Stellar atmospheres lecture (2003) S. Dreizler · 1 Stellar Atmospheres: Motivation 1 Stellar Atmospheres: Literature • Dimitri Mihalas – Stellar Atmospheres, W.H. Freeman, San

1

Welcome to the

Lecture on stellar atmospheres

Stefan Dreizler

Universitäts-Sternwarte, University of Göttingen

Germany

Stellar Atmospheres

Outline:

• Introduction

• Radiation field

• Radiation transfer

• Emission and absorption

• Radiative equilibrium

• Hydrostatic equilibrium

• Stellar atmosphere models

1

Stellar Atmospheres: Motivation

1

Stellar Atmospheres: Literature

• Dimitri Mihalas– Stellar Atmospheres, W.H. Freeman, San Francisco

• Albrecht Unsöld– Physik der Sternatmosphären, Springer Verlag (in German)

• Rob Rutten– Lecture Notes Radiative Transfer in Stellar Atmospheres

http://www.fys.ruu.nl/~rutten/node20.html


2

Why physics of stellar atmospheres?

Physics

Stellar atmospheres as laboratories

Plasma-, atomic-, and molecularphysics, hydrodynamics, thermodynamics

Basic research

Technical application

Astronomy

Spectral analysis of stars

Structure and evolution of stars

Galaxy evolution

Evolution of the Universe

2


3

Magnetic fields in white dwarfs and neutron stars

Shift of spectral lines with increasing field strength


4

Hertzsprung Russell Diagram

L~R2 T4eff

100 R ¿¿¿¿

1 R ¿¿¿¿

=700000Km

0.01 R ¿¿¿¿

4 1026 W

5800K

3


5

Massive stars


6

Chemical evolution of the Galaxy

Carretta et al. 2002, AJ 124, 481

4


7

SN movie


8

SN Ia

5


9

SN Ia cosmology

ΩM, ΩΛ

0 , 1

0.5, 0.5

1 , 0

1.5, -.5

0 , 0

1 , 0

2 , 0

Redshift z


10

SN Ia Kosmologie

ΩΩ ΩΩΛΛ ΛΛ

6


11

Uranium-Thorium clock


12

Stellar atmosphere – definition

• From outside visible, observable layers of the star

• Layers from which radiation can escape into space– Dimension

• Not stellar interior (optically thick)

• No nebula, ISM, IGM, etc. (optically thin)

• But: chromospheres, coronae, stellar winds, accretiondisks and planetary atmospheres are closely related topics

7


13

Sonne


14

Fraunhofer lines

8


15

Spectrum - schematically

Inte

nsit

y

Wavelength / nm


16

Spectrum formation

9


17

Formation of absorption lines

Interior outerboundary

observer

continuum

line center

continuum

stellar atmosphere intensity


18

Line formation / stellar spectral types

spectral line temperaturestructure

interior

flux

wavelength

tempera

ture

depth / km

10


19

The spectral types on the main sequence

O B A F G K M

O5

B4

O7

B6

A1

A5

A8

A9

F8

G2

G5

G8

A7

F3


20

Die Spektraltypen der Hauptreihe

O B A F G K M F6

F8

G2

G5

G8

A7

F3

G6

G9

K4

K5

F8

G1

11


21

Classification scheme

M9

L3

L5

L8


22

Classification scheme

T dwarfs

12


23








24

Optical telescopes

Calar Alto (Spain)3.5m telescope

13


25

Optical telescopes

ESO/VLT


26

Why is it important?

UV / EUV observations

flux

flux

wavelength / Å

wavelength / Å

14


27

UV/opticaltelescopes

HST


28

X-ray telescopes

XMM

15


29

Gamma-ray telescopes

INTEGRAL


30

Infrared observatories

ISO

JWST

16


31

Sub-mm telescopes


32

Radio telescopes

100m dish at Effelsberg

17


33








34

PN – NGC6751 - HST

18


35

Planetary nebula spectrum


36

ISM spectrum

19


37

Quasar + IGM spectrum


38







20


39

Eta Carinae - HST


40

Stellar wind spectrum

21


41

Formation of wind spectrum (P Cygni line profiles)


42

Stellar winds – P Cyg profiles

22


43

Accretion disks


44

AM CVn disk spectrum

models

23


45

Temperature structure of an accretion disk

Distance from star [km]

Heig

ht

[km

]


46

Planetary atmospheres

24


47

Quantitative spectral analyses – what can we learn?

Shape of line profile:Temperature Film

Density Film

Abundance Film

Rotation

Turbulence

Magnetic field

Line position:Chemical composition

Velocities

Redshift

Temporal variation:Companion

Surface structure

Spots

Pulsation


48

Zeeman effect

25


49

Magnetic fieldsL

l

optical spectrum

circular polarization

position of line components

spectrum of a white dwarf (PG 1658+440) with fieldstrength of about 5 MG


50

Magnetic fieldsL

White dwarf Grw+70 8247B=300MG

optical spectrum

Circular polarization

positionof line components

26


51

Extrasolar planets


52

Velocity fields

Wavelength / Å

Dis

tance

/ 1

000 k

m

~ 0.01Å

Solar disk

time / min

dis

tance

/ 1

000km

27


53

Non-radial pulsation modes


54

Time resolved spectroscopy

28


55

Time dependent line profiles

Tim

e

Flu

x


56

Doppler tomography

29


57

Summary – stellar atmospheres theory

The atmosphere of a star contains less than one billionth of its total mass, so, why do we care at all?

• The atmosphere of a star is that what we can see, measure, and analyze.• The stellar atmosphere is therefore the source of information in order to put a

star from the color-magnitude diagram (e.g. B-V,mv) of the observer into the HRD (L,Teff) of the theoretician and, hence, to drive the theory of stellar evolution.

• Atmosphere analyses reveal element abundances and show us results of cosmo-chemistry, starting from the earliest moments of the formation of the Universe.

• Hence, working with stellar atmospheres enables a test for big-bang theory.• Stars are the building blocks of galaxies. Our understanding of the most

distant (hence most early emerged) galaxies, which cannot be resolved in single stars, is not possible without knowledge of processes in atmospheres of single stars.

• Work on stellar atmospheres is a big challenge. The atmosphere is that region, where the transition between the thermodynamic equilibrium of the stellar interior into the empty blackness of space occurs. It is a region of extreme non-equilibrium states.


58

Summary – stellar atmospheres theory

Important source of information for many disciplines in astrophysics– research for pure knowledge, contribution to our culture

– ambivalent applications (e.g. nuclear weapons)

Application of diverse disciplines– physics

– numerical methods

Still a very active field of research, many unsolved problems– e.g. dynamical processes

1

Stellar Atmospheres: The Radiation Field

1

The Radiation Field


2

Description of the radiation field

Macroscopic description:

Specific intensity

as function of frequency, direction, location, and time; energy

of radiation field (no polarization)

•in frequency interval (ν,ν+dν)

•in time interval (t,t+dt)

•in solid angle dω around n

•through area element dσ at location r ⊥ n

( , , , )I n r tν νr r

σωννν

dddtd

EdtrnI

:),,,(

4

=rr

2


3

The radiation field

dA dσσσσ

ϕϕϕϕ

ϑϑϑϑ

dωωωω

nr


4

Relation

Energy in frequency interval (ν,ν+∆ν)

Energy in wavelength interval (λ,λ+∆λ)

i.e.

thus

with

ωλϑλ dddtdAIEd cos 4 =

νI→

λI→

|||| λν λν dIdI =

222 νλλν λνλ

νλ Ic

IIc

Ic

dȜdȞ

c ==−=⇒=

I I

energy energyDimension

area time freq. solid angle area time wavelength solid angle

Unit

ν λ

22

erg erg

cm s Hz sterad cm s A sterado

λν II ↔

3


5

Invariance of specific intensity

2

2

2

Irradiated energy:

( , ) cos

as seen from dA subtends solid angle

cos /

cos cos , )

now, as seen from

cos cos , )

if no

=

′

′ ′=

′ ′→ =

′

′ ′′ ′=

Ȟ

Ȟ

dE I d dA d

dA d

d dA d

dA dAdE I ( dȞ

d

dA dA

dA dAdE I ( dȞ

d

ν ν ϑ ν ϑ ω

ω

ω ϑ

ϑ ϑν ϑ

ϑ ϑν ϑ

sources or sinks along d:

′ ′= ⇔ =dE dE I Iν ν

The specific intensity is distance independent if no

sources or sinks are present. G


6

Irradiance of two area elements

dA dA´

dϑϑϑϑ ϑϑϑϑ´

4


7

Specific Intensity

Specific intensity can only be measured from extended

objects, e.g. Sun, nebulae, planets

Detector measures energy per time and frequency interval

2

cos

e.g. is the detector area

~ (1 ) is the seeing disk

dE I d A

A

d

ν ϑ ω

ω

=

′′


8

Special symmetries

• Time dependence unimportant for most problems

• In most cases the stellar atmosphere can be described in

plane-parallel geometry

Sun: atmosphere 200 km 1

1radius 700000 km 3500

: cos ( , , )ȞI I zνµ ϑ ν µ

= = <<

= =

z

ϑϑϑϑ

5


9

• For extended objects, e.g. giant stars (expanding

atmospheres) spherical symmetry can be assumed

spherical coordinates: Cartesian coordinates:

( , , ) ( , , )I r I p zν νν µ ν

r=R outer boundary

p r

z ϑϑϑϑ

ϑϑϑϑ


10

Integrals over angle, moments of intensity

• The 0-th moment, mean intensity

• In case of plane-parallel or spherical geometry

4

2 /2

0 / 2

2 1

0 1

1( ) with spherical coordinates

4

1sin with : cos

41

4

J I n d

I d d

I d d

ν ν

π

π π

νπ

π

ν

ωπ

ϑ ϑ φ µ ϑπ

µ φπ

−

−

=

= =

=

∫

∫ ∫

∫ ∫

r

1

1

2

1

2

energy erg

area time frequency cm s Hz

J I dν ν µ−

=

//

∫

6


11

Jν is related to the energy density uν

4

radiated energy through area element during time :

hence, the energy contained in volume element per frequency interval is

4

Ȟ

dA dt

dE I d dt d dA

l c dt dV l dA c dt dA

dV

u dV d d t dAI d dνπν ω π

ν ω

ν ν

=

= ⇒ = =

= =∫4

3

430 0

energy erg

volume frequency cm Hz

total radiation energy in volume element:

energy erg

volume cm

/

c

c

dȞ

u J

u u d J d

dJ V c

πν ν

π

ν

ν νν ν∞ ∞

=

= =∫ ∫

l

dV

dA

dωωωω


12

The 1st moment: radiation flux

4

,

, , ,

( )

propagation vector in

spherical coordinates:

sin cos

sin sin

cos

( , )sin cos sin

in plane-parallel or spherical geometry:

0, 2 ( )

x

x y z

F I n n d

n

F I d d

F F F F I d

ν νπ

ν

ν ν ν ν ν

ω

ϑ φ

ϑ φ

ϑ

ϑ φ ϑ φ ϑ ϑ φ

π µ µ

=

=

⇒ =

= = = =

∫

∫∫

r r r

r

1

1

2

energy erg

area time frequency cm s Hz

µ−

//

∫

ϕϕϕϕ

ϑϑϑϑ

x

y

z

7


13

Meaning of flux:

Radiation flux = netto energy going through area ⊥ z-axis

Decomposition into two half-spaces:

Special case: isotropic radiation field:

Other definitions:

1

1

1 0

0 1

1 1

0 0

2 ( )

2 ( ) 2 ( )

2 ( ) 2 ( )

netto outwards - inwards

F I d

I d I d

I d I d

F F

π µ µ µ

π µ µ µ π µ µ µ

π µ µ µ π µ µ µ

−

−

+ −

=

= +

= − −

= −

=

∫∫ ∫∫ ∫

0=F

F astrophysical flux

Eddington flux

F 4

H

F H

ν

ν

ν ν νπ π= =


14

Idea behind definition of Eddington flux

In 1-dimensional geometry the n-th moments of intensity are

1

1

1

1

12

1

1n

1

10-th moment: ( )

21

1st moment: ( ) 21

2nd moment: ( ) 21

n-th moment: ( ) 2

J I d

H I d

K I d

I d

ν

ν

ν

µ µ

µ µ µ

µ µ µ

µ µ µ

−

−

−

−

=

=

=

=

∫

∫

∫

∫

8


15

Idea behind definition of astrophysical flux

Intensity averaged over stellar disk = astrophysical flux

2 2 2

2

2

sin

(1 )

2 2

p R

p R

dpp R

d

pdp R d

ϑ

µ

µµ

µ µ

=

= −

= −

= −

R

p=Rsinϑϑϑϑ

ϑϑϑϑnr


16

Idea behind definition of astrophysical flux

Intensity averaged over stellar disk = astrophysical flux

2 0

1 2

2 0

1( )2

1

( )2

/ F

F 0 no inward flux at stellar surface

F

R

I I p pdpR

I R dR

F

I

ν ν

ν

ν ν

ν

ν ν

ππ

µ π µ µπ

π+ +

−

=

=

= =

=

=

∫

∫

dp

p

9


17

Flux at location of observer

4

2 22 2 * *

2 2

( )

Flux at distant observer's detector normal to the line of sight:

/ Fv v

F I n n d

R Rf I d I R d F

d d

ν νπ

ν ν ν

ω

ω π π∗

=

= = = =

∫r r r

νI

R∗d

νf

dω

G


18

Total energy radiated away by the star, luminosity

Integral over frequency at outer boundary:

Multiplied by stellar surface area yields the luminosity

0 0F F d F dν νν ν

∞ ∞+= =∫ ∫

2 2 2 2 2* * *4 4 F 16

energy erg

time s

L R F R R Hπ π π= = =

G

10


19

The photon gas pressure

Photon momentum:

Force:

Pressure:

Isotropic radiation field:

cEp /νν =

ϑνν cos1

dt

dE

cdt

dpF == ⊥

νν

π

ν

ν

νν

πµµ

πωϑν

νωϑ

ϑ

Kc

dIc

dIc

P

ddIc

dAdt

dE

cdA

FdP

42 cos

1)(

cos1

cos1

1

1

2

4

2

2

===

=

==

∫∫−

I ( ) I J

I4 4 1P( ) u I P( ) u J 3K

c 3 c 3

ν ν ν

νν ν ν ν ν

µ = =

π πν = = ⇒ ν = =

G


20

Special case: black body radiation (Hohlraumstrahlung)

Radiation field in

Thermodynamic

Equilibrium with matter

of temperature T13

2

1

3

12

5

5

( , , , ) ( )

( , ) bzw. ( , )

in cavity: 0

2( , ) exp 1

2( , ) exp 1

2( , ) exp 1

2( , )

−

−

−

=

= =

= = =

= −

= −

= −

=

r r

r

I n r t I

I B T I B T

F J I B

h hB T

c kT

hc hcB T

kT

hc hcB T

kT

hB T

c

ν ν

ν ν λ λ

ν ν ν

ν

ν

λ

λ

ν ν

ν ν

ν νν

λλ λ

λλ λ

νν

1

3exp 1

−

−

h

kT

ν

11


21

Hohlraumstrahlung


22

Asymptotic behaviour

• In the „red“ Rayleigh-

Jeans domain

• In the „blue“ Wien domain

4

2

2

2 ),(

2 ),(

1exp 1

λλ

νν

ννν

ν

ν

ckTTB

c

TkTB

kT

h

kT

h

kT

h

=

=

+≈

<<

−=

−=

≈−

>>

kT

hchcTB

kT

h

c

hTB

kT

h

kT

h

kT

h

λλλ

ννν

ννν

ν

ν

exp2

),(

exp2

),(

exp1exp 1

3

2

3

12


23

Wien´s law

( )

( )

max max

max

13

2

x

x

x xmax

xmax

maxmax

2( , ) exp 1

3 1 e

e 1

0 3 e / e 1 0

3 1 e 0

numerical solution: 2.821

−

−

= −

− = + −

= → − − =

→ − − =

= =

d d h hB T

d d c kT

xB

dB x

d

x

hx

kT

ν

ν

ν

ν νν

ν ν

ν ν

ν

νλ

( )max

max

xmax

max max

max

0.5100cm deg

0 5 1 e 0

numerical solution: 4.965 0.2897 cm deg

−

=

= → − − =

= = =

T

dB x

d

hcx T

kT

λλ

λλ

x:=hv/kT


24

Integration over frequencies

Stefan-Boltzmann law

Energy density of blackbody radiation:

13

20 0

4 3 4 44 4

2 3 2 30

4

5 44 5 -2 -1 -4

2 3

2( ) ( ) exp 1

2 2

1 15

15

2 with 5.669 10 erg cm s deg

15

x

h hB T B T d d

c kT

k x kT dx T

c h e c h

kT ı

c h

ν

ν νν ν

π

π

σ π

π

−∞ ∞

∞

−

= = −

= =−

=

= = = ⋅

∫ ∫

∫

4

0

4)(

4)(

4T

cTB

cdJ

cu

σπνν

πν === ∫

∞

13


25

Stars as black bodies – effective temperature

Surface as „open“ cavity (... physically nonsense)

Attention: definition dependent on stellar radius!

( )

4

2 2 2 4* *

1/ 4 1/ 4 1/ 2eff *

, 0

for 0

0 for 0

with F and F ( ) T

luminosity: 4 F 4

hence, eff. temperature: 4

+ −

− − −

= =

>=

≤

= = =

= =

=

I B I

BI

B B T

L R R T

T L R

ν ν ν

νν

ν ν

µ

µ

σ

π

π σπ

σπ

F

G


26

Stars as black bodies – effective temperature

14


27

Examples and applications

• Solar constant, effective temperature of the Sun

• Sun‘s center

2 -1 -2

0

213 10

2*

10 -1 -2

4eff eff

( ) 1.36 kW/m 1.36 erg s cm

F with 1.5 10 cm 6.69 10 cm

F 2.01 10 erg s cm flux at solar surface

F 5780 K

f d f

df d R

R

T T

ν ν ν

π

π

σ

∞

= = =

= = ⋅ = ⋅

= ⋅

= ⇒ =

∫

7

max

max

1.4 10 K

Planck maximum at 3.4Å ( )

or 2.1Å ( )

with 1Å 12.4 keV maximum 4 keV

cT

B

B

ν

λ

λ

λ

= ⋅

⇒ =

=

≈

F


28

Examples and applications

• Main sequence star, spectral type O

• Interstellar dust

• 3K background radiation

** eff

4 2*6eff* *

*

eff

max max

10 , 60000 K

10

882Å ( ) or 501Å ( )

R R T

TL RL L

L T R

B Bν λλ λ

= =

= ⇒ =

= =

)( mm3.0 ,K 20 max νλ BT ==

)( mm9.1 ,K 7.2 max νλ BT ==

15


29

Radiation temperature

... is the temperature, at which the corresponding blackbody

would have equal intensity

Comfortable quantity with Kelvin as unit

Often used in radio astronomy

1

3rad

1

rad3

12

ln1exp2

)(

−−

+=⇒

−

=

νν

λλλλ

I

hc

k

hcT

ȜkT

hchcI


30

The Radiation Field

- Summary -

16


31

Summary: Definition of specific intensity

dA dσσσσ

ϕϕϕϕ

ϑϑϑϑ

dωωωω

nr

4d EI ( ,n, r, t) :

d dt d dν ν =ν ω σ

r r


32

Summary: Moments of radiation field

In 1-dim geometry (plane-parallel or spherically symmetric):

1

1

1

1

12

1

10-th moment: ( )

21

1st moment: ( ) 21

2nd moment: ( ) 2

−

−

−

=

=

=

∫

∫

∫

J I d

H I d

K I d

ν

ν

ν

µ µ

µ µ µ

µ µ µ

Mean intensity

Eddington flux

K-integral

F =astrophysical flux =Eddington flux flux F 4 = = =H F F Hν ν ν ν ν νπ π

4

0 0energy density

∞ ∞

= =∫ ∫cu u d J dπ

ν νν ν

0 0total flux at stellar surface

∞ ∞+= =∫ ∫F F d F dν νν ν

2 2 2 2 2* * *stellar luminosity 4 4 F 16= = =L R F R R Hπ π π

17


33

Summary: Moments of radiation field

4pressure of photon gas P( ) K

cν

πν =

13

2

2blackbody radiation ( , ) exp 1

−

= −

h hB T

c kTν

ν νν

44energy density of blackbody radiation u T

c

σ=

4

0

Stefan-Boltzmann law ( ) ( )∞

= =∫B T B T d Tν

σν

π

2 2 2 2 2 4* * * effeffective temperature 4 F 4 4= = =L R R B R Tπ π σπ

maxWien´s law constant=Tλ

1

Stellar Atmospheres: Radiation Transfer

1

Radiation Transfer


2

Interaction radiation – matter

Energy can be removed from, or delivered to, the radiation field

Classification by physical processes:

True absorption: photon is destroyed, energy is transferred

into kinetic energy of gas; photon is

thermalized

True emission: photon is generated, extracts kinetic energy

from the gas

Scattering: photon interacts with scatterer

→ direction changed, energy slightly

changed

→ no energy exchange with gas

2


3

Examples: true absorption and emission

• photoionization (bound-free) excess energy is transferred

into kinetic energy of the released electron → effect on

local temperature

• photoexcitation (bound-bound) followed by electron

collisional de-excitation; excitation energy is transferred to

the electron → effect on local temperature

• photoexcitation (bound-bound) followed by collisional

ionization

• reverse processes are examples for true emission


4

Examples: scattering processes

• 2-level atom absorbs photon

with frequency ν1, re-emits photon

with frequency ν2; frequencies not

exactly equal, because

– levels a and b have non-vanishing energy width

– Doppler effect because atom moves

• Scattering of photons by free electrons: Compton- or

Thomson scattering, (anelastic or elastic) collision of a

photon with a free electron

ν1 ν2

b

a

3


5

Fluorescence

Neither scattering nor true absorption process

c-b: collisional de-excitation

b-a: radiative

c

b

a


6

Change of intensity along path element

generally:

plane-parallel geometry:

spherical geometry:

ds

dIν

with dI dI

dt µdsds dt

ν νµ= − = −

r

rds

d

d

d

ds

d

ds

rd)d(

ds

drdsdr

/)1(

)sin(1

sin

sinsin

cos

2µ

ϑϑϑ

ϑ

µµ

ϑϑϑϑ

µϑ

−=

−−==⇒

−=≈+

=⇒=

r

I

r

I

ds

dI

ds

dI

ds

dr

r

I

ds

dI

21

µ

µµ

µ

µ

ννν

ννν

−

∂

∂+

∂

∂=⇒

∂

∂+

∂

∂=

4


7

Plane-parallel geometry

outer boundary

geom

etr

icaldepth

t

dt

ds

ϑ

withdI dI

dt µdsds dt

ν νµ= − = −


8

Spherical geometry

dr

ds

ϑϑϑϑ

ϑϑϑϑ+dϑϑϑϑ

-dϑϑϑϑ

r

I

r

I

ds

dI

ds

dI

ds

dr

r

I

ds

dI

21

µ

µµ

µ

µ

ννν

ννν

−

∂

∂+

∂

∂=⇒

∂

∂+

∂

∂=

r

rds

d

d

d

ds

d

ds

rd)d(

ds

drdsdr

/)1(

)1

sin(sin

sinsin

cos

2µ

ϑϑϑ

ϑ

µµ

ϑϑϑϑ

µϑ

−=

−−==⇒

−=≈+

=⇒=

-rdϑϑϑϑ

5


9

Change of intensity along path element

generally:

plane-parallel geometry:

spherical geometry:

ds

dIν

with dI dI

dt µdsds dt

ν νµ= − = −

r

rds

d

d

d

ds

d

ds

rd)d(

ds

drdsdr

/)1(

)sin(1

sin

sinsin

cos

2µ

ϑϑϑ

ϑ

µµ

ϑϑϑϑ

µϑ

−=

−−==⇒

−=≈+

=⇒=

r

I

r

I

ds

dI

ds

dI

ds

dr

r

I

ds

dI

21

µ

µµ

µ

µ

ννν

ννν

−

∂

∂+

∂

∂=⇒

∂

∂+

∂

∂=


10

Right-hand side of transfer equation

• No absorption (vacuum)

• Absorption only, no emission

.const0 =⇒= νν I

ds

dIinvariance of intensity

energy removed from ray:

is proportional to energy content in ray:

and to the path element:

ννν dIIdsI +

σωνν dddtddIdE −=

σωνν dddtdI

ds

F

6


11

Absorption coefficient

• thus:

κ absorption coefficient, opacity

• dimension: 1/length unit: cm-1

• but also often used: mass absorption coefficient, e.g., per

gram matter

κ in general complicated function of physical quantities T, P,

and frequency, direction, time...

• often there is a coordinate system in which κ isotropic, e.g.

co-moving frame in moving atmospheres

• counter-example: magnetic fields (Zeeman effect)

dsIdI νν κ−=

( )tnr ,,, νκκrr

=

( )νκκ ,rr

=

G


12

only absorption, plane-parallel geometry

outer boundary

geom

etr

icaldepth

t

dt ds

ϑ

0

/

optical

( , )( , ) ( , ) ( , ) ( , ) ( , )

with : ( , ) ( , ) with 0 at 0

( , ) 1( , )

( , ) integration constant, fixed by

depth

boun

=

= − ⇒ − = −

′ ′= → = = =

⇒ =

⇒ = ⋅

∫t

t

dI t dIt I t µ t t I t

ds dt

d dt t t dt t

dII

d

I c e c

ν νν ν

νν

τ µν

µκ ν µ µ κ ν µ

τ κ τ ν κ ν τ

µ τµ τ

τ µ

µ τ dary values

7


13

Schuster boundary-value problem

τ=0 outer boundary

µττνν

µτν

µτνν

µν

ττµτµ

ττµµ

τµτµ

τµµ

/)(

max

/

max

/

/0

max

max

),(),(

),(:0

)0,(),(

)0,(:0

−−++

+

−−−

−

==

⋅==>

==

=⋅==<

eII

ecI

eII

cecI

τ=τmax inner boundary

−νI

+νI

G

G


14

Example: homogeneous medium

e.g. glass filter

max

( 0) /

max

/

max

( , ) = thickness of filter

( , 0) ( , )

( , ) ( , 0)

d

d

t t d d

I I e

I I e

κ µν ν

κ µν ν

κ µ κ τ κ τ κ

µ τ µ τ τ

µ τ τ µ τ

+ + − −

−− −

= → = → =

= = = ⋅

= = = ⋅

s

I+

d/µs

I -

d/|µ|

8


15

Half-width thickness

2/1 : 2/1

2/1 =− sesκ

200000Solar atmosphere

6600Glas fiber

330City air

0.066Window glass

0.033River water

S1/2 / meterMaterial


16

Physical interpretation of optical depth

What is the mean penetration depth of photons into medium?

