Emission and Absorption - mps.mpg.de · classical radiation damping constant ... Profile function...

1

Stellar Atmospheres: Emission and Absorption

1

Emission and Absorption


2

Chemical composition

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

Abundances, e.g., given as mass fractions βk• Solar abundances

M

M

001.0

009.0001.0004.028.071.0

=

=====

Fe

O

N

C

He

H

β

βββ

ββ

Universal abundance for Population I stars

2


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

β ββ ββ β

=

=

= L

0.0020.964

0.029

0.003

0.002

H H

He He

C C

N N

O O

β ββ ββ ββ ββ β

≤ >

= >>

= ≈

= <

0.05

0.25

0.550.02

0.15

H H

He He

C C

N

O O

β ββ ββ βββ β

≤ >

= >><

= >>


4

Other definitions

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

∑ ′kk

NN

00.12)/log( += Hkk NNε

3


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, quantizedE1 = E(ground state) = 0Eion = ionization energy

bound states, „levels“

free states

ionization limit

1

65432

Ene

rgy

Eion


6

Transitions in atoms/ions

1. bound-bound transitions = lines2. bound-free transitions = ionization and

recombination processes3. free-free transitions = Bremsstrahlung

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

Ene

rgie

Eion

Photon absorption cross-sections

1 2

3

)(),( νηνκ ν

4


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

ffE∆E∆

( )2ebf 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 ω0

Damping constant γ• Periodic excitation with frequency ω by E-fieldEquation of motion:

inertia + damping + restoring force = excitationUsual Ansatz for solution:

tieeExmxmxm ωωγ 020 =++ &&&

tiextx ω0)( =

( ) tiem

eExi ωωωγω 0202 =++−

5


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) xc

eEx(t) t tm

eE(x(t)) t t t tm N N N

ω ω γωω ω ω ωω ω ω γ ω ω ω γ

ω ω ω γ ω ω ω γ ωω ω ω ω

=

−= − + − − + − + − − = + +

&&

&&

&&

( )

( )( )

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) em

eEx(t) em i

eEx(t) em

eE(x(t)) t t

i

m

ω

ω

ω

ω ωγ ω

ω ω ωγ

ω ω ωγω ω ω γ

ω ω γωω ωω ω ω γ ω ω ω γ


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

ω ω ω ω

ω ω γ ωω

ω ω γ ω

ωω ω γ ω

( )

( )

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 )

= = /2

=−

− +

+

=

C

e

e Epm c

C

Em c

ωω ω γ

ν ω π

νϕ νν ν γ π ν

ω

ϕ ν

6


11


0

0

0 0 02 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: : ( )

(

+∞

−∞

∆ =

7


13

The damping constant

• Radiation damping, classically (other damping mechanisms later)• Damping force (“friction“)

power=force ⋅velocityelectrodynamics

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

averaged over one period:

classical radiation damping constant

)(txmF &γ=( )2)()( txmtp &γ=

( )232

)(32)( tx

cetp &&=

22 4

03

2 (where we used ( ) )3

= = i tem x t x ec

ωγ ω ω

3

20

2

0 32

mceωω ωγ =⇒≈


14

Half-width

Insert into expression for FWHM:2 2

0FWHM 3

24FWHM FWHM

FWHM FWHM2 2

42 3

4 1.18 10 Å3

emc

c emc

π νγνπ

ν λ πλ νν λ ν

−

∆ = =

∆ ∆= ⇒ ∆ = ∆ = = ⋅

8


15

The absorption cross-section

Definition absorption coefficient κwith nlow = number density of absorbers:

absorption cross-section (definition), dimension: areaSeparating off frequency dependence: Dimension : area ⋅ frequency

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

dsIdI νν νκ )(−=low)()( nνσνκ =

)(νσ)()( 0 νϕσνσ =

0σ

)1()(8

),( 20

0

−′−=′

=

µδννδπ

µνν

ω

EcI

eEE tix


16

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

On the other hand:

Equating:

Classically: independent of particular transitionQuantum 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 epm c mce E mc e Epm c e m

π ν ωϕ ν γ γγ

π ν ϕ ν ϕ νπ ν

20

2 22 00

22

0

( ) ( , ) ( )8

( ) ( )8 8

( ) ( ) 0.026537 cm Hz

cp I d d E

e Ec Em

emc

ν µ

σ ν ν µ ν µ σ νπ

σ ν ϕ νππσ ν ϕ ν σ

′

′ ′= =

=

= ⇒ =

∫ ∫

)()( lu2

lowlu

2

lu νϕπνκπσ fmcenf

mce ==

index “lu” stands for transition lower→upper level

9


17

Oscillator strengthsOscillator strengths flu are obtained by:• Laboratory measurements• Solar spectrum• Quantum mechanical computations (Opacity Project etc.)

