Stellar Atmospheres: Emission and Absorption 1 Emission and Absorption

Stellar Atmospheres: Emission and Absorption

1

Emission and Absorption


2

Chemical composition

Stellar atmosphere = mixture, composed of many chemical elements, present as atoms, ions, or molecules

Abundances, e.g., given as mass fractions k

• Solar abundances

001.0

009.0

001.0

004.0

28.0

71.0

Fe

O

N

C

He

H

Universal abundance for Population I stars


3

Chemical composition

• Population II stars

• Chemically peculiar stars,

e.g., helium stars

• Chemically peculiar stars,

e.g., PG1159 stars

0.1 0.00001

H H

He He

Z Z

¤

¤

¤

0.002

0.964

0.029

0.003

0.002

H H

He He

C C

N N

O O

¤

¤

¤

¤

¤

0.05

0.25

0.55

0.02

0.15

H H

He He

C C

N

O O

¤

¤

¤

¤


4

Other definitions

• Particle number density Nk = number of atoms/ions of element k per unit volume

relation to mass density:

with Ak = mean mass of element k in atomic mass units (AMU)

mH = mass of hydrogen atom

• Particle number fraction

• logarithmic• Number of atoms per 106 Si atoms (meteorites)

kHkk NmA

k

k

N

N

00.12)/log( Hkk NN


5

The model atom

The population numbers (=occupation numbers)

ni = number density of atoms/ions of an element, which are in the level i

Ei = energy levels, quantized

E1 = E(ground state) = 0

Eion = ionization energy

bound states, „levels“

free states

ionization limit

1

65432

En

erg

y

Eion


6

Transitions in atoms/ions

1. bound-bound transitions = lines

2. bound-free transitions = ionization and

recombination processes

3. free-free transitions = Bremsstrahlung

We look for a relation between macroscopic quantities and microscopic (quantum mechanical) quantities, which describe the state transitions within an atom

En

erg

ie

Eion

Photon absorption cross-sections

1 2

3

)(),(


7

Line transitions:Bound-free transitions: thermal average of electron velocities v(Maxwell distribution, i.e., electrons in thermodynamic equilibrium)

Free-free transition: free electron in Coulomb field of an ion, Bremsstrahlung, classically: jump into other hyperbolic orbit, arbitrary

For all processes holds: can only be supplied or removed by:– Inelastic collisions with other particles (mostly electrons), collisional

processes– By absorption/emission of a photon, radiative processes– In addition: scattering processes = (in)elastic collisions of photons with

electrons or atoms- scattering off free electrons: Thomson or Compton scattering- scattering off bound electrons: Rayleigh scattering

Photon absorption cross-sections

ffEE

2e

bf th ion low

unbound state ion free electron 1/ 2 m v

E E E E

lowupbb EEE

+


8

The line absorption cross-section

Classical description (H.A. Lorentz)

Harmonic oscillator in electromagnetic field

• Damped oscillations (1-dim), eigen-frequency 0

Damping constant • Periodic excitation with frequency by E-field

Equation of motion:

inertia + damping + restoring force = excitation

Usual Ansatz for solution:

tieeExmxmxm 020

tiextx 0)(

tiem

eExi 02

02


9


22

3

2 22 20 0

2 2 2 2 2 2 2 2 2 20 0

22 2 4 2 2 52 2 60 02 2 20

2 2 2

Electrodynamics:

2( )

3

( ) cos ( )sin( ) ( )

2cos co

radiated p

s sin s

ower

in

ep(t) x

c

eEx(t) t t

m

eE(x(t)) t t t t

m N N N

2 2 00

02 20

2 200

2 2 2 2 20

2 20 0

2 2 2 2 2 2 2 2 2 20 0

1

expand ( )

real part Re cos sin( ) ( )

i t

t

ti

i

eEi x(t) e

meE

x(t) em i

eEx(t) e

m

eE(x(t)) t t

i

m


10

The line absorption coss-section

2 2

22 2 2 2 2202 0

222 2 2 20

2 42 0

22 2 2

2

0

3

2

average over one period

power, averaged ove

cos sin 1/ 2,

r one

cos sin 0

1(

2

1

perio

(2

3

d

2(

ep x

t t t t

eEx)

