5. Atomic radiation processes

Einstein coefficients for absorption and emission line profiles: damping profile, collisional broadening, Doppler broadening continuous absorption and scattering

1 2 A. Line transitions

Einstein coefficients

probability that a in frequency interval in the solid angle range is absorbed by an in the El with a resulting transition El à Eu per second:

atomic property ~ no. of incident probability for probability for absorption of with photon with

probability for absorption transition l à u profile

Blu: Einstein coefficient for absorption

3

Einstein coefficients

similarly for

Bul: Einstein coefficient for stimulated emission

and for

Aul: Einstein coefficient for spontaneous emission

4 Einstein coefficients, absorption and emission coefficients

Number of absorptions & stimulated emissions dF dV = dF ds Iν in dV per second:

ds absorbed energy in dV per second: stimulated emission counted as negative absorption and also (using definition of intensity):

Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening

for the spontaneously emitted energy: 5

Relations between Einstein coefficients

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 6 Relations between Einstein coefficients

for T à ∞ 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.

only one Einstein coefficient needed 7 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 iωt electromagnetic field (light wave) E(t) = E0 e . The potential of the time dependent perturbation on the atom is:

d: dipol operator

with transition probability 8 Oscillator strength

The result is

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):

ma = damping force + restoring force + EM force the electron oscillates preferentially at resonance (incoming radiation ν = ν ) damping 0 constant resonant The damping is caused, because the de- and accelerated electron radiates (natural) frequency

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 σ cl: classical ¼e nl: number tot L,cl cl density of cross section integrated over the line profile ·º dº = nl = nl ¾tot m c absorbers (cm2/s) Z e

1 °/4¼ '(º)dº = [Lorentz (damping) line profile] ¼ (º º )2 + (°/4¼)2 ¡ 0 oscillator strength flu is quantum mechanical correction to classical result

(effective number of classical oscillators, ≈ 1 for strong resonance lines) From (neglecting stimulated emission)

10 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 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 11 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

12

1. Natural damping profile

finite lifetime of atomic levels à line width

NATURAL LINE BROADENING OR RADIATION DAMPING

-8 t = 1 / Aul (¼ 10 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 2 Δλ1/2 = Δν1/2λ /c line broadening -4 e.g. Ly α: Δλ1/2 = 1.2 10 A -4 1 ¡/4¼ Hα: Δλ1/2 = 4.6 10 A '(º) = ¼ (º º )2 + (¡/4¼)2 ¡ 0 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 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

n ΔE(t) = h Δν = C/r (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 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 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 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 Γ à ΔE ~ 1/r3 à e ~ ne

16 Collisional broadening

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

n = 6 van der Waals broadening atom A perturbed by atom B Γ important in cool stars, e.g. Na perturbed by H in the Sun à e ~ ne

17 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 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

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 à differential equation

relative probability for particle in shell (r,r+dr) probability for no particle in (0,r) N: 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 à large F define: ¯ = = 0 0 F { r } à stronger broadening 0 from W(r) dr à 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), 2¯ 1 y3/2 W(¯) = e¡ y sin(¯y)dy Chandrasekhar (1943, Phys. Rev. 15, 1) ¼ Z0 Ecker (1957, Zeitschrift f. Physik, 148, 593 & 149,245)

4/3 2¯± 1 ±g(y) 2/3 W(¯) = e¡ y sin(± ¯y)dy ¼ 0 number of particles Z inside Debye sphere 2 3/2 1 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 4¼ ± = D3N number of particles inside Debye sphere Mihalas, 78! 3 22 3. Doppler broadening

radiating atoms have thermal velocity Maxwellian distribution:

Doppler effect: atom with velocity v emitting at frequency , observed at frequency :

23 Doppler broadening

Define the velocity component along the line of sight: The Maxwellian distribution for this component is:

thermal velocity

if we observe at v, an atom with velocity component ξ absorbs at ν′ in its frame if v/c <<1 à line center

24 Doppler broadening

line profile for v = 0 à profile for v ≠ 0

New line profile: convolution

profile function in rest velocity distribution frame function 25 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