• Allowed lines: flu≈1, • Forbidden:

10


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

ν ν

ν

νη ψ νπνη ψ νπ

=

=

0low lu

20

lu lu

hv( ) n B ( )4hv e i.e. B f 4 mc

κ ν = ϕ νπ

π=π

)(νψ )()( νψνϕ =


20

Relations between Einstein coefficients

Derivation in TE; since they are atomic constants, theserelations 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 TB n B

ν ν ν ν

ν ν

ν

ν

ν ν νπ π π

−

+ = =

+ =

− =

= −

=

11


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

3 102

( ) 1 with Boltzmann equation:

( ) 1 comparison with Planck blackbody radiation:

2( ) 1

h kT

h kT

h kT

n gA n BB T eB n B n g

A g BB T eB g B

hB T ec

νν

νν

νν

ν

−

−

−

−

= − =

= −

= −

⇒3

ul 02

ul

low lulow lu up ul

up ul

2

1

A hB cg B g B g Bg B

ν=

⇒ = ⇒ =


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 8 3

eB fmchvg g eB B fg g mchv

g ghv e vA B f fc g mc g

π

π

π γ

=

= =

= = = ulA 1 time

ul1 A

ulul γ≈A18 s10 −

s10 8−

12


23

Comparison induced/spontaneous emission

When is spontaneous or induced emission stronger?

At wavelengths shorter than λ∗ spontaneous emission isdominant

( )***

*

spontaneous 3 2ul * up ul *

induced * * 2 3ul * up ul *

**

*

*

with ( ) 4 21 1

( ) ( ) 4 ( ) 2

: 1 2 ln 2

e.g. 10000K : 20000 A

50000K : 4160 A

v v

h kTv

v v

h kT

*

*

I BA h n A h c e

B T B h n B B T c h

e h kT

T

T

ν

ν

ν

ν ψ ν πη νη ν ψ ν π ν

ν

λ

λ

=

= = = −

= ⇒ = ⇒ =

= =

= =

o

o


24

Induced emission as negative absorptionRadiation transfer equation:

Useful definition: κ corrected for induced emission:

spontaneous induced

spontaneous induced

induced0 0lu lu low lu ul up

with

( ), ( )4 4

= − = +

= + −

= = v

dI IdsdI Ids

h hB n B n I

νν ν ν ν ν

νν ν ν

η κ η η η

η η κ

ν νκ ϕ ν η ϕ νπ π

( )spontaneous 0ul up lu low2

lowlu lu low up

up

3 2spontaneous 0 lowlu lu up2

up

( ) 4

( )

2 ( )

dI hB n B n Ids

ge f n nmc g

h ge f nc mc g

νν ν

νη ϕ νπ

πκ ϕ ν

ν πη ϕ ν

= + −

= −

=

transition low→up

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

13


25

The line source functionGeneral source function:Special case: emission and absorption by one line transition:

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

κηvvS =

( )1

up

low

low

up2

30lu

uplowlow

up

up2

30

0upullowlu

0upul

lu

lulu

12

-

2

)(4

)(4

−

−=

=−

==

nn

gg

chvS

nngg

nchv

vhvnBnB

vhvnAS

v

vv

ϕπ

ϕπ

κη

[ ] ),(12 01230lu 0 TvBe

chvS v

kThvv =−=

−


26

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

Heisenberg uncertainty principle:Energy level not infinitely sharpq.m. ⇒ profile function = Lorentz profile

Simple case: resonance lines (transitions to ground state)example Lyα (transition 2→1):example Hα (3→2):

Line broadening: Radiation damping

∑<

=ul

ul1 Aτ

h=⋅∆ τE

∑∑

14


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 ω


28

Line broadening: Pressure broadening

Reason: collision of radiating atom with other particles⇒Phase changes, disturbed oscillation