[ ]

(mathematically: expectation value of probability

/

function ( ) )

0

( ) / ( 0)

( ) : probability for absorption in interval ,

(

note nor

)

m

1alizati

p

I I

p d

p

d

e

d

d

ν ν

τ

τ τ

µ

ττ ττ τ

τ τ τ τ τ

τµ− −

−

∞

=

=

= +

= ⋅

∫

/

00 0

/

0 0

f

on: ( ) 1

1 e 1

mean penetration depth

1 1s mean free path

x

x

f

dp d e e

d xe dx

t

τ µ

τ µ

ττ τ

µ

τ τ τ µ µµ

τ µ

τ

µ µ κ κ

∞ ∞∞− −

∞ ∞− −

= = − =

= ⋅ = = ⋅

=

= = =

∫ ∫

∫ ∫

τ

0

1

9


17

The right-hand side of the transfer equation

• transfer equation including emission

emission coefficient ην

• dimension: intensity / length unit: erg cm-3 sterad -1

Energy added to the ray:

is proportional to path element:

ννν dIIdsI +

σωνν dddtddIdE +=

dsdI νν η=

ds


18

• Transfer equation including emission

η in general a complicated function of physical quantities

T,P,..., and frequency

η is not isotropic even in static atmospheres, but is usually

assumed to isotropic (complete redistribution)

if constant with time:

The right-hand side of the transfer equation

( )tnr ,,, νηηrr

=

( )νηη ,rr

=

10


19

The complete transfer equation

Definition of source function:

• Plane-parallel geometry

• Spherical geometry

ννν νκη I

ds

dI)(−=

( )ννν νκ IS

ds

dI−= )(

)(νκ

ηνν =S

( )),,(),,(),(),,(

tItStdt

tdIµνµννκ

µνµ νν

ν −=−

( )),,(),,(),(),,(1),,( 2

rIrSrrI

rr

rIµνµννκ

µ

µνµµνµ νν

νν −=∂

∂−+

∂

∂


20

Solution with given source function:

Formal solution

• Plane-parallel case

linear 1st-order differential equation of form

has the integrating factor

und thus the solution

(proof by insertion)

in our case:

( )ννν

µτSI

d

dI−=

1

)()( xgyxfy =+′

= ∫

x

x

dxxfxM

0

)(exp)(

0

x

0

x

1y(x) g(x)M(x)dx C C=y(x )

M(x)

= +

∫

)()(

)( /1)(

/1)(

νν

νν

ν

τ

τµ

µ

τ

Ixy

Sxg

xf

x

→

−→

−→

→

dI 1 1I S

d

νν ν+ =

τ −µ −µor:

11


21

Formal solution for I+

Reference point x0 : τ=τmax for I+ (µ> 0) outgoing radiation

−−+

′

−′−′=

+′

′−′

−−=

−=

′=

++

++

∫

∫

∫

µ

τττ

µ

τ

µ

ττττ

ττµ

τττ

µµ

τττ

µ

τττ

µτ

ν

τ

τ

νν

ν

τ

τ

νν

τ

τ

maxmax

maxmaxmax

max

exp)(exp)()(

)(exp)(1

-exp)(

exp1

-exp)(

max

max

max

Id

SI

IdSI

dM

weighted mean over

source function

exponentially absorbed ingoing radiation

from inner boundary

I+#

τ´τ τ+∆τ=τ+µ

pin point

e−∆τ/µHence, as rough approximation:

)()( µττ νν +≈+ SI

F

S


22

Formal solution for I -

Reference point x0 : τ=0 for I - (µ< 0) ingoing radiation

−+

′

′−−′=

+′

′′

−=

=

′=

−−

−−

∫

∫

∫

µ

τ

µ

τ

µ

ττττ

τµ

ττ

µµ

ττ

µ

ττ

µτ

ν

τ

νν

ν

τ

νν

τ

exp)0(exp)()(

)0(exp)(1

-exp)(

exp1

-exp)(

0

0

0

Id

SI

IdSI

dM

weighted mean over source

function

exponentially absorbed ingoing radiation

from outer boundary

F

12


23

Emergent intensity

Eddington-Barbier-Relation

max

max

maxmax

0

max

0

(0) ( ) exp ( ) exp

for semi-infinite atmospheres: :

(0) ( ) exp

hence, approximately: (0) ( )

dI S I

dI S

I S

τ

ν ν ν

τ

ν ν

ν ν

ττ ττ τ

µ µ µ

τ

τ ττ

µ µ

τ µ

+ +

+

+

′ ′ ′= − + −

→∞

′ ′ ′= −

≈ =

∫

∫

0 1

0 1 0 1

0 0

Relation is exactly valid if source function is linear in :

i.e. with ) and : / we have:

(0) ( )

∞ ∞+ − −

′= + ⋅ =

= + = + ⋅ =∫ ∫

Ȟ Ȟ Ȟ

x x

Ȟ Ȟ Ȟ Ȟ

S (Ĳ S S Ĳ x

I S e dx S xe dx S S Sν ν

τ

τ µ

µ µ µ


24

The source function

In thermodynamic equilibrium (TE): for any volume element it is:

absorbed energy = emitted energy

per second per second

The local thermodynamic equilibrium (LTE): we assume that

In stellar atmospheres TE is not fulfilled, because– System is open for radiation

– T(r) ≠const (temperature gradient)

νν

ν

νν

νν

κ

η

ηκ

νωσηνωσκ

BS

B

dddsddddsdI

==

=

=

))(,(),( rTBrSrr

νν νν =

Local temperature, unfortunately unknown at the outset

µ

τ

µ

ττ

τ

νν

′

′−′= ∫+ d

TBImax

0

exp))(()0( z.B.

Kirchhoff´s law

13


25

Source function with scatteringExample: thermal absorption + continuum scattering

(Thomson scattering of free electrons)

Inserting into formal solution:

( ) ( )

( ) ( )

0

( ) ( ) ( ) , ; , ,4

, ; , ,

(1 ) with /( )

dB R n n I n d

R n n

B J

B JS J B

I

ν ν ν

ν ν ν

ν νν ν ν

ωκ ν χ ν σ ν η χ σ ν ν ν ν

π

ν ν δ ν ν

η χ σ

χ σρ ρ ρ σ σ χ

χ σ

∞

′ ′ ′ ′ ′= + = +

′ ′ ′=

= +

+= = + − = +

+

∫ ∫r r r

r r

0 0

(0) (1 ) exp exp

( , )4

d dB J

dI

ν ν ν

ν

τ τ τ τρ ρ

µ µ µ µ

ωτ µ

π

∞ ∞+ ′ ′ ′ ′

= − − + −

∫ ∫

∫

true absorption

scattering

redistribution function

isotropic, coherent:

integral equation for Iν


26

The Schwarzschild-Milne equations

Expressions for moments of radiation field obtained by

integration of formal solution over angles µ

0-th moment1

1

1 0

0 1 0

1( ) ( , )

2

1( ) ( )exp ( ) exp

2

(written for semi-infinite atmosphere without irradiation from outside)

exchange integrals ( ,

−

∞

−

=

′ ′ ′ ′ − −′ ′= − + −

∫

∫ ∫ ∫ ∫

J I d

d dJ d S d S

S

ν ν

τ

ν ν ν

τ

τ τ µ µ

τ τ τ τ τ ττ µ τ µ τ

µ µ µ µ

( )( ) ( )( )

( )( ) ( )( )

2 2

1

2 2

0 1

1 1

1 1 independent of ) ,

1( ) ( ) exp ( ) exp

2

1( ) ( ) exp ( ) exp

2

∞ ∞

∞

∞ ∞ ∞

= = → =

′ ′ ′ ′ ′ ′= − − − + − −

′ ′ ′ ′ ′ ′= − − + − −

∫ ∫ ∫ ∫

∫ ∫

m mdw dw

w dd w

dw dwJ d S w w d S w w

w w

dw dwJ d S w d S w

w w

τ

ν ν ν

τ

ν ν ν

τ

τ µ µµ µ µ

τ τ τ τ τ τ τ τ τ

τ τ τ τ τ τ τ τ τ0

∫ ∫τ

14


27

The Schwarzschild-Milne equations

0-th moment

Karl Schwarzschild (1914)

( )( ) ( )( )

( ) ( )

1 0 1

1 1

0

-1

1

1

1( ) ( ) exp ( ) exp

2

1( ) ( ) ( )

2

with (x): t exponential integral of 1st order

( )

xt

dw dwJ d S w d S w

w w

J S E d S E d

E e dt

J

τ

ν ν ν

τ

τ

ν ν ν

τ

ν



τ

∞ ∞ ∞

∞

∞−

′ ′ ′ ′ ′ ′= − − + − −

′ ′ ′ ′ ′ ′= − + −

=

=

∫ ∫ ∫ ∫

∫ ∫

∫

( )1

0

1( )

2S E dν τ τ τ τ

∞

′ ′ ′−∫ G


28

The Lambda operator

Definition

In analogy, we obtain the Milne equations for the

1st moment

2nd moment

[ ] ( )

( )

1

0

1( ) ( )

2

( )

f t f t E t dt

J Sν ν

τ

τ

∞

Λ = −

⇒ = Λ

∫

( ) ( ) ( )2 2

0

1 1 1( ) ( ) ( )

2 2 4H S t E t dt S t E t dt S

τ

ν ν ν ν

τ

τ τ τ∞

= − − − = Φ∫ ∫

( ) ( )3

0

1

1 1( ) ( )

2 4

with ( )xt

n n

K S t E t dt S

eE x dt

t

ν ν ντ τ∞

∞ −

= − = Χ

=

∫

∫

GG

15


29

LTE

Strict LTE

Including scattering

Integral equation for

Solve

( ))()( ττ νν TBJ Λ=

( )( ))()1()(

)()1(

τρρτ

τρρ

ννν

ννν

TBJJ

TBJS

−Λ+Λ=

−+=

)(τνJ

)(41)(

)(41)(

)()()(

ττ

ττ

τττ

νν

νν

ννν

SK

SH

ISJ

Χ=→

Φ=→

→→


30

Excursion: exponential integral function

see Chandrasekhar: Radiative Transfer III.18

• For classical LTE atmosphere models, >50% of

computation time is needed to calculate En(x)

• In non-LTE models, En(x) is needed to calculate electron

collisional rates

• Recursion formula

1

( 1) ( 1)

1 1

( 1)( 1)

1

1

1

integration by parts ( )

1 1with product rule ( ) ( )

1 1

1 10

1 1

1( ) ( ) for 1

1

( ) ( )

n xt

n

n xt n xt

n

nx n xt

x

n n

n

E x t e dt

E x t e t x e dtn n

e x t e dtn n

E x e xE x nn

E x E x

∞− −

∞ ∞− − − − − −

∞− −− − − −

−−

=

= − − − −− −

= + −− −

= − > −

→

∫

∫

∫

G

16


31


• differentiation

( 1)

1 1 1

1

1 1

1

11 1

1

( ) ( ) ( )

( ) ( ) 1

1 ( ) ( ) ( )

( )

∞ ∞ ∞− − − − − − −

−

∞∞ ∞ −− − − − −

−

= = − = −

= − >

= = − = = −

= −

∫ ∫ ∫

∫ ∫

n xt n xt n xt

n

n n

xxt xt xt

x

d dE x t e dt t t e dt t e dt

dx dx

dE x E x n

dx

d d eE x t e dt t t e dt e

dx dx x x

d eE x

dx x


32


• integrals0

1 1

0 00

1 1

1 1

00

1 2

1

repeated integration by parts

1 1

d etc. until ( )

1 1 dx

1 ( 1)( 2) ( 1

s

l

n

ss sl ll

n n n

s sl l x

n n

l l l n

n n

x E (x)dx

x xx E (x)dx E (x) E (x)dx

l l

x x eE (x) E (x)dx E x

l l x

s s xE (s) E (s)

l l l l

+ +

+ + −

−

+ + +

−

′= −+ +

= + = −+ +

= + + ++ + + +

∫

∫ ∫

∫

L 1

1

0

1

0 0

( ))( 2) ( )

1

( 1)( 2) ( )

for

1 ( 1)! ( 1)! ! !

( 1)( 2) ( ) ( 1)( 2) ( ) ( )!

s

l n x

l l n x

n

E sl l n

x e dxl l l n

s

l n l n l lx E (x)dx x e dx

l l l n l l l n l n l n

+ − −

∞ ∞+ − −

+ +

++ + +

→∞

+ − + −= = = =

+ + + + + + + +

∫

∫ ∫

L

L

L L

17


33


• asymptotic behaviour

1

1

1 1

1 0 0

1

(1 ) (1 )

( ) ln (1

∞ ∞− − −

∞− − −

−

= +

= − − + + −

= − − + −

∫ ∫ ∫

∫ ∫ ∫ ∫

u u u

x x

x

u u u

x

du du due e e

u u u

du du du due e e

u u u u

E x x eγ0

1

1

1

0

1 2 3

)

0.5772156 Euler s constant

series expansion for the integral:

( ) ln ( 1)!

1 1 1Values at 0 : (0) 0 (0) 1 (0) 1, (0)

1 1 2

∞−

=

−−

=

= − − + −

= = − ⋅ = > = = − −

∫

∑

L

x

u

nn

n

n n

du

u

xE x x

n n

x E e E n E En n

γ

γ

1 2 2 3

1 1

1

1

1 1 1 2 6: ( ) 1

10 : ( )

∞ ∞− − −−

∞−

→∞ = = + = = − + − +

→ = =

∫ ∫

∫

L Lx xt x

xt

xt

e e ex E x e dt dt

t x x t x x x x

x E x e dtt


34

Example: linear source function

... one can show this

Conclusions:

The mean intensity approaches the local source function

The flux only depends on the gradient of the source function

( )

( ) ( )

[ ]

[ ]

1

0

1 1

0 0

3 2

3 4

S( ) a b

1J( ) S (a b )E d

2

1 1a E d b E d

2 2

1J( ) a b bE ( ) aE ( )

2

1 1H( ) b aE ( ) bE ( )

3 2

∞

∞ ∞

τ = + τ

′ ′ ′τ = Λ = + τ τ − τ τ

′ ′ ′ ′ ′= τ − τ τ + τ τ − τ τ

τ = + τ+ τ − τ

τ = + τ − τ

∫

∫ ∫

νν ττ SbaJxeE x

n =+→→≈>> − 0/:1

3/bH →ν

G….

