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5. Atomic radiation processes
Einstein coefficients for absorption and emission
oscillator strength
line profiles: damping profile, collisional broadening, Doppler broadening
continuous absorption and scattering
2
3
A. Line transitions 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 à Eu per second:
atomic property ~ no. of incident photons
probability for absorption of photon with
absorption profile
probability for with
probability for transition l à u
Blu: Einstein coefficient for absorption
4
Einstein coefficients
similarly for stimulated emission
Bul: Einstein coefficient for stimulated emission
and for spontaneous emission
Aul: Einstein coefficient for spontaneous emission
5
Einstein coefficients, absorption and emission coefficients
Iν dF
ds
dV = dF ds
absorbed energy in dV per second:
and also (using definition of intensity):
for the spontaneously emitted energy:
Number of absorptions & stimulated emissions in dV per second:
stimulated emission counted as negative absorption
Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening
6
Einstein coefficients are atomic properties à do not depend on thermodynamic state of matter
We can assume TE:
From the Boltzmann formula:
for hν/kT << 1:
for T à ∞ 0
Relations between Einstein coefficients
7
Relations between Einstein coefficients
only one Einstein coefficient needed
Note: Einstein coefficients atomic quantities. That means any relationship that holds in a special thermodynamic situation (such as T very large) must be generally valid.
for T à ∞
8
Oscillator strength
Quantum mechanics
The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation.
Eigenvalue problem using using wave function:
Consider a time-dependent perturbation such as an external electromagnetic field (light wave) E(t) = E0 eiωt.
The potential of the time dependent perturbation on the atom is:
d: dipol operator
with transition probability
9
Oscillator strength
flu: oscillator strength (dimensionless) classical result from electrodynamics
= 0.02654 cm2/s
The result is
Classical electrodynamics
electron quasi-elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion (damped harmonic oscillator):
damping constant resonant
(natural) frequency
ma = damping force + restoring force + EM force
the electron oscillates preferentially at resonance (incoming radiation ν = ν0)
The damping is caused, because the de- and accelerated electron radiates
10
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:
nl: number density of absorbers
oscillator strength flu is quantum mechanical correction to classical result
(effective number of classical oscillators, ≈ 1 for strong resonance lines)
From (neglecting stimulated emission)
integrated over the line profile
σtotcl: classical
cross section (cm2/s)
Z·L,clº dº = nl
¼e2
mec= nl ¾
cltot
'(º)dº =1
¼
°/4¼
(º ¡ º0)2 + (°/4¼)2[Lorentz (damping) line profile]
11
Oscillator strength
absorption cross section; dimension is cm2
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
g: Gaunt factor
Hα: f=0.6407
Hβ: f=0.1193
Hγ: f=0.0447
12
Line profiles
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
13
1. Natural damping profile
finite lifetime of atomic levels à line width
NATURAL LINE BROADENING OR RADIATION DAMPING
line broadening
Lorentzian profile
'(º) =1
¼
¡/4¼
(º ¡ º0)2 + (¡/4¼)2
t = 1 / Aul (¼ 10-8 s in H atom 2 à 1): finite lifetime with respect to spontaneous emission
Δ E t ≥ h/2π uncertainty principle
Δν1/2 = Γ / 2π
Δλ1/2 = Δν1/2λ2/c
e.g. Ly α: Δλ1/2 = 1.2 10-4 A
Hα: Δλ1/2 = 4.6 10-4 A
14
Natural damping profile
resonance 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
excited line
15
2. Collisional broadening
a) impact approximation: radiating atoms are perturbed by passing particles at distance r(t). Duration of collision << lifetime in level à lifetime shortened à 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 à consider stationary distribution of perturbers
r: distance to perturbing atom
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 à line shifts, asymmetries & broadening
ΔE(t) = h Δν = C/rn(t)
16
Collisional broadening
n = 2 linear Stark effect ΔE ~ F for levels with degenerate angular momentum (e.g. HI, HeII)
field strength F ~ 1/r2
à 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) à 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
à ΔE ~ 1/r3 à Γe ~ ne
17
Collisional broadening
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
n = 4 quadratic Stark effect ΔE ~ F2 field strength F ~ 1/r2
à ΔE ~ 1/r4 (no dipole moment)
important for metal ions broadened by e- in hot stars à Lorentz profile with Γe ~ ne
à Γe ~ ne
18
Quasi-static approximation
tperturbation >> τ = 1/Aul
à 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 à field at location of absorbing or emitting atom
statistical frequency of particle distribution
à probability of fields of different strength (each producing an energy shift ΔE = h Δν)
à field strength distribution function
à line broadening
Linear Stark effect of H lines can be approximated to 0st order in this way
19
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
For calculating W(F)dF we use as a first step the nearest-neighbor approximation: main effect from nearest particle
20
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)
relative probability for particle in shell (r,r+dr) N: particle density probability for no particle in (0,r)
Integral equation for W(r) differentiating à differential equation
Normalized solution
Differential equation
21
Quasi-static approximation
mean interparticle
distance:
normal field strength:
define:
from W(r) dr à W(β) dβ:
Stark broadened line profile in the wings, not Δλ-2 as for natural or impact broadening
note: at high particle density à large F0
à stronger broadening ¯ =
F
F0= {r0
r}2
Linear Stark effect:
22
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)
number of particles inside Debye sphere
W (¯) =2¯
¼
Z 1
0
e¡y3/2
y sin(¯y)dy Holtsmark (1919),
Chandrasekhar (1943, Phys. Rev. 15, 1)
W (¯) =2¯±4/3
¼
Z 1
0
e¡±g(y)y sin(±2/3¯y)dy
g(y) =2
3y3/2
Z 1
y
(1¡ z¡1sinz)z¡5/2dz
± =4¼
3D3N
D = 4.8T
ne
1/2
cm
number of particles inside Debye sphere
Debye length, field of ion vanishes beyond D
Ecker (1957, Zeitschrift f. Physik, 148, 593 & 149,245)
Mihalas, 78!