E)

m

c

m

ex

4 20

2

4 22

04

22 2 2 2

3

4

22 2 2

3

0

0

2

2

C=normalization constant ( )3

( ) profi

)3

le functi

( ) /

/

n( 2

o )

C

e

e Ep

m c

C

E

m c


11


0

0

0 0 0

2 2 2 2 2 20 0 0 0 0

2 20 0

2 2 2 20 0

0

since - , :

( ) (( )( )) 4 ( )

( )4( ) ( / 2 ) 4 ( ) ( / 4 )

now: calculating the normalization constant

( ) 1

4substitution: : ( )

(

C C

d

x

0

0

2 22 00 2 2 2

0

4)

4 1

=

C dxd C C

x


12

Profile function, Lorentz profile

properties:• Symmetry:

• Asymptotically:

• FWHM:


220

2

)4/()(

4/)(

/4Max

Max2

1

0

)(

FWHM

))(())(( 00

220 1)(1)(

24

2

)4/()2(

4/2FWHM22

FWHM

2


13

The damping constant

• Radiation damping, classically (other damping mechanisms later)

• Damping force (“friction“)

power=force velocity

electrodynamics

• Hence, Ansatz for frictional force is not correct• Help: define such, that the power is correct, when time-

averaged over one period:

classical radiation damping constant

)(txmF 2)()( txmtp

2

3

2

)(3

2)( tx

c

etp

22 4

03

2 (where we used ( ) )

3 i te

m x t x ec

3

20

2

0 3

2

mc

eωω


14

Half-width

Insert into expression for FWHM:2 2

0FWHM 3

24FWHM FWHM

FWHM FWHM2 2

4

2 3

41.18 10 Å

3

e

mc

c e

mc


15

The absorption cross-section

Definition absorption coefficient with nlow = number density of absorbers:

absorption cross-section (definition), dimension: area

Separating off frequency dependence:

Dimension : area frequency

Now: calculate absorption cross-section of classical harmonic oscillator for plane electromagnetic wave:

dsIdI )(low)()( n

)()()( 0

0

)1()(8

),( 20

0

Ec

I

eEE tix


16

Power, averaged over one period, extracted from the radiation field:

On the other hand:

Equating:

Classically: independent of particular transition

Quantum mechanically: correction factor, oscillator strength

4 2 2 2 2 20 0 0

class.2 3 3

4 2 2 2 3 2 20 0 0

2 3 2 2 20

2( ) with

3 3

3( ) ( )

3 2 4 8

e E ep

m c mc

e E mc e Ep

m c e m

20

2 22 00

22

0

( ) ( , ) ( )8

( ) ( )8 8

( ) ( ) 0.026537 cm Hz

cp I d d E

e EcE

m

e

mc

)()( lu

2

lowlu

2

lu fmc

enf

mc

e

index “lu” stands for transition lower→upper level


17

Oscillator strengthsOscillator strengths flu are obtained by:

• Laboratory measurements• Solar spectrum• Quantum mechanical computations (Opacity Project etc.)

• Allowed lines: flu1,

• Forbidden: <<1 e.g. He I 1s2 1S1s2s 3S flu=210-14

/Å Line flu glow gup

1215.7 Ly 0.41 2 8

1025.7 Ly 0.07 2 18

972.5 Ly 0.03 2 32

6562.8 H 0.64 8 18

4861.3 H 0.12 8 32

4340.5 H 0.04 8 50


18

Opacity status report

Connecting the (macroscopic) opacity with (microscopic) atomic physics

View atoms as harmonic oscillator– Eigenfrequency: transition energy– Profile function: reaction of an oscillator to extrenal driving (EM wave)– Classical crossection: radiated power = damping

,

2

low low,up( ) ( )

low up

en f

mc

Population number of lower level

QM correction factorProfile function

Classical crossection


19

Extension to emission coefficient

Alternative formulation by defining Einstein coefficients:

Similar definition for emission processes:

profile function, complete redistribution:

induced 0up ul

spontaneous 0up ul

( )4

( )4

hn B I

hn A

0low lu

20

lu lu

hv( ) n B ( )