18


35

Moments of transfer equation

• Plane-parallel geometry

• 0-th moment

• 1st moment

ννν

τµ SI

d

dI−=

∫−

1

12

1µdL

(I)

2

1)(

2

1)(

2

11

1

1

1

1

1

ννν

ννν

τ

µµµµµµτ

SJHd

d

dSdIdId

d

−=

−=

∫∫∫−−−

∫−

1

12

1µµdL

(II)

2

1)(

2

1)(

2

11

1

1

1

1

1

2

νν

ννν

τ

µµµµµµµµτ

HKd

d

dSdIdId

d

=

−=

∫∫∫−−−

G


36

( )

( ) ( ) (I) 1

20

2

2

11

2

1

2

1

2

11

2

1

2

1

2

2

1

1

1

1

2

1

1

1

1

1

1

21

1

ννν

νννν

ννννν

ννν

ν

κ

κ

κκµµµ

µκµκµµ

µµµ

JSHrrr

JSHr

Hr

JSdIr

Ir

Hr

dIdSdI

rdI

r

−=∂

∂

−=++∂

∂

−=−

−

−+

∂

∂

−=∂

∂−+

∂

∂

∫

∫∫∫∫

−−

−−−−


• Spherical geometry

• 0-th moment∫−

1

12

1µdL

( )),,(),,(),(),,(1),,( 2

rIrSrrI

rr

rIµνµννκ

µ

µνµµνµ νν

νν −=∂

∂−+

∂

∂

19


37

( )

(II) 3

31

0

031

2

1

2

1

2

1

2

1

2

1

2

1

1

1

21

1

3

1

1

1

1

1

1

31

1

2

ννν

ν

νννν

νννν

ννν

ν

κ

κ

κµµµµ

µµκµµκµµ

µµµµ

Hr

J

r

KK

r

HKJr

Kr

HdIr

Ir

Kr

dIdSdI

rdI

r

−=−+∂

∂

−=−−+∂

∂

−=−

−

−+

∂

∂

−=∂

∂−+

∂

∂

∫

∫∫∫∫

−−

−−−−


• 1st moment ∫−

1

12

1µµdL


38

Solution of moment equations

Problem: n-th momentum equation contains (n+1)-st moment

→always one more unknowns than differential equations

→ to close the system, another equation has to be found

Closure by introduction of variable Eddington factors

Eddington factor, is found by iteration

ννν JfK ⋅=

νf

new

starting estimate for ( ) ( ), solve

/

f I II K

f K J

ν ν

ν ν ν

→ + →

→ =

20


39

Solution of moment equations

2 differential eqs. for

Start: approximation for , assumption: anisotropy small, i.e.

substitute by (Eddington approximation)

ννν

ννν

τ

τ

Hd

JfdII

SJd

dHI

=

−=

)( )(

)(

νν HJ ,

νf

νI νJ

3

1

3

1

3

1

3

1

2

1

2

1

2

1)(

1

1

3

1

1

2

1

1

2

=→

=→

=

=≈=−−−

∫∫

ν

νν

ννννν µµµµµτ

f

JK

JJdJdIK


40

Eddington approximation

Is exact, if Iν linear in µ

(one can show by Taylor expansion of S in terms of B that this linear

relation is very good at large optical depths)

0 1

1

0

1

11 3

1 1

1 1

11 32

0 0

1 1

I ( ) I I

1J I ( )d I

2

1 1 1H I ( ) d I I

2 2 3 3

1 1 1K I ( ) d I I

2 2 3 3

1K J

3

ν ν ν

ν ν ν

−

ν ν ν ν

− −

ν ν ν ν

− −

ν ν

µ = +µ

= µ µ =

µ= µ µ µ = =

µ= µ µ µ = =

⇒ =

∫

∫

∫

G

21


41

Summary: Radiation Transfer


42

ννν νκη I

ds

dI)(−=Transfer equation

, ( )νη κ νEmission and absorption coefficients

Definitions: source function

optical depth

/ ( )Sν νη κ ν=

d dsτ κ= ⋅

Formal solution of transfer equation

max

maxmax( ) ( ) exp ( ) exp

dI S I

τ

ν ν ν

τ

τ ττ τ ττ τ τ

µ µ µ+ +′ ′ −−

′= − + −

∫

(0) ( )+ ≈ =I Sν ν τ µEddington-Barbier relation

LTE Local Thermodynamic Equilibrium

))(,(),( rTBrSrr

νν νν = ( )T rr

local temperature

Including scattering: (1 ) with /( )= + − = +S J Bν ν νρ ρ ρ σ σ χ

22


43

Schwarzschild-Milne equations

Moment equations of formal solution

( )( ) = ΛJ Sν ντ ( )1

( )4

= ΦH Sν ντ integral operatorsΛ, Φ

Moments of transfer equation (plane-parallel)( )

dH d K

J S Hd d

ν νν ν ν

τ τ= − =

Differential equation system (for J,H,K),

closed by variable Eddington factor : /=f K Jν ν ν


44

Summary: How to calculate I and the moments J,H,K

(with given source function S)?

( )1dI

I Sdτ µ= −Solve transfer equation

(no irradiation from outside, semi-infinite

atmosphere, drop frequency index)

Formal solution: ( ) ( )exp ( , analogous d

I S Iτ

τ τ ττ τ µ

µ µ

∞+ −′ ′ −

′= − > 0)

∫

How to calculate the higher moments? Two possibilities:

1. Insert formal solution into definitions of J,H,K:1

n

1

1

2 −∫ I dµ µ

ĺ ( )( ) = ΛJ Sτ ( )1

( )4

= ΦH Sτ ( )1

( )4

= ΧK Sτ Schwarzschild-Milne equations

2. Angular integration of transfer equation, i.e. 0-th & 1st moment1

n

1

1...

2 −∫ dµ µ

ĺ( )

dH d K

J S Hd dτ τ= − = 2 moment equations for 3 quantities J,H,K

Eliminate K by Eddington factor f: K f J= ⋅

ĺ ( )

dH d f JJ S H

d dτ τ

⋅= − = solve: J,H,K F new f (=K/J) iteration

1

Stellar Atmospheres: Emission and Absorption

1

Emission and Absorption

F


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

009.0

001.0

004.0

28.0

71.0

=

=

=

=

=

=

Fe

O

N

C

He

H

β

β

β

β

β

β

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 β

β β

β β

=

=

= L

¿

¿

¿

0.002

0.964

0.029

0.003

0.002

H H

He He

C C

N N

O O

β β

β β

β β

β β

β β

≤ <<

= >>

= >>

= ≈

= <

¿

¿

¿

¿

¿

0.05

0.25

0.55

0.02

0.15

H H

He He

C C

N

O O

β β

β β

β β

β

β β

≤ <<

= >>

= >>

<

= >>

¿

¿

¿

¿


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=ρβ

∑ ′k

k

N

N

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, quantized

E1 = E(ground state) = 0

Eion = ionization energy

bound states, „levels“

free states

ionization limit

1

65432

Energ

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, whichdescribe the state transitions within an atom

Energ

ie

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∆

( )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 ω0

Damping constant γ

• Periodic excitation with frequency ω by E-field

Equation of motion:

inertia + damping + restoring force = excitation

Usual Ansatz for solution:

tieeExmxmxm

ωωγ 020 =++ &&&

tiextx ω0)( =

( ) tiem

eExi ωωωγω 02

02 =++−

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) x

c

eEx(t) t t

m

eE(x(t)) t t t t

m N N N

ω ω γωω ω ω ω

ω ω ω γ ω ω ω γ

ω ω ω γ ω ω ω γ ωω ω ω ω

=

−= − + −

− + − + − − = + +

&&

&&

&&

( )

( )( )

2 2 00

0

2 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

m

eEx(t) e

m i

eEx(t) e

m

eE(x(t)) t t

i

m

ω

ω

ω

ω ωγ ω

ω ω ωγ

ω ω ωγ

ω ω ω γ

ω ω γωω ω

ω 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(

= = =

− +

= − +

=

=

− +

&

&

&

&&

&e

p x

t t t t

eEx)

E)

m

c

m

ex

ω ω ω ω

ω ω γ ωω

ω ω γ ω

ω

ω 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 Ep

m c

C

E

m c

ω

ω ω γ

ν ω π

νϕ ν

ν ν γ π ν

ω

ϕ ν

G

6


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

ν

ν

ν ν ν ν ν ν ν

ν ν ν ν ν ν ν ν ν

ν νϕ ν

ν ν γ π ν ν γ π

ϕ ν ν

πν ν

γ

ϕ0

0

2 22 00 2 2 2

0

4)

4 1

=

+∞ +∞

−∞ −∞

= = ⇒ =+∫ ∫

C dxd C C

x

ν

ν

ν ππ γν ν ν

γ γ ν π

π


12

Profile function, Lorentz profile

properties:

• Symmetry:

• Asymptotically:

• FWHM:


220

2

)4/()(

4/)(

πγνν

πγνϕ

+−=

γ/4Max =

Max2

1

0ν ν

)(νϕ

FWHM

))(())(( 00 ννϕννϕ −−=−+

220 1)(1)( ννννϕ ∆=−→

π

γ

π

γν

πγν

πγ

γ 24

2

)4/()2(

4/2FWHM22

FWHM

2

==∆⇒+∆

=

7


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 γ such, that the power is correct, when time-averaged over one period:

classical radiation damping constant

)(txmF &γ=

( )2)()( txmtp &γ=

( )2

3

2

)(3

2)( tx

c

etp &&=

22 4

03

2 (where we used ( ) )

3= = i te

m x t x ec

ωγ ω ω

3

20

2

0 3

2

mc

eȦȦ ωγ =⇒≈

G


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

π νγν

π

ν λ πλ ν

ν λ ν−

∆ = =

∆ ∆= ⇒ ∆ = ∆ = = ⋅

8


15

The absorption cross-section

Definition absorption coefficient κ

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 harmonicoscillator for plane electromagnetic wave:

dsIdI νν νκ )(−=

low)()( nνσνκ =)(νσ

)()( 0 νϕσνσ =

0σ

)1()(8

),( 20

0

−′−=′

=

µδννδπ

µνν

ω

Ec

I

eEEti

x


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

π ν ωϕ ν γ γ

γ

π νϕ ν ϕ ν

π ν

20

2 22 00

22

0

( ) ( , ) ( )8

( ) ( )8 8

( ) ( ) 0.026537 cm Hz

cp I d d E

e EcE

m

e

mc

ν µ

σ ν ν µ ν µ σ νπ

σ ν ϕ νπ

πσ ν ϕ ν σ

′

′ ′= =

=

= ⇒ =

∫ ∫

)()( lu

2

lowlu

2

lu νϕπ

νκπ

σ fmc

enf

mc

e==

F

index “lu” stands for transition lowerĺupper level

9


17

Oscillator strengths

Oscillator strengths flu are obtained by:

• Laboratory measurements

• Solar spectrum

• Quantum mechanical computations (Opacity Project etc.)

• Allowed lines: flu≈1,

• Forbidden: <<1 e.g. He I 1s2 1S→1s2s 3S flu=2 10-14

3280.12H β4861.3

gupglowfluLineλ/Å

820.41Ly α1215.7

5080.04H γ4340.5

1880.64H α6562.8

3220.03Ly γ972.5

1820.07Ly β1025.7


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

mcσ

πκ ν ϕ ν=

1442443

Population number of lower level

QM correction factorProfile function

Classical crossection

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 ν

πν

η ψ νπ

=

=

0low lu

20

lu lu

hv( ) n B ( )

4

hv e i.e. B f

4 mc

κ ν = ϕ νπ

π=

π

)(νψ )()( νψνϕ =

G


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 T

B n B

ν ν ν ν

ν ν

ν

ν

ν ν ν

π 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

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

νν

νν

νν

ν

−

−

−

−

= − =

= −

= −

⇒3

ul 02

ul

low lulow lu up ul

up ul

2

1

A h

B c

g Bg B g B

g B

ν=

⇒ = ⇒ =

G


22

Relation to oscillator strength

dimension

Interpretation of as lifetime of the excited state

order of magnitude:

at 5000 Å:

lifetime:

2 2

lu lu

0

2 2up up

ul lu lu

low 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

π

π

πγ

=

= =

= = = ulA 1 time

ul1 A

ulul γ≈A18 s10 −

s10 8−

F

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

( ) 4 211

( ) ( ) 4 ( ) 2

: 1 2 ln 2

e.g. 10000K : 20000 A

50000K : 4160 A

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

ν

ν

ν

ν ψ ν πη ν

η ν ψ ν π ν

ν

λ

λ

=

= = = −

= ⇒ = ⇒ =

= =

= =

o

o


24

Induced emission as negative absorption

Radiation transfer equation:

Useful definition: κ corrected for induced emission:

spontaneous induced

spontaneous induced

induced0 0lu lu low lu ul up

with

( ), ( )4 4

= − = +

= + −

= = v

dII

ds

dII

ds

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 ν

π π

( )spontaneous 0ul up lu low

2low

lu lu low up

up

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

νν ν

νη ϕ ν

π

πκ ϕ ν

ν πη ϕ ν

= + −

= −

=

transition lowĺup

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

13


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:

κηvvS =

( )

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

nA

S

v

vv

ϕπ

ϕπ

κ

η

[ ] ),(12

0

1

2

30lu 0 TvBe

c

hvS v

kThv

v =−=−


26

Every energy level has a finite lifetime τ 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 Lyα (transition 2→1):

example Hα (3→2):

Line broadening: Radiation damping

∑<

=ul

ul1 Aτ

h=⋅∆ τE

∑∑<<

+=+=l

l

l

AAj

juk

uku

11

ττγ

21 cl 1 2 12A 3 g g fγ = = γ cl cl3 2 8 0.41 0.31= γ ⋅ = γ

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

γ = γ + + = γ + + = γ

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


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

2

1

2

02/ 2

10/ 2

~ Fourier transform

sin2

( ) ~

2−

− =

−

∫t

i t i t

t

I

t

I e e dtω ω

ω ω

ωω ω

0( ) ~ i tE t e ω

15


29


Probability 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

∫∞

−

−

−

⋅=0

0/

2

00 /22

sinconst)( 0 τωωωω

ω τdtetI

t

v

( )0 /0 0 0( ) average time between two collisionst

W t dt e dtτ τ τ−= =

( ) ( )220 1

1)(

τωω

πτωϕ

+−=

12 , ~ = particle density of colliders

approximately constant

-N N

N Ȗ Ȗγ τ τ

γ

=

′ ′= ⋅

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


30


• 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 ions

neutral atoms with each other, H+H

ions

metals + H

linear Stark effect

resonance broadening

quadratic Stark effect

van der Waals broadening

2

3

4

6

dominant atnamep=

Film logg

16


31

Thermal broadening

Thermal 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:

kTm

xxxew2

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

x

x

x

kT m T

w C e w d

e d e d C C

w

ππ

∞−

∞ ∞− −

= =

= = =

= ⋅ =

⋅ = ⋅ = ⇒ = ⇒ =

=

∫

∫ ∫2 2

thv v

th

1

vxe

π−


32

Line profile

Doppler effect:profile function:

Line profile = Gauss curve– Symmetric about ν0

– Maximum:– Half width:– Temperature dependency:

cc

th

0

th

0

v ,

v=

∆=

∆

ν

ν

ν

ν

02 2

th

0

2 20 th

01

th

( )

th

(v ) ( ) , with 1 we obtain:

1( )

Ȟ

x x

Ȟ

Cw e ( )dȞ

cȞ

eȞ

ν ν

ν ν ν

νϕ ν ϕ ν

π

ϕ νπ

+∞

−∆ ∆

−∞

− − ∆

⇒ = =∆

=∆

∫πth

1Max

v∆=

Max2

1

0ν ν

)(νϕ

FWHM

πth1 v∆

ththFWHM 67.12ln2 vvv ∆=∆=∆

Tv ~th∆

17


33

Examples

At λ0=5000Å:

T=6000K, A=56 (Fe): ∆ λth=0.02Å

T=50000K, A=1 (H): ∆ λth=0.5Å

Compare 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 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

−=π

18


35

Joint effect of different broadening mechanisms

Mathematically: convolution

commutative:

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 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-transformation

Convolution 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 B

A B

x C

A 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 x

x A x B

CL x L x L x C A B

x C

π π

π

= =+ +

= ∗ = = ++

19


37

Application to profile functions

Convolving Gauss and Lorentz profile (thermal broadening + damping)

( )

2 20

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 ǻ V( ) G L(v )d

( ) ǻ a /( ʌǻ ) y ( ) ǻ

aG y e L(y)

y

ν ν ν γ πν ν

ν π ν ν γ π

ν ν γ ν ν ν ν ν

ν ν ν γ ν ν ν ν

ν π

ν 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 dy

a y a

a eV H a H a dy

y 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 lines

Example: 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 + 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

ν ν ν ν, +

+

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

dtdIPn vv νlow

v vv21 2ion dhmdvmEhv =→+=

G

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 e

G n n hG c

F h

G c

n P me

n n hG

n n

ν ν ν ν

ν ν

νν

ν

ν

ν

−

−

= + =

= +

= − = −

⇒ =

⇒ =

• ion

2

3/ 2uplow

2up e low

3/ 2

v 2 2e 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

π

ππ

−

−

=

• =

G


42

Einstein-Milne relations

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

2ion

3/ 2up 3/ 2

up

low

3

2

2low

2up 2

3lo

/ 2up

2e low

e

3/ 2

v 2 2e

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

π

ππ

π

π

− −

=

=

=

=

F

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 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

ν

νν ν

ν

κ ν ν ν

ν σ −

= −

= − = = −

[ ]

vv

v

kThv

v

Bvv

mhFnnv

ehvPnv

)()(

/v)v()v()(

1)(2

eup

low

κη

η

κ

=

=

−= −

M

*

up /up3

low

2( ) ...

ch kT

nhn e

n

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

( )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 ce

n

σ ν ν ν νσ ν

ν σπ

−

>=

= = = ⋅

( )3

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

∑=

=max

1bf

totbf )()(

n

n

n

nnvv σκ

23


45

Continuum absorption cross-sections

Optical continuum dominatedby Paschen continuum


46

The solar continuum spectrum and the H-ion

H- 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

+ −− −= = = =

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

24


47


Ionized metals deliver free electrons to build H

-


48


G

25


49



50

Scattering processes

Thomson scattering at free electrons

Absorption coefficient follows from power of harmonic oscillator ( Thomson cross-section)

Thomson cross-section is wavelength-independent

eeσκ n=

eσ

( ) ( )

4 2 40

22 3 22 2 20

0

4 220

e 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

86.65 10 cm

3

e Ep

m c

e E cp p E

m c

e

m c

ν

ν 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 40

22 3 22 2 20

0

4 2 420

R 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

8

3

( )

lu

lu

lu lu

l

l lu

e Ep

m c

e E cp p E

m c

ef f

m c

n f

ν

ν ν γ π ν

ν ν ν

νσ

ν π

π ν νσ σ

ν ν

νκ ν σ

ν

= − + << =

→ = =

= =

→ =∑

(here we have included the oscillator strengthas the quantum mechanical correction)


52

Raman scattering

Discovered in symbiotic nova RR Tel

Raman scattering of O VI resonance line (Schmid 1987)

n=1

n=2

n=3virtuallevel

1215Å1026Å 1032/38Å

Raman-scattered line 6825/7082Å

ĮLy OVI

111

λλλ−=

Schmid 1989,

Espey et al. 1995

27


53

Two-photon processes


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, becauseless and less bound-free processes present

• Free-free opacity important at high temperatures

),()(/)()( ffffff TvBvvvS vvv == κη

( )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

− νκ ν = σ ν −

π

νσ ν = ⋅ ⋅ ν

π

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 v

n n T Iρ=

( , )i i

n 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 ge

En g

− −=

kTEii ieg

g

n

n /

11

−=

1 1

1 1 1

1

1

1

/

/partition function

(

1

, with ( ) :)

−

−

= = =

= =→

∑

∑

∑ ∑ ∑ j

jE kT

j

i i i i

jj j

i i

E kT

j

n n n nn g

nn n n n n

n

g e

U T

n n gU g

nT

Ne

G

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 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 ii

g pg p i


58

Hummer-Mihalas occupation probabilities

30


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 nπ π π

Ω=

Ω = ⋅ = ⋅ = ⋅ → =

F

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

π

π

π π

π

∞− − −

∞− − −

− −

−

= =

=

=

=

∫

∫

low ) /E kT−

33/216/)(

low

up2/3

low

eup cm K 1007.2 )( lowup −−−⋅=== Ce

g

g

C

TTf

n

nn 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 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

− =

⇒ = − + +

⇒ =

G


62

Hydrogen ionization

Ioniz

atio

n d

egre

e x

Temperature / 1000 K

32


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

1j

2/3

e

3

e1j1j1

jIon

22

1

+

+

=

π

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

kTEEkTEEeTCnN

U

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

1j

1j

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

Φ==

==

==

−

++

−

+

+−−

+

+

−−+

+

∑

∑∑

F

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 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 NN

N N N N NN N N

N N N N N

n TN

Nn T

+

+ +

= =

+

+ +

=

=

= ⋅ ⋅ ⋅

= = = + + ⋅ + +

⋅ ⋅ ⋅

= =+ + ⋅ + +

Φ

=

+ Φ

∑ ∑

∏

K

K L

K

K L

J-1J

m 1 m=∑∏


66

Ionization fractions

34


67

Summary: Emission and Absorption

F


68

Ɣ Line absorption and emission coefficients (bound-bound)

32 2low 0 low

lu lu low up lu lu up2up up

( ) 2

( ) ( ) ( )

= − =

g h ge ef n n f n

mc g c mc g

νπ πκ ν η νϕ ν ϕ ν

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

2 2

1( , v)

(v )

∞ −

−∞

=− +∆ ∫

y

D

eV a dy

y aν π

Ɣ Continuum (bound-free)*

up h / kTlow up

low

n(v) n n e

n− ν

ν

κ = σ −

*

up h /kTup3

low

n2h( ) n e

c n− ν

ν ν

νη ν = σ

Ɣ 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=Ɣ 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 low

up up

n ge

n g

up low

3/2( ) /up upe

3low e low

22 − − =

E E kTn gm kT

en n h g

π

Boltzmann

Saha

1

Stellar Atmospheres: Hydrostatic Equilibrium

1

Hydrostatic Equilibrium

Particle conservation


2

Ideal gas

P

P+dP

r

dr

dA

H

2 2

mass density

atomic weight

forces acting on volume element:

buoyancy:

r rg

P

P pressurek

P TAm

A

dV dAdr dm dV

GM dm GMdF dAdr

r r

dF dPdA

ρ ρ

ρ

ρ

=

= ⋅ =

=

= =

= − = −

= − (pressure difference * area)

2


3

Ideal gas

In stellar atmospheres:

log g is besides Teff the 2nd

fundamental parameter of

static stellar atmospheres

2

2

2

-

mass of atmosphere negligible

thickness of atmosphere stellar radius

with surface gravity

usually written as log( / cm s

:

)

∗

∗

∗

∗

∗

∗

=

= <<

→ = − =

=

r

g

M M

r R

dF dAdr dAdrGM

gR

GMg

R

g

ρρ

4.0 .... 4.5

4.44

0 .... 1

~8

~15

3.0

Main sequence star

Sun

Supergiants

White dwarfs

Neutron stars

Earth

log gType


4

Hydrostatic equilibrium, ideal gas

buoyancy = gravitational force:

example:

H

( )

( )( )

0

0

eliminate with ideal gas equation ( )(

)

:

+ =

− − =

= −

= −

P gdF dF

dPdA g dAdr

dPg

drr

A r mr P r

k

dPg

d rr T

ρ

ρ

ρ

0 H

0

H H

( )

0

( )/

0

/

H

( ) const , ( ) const (i.e., no ionization or dissociation)

1 ( )

solution:

P(r)

press

P(r )

P(r

: ure scale height

) P(r )

− −

− −

= = = =

= − ⇒ = −

=

=

= r r gAm

Hr

T

r

k

T r T A r A

Am AmdP dPg P r g

dr k

kTH

gA

T P d

m

r kT

e

e

3


5

Atmospheric pressure scale heights

Earth:

Sun:

White dwarf:

Neutron star:

km 9

3log

K 300

)(N 28 2

=

=

≈

≈

H

g

T

A

km 180

44.4log

K 6000

)(H 1

=

=

≈

=

H

g

T

A

km 25.0

8log

K 15000

)(H 5.0

=

=

=

+= +

H

g

T

nA e

! mm 6.1

15log

K 10

)(H 5.06 =

=

=

+= +

H

g

T

nA e

H

=kT

HgAm


6

Effect of radiation pressure

2nd moment of intensity

1st moment of transfer equation (plane-parallel case)

F

F

vR Kc

vPπ4

)( =

0

( )

4 with ( ) ( )

( )

4( )

integration over frequencies:

4( )

∞

=

= =

=

= ∫

vv

Rv

Rv

Rv

dKH

d v

dPH d v v dr

d v c

dPv H

dr c

dPv H dv

dr c

τ

πτ κ

τ

πκ

πκ

4


7

Effect of radiation pressure

Extended hydrostatic equation

In the outer layers of many stars:

Atmosphere is no longer static, hydrodynamical equation

Expanding stellar atmospheres, radiation-driven winds

ef

0

eff rad

f

0

effective gravity

( (depth d

( ) ( )

( ) ( )

definition:

ependent4 1

: ( )(

!))

( )

)

4

∞

∞

= −

−

= −

=

= − =

∫

∫ v

Rv

dPv

g

dPg r g r

dr

g r r

g v H

H dvd

dv

c

gc

r

rr

g

ρ ρ

ρ

πκ

ρ

πκ

eff rad

0

4 10 i.e. ( )

( )

∞

< = >∫ vg g v H dv gc r

πκ

ρ


8

The Eddington limit

Estimate radiative acceleration

Consider only (Thomson) electron scattering as opacity

(Thomson cross-section)

number of free electrons per atomic mass unit

Pure hydrogen atmosphere, completely ionized

Pure helium atmosphere, completely ionized

Flux conservation:

e)( σσ =v

=q

1=q

5.04/2 ==q

Hm

q

cdvH

m

q

cdvHn

qmncg

H

v

H

v

H

e

0

e

0

ee

e

e

rad

44

/

14 σπσ

πσ

π=== ∫∫

∞∞

4

eff4

H Tσ

π= F

e 2 44rad e effe

e eff 2

H H

4.51ee

H

414 1

m 4g m 4

/10

4 m /

g q R Tq MT G

c R c G M

q L LLq

c G M M M

σ πσσπ σ

π π

σ

π−

Γ = = =

Γ = =

5


9

The Eddington limit

Consequence: for given stellar mass there exists a maximum

luminosity. No stable stars exist above this luminosity limit.

Sun:

Main sequence stars (central H-burning)

Mass luminosity relation:

Gives a mass limit for main sequence stars

Eddington limit written with effective temperature

and gravity

Straight line in (log Teff,log g)-diagram

4.51

max 10 1L L q M M−= ⋅ ⋅

1e <<Γ

( )3

max/ / 180L L M M M M≈ → =

0loglog4log12.15

1/10

eff

4

eff

12.15

e

=−++−

==Γ −

gTq

gqT


10

The Eddington limit

Positions of analyzed

central stars of planetary nebulae

and

theoretical stellar evolutionary tracks

(mass labeled in solar masses)

6


11

Computation of electron density

At a given temperature, the hydrostatic equation gives the

gas pressure at any depth, or the total particle density N:

NN massive particle density

The Saha equation yields for given (ne,T) the ion- and atomic

densities NN.

The Boltzmann equation then yields for given (NN,T) the

population densities of all atomic levels: ni.

Now, how to get ne?