Intensity spectrum (=power spectrum) of the cut wave train:

t0 = time between two collisions

0

0

0

21

202/ 2

10/ 2

~ Fourier transform

sin2( ) ~

2−

− = −

∫t

i t i t

t

I

tI e e dtω ω

ω ω

ω ω ω

0( ) ~ i tE t e ω

15


29

Line broadening: Pressure broadeningProbability distribution for t0

Averaging over all t0 gives

Performing integration and normalization gives profile function of intensity spectrum:

i.e. profile function for collisional broadening is a Lorentz profile with

∫∞

−

−

−⋅=

00

/2

00 /22

sinconst)( 0 τωωωωω τ dtetI tv

( )0 /0 0 0( ) average time between two collisionstW t dt e dtτ τ τ−= =

( ) ( )220 11)(

τωωπτωϕ

+−=

12 , ~ = particle density of colliders approximately constant

-N NN γ γ

γ τ τγ

=′ ′= ⋅

(to calculate γ´: calculation of τ necessary; for that: assumption about phase shift needed, e.g., given by semi-classical theory)


30

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

Phase shifts ∆ω:

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∆ω = = =

hydrogen-like ionsneutral atoms with each other, H+Hionsmetals + H

linear Stark effectresonance broadeningquadratic Stark effectvan der Waals broadening

2346

dominant atnamep=

Film logg

16


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 1v

(v )

x

x

x x x

xx

x

kT m T

w C e w d

e d e d C C

w

ππ

∞−

∞ ∞− −

= =

= = =

= ⋅ =

⋅ = ⋅ = ⇒ = ⇒ =

=

∫

∫ ∫2 2

thv v

th

1v

xeπ

−


32

Line profile

Doppler effect:profile function:

Line profile = Gauss curve– Symmetric about ν0– Maximum:– Half width:– Temperature dependency:

ccth

0

th

0

v , v =∆=∆νν

νν

02 2

th

0

2 20 th

01

th

( )

th

(v ) ( ) , with 1 we obtain:

1( )

ν

x xν

Cw e ( )dνcν

eν

ν ν

ν ν ν

νϕ ν ϕ νπ

ϕ νπ

+∞−∆ ∆

−∞

− − ∆

⇒ = =∆

=∆

∫πth

1Maxv∆

=

Max21

0ν ν

)(νϕ

FWHM

πth1 v∆ththFWHM 67.12ln2 vvv ∆=∆=∆

Tv ~th∆

17


33

Examples

At λ0=5000Å:T=6000K, A=56 (Fe): ∆ λth=0.02ÅT=50000K, A=1 (H): ∆ λth=0.5ÅCompare with radiation damping: ∆ λFWHM=1.18 10-4Å

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

In the line wings the Lorentz profile is dominant

210 43th

2 6rad

Gauss (10 ) : e 10 Lorentz (1000 ) : 1 1000 10

− −

−

∆λ ≈≈

∆λ ≈


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 broadeningvmicro is a free parameter, to be determined empirically

Solar photosphere: vmicro =1.3 km/s

2micro

2 vv

microx v

1)v( xew x−=

π

18


35

Joint effect of different broadening mechanisms

Mathematically: convolutioncommutative:multiplication of areas:

Fourier transformation:

y y xx

xprofile A + profile B = joint effect

dxxfdxxfdxxff BABA ∫∫∫∞

∞−

∞

∞−

∞

∞−

⋅=∗ )()())((

ABBA ffff ∗=∗∫∞

∞−

−=∗ dyyxfyfxff BABA )()())((

BABA ffff ⋅∗ =~~

2~

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


36

Application to profile functionsConvolution 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

π π

π

− −

−

= =

= ∗ = = +

2 2 2 2

2 2

/ /( ) ( )

/( ) ( ) ( ) with

A B

C A B

A BL x L xx A x B

CL x L x L x C A Bx C

π π

π

= =+ +

= ∗ = = ++

19


37

Application to profile functionsConvolving 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 ∆ V( ) G L(v )d

( ) ∆ a /( π∆ ) y ( ) ∆

aG y e L(y)y

ν ν ν γ πν νν π ν ν γ π

ν ν γ ν ν ν ν ν

ν ν ν γ ν ν ν ν

ν πν π

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 dya y a

a eV H a H a dyy a

πν π

πν π

ization: ( , v) v∞

−∞

=∫ H a d π


38

Voigt profile, line wings

20


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: photoionizationand recombination