4

hv e i.e. B f

4 mc

)( )()(


20

Relations between Einstein coefficients

Derivation in TE; since they are atomic constants, these relations are valid independent of thermodynamic state

In TE, each process is in equilibrium with its inverse, i.e., within one line there is no netto destruction or creation of photons (detailed balance)

0 0 0ul up ul up lu low

ul ul up lu low

low lu up ul up ul

1

ul low lu

ul up ul

TE: ( )4 4 4

( ) ( )

(

emitted intensity absorbed inte

)

nsity

( ) 1

h h hB I n A n B I n I B T

B B T A n B B T n

B T n B n B n A

A n BB T

B n B


21

Relations between Einstein coefficients

0

0

0

1

up upul low lu

ul up ul low low

1

ul low lu

ul up ul

31

02

( ) 1 with Boltzmann equation:

( ) 1 comparison with Planck blackbody radiation:

2( ) 1

h kT

h kT

h kT

n gA n BB T e

B n B n g

A g BB T e

B g B

hB T e

c

3

ul 02

ul

low lulow lu up ul

up ul

2

1

A h

B c

g Bg B g B

g B


22

Relation to oscillator strength

dimension

Interpretation of as lifetime of the excited state

order of magnitude:

at 5000 Å:

lifetime:

2 2

lu lu0

2 2up up

ul lu lulow low 0

3 2 2 2up up0 0

ul ul lu ul lu2 3low low

4

4

2 83

eB f

mchv

g g eB B f

g g mchv

g ghv e vA B f f

c g mc g

ulA 1 time

ul1 A

ulul A18 s10

s10 8


23

Comparison induced/spontaneous emission

When is spontaneous or induced emission stronger?

At wavelengths shorter than spontaneous emission is dominant

**

**

spontaneous 3 2ul * up ul *

induced * * 2 3ul * up ul *

**

*

*

with

( ) 4 211

( ) ( ) 4 ( ) 2

: 1 2 ln 2

e.g. 10000K : 20000A

50000K : 4160A

v v

h kTv

v v

h kT

*

*

I B

A h n A h ce

B T B h n B B T c h

e h kT

T

T


24

Induced emission as negative absorptionRadiation transfer equation:

Useful definition: corrected for induced emission:

spontaneous induced

spontaneous induced

induced0 0lu lu low lu ul up

with

( ), ( )4 4

v

dII

dsdI

Ids

h hB n B n I

spontaneous 0ul up lu low

2low

lu lu low upup

3 2spontaneous 0 lowlu lu up2

up

( ) 4

( )

2 ( )

dI hB n B n I

ds

gef n n

mc g

h gef n

c mc g

transition low→up

So we get (formulated withoscillator strength insteadof Einstein coefficients):


25

The line source function

General source function:

Special case: emission and absorption by one line transition:

• Not dependent on frequency• Only a function of population numbers• In LTE:

vvS

1

up

low

low

up

2

30lu

uplowlow

up

up

2

30

0upullowlu

0upul

lu

lulu

12

-

2

)(4

)(4

n

n

g

g

c

hvS

nng

g

n

c

hv

vhv

nBnB

vhv

nAS

v

vv

),(12

0

1

2

30lu 0 TvBe

c

hvS v

kThvv


26

Every energy level has a finite lifetime against radiative decay (except ground level)

Heisenberg uncertainty principle:

Energy level not infinitely sharp

q.m. profile function = Lorentz profile

Simple case: resonance lines (transitions to ground state)example Ly (transition 21):

example H (32):

Line broadening: Radiation damping

ul

ul1 A

E

l

ll

AAj

juk

uku

11

21 cl 1 2 12A 3 g g f cl cl3 2 8 0.41 0.31

1 2 1cl 12 23 13 cl cl

2 3 3

g g g 2 8 23 f f f 3 0.41 0.64 0.07 1.18

g g g 8 18 18


27

Line broadening: Pressure broadening

Reason: collision of radiating atom with other particles

Phase changes, disturbed oscillation

t0 = time between two collisions

0( ) ~ i tE t e


30

Line broadening: Pressure broadening• Semi-classical theory (Weisskopf, Lindholm), „Impact Theory“