23
3. Doppler broadening
radiating atoms have thermal velocity
Maxwellian distribution:
Doppler effect: atom with velocity v emitting at frequency , observed at frequency :
24
Doppler broadening
Define the velocity component along the line of sight:
The Maxwellian distribution for this component is:
thermal velocity
if v/c <<1 à line center
if we observe at v, an atom with velocity component ξ absorbs at ν′ in its frame
25
Doppler broadening
line profile for v = 0 à profile for v ≠ 0
New line profile: convolution
profile function in rest frame
velocity distribution function
26
Doppler broadening: sharp line approximation
: Doppler width of the line
thermal velocity
Approximation 1: assume a sharp line – half width of profile function <<
delta function
27
Doppler broadening: sharp line approximation
Gaussian profile – valid in the line core
28
Doppler broadening: Voigt function
Approximation 2: assume a Lorentzian profile – half width of profile function >
Voigt function (Lorentzian * Gaussian): calculated numerically
'(º) =1
¼
¡/4¼
(º ¡ º0)2 + (¡/4¼)2
29
Voigt function: core Gaussian, wings Lorentzian
normalization:
usually α <<1
max at v=0:
H(α,v=0) ≈ 1-α
~ Gaussian ~ Lorentzian
Unsoeld, 68!
30
Doppler broadening: Voigt function
Approximate representation of Voigt function:
line core: Doppler broadening
line wings: damping profile 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
31
General case: two broadening mechanisms
two broadening functions representing
two broadening mechanisms
Resulting broadening function is convolution of the two individual broadening functions
32
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
33
5. Rotational broadening
If the star rotates, some surface elements move towards the observer and some away
to observer
Rotational velocity at equator: vrot = Ω•R
Observer sees vrot sini
redshift blueshift
stripes of constant wavelength shift
Intensity shifts in wavelength along stripe
Gray, 1992
blueshift redshift
��
�
= �v
rot
sin(i)
c
x
R
��
�
= +v
rot
sin(i)
c
x
R
34
stellar spectroscopy uses mostly normalized spectra à Fλ / Fcontinuum
Rotation and observed line profile
integral over stellar disk towards observer, also integral over all solid angles for one surface element
observed stellar line profile
Spri
ng 2
013
35
M33 A supergiant Keck (ESI)
U, Urbaneja, Kudritzki, 2009, ApJ 740, 1120
Fλ /Fcontinuum
36
stellar spectroscopy uses mostly normalized spectra à Fλ / Fcontinuum
Rotation changes integral over stellar surface
integral over stellar disk towards observer
observed stellar line profile
Correct treatment by numerical integral over stellar surface with intensity calculated by model atmosphere
Gray, 1992
��
�
= �v
rot
sin(i)
c
x
R
��
�
= +v
rot
sin(i)
c
x
R
37
Approximation: assume intrinsic Pλ const. over surface
independent of cosθ
integral of Doppler shifted profile over stellar disk and weighted by continuum intensity towards obs.
continuum limb darkening
P (�) = I(�, cos✓)/Ic(�, cos✓)
��
�
= +v
rot
sin(i)
c
x
R
��
�
= �v
rot
sin(i)
c
x
R
��
�
=v
rot
sin(i)
c
x
R
38
Prot
(�) =F (�)
Fc
P
rot
(�) =
R 1�1 P [����(x)]A(x)dx
R 1�1 A(x)dx
G[��(x] =2⇡
p1� x
2 + �2 (1� x
2)
1 + 23�
P
rot
(�) =
Z 1
�1P [����(x)]G[��(x)]dx
Z pa
2 � x
2dx =
1
2(xpa
2 � x
2 + a
2arcsin
x
a
)
using
and normalization à
rotational line broadening
profile function
A(x) =
Z p1�x
2
0(1 + �
p1� x
2 + y
2)dy
39
Rotational line broadening
Rotationally broadened line profile
convolution of original profile with rotational broadening function
G[��(x] =2⇡
p1� x
2 + �2 (1� x
2)
1 + 23� �� = �
v
rot
sin(i)
c
x
R
40
Rotational broadening profile function
G(x)
Unsoeld, 1968
41
Rotationally broadened line profiles
v sini km/s
Note: rotation does not change equivalent width!!!