We have k different species with abundances αkParticle density of species k:

gasP NkT=

atoms ions e N eN N N n N n= + + = +

( )K

N e N

k 1

, and it is k k k kN N N n N Nα α=

= = − =∑


12

Charge conservation

Stellar atmosphere is electrically neutral

Charge conservation electron density=ion density * charge

Combine with Saha equation (LTE)

by the use of ionization fractions:

We write the charge conservation as

Non-linear equation, iterative solution, i.e., determine zeros of

e

1 1

, density of j-th ionization stage of species k jkK

jk jk

k j

n j N N= =

= ⋅ =∑∑

∑∏

∏

= =

=

+

==jk

m

jk-

ml

lk

jk-

jl

lk

k

jk

jk

(T)ĭn

(T)ĭn

N

Nf

1

1

e

1

e

1

)(),()(

),()(),(

e

1 1

eee

1 1

ee

1

e

1

e

nFTnfjnNn

TnfjnNTnfNjn

K

k

jk

j

jkk

K

k

jk

j

jkk

K

k

jk

j

jkk

=⋅−=

⋅−=⋅=

∑ ∑

∑ ∑∑∑

= =

= == =

α

α

0)( ee =− nnF use Newton-Raphson, converges after 2-4 iterations;yields ne and fij, and with Boltzmann all level populations

7


13

Summary: Hydrostatic Equilibrium


14

Summary: Hydrostatic Equilibrium

Hydrostatic equation including radiation pressure

Photon pressure: Eddington Limit

Hydrostatic equation → N

Combined charge equation + ionization fraction → ne

→ Population numbers nijk (LTE) with Saha and Boltzmann

equations

0

4( ) ( ) ( )R

v

dPdPg r g r v H dv

dr dr c

πρ ρ κ

∞

= − = − ∫

8


15

3 hours

Stellar

Atmospheres……per day… …is too much!!!

1

Stellar Atmospheres: Radiative Equilibrium

1

Radiative Equilibrium

Energy conservation


2


Assumption:

Energy conservation, i.e., no nuclear energy sources

Counter-example: radioactive decay of Ni56 →Co56 →Fe56 in

supernova atmospheres

Energy transfer predominantly by radiation

Other possibilities:

Convection e.g., H convection zone in outer solar layer

Heat conduction e.g., solar corona or interior of white dwarfs

Radiative equilibrium means, that we have at each location:

Radiation energy absorbed / sec

=

Radiation energy emitted / sec

integrated over all

frequencies and

angles

2


3


Absorption per cm2 and second:

Emission per cm2 and second:

Assumption: isotropic opacities and emissivities

Integration over dω then yields

Constraint equation in addition to the radiative transfer

equation; fixes temperature stratification T(r)

∫ ∫∞

π

κω4 0

)( vIvdvd

∫ ∫∞

π

ηω4 0

)(vdvd

( ) 0)( )()(000

=−⇒= ∫∫∫∞∞∞

dvSJvvdvJvdv vvv κηκ


4

Conservation of flux

Alternative formulation of energy equation

In plane-parallel geometry: 0-th moment of transfer equation

Integration over frequency, exchange integration and

differentiation:

( )vvv SJ

dt

dH−= κ

( )0 0

4

eff

0

4

eff

0 0

0 because of radiative equilibrium

const for all depths. Alternatively written:4

(1st moment of transfer equat4

∞ ∞

∞

∞ ∞

= − =

⇒ = = =

= =

∫ ∫

∫

∫ ∫

v v v

v

v

v

dH dv J S dv

dt

H H dv T

dv T dvdK

Hd

κ

σ

π

σ

π τ

4

eff

0

ion)

( ) (definiton of Eddington factor)

4

∞

⇒ =∫ v vddv T

d

f J σ

τ π

F

F

3


5

Which formulation is good or better?

I Radiative equilibrium: local, integral form of energy

equation

II Conservation of flux: non-local (gradient), differential form

of radiative equilibrium

I / II numerically better behaviour in small / large depths

Very useful is a linear combination of both formulations:

A,B are coefficients, providing a smooth transition between

formulations I and II.

( ) 0)(

00

=

−⋅+

−⋅ ∫∫

∞∞

Hdvd

JfdBdvSJA vv

vv τκ


6

Flux conservation in spherically symmetric geometry

0-th moment of transfer equation:

( ) ( )

( )

HRLLdvHr

dvJSrdvHrr

JSHrrr

v

vvv

vvv

22

2

0

2

0

2

0

2

2

2

16 because 16

1const

0

1

ππ

κ

κ

===

=−=

∂

∂⇒

−=∂

∂

∫

∫∫∞

∞∞

4


7

Another alternative, if T de-couples from radiation field

Thermal balance of electrons

∑

∑

∑ ∫

∑ ∫

∑ ∫

∑ ∫

=

=

+

−=

−=

+=

=

=−

∞−

∞

∞−

∞

ml

lmlml

C

ml

lmlmm

H

kl

kThv

vlk

vlkk

C

kl

lkvlkl

H

j

kThv

vjj

C

j

vjj

H

CH

hvTqnnQ

hvTqnnQ

dvec

hvJ

v

vJvnQ

dvv

vJvnQ

dvec

hvJTvNnQ

dvJTvNnQ

QQ

,

ec

,

ec

, 0

2

3

bf,bf

, 0

bf,bf

0

2

3

ff,eff

0

ff,eff

)(

)(

21)( 4

1)( 4

2

),( 4

),( 4

0

απ

απ

απ

απ


8

The gray atmosphere

Simple but insightful problem to solve the transfer equation

together with the constraint equation for radiative equilibrium

Gray atmosphere: κκν =

( ) ( )

( ) ( )

( )


with

Integration over frequency

Radiative equilibrium ( ) ( ) 0

and because

= − = =

= − =

− = − = − =

=⇒∫ ∫

v vv v v

v v v v

dH dKI J S II H dt

d d

dH dKI

J

J S II Hd d

J S dv J S dv J

S

S

I

τ κτ τ

τ τ

κ κ

( ) ( )1 2 1 2

2

2

of conservation of flux 0

from follows , see below0

=

⇒ ⇒ == + = =

dH

d

dKII K c c II c H

d

dc

K

dτ

τ

ττ

5


9

The gray atmosphere

Relations (I) und (II) represent two equations for three

quantities S,J,K with pre-chosen H (resp. Teff)

Closure equation: Eddington approximation

Source function is linear in τ

Temperature stratification?

In LTE:

( )2K 1 3J J 3K IIIS 3H 3c= → = = = τ +

( )

( )

4

4

2

4

eff

4 4

eff 2 2

( ) ( ( ))

insert into : 3 3

with we get: 4

3( ) 3 is now determined from boundary condition ( =0)

4

= =

= +

=

= +

S B T T

III T H c

H T

T T c IV c

στ τ

πσ

τπ

σ

πσ

τ σ τ τπ π

F


10

Gray atmosphere: Outer boundary condition

Emergent flux:

( )

( )

[ ]

3

23 ,

3

2

4

3

3

2

2

1

3

1

2

3)0(

)(12

1)( and

!)(with

)()( 2

3

)(332

1

from with )()(2

1)0(

4

eff

4

22

12

0

0

22

0

2

0

22

0

2

+=

+=⇒

=→

+=

−−

=+

=

′′+′′′=

′′+′=

′′′=

−∞

∞∞

∞

∞

∫

∫∫

∫

∫

ττ

τττττ

τττ

τττ

HSTT

HccHH

ttEetEnl

ldttEt

dEcdEH

dEcH

IIISdESH

t

n

l

F

F

from (IV): (from III)

6


11

Avoiding Eddington approximation

Ansatz:

Insert into Schwarzschild equation:

Approximate solution for J by iteration (“Lambda iteration“)

( )

))((4

3)(

function Hopf )(

oftion generaliza ))((3)(

4

eff ττπ

στ

τ

τττ

qTJ

q

IIIqHJ

+=

=

+=

F

( ) ( )∫∞

′−′′+′=+⇒

Λ=Λ=

0

1)(2

1)(

Jfor equation integral )(

τττττττ

τ

dEqq

JSJ

( )

+−+=+Λ=Λ=

+=

)(2

1)(

3

1

3

23)32(3

)32(3

32

)1()2(

)1(

ττττ

τ

EEHHJJ

HJ

F

(*) integral equation for q, see below

i.e., start with Eddington approximation

(was result for linear S)


12

At the surface

At inner boundary

Basic problem of Lambda Iteration: Good in outer layers, but

does not work at large optical depths, because exponential

integral function approaches zero exponentially.

Exact solution of (*) for Hopf function, e.g., by Laplace

transformation (Kourganoff, Basic Methods in Transfer Problems)

Analytical approximation (Unsöld, Sternatmosphären, p. 138)

( )358.034

1

3

1

3

23

2

1)0( , 1)0( , 0

)2(

32

+=

+−+=

===

ττ

τ

HHJ

EE

2 3

(2)

, ( ) 0 , ( ) 0

23

3

E E

J H

τ

τ

= ∞ ∞ = ∞ =

= +

ττ 972.11167.06940.0)( −−≈ eq

exact: q(0)=0.577….

7


13

Gray atmosphere: Interpretation of results

Temperature gradient

The higher the effective temperature, the steeper the

temperature gradient.

The larger the opacity, the steeper the (geometric) temperature

gradient.

Flux of gray atmosphere

τκ

τ

ττ

d

dT

dt

dT

Td

dT

Td

dTTT

d

d

=

==

4

eff

4

eff

34

~

4

34

[ ]

2 2

0

1/ 4

eff eff

LTE: ( ( ))

1 1( ) ( ( )) ( ) ( ( )) ( )

2 2

with , 3 4 ( ( )) ( ) ( )

and

v v

v v v

v

S B T

H B T E t dt B T E t dt

hv kT T T q p hv kT p

H d H dv H

τ

τ

α

τ

τ τ τ τ τ

α τ τ τ α τ

σα

∞

=

= − − −

= = + = → =

= =

∫ ∫

3 3 3

2 3 3

4

eff

43eff 2 2

4 3 2

eff 0

1 2 4

2

4

( ) ( )4 4( ) /

exp( ( )) 1 exp( ( )) 1

vv

hv k k

hc h v

T

H kT E t E tdv kH H H dt dt

H d T h h c p p

τ

α

τ

π α

σ

π

τ τπ πτ α

α σ σ α τ α τ

∞ − −→ = = = −

− − ∫ ∫


14

Gray atmosphere: Interpretation of results

Limb darkening of total radiation

i.e., intensity at limb of stellar disk smaller than at center by

40%, good agreement with solar observations

Empirical determination of temperature stratification

Observations at different wavelengths yield different T-

structures, hence, the opacity must be a function of

wavelength

4 4

eff

3 2I( 0, ) S( ) B(T( )) T ( ) T

4 3

I(0, ) 2 / 3 2 3(1 cos )

I(0,1) 1 2 / 3 5 2

σ σ τ = µ = τ = µ = τ = µ = τ = µ = µ +

π π µ µ +

→ = = + ϑ+

TTBSSI →=→=→= ))(()()( ),0( measure ττµτµτ

F

8


15

The Rosseland opacity

Gray approximation (κ=const) very coarse, ist there a good mean value ? What choice to make for a mean value?

For each of these 3 equations one can find a mean , with which the equations for the gray case are equal to the frequency-integrated non-gray equations.

Because we demand flux conservation, the 1st momentequation is decisive for our choice: → Rosseland mean of opacity

transfer equation

0-th moment

1st moment

non-graygray

)( ISdz

dI−= κµ ))(( vv

v ISvdz

dI−= κµ

))(( vvv JSv

dz

dH−= κ

vv Hv

dz

dK)(κ−=

0)( =−= JSdz

dHκ

Hdz

dKκ−=

κ

κ


16


Definition of Rosseland mean of opacity

3

0

340

0

00

4

)(

1

1

4 and with

)(

1

1

: LTE and 3/1ion approximatEddington with )(

1

1

1

)(

1const

T

dvdT

dB

v

dz

dTTT

dz

d

dz

dB

dz

dT

dT

dB

dz

dB

dz

dB

dvdz

dB

v

BJJK

dz

dK

dvdz

dK

v

dz

dKdv

dz

dK

vdvH

v

R

vv

v

R

v

R

R

vv

π

σ

κ

κ

π

σ

π

σκ

κ

κ

κ

κκ

∫

∫

∫

∫∫

∞

∞

∞

∞∞

=

=

===

===

===

9


17


The Rosseland mean is a weighted mean

of opacity with weight function

Particularly, strong weight is given to those frequencies, where the radiation flux is large.

The corresponding optical depth is called Rosseland depth

For the gray approximation with is very good,

i.e.

Rκ

1

)(

1

vκ dT

dBv

Rκ1>>Rossτ

∫ ′′=z

RRoss zdzz0

)()( κτ

))((4

3)(

4

eff

4

RossRossRoss qTT τττ +=


18

Convection

Compute model atmosphere assuming

• Radiative equilibrium (Sect. VI) → temperature stratification

• Hydrostatic equilibrium → pressure stratification

Is this structure stable against convection, i.e. small

perturbations?

• Thought experiment

Displace a blob of gas by ∆r upwards, fast enough that no heat

exchange with surrounding occurs (i.e., adiabatic), but slow

enough that pressure balance with surrounding is retained (i.e.

<< sound velocity)

10


19

Inside of blob outside

Stratification becomes unstable, if temperature gradient

rises above critical value.