Phase shifts :

find constants Cp by laboratory measurements, or calculate

• Good results for p=2 (H, He II): „Unified Theory“– H Vidal, Cooper, Smith 1973– He II Schöning, Butler 1989

• For p=4 (He I)– Barnard, Cooper, Shamey; Barnard, Cooper, Smith; Beauchamp et al.

ppAnsatz: C r , p 2,3,4,6 , r(t) distance to colliding particle

p= name dominant at

2

3

4

6

linear Stark effect

resonance broadening

quadratic Stark effect

van der Waals broadening

hydrogen-like ions

neutral atoms with each other, H+H

ions

metals + H

Film logg


31

Thermal broadeningThermal motion of atoms (Doppler effect)

Velocities distributed according to Maxwell, i.e.

for one spatial direction x (line-of-sight)

Thermal (most probable) velocity vth:

kTmxx

xew2

A v21~)v(

2 2th

2 2 2th

1/ 24th A

th

v vx x

0

v vth th

0 0 th

x

v 2 12.85 10 A km/s

example: T 6000K, A 56 (iron): v 1.33 km/s

i.e. (v ) , with (v ) v 1 we obtain:

1C v C v x 1 v 1

v

(v )

x

x

x x x

xx

x

kT m T

w C e w d

e d e d C C

w

2 2

thv v

th

1

vxe


32

Line profile

Doppler effect:profile function:

Line profile = Gauss curve– Symmetric about 0

– Maximum:– Half width:– Temperature dependency:

ccth

0

th

0

v ,

v

02 2

th

0

2 20 th

01

th

( )

th

(v ) ( ) , with 1 we obtain:

1( )

ν

x x

ν

Cw e ( )dν

cν

eν

th

1Max

v

Max2

1

0

)(

FWHM

th1 v

ththFWHM 67.12ln2 vvv Tv ~th


33

Examples

At 0=5000Å:

T=6000K, A=56 (Fe): th=0.02Å

T=50000K, A=1 (H): th=0.5Å

Compare with radiation damping: FWHM=1.18 10-4Å

But: decline of Gauss profile in wings is much steeper than for Lorentz profile:

In the line wings the Lorentz profile is dominant

210 43th

2 6rad

Gauss (10 ) : e 10

Lorentz (1000 ) : 1 1000 10


34

Line broadening: Microturbulence

Reason: chaotic motion (turbulent flows) with length scales smaller than photon mean free path

Phenomenological description:

Velocity distribution:

i.e., in analogy to thermal broadening

vmicro is a free parameter, to be determined empirically

Solar photosphere: vmicro =1.3 km/s

2micro

2 vv

microx v

1)v( xew x


35

Joint effect of different broadening mechanisms

Mathematically: convolution

commutative:

multiplication of areas:

Fourier transformation:

y y xx

x profile A + profile B = joint effect

dxxfdxxfdxxff BABA

)()())((

ABBA ffff

dyyxfyfxff BABA )()())((

BABA ffff ~~

2~

i.e.: in Fourier space the convolution is a multiplication


36

Application to profile functions

Convolution of two Gauss profiles (thermal broadening + microturbulence)

Result: Gauss profile with quadratic summation of half-widths; proof by Fourier transformation, multiplication, and back-transformationConvolution of two Lorentz profiles (radiation + collisional damping)

Result: Lorentz profile with sum of half-widths; proof as above

2 2 2 2

2 2 2 2 2C

( ) 1 ( ) 1

G ( ) ( ) ( ) 1 with C

x A x BA B

x CA B

G x A e G x B e

x G x G x C e A B

2 2 2 2

2 2

/ /( ) ( )

/( ) ( ) ( ) with

A B

C A B

A BL x L x

x A x BC

L x L x L x C A Bx C


37

Application to profile functions

Convolving Gauss and Lorentz profile (thermal broadening + damping)

2 2

0

2

2( )

220

0 0

2

1 / 4( )

( ) / 4

depends on , , : ( )́ ´ ´

Transformation: v: : 4 : ´

/1( )

D

D

D

D D D

y D

D

G e L( )

V G L Δ V( ) G L(v )d

( ) Δ a /( πΔ ) y ( ) Δ

aG y e L(y)

y

2

2

2 2 2

2 2

Voigt fu

1

(v )

1Def: ( , v) with ( , v)

(v )

, no analytical representation possible.

