions and finally with Doppler broadening
30 General case: two broadening mechanisms
two broadening functions representing two broadening mechanisms
Resulting broadening function is convolution of the two individual broadening functions
31 4. Microturbulence broadening
In addition to thermal velocity: Macroscopic turbulent motion of stellar atmosphere gas within optically thin volume elements. This is approximated by an additional Maxwellian velocity distribution.
Additional Gaussian broadening function in absorption coefficient
32 If the star rotates, some blueshift redshift surface elements move towards the observer and 5. Rotational broadening some away
blueshift redshift
Gray, 1992
to observer
Rotational velocity at stripes of constant wavelength shift equator: vrot = Ω•R
Observer sees vrot sini
Intensity shifts in wavelength along stripe
33 Rotation and observed line profile
continuum stellar spectroscopy uses mostly normalized spectra Fλ / F
integral over stellar disk towards observer, also integral over all solid angles for one surface element observed stellar line profile
34 U, Urbaneja, M33 Asupergiant ApJ Kudritzki, 2009, Keck (ESI) Spring 2013 740, 1120 740, F
λ /F continuum
35 Rotation changes integral over stellar surface
continuum stellar spectroscopy uses mostly normalized spectra Fλ / F
Gray, 1992
integral over stellar disk towards observer
observed stellar line profile
Correct treatment by numerical integral over stellar surface with intensity calculated by model atmosphere
36 Approximation: assume Pλ const. over surface
integral of Doppler shifted profile over stellar disk and weighted by continuum intensity towards obs.
continuum limb darkening
37 Approximation: assume Pλ const. over surface
Rotational broadening function
Rotationally broadened line profile convolution of original profile with rotational broadening function 38 Rotational broadening
Note: Rotational line broadening is not caused by physical processes affecting the absorption coefficient. It is not the result of radiative transfer. It is caused by the macroscopic motion of the stellar surface elements and the Doppler-effect.
39 Rotational broadening profile function
G(x)
Unsoeld, 1968
40 Note: rotation does not change equivalent width!!!
Rotationally broadened line profiles
v sini km/s
Gray, 1992 41 observed stellar line profiles
v sini large
v sini small
Gray, 1992
42 Note: rotation does not change equivalent width!!!
Rotationally broadened line profiles
v sini km/s
Gray, 1992 43 observed stellar line profiles
θ Car, B0V vsini~250 km/s
Schoenberner, Kudritzki et al. 1988 44
6. Macro-turbulence
The macroscopic motion of optically thick surface volume elements is approximated by a Maxwellian velocity distribution. This broadens the emergent line profiles in addition to rotation
45 Rotation and macro-turbulence
rotation
rotation + macro-turb. 5 Å
Gray, 1992
46 B.B. ContinuousContinuous transitionstransitions
1. Bound-free and free-free processes
absorption
hν + El Eu
emission
spontaneous Eu El + hν (isotropic)
hν stimulated hν + Eu El +2hν lk (non-isotropic)
bound-bound: spectral lines
bound-free cont κν photoionization - recombination free-free
consider photon hν≥hνlk (energy > threshold): extra-energy to free electron e.g. Hydrogen 1 R 5 -1 mv2 = hν hc R = Rydberg constant = 1.0968 10 cm 2 − n2 47
b-f and f-f processes
Hydrogen l continuum Wavelength (A) Edge
1 continuum 912 Lyman
2 continuum 3646 Balmer
3 continuum 8204 Paschen
4 continuum 14584 Brackett
5 continuum 22790 Pfund
48 b-f and f-f processes
b f - for a single transition κν− = nl σlk (ν)
b f - for all transitions: κν− = nl σlk(ν) elements, l ionsX X Gaunt factor ≈ 1
3 νl for hydrogenic ions σ (ν)=σ (n) g (ν) Kramers 1923 lk 0 ν bf ³ ´ Gray, 92 -18 2 for H: σ0 = 7.9 10 n cm
ν = 3.29 1015 / n2 Hz l
absorption per particle 49 b-f and f-f processes Gray, 92
late A
4 b-f 29 Z σ (ν)=2.815 10 gbf (ν) n × n5ν 3
peaks increase with n: 2 hνn = E En =13.6/n eV ∞ −
3 6 = ν − n ⇒ → late B for non-H-like atoms no ν-3 dependence peaks at resonant frequencies free-free absorption much smaller 50 b-f and f-f processes
Hydrogen dominant continuous absorber in B, A & F stars (later stars H-) Energy distribution strongly modulated at the edges:
Balmer Paschen Vega
Brackett
51 b-f and f-f processes : Einstein-Milne relations
Generalize Einstein relations to bound-free processes relating photoionizations and radiative recombinations
hν line transitions σ (ν)= ϕ(ν) B glBlu = guBul lu 4π lu
2h ν3 A = B ul c2 ul
stimulated b-f emission b-f transitions
2 3/2 b f Ion gi h Ei /kT hν /kT κ − = σ (ν) n n n e Ion e− ν ik i − e 1 2g 2πmkT elements, i " 1 µ ¶ # Xions X ni∗ LTE occupation number 52 2. Scattering
In scattering events photons are not destroyed, but redirected and perhaps shifted in frequency. In free-free process photon interacts with electron in the presence of ion’s potential. For scattering there is no influence of ion’s presence.
sc in general: κ = ne σe
Calculation of cross sections for scattering by free electrons: - very high energy (several MeV’s): Klein-Nishina formula (Q.E.D.)