Recombination spontaneous + inducedTransition probabilities:

I) number of photoionizationsII) number of recombinationsPhoton energyIn 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 dG

ν ν ν ν, ++

[ ] dtdIGFnn v vv)v()v()v(eup +dtdIPn vv νlow

v vv21 2ion dhmdvmEhv =→+=

21


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 eG n n hG c

F hG c

n P m en n hG

n n

ν ν ν ν

ν ν

νν

ν

ν

ν

−−

= + =

= +

= − = −

⇒ =

⇒ =

• 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 mkT en n h g

mn n d n e dkT

π

ππ

−

−

=

• =


42

Einstein-Milne relations

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

2ion

3/ 2up 3/ 2

up

low

3

22

low2

up 23

lo

/ 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 ekT

P h eG m

h em

gh m m

nn

gmkT

m h ggP m

G h g

en h g

π

ππ

π

π

− −

=

=

=

=

22


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 σκ lowlow)( ==

[ ] mvhIGFnnv vv /)v()v()v()( 2eup +=η

2low up e

*up up /

low e low uplow low

( ) (v) (v) /

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

n P h n n G h m

n nG hn P h n n n en P m n

ν

νν ν

ν

κ ν ν ν

ν σ −

= −

= − = = −

[ ]

vv

v

kThvv

Bvv

mhFnnvehvPnv

)()(

/v)v()v()(

1)(2

eup

low

κη

ηκ

=

=

−= −

M

*up /

up3low

2( ) ...c

h kTnh n en

νν ν

νη ν σ −

= =

definition. of cross-section σ


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

( )3th th th

318 2

th th 2 22 2 2

for ( )0 else

8 threshold frequency, 7.906 10 cm3 3

principal quantum number, Z nuclear charge

h n nZ Zm ce

n

σ ν ν ν νσ ν

ν σπ

−

>=

= = = ⋅

( )3th th bf bf( ) ( , ) , ( , ) Gaunt factorg n g nσ ν σ ν ν ν ν=

∑=

=max

1bf

totbf )()(

n

nn

n nvv σκ

23


45

Continuum absorption cross-sections

Optical continuum dominatedby 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 orderH- 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+ −− −= = = =

500106103

)3()1(

)1()3(

1062

18)1()3(

10

8

104.23/eV1.12

1

3

0

0

00

0

0

=⋅⋅=

==

==

=

⋅=====

−

−

−−−

−−

nnnn

nnn

nnn

eegg

nnnn

H

H

H

H

H

H

kT

H

H

24


47

The solar continuum spectrum and the H- ion

Ionized metals deliver free electrons to build H-


48


25


49



50

Scattering processes

Thomson scattering at free electronsAbsorption coefficient follows from power of

harmonic oscillator ( Thomson cross-section)

Thomson cross-section is wavelength-independent

eeσκ n=eσ

( ) ( )4 2 4

022 3 22 2 2

0

04 2

20e 02 3

425 2

e 2 4

3 2

free electrons: no resonance frequency, no friction: 0; 0

, on the other hand we had 3 8

8 6.65 10 cm3

e Epm c

e E cp p Em c

em c

νν ν γ π ν

ν γ

σπ

πσ −

= − + = =

→ = =

= = ⋅

26


51

Scattering processes

Rayleigh scattering of photons on electrons bound in atomsor molecules

Rayleigh scattering on Lyα important for stellar spectral typesG and K

( ) ( )4 2 4

022 3 22 2 2

0

04 2 4

20R 02 3 4

4 4 4

R lu e lu2 4 4 4

4

R e lu 4

3 2

semi-classical:

on the other hand we had 3 8

83

( )

lu

lu

lu lu

ll lu

e Epm c

e E cp p Em c

e f fm c

n f

νν ν γ π ν

ν ν νν σν π

π ν νσ σν ν

νκ ν σν

= − +

27


53

Two-photon processes


54

Free-free absorption and emissionAssumption (also valid in non-LTE case):Electron velocity distribution in TE, i.e. Maxwell distribution

Free-free processes always in TESimilar 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 == κη