(approximate formulae or numerical evaluation)

Norm

nc

a

o

l

ti n

y

D

y

D

a eV dy

a y a

a eV H a H a dy

y a

ization: ( , v) v

H a d


38

Voigt profile, line wings


39

Treatment of very large number of linesExample: bound-bound opacity for 50Å interval in the UV:

Direct computation would require very much frequency points• Opacity Sampling• Opacity Distribution Functions ODF (Kurucz 1979)

MöllerDiploma thesisKiel University 1990


40

Bound-free absorption and emission

Einstein-Milne relations, Milne 1924: Generalization of Einstein relations to continuum processes: photoionization and recombination

Recombination spontaneous + induced

Transition probabilities:

I) number of photoionizations

II) number of recombinations

Photon energy

In TE, detailed balancing: I) = II)

: probability for photoionization in

(v) : spontaneous recapture probability of electron in v, v v

(v) : corresponding induced probability v=electron velocity

P d

F d

G

dtdIGFnn v vv)v()v()v(eup

dtdIPn vv low

v vv21 2ion dhmdvmEhv


41

Einstein-Milne relations

low up e

low up e

13

1low

2up e

3

2

low

up e

low up

(v) (v) (v) with

(v) (v) (v)

(v) 21 1

(v) (v) (v)

(v) 2

(v)

(v) (v)

f

v

v

h kTv

h kTv

n P I dvdt n n F G I h m dvdt I B

n P B n n F G B h m

n P mF hB e

G n n hG c

F h

G c

n P me

n n hG

n n

ion

2

3/ 2uplow

2up e low

3/ 2v 2 2

e e e

2 2rom Saha equation:

(v) : Maxwell distribution: (v) v 4 v v2

E kT

m kT

gn mkTe

n n h g

mn n d n e d

kT


42

Einstein-Milne relations

Einstein-Milne relations, continuum analogs to Aji, Bji, Bij

2ion

3/ 2up 3/ 2

up

low

3

22

low

2up 2

3lo

/ 2up

2e low

e

3/ 2v 2 2

e

w

(v)

2 4 v

8

(v)

4 v2

v(v

2 2

)

hv kT

m kTE kT

v

hv kT

v

n

mn e

kT

P he

G m

he

m

gh mm

n

n

gmkT

m h g

gP m

G h g

en h g


43

Absorption and emission coefficients

absorption coefficient (opacity)

emission coefficient (emissivity)

And again: induced emission as negative absorption

and (using Einstein-Milne relations)

LTE:

vv nhvPnv lowlow)(

mvhIGFnnv vv /)v()v()v()( 2eup

2low up e

*

up up /low e low up

low low

( ) (v) (v) /

(v)1 (v) ... h kT

n P h n n G h m

n nG hn P h n n n e

n P m n

vv

v

kThvv

Bvv

mhFnnv

ehvPnv

)()(

/v)v()v()(

1)(2

eup

low

*

up /up3

low

2( ) ...

ch kTnh

n en

definition. of cross-section


44

Continuum absorption cross-sections

H-like ions: semi-classical Kramers formula

Quantum mechanical calculations yield correction factors

Adding up of bound-free absorptions from all atomic levels: example hydrogen

3

th th th

318 2

th th 2 22 2 2

for ( )

0 else

8 threshold frequency, 7.906 10 cm

3 3 principal quantum number, Z nuclear charge

h n n

Z Zm cen

3

th th bf bf( ) ( , ) , ( , ) Gaunt factorg n g n

max

1bf

totbf )()(

n

nn

n nvv


45

Continuum absorption cross-sections

Optical continuum dominated by Paschen continuum


46

The solar continuum spectrum and the H- ionH- ion has one bound state, ionization energy 0.75 eVAbsorption edge near 17000Å,hence, can potentially contribute to opacity in optical band