- high energy photons (electrons): Compton (inverse Compton) scattering - low energy (< 12.4 kEV ' 1 Angstrom): Thomson scattering
53 Thomson scattering
THOMSON SCATTERING: important source of opacity in hot OB stars
4 8π 2 8π e 25 2 σe = r0 = 2 4 =6.65 10− cm 3 3 me c ×
independent of frequency, isotropic
Approximations:
2 - angle averaging done, in reality: σe σe (1+cos θ) for single scattering
- neglected velocity distribution and Doppler shift (frequency-dependency)
54 Simple example: hot star -pure hydrogen atmosphere total opacity
Total opacity
N N line gi κν = σij (ν) ni nj line absorption − gj i=1 j=i+1 µ ¶ X X N hν/kT + σik(ν) ni n∗e− bound-free − i i=1 X ³ ´ hν /kT free-free +nenpσkk(ν,T) 1 e− − ³ ´ +ne σe Thomson scattering
55 Simple example: hot star -pure hydrogen atmosphere total emissivity Total emissivity
2hν 3 N N g line i line emission ²ν = 2 σij (ν) nj c gj i=1 j=i+1 X X 3 N 2hν hν/kT + n∗σik(ν)e− bound-free c2 i i=1 X 2hν3 + n n σ (ν,T)e hν/kT free-free c2 e p kk −
Thomson scattering +neσe Jν
56 Rayleigh scattering – important in cool stars
RAYLEIGH SCATTERING: line absorption/emission of atoms and molecules far from resonance frequency: ν << ν0
from classical expression of cross section for oscillators:
ω4 σ(ω)=f σ (ω)=f σ ij kl ij e (ω2 ω2 )2 + ω2γ2 − ij ω4 for ω << ωij σ(ω) fij σe 4 2 2 ≈ ωij + ω γ important in cool G-K stars 4 ω for strong lines (e.g. Lyman for γ << ωij σ(ω) fij σe 4 ≈ ωij series when H is neutral) the λ-4 decrease in the far wings 4 4 σ(ω) ω λ− can be important in the ∼ ∼ optical 57 What are the dominant elements for the continuum opacity? - hot stars (B,A,F) : H, He I, He II - cool stars (G,K): the bound state of the H- ion (1 proton + 2 electrons)
only way to explain solar continuum (Wildt 1939) - The H ion - important in cool stars
ionization potential = 0.754 eV
λion = 16550 Angstroms H- b-f peaks around 8500 A
H- f-f ~ λ3 (IR important)
He- b-f negligible,
He- f-f can be important
in cool stars in IR
requires metals to provide source of electrons dominant source of b-f opacity in the Sun
Gray, 92 58 Additional absorbers
Hydrogen molecules in cool stars
H2 molecules more numerous than atomic H in M stars H + absorption in UV 2 - H2 f-f absorption in IR Helium molecules He- f-f absorption for cool stars
Metal atoms in cool and hot stars (lines and b-f) C,N,O, Si, Al, Mg, Fe, Ti, …. Molecules in cool stars
TiO, CO, H2O, FeH, CH4, NH3,…
59 Examples of continuous absorption coefficients
Unsoeld, 68
Teff = 5040 K B0: Teff = 28,000 K 60
Modern model atmospheres include
● millions of spectral lines (atoms and ions)
● all bf- and ff-transitions of hydrogen helium and metals
● contributions of all important negative ions
● molecular opacities (lines and continua) 61 Concepcion 2007 complex atomicmodelsfor O- stars (Pauldrachetal.,2001) 62 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
4 106 b-b line transitions Auger-ionization
recently improved models are based on Superstructure Eisner et al., 1974, CPC 8,270
AWAP 05/19/05 63 Concepcion 2007 64 Recent Improvements on Atomic Data
• requires solution of Schrödinger equation for N-electron system • efficient technique: R-matrix method in CC approximation
• Opacity Project Seaton et al. 1994, MNRAS, 266, 805
• IRON Project Hummer et al. 1993, A&A, 279, 298
accurate radiative/collisional data to 10% on the mean 65 Confrontation with Reality
Photoionization Electron Collision
Nahar 2003, ASP Conf. Proc.Ser. 288, in press Williams 1999, Rep. Prog. Phys., 62, 1431
high-precision atomic data 66
Improved Modelling for Astrophysics e.g. photoionization cross-sections for carbon model atom
67 Przybilla, Butler & Kudritzki 2001b, A&A, 379, 936 Red supergiants, NLTE model atom for FeI
Bergemann, Kudritzki et al., 2012
68 Red supergiants, NLTE model atom for TiI
Bergemann, Kudritzki et al., 2012
69 TiII
Bergemann, Kudritzki et al., 2012
TiI
70 USM 2011 MARCS modelatmospheres, Gustafsson TiO in redsupergiants
et al.,2009
71 USM 2011 TiO in redsupergiants
atmospheres MARCS model abundances… carbon A smallchangein 72 CO molecule in red supergiants
The Scutum RSG clusters
Davies, Origilia, Kudritzki et al., 2009 73
Rayner, Cushing, Vacca, 2009: molecules in Brown Dwarfs
74 Exploring the substellar temperature regime down to ~550K Burningham et al. (2009)
Jupiter
T9.0 ~ 550K
T8.5 ~ 700K
75