Gray, 1992
42
observed stellar line profiles
v sini large
v sini small
Gray, 1992
43
Rotationally broadened line profiles
v sini km/s
Note: rotation does not change equivalent width!!!
Gray, 1992
44
observed stellar line profiles
θ Car, B0V vsini~250 km/s
Schoenberner, Kudritzki et al. 1988
45
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
46
Rotation and macro-turbulence
Gray, 1992
5 Å
rotation
rotation + macro-turb.
47
B. Continuous transitions 1. Bound-free and free-free processes
hνlk
consider photon hν ≥ hνlk (energy > threshold): extra-energy to free electron
e.g. Hydrogen
R = Rydberg constant = 1.0968 105 cm-1
absorption
hν + El à Eu
emission spontaneous Eu à El + hν(isotropic) stimulated hν + Eu à El +2hν(non-isotropic)
photo
ioniz
atio
n -
recom
binat
ion!
bound-bound: spectral lines
bound-free
free-free
κνcont
48
b-f and f-f processes
l à continuum Wavelength (A) Edge
1 à continuum 912 Lyman
2 à continuum 3646 Balmer
3 à continuum 8204 Paschen
4 à continuum 14584 Brackett
5 à continuum 22790 Pfund
Hydrogen
49
b-f and f-f processes
for hydrogenic ions
Gaunt factor ≈ 1
for H: σ0 = 7.9 10-18 n cm2
νl = 3.29 1015 / n2 Hz
- for a single transition
- for all transitions:
Kramers 1923
absorption per particle à
Gray, 92!
50
b-f and f-f processes
for non-H-like atoms no ν-3 dependence
peaks at resonant frequencies
free-free absorption much smaller
peaks increase with n:
late A
late B
Gray, 92!
51
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:
Vega
Balmer
Paschen
Brackett
52
b-f and f-f processes : Einstein-Milne relations
Generalize Einstein relations to bound-free processes relating photoionizations and radiative recombinations
line transitions
b-f transitions stimulated b-f emission
53
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.
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
in general: κsc = ne σe
54
Thomson scattering
THOMSON SCATTERING: important source of opacity in hot OB stars
independent of frequency, isotropic
Approximations:
- angle averaging done, in reality: σe à σe (1+cos2 θ) for single scattering
- neglected velocity distribution and Doppler shift (frequency-dependency)
55
Simple example: hot star -pure hydrogen atmosphere total opacity
Total opacity
line absorption
bound-free
free-free
Thomson scattering
56
Total emissivity
line emission
bound-free
free-free
Thomson scattering
Simple example: hot star -pure hydrogen atmosphere total emissivity
57
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:
important in cool G-K stars
for strong lines (e.g. Lyman series when H is neutral) the λ-4 decrease in the far wings can be important in the optical
58
The H- ion - important in cool stars
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)
ionization potential = 0.754 eV
à 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!
59
Additional absorbers
Hydrogen molecules in cool stars
H2 molecules more numerous than atomic H in M stars
H2+ absorption in UV
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,…
60
Examples of continuous absorption coefficients
Teff = 5040 K B0: Teff = 28,000 K
Unsoeld, 68!
61
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)
Conc
epci
on 2
007
62
complex atomic models for O-stars (Pauldrach et al., 2001)
AWAP 05/19/05 63
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 ) 20,000 diel. rec. transitions 4 106 b-b line transitions Auger-ionization
recently improved models are based on Superstructure Eisner et al., 1974, CPC 8,270
Conc
epci
on 2
007
64
65
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
66
Confrontation with Reality
Photoionization Electron Collision!
Nahar 2003, ASP Conf. Proc.Ser. 288, in press Williams 1999, Rep. Prog. Phys., 62, 1431 !
ü high-precision atomic data ü
67
Improved Modelling for Astrophysics
Przybilla, Butler & Kudritzki 2001b, A&A, 379, 936!
e.g. photoionization cross-sections for carbon model atom !
68
Bergemann, Kudritzki et al., 2012
Red supergiants, NLTE model atom for FeI !
69
Red supergiants, NLTE model atom for TiI !
Bergemann, Kudritzki et al., 2012
70
TiI
TiII
Bergemann, Kudritzki et al., 2012
USM
201
1
71
TiO in red supergiants!
MARCS model atmospheres, Gustafsson et al., 2009!
USM
201
1
72
A small change in !carbon abundances… !!MARCS model!atmospheres!
TiO in red supergiants!
73
The Scutum RSG clusters
CO molecule in red supergiants
Davies, Origilia, Kudritzki et al., 2009
74
Rayner, Cushing, Vacca, 2009: molecules in Brown Dwarfs
75
Exploring the substellar temperature regime down to ~550K Burningham et al. (2009)
T9.0 ~ 550K T8.5 ~ 700K
Jupiter