( ), ( )T r rρ

ad ad

ad ad

( )

( )

T T T r r

r rρ ρ ρ

+ ∆ = + ∆

+ ∆ = + ∆

r∆

( ), ( )T r rρ

rad rad

rad rad

( )

( )

T T T r r

r rρ ρ ρ

+ ∆ = + ∆

+ ∆ = + ∆

ad rad

ad rad

ad rad

a

sta

( ) ( ) further buoyancy,

( ) ( ) gas blob falls back,

i.e.

with ideal gas equation p= and pressure balance

ble

stable

unstable

unsta

ble

+ ∆ < + ∆ →

+ ∆ > + ∆ →

<

>

H

r r r r

r r r r

d d

dr dr

kT

Am

ρ ρ

ρ ρ

ρ ρ

ρ ρ d ad rad rad

ad radunstable

s

=

table

>

<

T T

dT dT

dr dr

ρ

addT dr


20

Alternative notation

Pressure as independent depth variable:

Schwarzschild criterion

Abbreviated notation

eff eff

eff

eff eff

ad rad

hydrostatic equation: ( )

(ln )

(ln )

(ln ) (ln )

(ln ) (

ide

unstabl

al gas

stabln

e

le )

= − = −

→ = −

= − = −

<

>

H

H

H H

dp g dr g dr

kTdpAm g p

Am AmdT dT T d Tg g

k dp p k d p

d T d T

d

d

Am p

k T

p d

r

dr

p

ρ

ad rada

ad d

d rad

ra

(ln ) (ln );

(ln ) (l

sta

n

b

)

le

∇ = ∇ =

∇ > ∇

d T d T

d p d p

11


21

The adiabatic gradient

Internal energy of a one-atomic gas excluding effects of

ionisation and excitation

But if energy can be absorbed by ionization:

Specific heat at constant pressure

V V

0 (no heat exchange)

(1st law of therm

intern 0 (al

ody

energy

namic

*)

s)

=

= +

= ⇒ + =

dQ

dQ pdV

c d

dE

dE c dT T pdV

V

3 3

2 2E NkT c Nk= → =

V

3

2c Nk>>

p V V

t t

p V

( )

p cons p cons

c c

Q dE dV d NkT p Nkc p c p c p

T dT dT dT

Nk

p= =

∂

−

= = + = + = +∂

=→


22


( )

p V

p VV

p V

V

p

p V

V

V

p

V

p

V

Ideal gas:

(**)

/

from(*) with (**) 0

0

0

(ln ) (l

pV

n )

defi

= ⇒ + = =

+=

−

+−→ + =

⋅−

−+ + =

+ =

= −

−pV NkT Vdp pdV dT dT

Vdp pdVdT

c c

V pc cc pdV

c cc

c cdV

p V c

cdp dV

dp d

p V c

cd V d p

V

d

Nk c

p d

V

c

c

V

p

V

(ln ) 1nition: :

(ln )= = −c d V

c d pγ

γ

12


23


ad

ad

rad

(ln )needed:

(ln )

/

ln ln ln ln( )

(ln ) (ln )1

(ln ) (ln )

(ln ) 1 1

1 st

1(ln )

b e

1

a l

=

= + −

=

−∇ <

+

−= − =

−∇ =

d T

d p

T pV Nk

T p V Nk

d T d V

d p d p

d T

d p

γ

γ γ

γ

γ

γ

γ

Schwarzschild criterion


24


• 1-atomic gas

• with ionization

• Most important example: Hydrogen (Unsöld p.228)

V p V

ad

3 2 5 2

5 3 2 5 0.4

c Nk c c Nk Nk

γ

= = + =

= ∇ = =

ad1 0 convection starts effect−→ ∇ → γγ

( ) ( )

( ) ( )

2

Ion

ad 22

Ion

2

2 5 2

5 5 2

( ) ( ) ( )with ionization degree

2 2

+ − +∇ =

+ − +

= − + +

x x E kT

x x E kT

f T f T f Tx

N N NF

13


25


( )( )

( ) ( )

2

Ion

ad 22

Ion

2

2 5 2

5 5 2

( ) ( ) ( )

2 2

x x E kT

x x E kT

f T f T f Tx

N N N

+ − +∇ =

+ − +

= − + +


26

Example: Grey approximation

( )( ) ( )

( )( )( )

4 4

eff

4

eff

1

3 2( )4 3

3 24ln ln ln4 3

2ln3

4

hydrostatic equation: Ansatz: ( here a mass absorption coefficient)

(ln ) 1

243

1 integrate

1

+

= +

= + +

+= =

= =

→ = → = →+

+

b

b b

b

T T

T T

d

d

dp gAp

d

g g gp p

A b A

d T

Ap

dp

d

d

τ τ

τ

τ

τ

κ κτ κ

τ τ

ττ

( )

1

1

rad

rad

1

( 1)

1 1

( 1)

243

becomes large, if opacity strongly increases with depth (i.e

(ln ) 1

(

. exponent b large).

The absolute value of is

1)

l

not

l

essenti

n

n

+

+

=+

= = = =

+= =

+

∇

+

∇

b b

b

g g

p p

d T

A

d

d p

d b

d p

p

p

d p

d

d

A

b

τ

ττ

τ

τ τ

τ

κ

τ

rad ad

a

l but

large

the change o

(> ): c

f with depth (g

onvection starts,

radient

-Ef ekt

)

f∇ ∇

κ

κ

14


27

Hydrogen convection zone in the Sun

κ-effect and γ-effect act together

Going from the surface into the interior: At T~6000K ionization of

hydrogen begins

∇ad decreases and κ increases, because a) more and more

electrons are available to form H−

and b) the excitation of H is

responsible for increased bound-free opacity

In the Sun: outer layers of atmosphere radiative

inner layers of atmosphere convective

In F stars: large parts of atmosphere convective

In O,B stars: Hydrogen completely ionized, atmosphere radiative;

He I and He II ionization zones, but energy transport by

convection inefficient

Video


28

Transport of energy by convection

Consistent hydrodynamical simulations very costly;

Ad hoc theory: mixing length theory (Vitense 1953)

Model: gas blobs rise and fall along distance l (mixing length).

After moving by distance l they dissolve and the surrounding

gas absorbs their energy.

Gas blobs move without friction, only accelerated by buoyancy;

detailed presentation in Mihalas‘ textbook (p. 187-190)

( ) = pressure scale height

mixing length parameter

=0.5 2

=

L

l H r Hα

α

α

15


29

Transport of energy by convection

Again, for details see Mihalas (p. 187-190)

For a given temperature structure

4

rad eff

conv

conv

ad rad

( )

compute ( )

flux conservation including convective flux

new temperature stratification ( )

wit

(

h

)

= −

∇ < ∇ < ∇

→

→

→

F r T F r

F r

T r

σ

πiterate


30

Summary: Radiative Equilibrium

16


31

Radiative Equilibrium:

Schwarzschildt Criterion:

Temperature of a gray Atmosphere

( ) 0)(

00

=

−⋅+

−⋅ ∫∫

∞∞

Hdvd

JfdBdvSJA vv

vv τκ

4 4

eff

3 2

4 3T T τ

= +

ad rad(ln ) (ln )

(

unst

stable ln ) (

bl

ln

e

)

ad T d T

d p d p

>

<


32

3 hours

Stellar

Atmospheres……per day… …is too much!!!

1

Stellar Atmospheres: Non-LTE Rate Equations

1

The non-LTE Rate Equations

Statistical equations


2

Population numbers

LTE: population numbers follow from Saha-Boltzmann equations, i.e. purely local problem

Non-LTE: population numbers also depend on radiation field. This, in turn, is depending on the population numbers in all depths, i.e. non-local problem.

The Saha-Boltzmann equations are replaced by a detailed consideration of atomic processes which are responsible for the population and de-population of atomic energy levels:

Excitation and de-excitation

by radiation or collisions

Ionization and recombination

),(**

eii nTnn =

),,( JnTnn eii =

2


3

Statistical Equilibrium

Change of population number of a level with time:

= Sum of all population processes into this level

- Sum of all de-population processes out from this level

One such equation for each level

The transition rate comprises radiative rates

and collision rates

In stellar atmospheres we often have the stationary case:

These equations determine the population numbers.

∑∑≠≠

−=ij

iji

ij

jiji PnPnndt

d∑

∑

≠

≠

−

=

ij

iji

ij

jij

i

Pn

Pn

ndt

d

0 hence for all levels ≠ ≠

= =∑ ∑i j ji i ij

j i j i

dn n P n P i

dt

ijPijR

ijC


4

Radiative rates: bound-bound transitions

Two alternative formulations:a) Einstein coefficientsb) Line absorption coefficients advantage a): useful for analytical expressions with simplified

model atomsadvantage b): similar expressions in case of bound-free

transitions: good for efficient programmingNumber of transitions i→j induced by intensity Iν in frequency

interval dν und solid angle dω

Integration over frequencies and angles yields

Or alternatively

jijiij ABB

ij( )σ ν

/ 4 (absorbed Energy / )i ij v vn B I dvd hvϕ ω π

∫∞

=0

dvJBnRn vvijiiji ϕ

∫∞

==0

)(4 4/)()(with dvJ

hv

vnRnhvvBv v

ij

iijiijij

σππϕσ

F

3


5

Radiative rates: bound-bound transitions

In analogy, number of stimulated emissions:

Number of spontaneous emissions:

Total downwards rate:

∫

∫∫∞

∞∞

=′

==′

0

00

4 dvJhvg

gnRn

dvJg

gBndvJBnRn

v

ij

j

ijjij

vv

j

iijjvvjijjij

σπ

ϕϕ

∫

∫∫∞

∞∞

=′′

==′′

0

2

3

0

2

3

0

24

2

dvc

hv

hvg

gnRn

dvc

hvBndvAnRn

ij

j

ijjij

vjijvjijjij

σπ

ϕϕ

( )

[ ]

3

2

0

* *3

2

0

24

24

ijij ji j ji ji j v

j

ij hv kTi ij ji j ji j v

j j

g hvn R n R R n J dv

g hv c

n n hvn R n R n J e dv

n n hv c

σπ

σπ

∞

∞−

′ ′′= + = +

= = +

∫

∫


6

Radiative rates: bound-free transitions

Also possible: ionization into excited states of parent ion

Example C III:

Ground state 2s2 1S

Photoionisation produces C IV in ground state 2s 2S

C III in first excited state 2s2p 3Po

Two possibilities:

Ionization of 2p electron ĺ C IV in ground state 2s 2S

Ionization of 2s electron ĺ C IV in first excited state 2p 2P

C III two excited electrons, e.g. 2p2 3P

Photoionization only into excited C IV ion 2p 2P

4


7


Number of photoionizations = absorbed energy in dν, divided by

photon energy, integrated over frequencies and solid angle

Number of spontaneous recombinations:

∫∫ ∫∞∞

=→00

)(4 dvJ

hv

vnRndvdIpn v

ij

iijivvi

σπω FF

3

2

0 0

3

2

0

*3

2

*

0

2(v) (v) v v 4 (v)

2

(v

4 (v)

( ) 24

)

1

(v)

j e j ji j e

j ji j e

ij hv kTij ji j

hv k

v

e

j

T i

j

G

nmp e

hv hn n F d d n R n n dv

c m

hv hn R n n dv

c m

vn hvn R n e dv

n hv c

h n n

ω π

π

σπ

∞ ∞

∞

∞−

−

→ =

=

=

∫ ∫ ∫

∫

∫

F


8


Number of induced recombinations

Total recombination rate

∫

∫

∫∫ ∫

∞−

∞−

∞∞

=

=

=→

0

*

0

*

00

)(4

v)(

1v)(4

)v(v)(4vv)v(v)(

dveJhv

v

n

nnRn

dvm

hJ

nn

ne

h

mpnnRn

dvm

hJGnnRnddIGnn

kThv

v

ij

j

ijjij

v

ej

ikThv

vejjij

vejjijvej

σπ

π

πω

∫∞

−

+

=

0

2

3**

2)(4 dveJ

c

hv

hv

v

n

nnR

n

nn

kThv

v

ij

j

ijji

j

ij

σπ

5


9

Radiative rates

Upward rates:

Downward rates:

Remark: in TE we have

∫∞

−

+=

0

2

3

*

2)(4

with

dveJc

hv

hv

vR

Rn

nn

kThv

v

ij

ji

ji

j

ij

σπ

∫∞

=0

)(4

with

dvJhv

vR

Rn

v

ij

ij

iji

σπ

*

**

=⇒=⇒=

j

i

j

ijiijvv

n

n

n

nRRBJ


10

Collisional rates

Stellar atmosphere: Plasma, with atoms, ions, electrons

Particle collisions induce excitation and ionization

Cool stars: matter mostly neutral ⇒ frequent collisions with neutral hydrogen atoms

Hot stars: matter mostly ionized ⇒ collisions with ionsbecome important; but much more important becomeelectron collisions

Therefore, in the following, we only consider collisions of atoms and ions with electrons.