H almost exclusively neutral, but in the optical Paschen-continuum, i.e. population of H(n=3) decisive:

Bound-free cross-sections for H- and H0 are of similar order

H- bound-free opacity therefore dominates the visual continuum spectrum of the Sun

0 0

4 7.5eSun: 6000K, log 13.6 Saha equation: 10 , 10H H

H H

n nT n

n n

500106

103

)3(

)1(

)1()3(

1062

18

)1(

)3(

10

8

104.23/eV1.12

1

3

0

0

00

0

0

nn

nn

nn

n

nn

n

eeg

g

nn

nn

H

H

H

H

H

H

kT

H

H


47

The solar continuum spectrum and the H- ion

Ionized metals deliver free electrons to build H

-


49

The solar continuum spectrum and the H- ion


54

Free-free absorption and emission

Assumption (also valid in non-LTE case):

Electron velocity distribution in TE, i.e. Maxwell distribution

Free-free processes always in TE

Similar to bound-free process we get:

generalized Kramers formula, with Gauntfaktor from q.m.• Free-free opacity important at higher energies, because

less and less bound-free processes present• Free-free opacity important at high temperatures

),()(/)()( ffffff TvBvvvS vvv

ff h / kTff e k

2 2 6

ff ff3/ 2 3

( ) ( )n n 1 e

16 Z e( ) g (n, ,T)

hc(2 T

1

m)3 3

1

1/ 2 3/ 2ffff bf bf~ T , but ~ T (Saha), therefore: / T


55

Computation of population numbers

General case, non-LTE:

In LTE, just

In LTE completely given by:• Boltzmann equation (excitation within an ion)• Saha equation (ionization)

( , , )i i vn n T I

( , )i in n T


56

Boltzmann equationDerivation in textbooks

Other formulations:

• Related to ground state (E1=0)

• Related to total number density N of respective ion

( ) / statistical weight

excitation energy i jE E kT ii i

ij j

gn ge

En g

kTEii ieg

g

n

n /

11

1 1

1 1 1

1

1

1

/

/partition function(

1

, with ( ) :)

j

jE kTj

i i i i

jj j

i i

E kTj

n n n nn gnn n n n nn

g e

U T

n n gU g

nT

Ne


57

Divergence of partition function

e.g. hydrogen:

Normalization can be reached only if number of levels is finite.

Very highly excited levels cannot exist because of interaction with neighbouring particles, radius H atom:

At density 1015 atoms/cm3 mean distance about 10-5 cm

r(nmax) = 10-5 cm nmax ~43

Levels are “dissolved“; description by concept of occupation probabilities pi (Mihalas, Hummer, Däppen 1991)

i

2i i i Ion

E / kTi

g 2n g , E E

i.e. g e

i i

i

lim lim

lim

Nni

i

20)( nanr

with w0 hen i i iig pg p i


58

Hummer-Mihalas occupation probabilities


59

Saha equation

Derivation with Boltzmann formula, but upper state is now a 2-particle state (ion plus free electron)

Energy:

Statistical weight:

Insert into Boltzmann formula

Statistical weight of free electron =number of available states in interval [p,p+dp] (Pauli principle):

2ion eE E p 2m p=electron momentum)

)(up pGgg

2ion e low

2up low e

up up ( 2 ) /

low low

( ) /up up 2

low low 0

( ) ( )

( )

E p m E kT

E E kT p m kT

n p g G pe

n g

n ge G p e dp

n g

3

2 2 2 3e e

phase space volume( )( ) 2 2 spins

phase space cell

( ) 4 1 4 ( ) 8x y z

d pG p dp

h

d p dxdydz dp dp dp dV p dp n p dp G p p h n

weight of ion * weight of free electron

Summarize over all final statesBy integration over p


60

Saha equation

Insertion into Boltzmann formula gives:

Saha equation for two levels in adjacent ionization stages

Alternative:

2up low e

2up low

up low

up

( ) /up up 22e3

low low e0

3/ 2( ) /up 2e3

low e 0

3/ 2( ) /upe3

low e

3/ 2(up upe

3low e low

8 with / 2

82

82

4

22

E E kT p mkT

E E kT x

E E kT

E

n ge p e dp x p m kT

n g h n

ge m kT x e dx

g h n

ge m kT

g h n

n gm kTe

n n h g

low ) /E kT

33/216/)(

low

up2/3

low

eup cm K 1007.2 )( lowup Ceg

g

C

TTf

n

nn kTEE


61

Example: hydrogen

Model atom with only one bound state:

5

low I I

up II II

3/ 21.58 10 K/II

I

II I

II

22

(H I ground state) 2

(H II ) 1

1( )

2

pure hydrogen: ,

ionization degree:

(( )

1

Te

e II

e

n n n g

n n n g

n n Te f T

n C

n n N n n

n n x

N N

x N f Tf T x

x

2

) ( )0

( ) ( ) ( )

2 2

( , )

f Tx

N N

f T f T f Tx

N N N

x x T N


62

Hydrogen ionization

Ion

iza

tion

de

gre

e x

Temperature / 1000 K


63

More complex model atoms

j=1,...,J ionization stages

i=1,...,I(j) levels per ionization stage j

Saha equation for ground states of ionization stages j and j+1:

With Boltzmann formula we get occupation number of i-th level:

kTEeg

g

kTm

hnnn /

11j

1j2/3

e

3

e1j1j1

jIon

22

1

kTEE

kTEkTE

eTCnng

gn

eg

gTCnne

g

gn

n

nn

/)(2/31e1j1

11j

ijij

/

11j

1j2/31e1j1

/

1j

ij1j

1j

ijij

ji

jIon

jIon

ji


64

More complex model atoms

Related to total number of particles in ionization stage j+1

Nj/Nj+1

kTEEkTEE eTCnNU

gneTCnN

U

g

g

gn

NU

gn

U

g

N

n

U

g

n

n

N

n

/)(2/31e1j

1j

ijij

/)(2/31e1j

1j

11j

11j

ijij

1j1j

11j11j

1j

11j

1j

11j

1j

11j

11j

1ij

1j

1ij

ji

jIon

ji

jIon

1 1i

)(je/2/3

1e1j

j

1j

j

j/2/3

1e1j

1j

i

/ij

/2/31e

1j

1j

i

/)(2/31e1j

1j

ij

iijj

jIon

jIon

ji

jIon

ji

jIon

TneTCnU

U

N

N

UeTCnU

NegeTCn

U

N

eTCnNU

gnN

kTE

kTEkTEkTE

kTEE


65

Ionization fraction

j j j 1 J-1

J j 1 j 2 J

J Jj J-1 J-1 J-2 J-1 1

j J Jj 1 j 1 J J J J-1 J 2

j j 1 J-1

j j j 1 j 2 JJ

J-1 J-1 J-2 J-1 1

J J J-1 J 2

J-1

e kj k j

e kk

1

1

( )

1 ( )

J

N N N N

N N N N

N N N N N NN N N N

N N N N N N

N N N

N N N N NNN N N N NN N NN N N N N

n TN

Nn T

J-1J

m 1 m


66

Ionization fractions


67

Summary: Emission and Absorption


68

● Line absorption and emission coefficients (bound-bound)32 2

low 0 lowlu lu low up lu lu up2

up up

( ) 2

( ) ( ) ( )

g h ge ef n n f n

mc g c mc g

profile function, e.g., Voigtprofile( ) 2

2 2

1( , v)

(v )

y

D

eV a dy

y a

● Continuum (bound-free)*

up h / kTlow up

low

n(v) n n e

n

*

up h / kTup3

low

n2h( ) n e

c n

● Continuum (free-free), always in LTE

ff -h / kTff e k( ) ( )n n 1- e ff ff

e k( ) ( )n n B ( ,T)

ee n● Scattering (Compton, on free electrons)e en J

Total opacity and emissivity add up all contributions, then source function S ( )


69

Excitation and ionization in LTE

( ) / low upE E kTlow low

up up

n ge

n g

up low

3/ 2( ) /up upe

3low e low

22

E E kTn gm kTe

n n h g

Boltzmann

Saha

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Stellar Atmospheres: Emission and Absorption 1 Emission and Absorption

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