26 Doppler broadening: sharp line approximation

Gaussian profile – valid in the line core

27 Doppler broadening: Voigt function

Approximation 2: assume a Lorentzian profile – half width of profile function >

1 ¡/4¼ '(º) = ¼ (º º )2 + (¡/4¼)2 ¡ 0

Voigt function (Lorentzian * Gaussian): calculated numerically 28 Voigt function: core Gaussian, wings Lorentzian

normalization:

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:

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

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 surface elements move towards the observer and 5. Rotational broadening some away

blueshift redshift

Gray, 1992

to observer v sin(i) x v sin(i) x = rot =+ rot c R c R 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 à 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 A supergiant

ApJ 740, 1120ApJ 740, 2009, Kudritzki, Keck (ESI) Spring 2013 F λ /F continuum

35 Rotation changes integral over stellar surface

continuum stellar spectroscopy uses mostly normalized spectra à Fλ / F

Gray, 1992

integral over stellar disk towards observer

observed stellar line profile

vrotsin(i) x v sin(i) x = =+ rot c R c R Correct treatment by numerical integral over stellar surface with intensity calculated by model atmosphere

36 P ()=I(,cos✓)/Ic(,cos✓)

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.

v sin(i) x = rot c R

continuum limb vrotsin(i) x darkening = c R v sin(i) x =+ rot c R

37 F () Prot()= Fc

1 P [ (˜x)]A(˜x)dx˜ p1 x˜2 1 2 2 Prot()= 1 A(˜x)= (1 + 1 x˜ +˜y )dy˜ A(˜x)dx˜ 0 R 1 Z p R 1 Prot()= P [ (˜x)]G[(˜x)]dx˜ 1 using Z 1 x a2 x2dx = (x a2 x2 + a2arcsin ) 2 a Z p p and normalization à 2 p 2 2 ⇡ 1 x˜ + 2 (1 x˜ ) G[(˜x]= 2 rotational line broadening 1+ 3 profile function 38 Rotational line broadening

2 p 2 2 ⇡ 1 x˜ + 2 (1 x˜ ) G[(˜x]= 2 vrotsin(i) x 1+ 3 = c R

Rotationally broadened line profile

convolution of original profile with rotational broadening function 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. Continuous transitions

1. Bound-free and free-free processes

absorption

hν + El à Eu

emission

spontaneous Eu à El + hν (isotropic)

! hν stimulated hν + Eu à El +2hν lk (non-isotropic) recombination

-

bound-bound: spectral lines

cont κν photoionization bound-free free-free

consider photon hν ≥ hνlk (energy > threshold): extra-energy to free electron e.g. Hydrogen R = Rydberg constant = 1.0968 105 cm-1 47

b-f and f-f processes

Hydrogen 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

48

b-f and f-f processes

- for a single transition

- for all transitions:

Gaunt factor ≈ 1

for hydrogenic ions Kramers 1923

Gray, 92!

-18 2 for H: σ0 = 7.9 10 n cm 15 2 νl = 3.29 10 / n Hz

absorption per particle à 49

b-f and f-f processes Gray, 92!

late A

peaks increase with 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

line transitions

stimulated b-f emission b-f transitions

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

independent of frequency, isotropic

Approximations:

2 - angle averaging done, in reality: σe à 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

line absorption

bound-free

free-free

Thomson scattering

55 Simple example: hot star -pure hydrogen atmosphere total emissivity

Total emissivity

line emission

bound-free

free-free

Thomson scattering

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:

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 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

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! 58 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,…

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 complex atomic models (Pauldrach for et al.,O-stars 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 ) 20,000 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

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! ü high-precision atomic data ü 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

TiO

model

in atmospheres,

red

supergiants

Gustafsson

et !

al . ,

2009 ! 71 USM 2011

TiO

in

red

supergiants ! atmospheres MARCS ! carbon A

small

abundances

change model ! !

in ! 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 regime down to ~550K Burningham et al. (2009)

Jupiter

T9.0 ~ 550K

T8.5 ~ 700K

75