( )ff h / kTff e k2 2 6

ff ff3/ 2 3

( ) ( )n n 1 e

16 Z e( ) g (n, ,T)hc(2 T

1m)3 3

1

− νκ ν = σ ν −

πν

σ ν = ⋅ ⋅ νπ

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

− −σ σ κ κ

28


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 Iρ=( , )i in n Tρ=


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 g e En g

− −=

kTEii iegg

nn /

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 Tn n g U g

nT

Ne

29


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 cmr(nmax) = 10-5 cm → nmax ~43Levels 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 ei i

i

lim lim

lim −= → = ∞ =

= ∞→ ∞ → ∞

→ ∞

Nni

i =∑

20)( nanr =

with w0 hen → ∞→ →i i iig pg p i


58

Hummer-Mihalas occupation probabilities

30


59

Saha equationDerivation 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 gn g

e G p e dpn g

− + −

∞− − −

=

→ = ∫

3

2 2 2 3e e

phase space volume( )( ) 2 2 spinsphase space cell

( ) 4 1 4 ( ) 8x y z

d pG p dph

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

Ω=

Ω = ⋅ = ⋅ = ⋅ → =

weight of ion * weight of free electron

Summarize over all final statesBy integration over p


60

Saha equationInsertion 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

8 2

8 24

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 ng

e m kT x e dxg h n

ge m kT

g h n

n gm kT en n h g

π

π

π π

π

∞− − −

∞− − −

− −

−

= =

=

=

=

∫

∫

low ) /E kT−

33/216/)(

low

up2/3

low

eup cm K 1007.2 )( lowup −−− ⋅=== Cegg

CTTf

nnn kTEE

31


61

Example: hydrogen

Model atom with only one bound state:

5

low I I

up II II

3/ 21.5810 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 gn n n g

n n T e f Tn C

n n N n nn n xN N

x N f Tf T xx

− ⋅

= = == = =

= =

= = +

= =

⇒ = ⇒ +−

2

) ( ) 0

( ) ( ) ( )2 2

( , )

f TxN N

f T f T f TxN N N

x x T N

− =

⇒ = − + +

⇒ =


62

Hydrogen ionization

Ioni

zatio

n de

gree

x

Temperature / 1000 K

32


63

More complex model atoms

j=1,...,J ionization stagesi=1,...,I(j) levels per ionization stage jSaha equation for ground states of ionization stages j and j+1:

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

kTEegg

kTmhnnn /

11j

1j2/3

e

3

e1j1j1

jIon

221

++

=

π

kTEE

kTEkTE

eTCnngg

n

egg

TCnnegg

nnn

n

/)(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 eTCnNUg

neTCnNUg

gg

n

NUg

nUg

Nn

Ug

nn

Nn

/)(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

TneTCnUU

NN

UeTCnUN

egeTCnUN

eTCnNUg

nN

kTE

kTEkTEkTE

kTEE

Φ==

==

==

−

++

−

+

+−−

+

+

−−+

+

∑

∑∑

33


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 2J-1

e kj k j

e kk

1

1

( )

1 ( )

J

N N N NN N N N

N N N N N NN N N NN N N N N N

N N NN N N N NN

N N N N NN N NN N N N N

n TNN n T

+

+ +

= =

+

+ +

=

=

= ⋅ ⋅ ⋅

= = = + + ⋅ + +

⋅ ⋅ ⋅= =

+ + ⋅ + +

Φ=

+ Φ

∑ ∑

∏

K

K L

K

K L

J-1J

m 1 m=∑∏


66

Ionization fractions

34


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 nmc g c mc g

νπ πκ ν η νϕ ν ϕ ν

profile function, e.g., Voigtprofile( ) =ϕ ν2

2 21( , v)

(v )

∞ −

−∞

=− +∆ ∫

y

D

eV a dyy aν π

● Continuum (bound-free)*

up h / kTlow up

low

n(v) n n e

n− ν

ν

κ = σ −

*up h /kT

up3low

n2h( ) n ec n

− νν ν

νη ν = σ

● Continuum (free-free), always in LTE

( )ff -h / kTff e k( ) ( )n n 1-e νκ ν = σ ν ff ff e k( ) ( )n n B ( ,T)νη ν = κ ν ν

eeσκ n=● Scattering (Compton, on free electrons) e en Jν νη (ν) = σ

Total opacity and emissivity add up all contributions, then source function S ( )ν ν= η /κ ν

35


69

Excitation and ionization in LTE

( ) /− −= low upE E kTlow lowup up

n g en g

up low

3 /2( ) /up upe

3low e low

22 − − =

E E kTn gm kT en n h g

π

Boltzmann

Saha

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