Am

Am

e

H

ion

electron 43masselectron

massion

v

v2/12/1

≈

=

=

6


11

Electron collisional rates

Transition i→j (j: bound or free), σij (v) = electron collisioncross-section, v = electron speed

Total number of transitions i→j:

minimum velocity necessary for excitation (threshold)

velocity distribution (Maxwell)

In TE we have therefore

Total number of transitions j→i:

)(vv)v()v(

0v

TnndfnnCn ijeiijeiiji Ω== ∫∞

σ

v)v( df

0v

jiij CC *

j

*

i nn =

ij

j

iji C

n

nC

*

jj nn

=


12


We look for: collisional cross-sections σij (v)

• experiments

• quantum mechanical calculations

Usually: Bohr radius πa02 as unit for cross-section σij (v)

σij (v) = πa02 Q ij

Q ij usually tabulated as function of energy of colliding electron

2

0

0

0

1 2

2 v 2 2e

v

1 2

2 11

0 0 0

u

0

m( ) (v) (v)v v with 1 2 v and (v) v 4 v v

2 kT

8(ukT) u with and 5.456 10

( ) ( ) wit

u:

(

E

h

m kT

ij ij

u

ij

e

u

ij ij ij

T f d m

k

E f d e d

kC T Q ue d C aT

m

T C T e T

σ ππ

ππ

∞−

∞− −

−

=

Ω = = =

= = = ⋅

Ω = Γ Γ

∫

∫

0 0 0

0

) ( xkT)( ) x, x: ( E - E )x

ijT Q E x u e d kT kT∞

−= + + =∫

7


13


Advantage of this choice of notation:

Main temperature dependence is described by

only weakly varying function of T

Hence, simple polynomial fit possible

⇒ Important for numerical application

Now: examples how to compute the Cij

)()( 0

0 TeTCT ij

u

ij Γ=Ω −

0 kTEeT−

)(TijΓ

i ij i e ijn C n n (T)= Ω


14

Computation of collisional rates: Excitation

Van Regemorter (1962): Very useful approximate formula for allowed dipole transitions

There exist many formulae, made for particular ions and transitions, e.g., (optically) forbidden transitions between n=2 levels in He I (Mihalas & Stone 1968)

coefficients c tabulated for each transition

[ ]

=

=Γ==

Γ

= −

else 2.0

number quantum principal equal with levelsbetween tionsfor transi 7.0

)(276.0,max)( /

n transitioradiative ofstrength oscillator

energy ionizationhydrogen

)(5.14

010000

00

2

0

0

0

0

g

uEegukTEuhvE

f

E

ueuE

EfTnCC

u

ij

ij

H

uHijeij

( ) 2

210

0

logloglogwith

)(0

−−

−

++=Γ

Γ=

TcTcc

TeTnCCu

eij

8


15

Computation of collisional rates: Ionization

The Seaton formula is in analogy to the van-Regemorter formulain case of excitation. Here, the photon absorption cross-sectionfor ionization is utilized:

Alternative: semi-empirical formula by Lotz (1968):

For H und He specific fit formulae are used, mostly from Mihalas(1967) and Mihalas & Stone (1968)

>

=

=

=

=

⋅=−

2 charge with ionsfor 0.3



ionizationfor section -crossphoton threshold

1055.1

0

0

0

130

Z

Z

Z

g

u

e

T

ngC

u

eij

σ

σ

[ ]

atoms individual toadjusted ,quantities empirical

/)()(5.2

01

1110010

2

0

0

a,b,cc uu

uuEubeuEuE

EaTnCC cH

eij

+=

−

=


16

Autoionization and dielectronic recombination

b bound state, d doubly excited state, autoionization level

c ground state of next Ion

d → c: Autoionization. d decays into ground state of nextionization stage plus free electron

c → d → b: Dielectronic recombination. Recombination via a doubly excited state of next lower ionization stage. d auto-ionizes again with high probability: Aauto=1013...1014/sec! But sometimes a stabilizing transition d → b occurs, bywhich the excited level decays radiatively.

0

negative

positive

Energyb

d

c

ion I, e.g. He I

ion II, e.g. He II

ionization energy

9


17

Computation of rates

Number of dielectronic recombinations from c to b:

In the limit of weak radiation fields the reverse process can beneglected. Then we obtain (Bates 1962):

So, the number of dielectronic recombinations from c to b is:

n transitiogstabilizin sspontaneoufor y probabilit == ssdcbc AAnRn

tionautoionizafor y probabilitn transitio

)( with )/(/2/3

1

**dIon

=

Φ==+= −

a

cdec

kTE

ecdsaadd

A

TnneTCnnnAAAnn

)/()( saascdeccbc AAAATnnRn +Φ=


18

Computation of rates

There are two different regimes:

a) high temperature dielectronic recombination HTDR

b) low temperature dielectronic recombination LTDR

for the cases that the autoionizing levels are close to the ionization limit (b) or far above it (a)

a) Most important recombination process He II → He I in the solar corona (T~2⋅106K)

b) Very important for specific ions in photospheres (T< 105K) e.g. N III λ4634-40Å emission complex in Of stars

Reason: upper level is overpopulated, because a stabilizingtransition is going into it.

Because in case b) *

ddsa nnAA =→>>

10


19

LTDR

The radiation field in photospheres is not weak, i.e., the reverse process b ĺ d is induced

Recombination rate:

Reverse process:

These rates are formally added to the usual ionization and recombination rates and do not show up explicitly in the rate equations.

2

3( ) 1

2

mean intensity in stabilizing transition, i.e.,

given by continuum value (line very broad, because short lifetime)

c cb c e cd s

cn R n n T A J

hv

J

= Φ +

Jg

g

hv

cAnJBnRn

b

dsbbdbbcb 3

2

2==


20

Complete rate equations

For each atomic level i of each ion, of each chemical elementwe have:

In detail:

0i j

j i j i

jj iin n PP≠ ≠

− =∑ ∑

( )

( )

( )

( )

*

*

0

i

j i

j i

j

j i

j

j

iji ij

j

ji j

ij ij

j

i

i

j i

i

i

j

R C

nR

nR C

n

Cn

n

C

n

Rn

>

<

>

<

+

+

+

+

+

−

−

=

∑

∑

∑

∑

rates out of i

rates into i

excitation and ionization

de-excitation and recombination



11


21

Closure equation

One equation for each chemical element is redundant, e.g., the equation for the highest level of the highest ionization stage; to see this, add up all equations except for the final one: these rate equations only yield population ratios.

We therefore need a closure equation for each chemical species:

Abundance definition equation of species k, written for example as number abundance yk relative to hydrogen:

∑∑=

hydrogen of numbers population

k species of numbers populationky


22

Abundance definition equation

Notation:

Population number of level i in ionization stage l : nl,i

NLTE levels

LTE levels

E0

do not appear explicitly in the rate equations; populations depend on ground level of nextionization stage:

)(,1,1

*

, Tnnn illeil Φ= +

12


23

Abundance definition equation

Notation:

NION number of ionization stages of chemical element k

NL(l) number of NLTE levels of ion l

LTE(l) number of LTE levels of ion l

Also, one of the abundance definition equations is redundant, since abundances are given relative to hydrogen (otherdefinitions don‘t help) ⇒ charge conservation

Φ++=

Φ+

⇒

++=

+

∑∑∑ ∑∑

∑∑∑ ∑∑

=== =+

=

=== ==

)(

1

)(

1

,

1

)(

1

1,1

)(

1

,

)(

1

*)(

11

)(

1

*

,

)(

1

,

)(1)(HLTE

i

ieprotons

HNL

i

ilk

NION

l

lLTE

i

liel

lNL

i

il

protons

HLTE

i

i

HNL

i

ik

NION

l

lLTE

i

il

lNL

i

il

TnnnyTnnn

nnnynn


24

Charge conservation equation

Notation:

Population number of level i, ion l, element k: nkli

NELEM number of chemical elements

q(l) charge of ion l

e

NELEM

k

NION

l

lLTE

i

klielk

klNL

i

kli

e

NELEM

k

NION

l

klLTE

i

kli

klNL

i

kli

nTnnnlq

nnnlq

=

Φ+

⇒=

+

∑ ∑ ∑∑

∑ ∑ ∑∑

= = =+

=

= = ==

1 1

)(

1

1,1,

),(

1

1 1

),(

1

*),(

1

)()(

)(

13


25

Complete rate equations: Matrix notation

Vector of population numbers

One such system of equations per depth point

Example: 3 chemical elements

Element 1: NLTE-levels: ion1: 6, ion2: 4, ion3: 1

Element 2: NLTE-levels: ion1: 3, ion2: 5, ion3: 1

Element 3: NLTE-levels: ion1: 5, ion2: 1, hydrogen

Number of levels: NLALL=26, i.e. 26 x 26 matrix

( )1 2 NLALL, , , NLALL= total number of NLTE levels

rate equation in matrixA notation =

= L

n b

n n n n


26

1 1

2 2

charge c

ab

ion

undance def. 1 1 1 1 1 1 1 1 1

abundance def. 1 1 1 1 1 1

ons

1

i

erv

on 2

ion 1

ion

H

1

2

t

− −

− −

x x

x x

x x

x x

x x

x x

x x

x x

x x

z z z

z z

y y

y

x

x

x

x

x

x

x

x

x

x

xx x

y z

11 11 12 12 21 21 22 2

1

2

2

25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

n. 0 0 0 0 0

0

0

0

0

=

p ez z z z zq q q q q q q q

n

n

n

n n

LTE contributions

abundances

Ionization into excited states

14


27

Elements of rate matrix

For each ion l with NL(l) NLTE levels one obtains a sub-matrix with the following elements:

( )

( )

( ) ( )

*

*

m<i 1

lower left

upper right

diagonal >

− + < = − + >

+ + + =

∑ ∑

ji ji

iij ji ij

j

km

im mi im im

mi

R C j i

nA R C j i

n

nR C R C j i

n

1 ( ) 1 ( ) highest level in parent ion,

into which ion can ionize; does not have to be ( ) 1 !

= =

= +

K K Ki NL l j NL l k k

l NL l


28

Elements corresponding to abundance definition eq.

Are located in final row of the respective element:

1

1

1,

( 1)

( )

1 1 ( ) except of ground state of excited ions

1 ground state of excited ions

=

=

−= −

=

= …

= + Φ =

∑

∑∑

NION

l

NION

l

ij e l m

m LTE l

i NL l

j NL l

A n j

[ ] [ ]

( )

- ( ) 1

1 =

− = … −

− + Φ =

∑

k

k e m

m LTE H

y j NLALL NL H NLALL

y n j NLALL

15


29

Elements corresponding to charge conservation eq.

Are located in the very final row of rate matrix, i.e., in

1,

( 1)

( ) 1 , except of

ground state of excited ions

( ) ( 1) else −= −

=

= …

=+ − Φ∑

ij

l m

m LTE

i NLALL

q l j NLALL

Aq l q l

Note: the inhomogeneity vector b (right-hand side of statistical equations) contains zeros except for the very last element (i=NLALL): electron density ne (from charge conservation equation)


30

Solution by linearization

The equation system is a linear system for and can be solved if, are known. But: these quantities are in general unknown. Usually, only approximate solutions within an iterative process are known.

Let all these variables change by e.g. in order to fulfill energy conservation or hydrostatic equilibrium.

Response of populations on such changes:

Let with actual quantities

And with new quantities

Neglecting 2nd order terms, we have:

An b= n

, ,e vn T J

, ,e vn T Jδ δ δ

nδ

, ,e vn T J

An b=

( )( ) ( )A A n n b bδ δ δ+ + = +

An b n n A bδ δ δ− = − − +

16


31

Linearization of rate equations

Needed: expressions for:

One possibility:

If in addition to the variables are introduced as unknowns, then we have the

→ Method of Complete Linearization

Other possibility: eliminates from the equation system by expressing through the other variables :

As an approximation one uses

(and iterates for exact solution)

,A bδ δ

n , ,e kn T J

1

NF

e k

ke k

A A AA T n J

T n Jδ δ δ δ

=

∂ ∂ ∂= + +∂ ∂ ∂∑

kJ,en TkJ

( ), ,k eJ f n T n=~ ( , , )d d

k k eJ S n T n

Jνννν discretized in NF frequency points


32


Method of approximate Λ-operators (Accelerated Lambda Iteration)

1

1

NF

e k

ke k

NLALLk k k

k e j

je j

A A AA T n S

T n S

S S SS T n n

T n n

δ δ δ δ

δ δ δ δ

=

=

∂ ∂ ∂= + +∂ ∂ ∂

∂ ∂ ∂= + +∂ ∂ ∂

∑

∑

( )1

analogous, :

0, ,0,=

∂=∂ ∂∂

+ + =∂ ∂∑ Le e

e

NF

k

k k

b

bb bT S

Tb n

Sn

n

δ

δ δδ δδ

17


33


Linearized equation for response as answer on changes

Expressions show the complex coupling of all variables. A change in the radiation field and, hence, the source function at any frequency causes a change of populations of all levels, even if a particular level cannot absorb or emit a photon at that very frequency!

1

1

1 1

NFk

k k

NFk

e

ke k e

NLALL NFk

j

j k k j

A A SAn b nA T n n

T S T

A A Sn n n

n S n

A Sn n

S n

δ δ

δ

δ

=

=

= =

∂ ∂ ∂− = − + − − ∂ ∂ ∂

∂ ∂ ∂+ − − ∂ ∂ ∂

∂ ∂+ −

∂ ∂

∑

∑

∑ ∑

, ,e vn T Jδ δ δnδ

1

NFk

k k j

A Sn

S n=

∂ ∂∂ ∂∑


34


In order to solve the linearized rate equations we need to compute these derivatives:

All derivatives can be computed analytically!

Increases accuracy and stability of numerical solution. More details later.

k, , , with respect to A,b,S∂ ∂ ∂ ∂∂ ∂ ∂ ∂e j kn T n S

18


35

LTE or NLTE?

When do departures from LTE become important?

LTE is a good approximation, if:

1) Collisional rates dominate for all transitions

2) Jv =Bv is a good approximation at all frequencies

*

solution of rate equations LTE

because =

<< → = + ≈

→

ij ij ij ij ij ij

ij i

ji j

R C P R C C

C n

C n

*

solution of rate equations LTE

=

→

→ =

ij ji

i i

j j

R R

n n

n n


36

LTE or NLTE?

When do departures from LTE become important?

LTE is a bad approximation, if:

1) Collisional rates are small

2) Radiative rates are large

3) Mean free path of photons is larger than that of electrons

Example: pure hydrogen plasma

Departures from LTE occur, if temperatures are high and densities are low

~ / , ij e e ij

C n T n T C↓ ↑ ⇒ ↓

~ , >1 ↑ ⇒ ↑ij ij

R T T Rα α

3/ 23/ 2 / /

~ 1/ (density of neutral H)

Saha: ~ ~

,

− ∆ −∆

∆

→ ∆

↓ ↑ ⇒ ∆ ↑

H

E kT E kT

H e p

e p

e

z n

Tn n n T e z e

n n

n T z

19


37

LTE or NLTE?


38

LTE or NLTE?

20


39

LTE or NLTE?

DA white dwarf, Teff= 60000K, log g= 7.5


40

LTE or NLTE?

DA

O w

ith lo

g g

= 6

.5

DO

with

log g

= 7

.5

21


41

Summary: non-LTE Rate Equations


42

Complete rate equations

For each atomic level i of each ion, of each chemical elementwe have:

In detail:

0i j

j i j i

jj iin n PP≠ ≠

− =∑ ∑

( )

( )

( )

( )

*

*

0

i

j i

j i

j

j i

j

j

iji ij

j

ji j

ij ij

j

i

i

j i

i

i

j

R C

nR

nR C

n

Cn

n

C

n

Rn

>

<

>

<

+

+

+

+

+

−

−

=

∑

∑

∑

∑

rates out of i

rates into i





1

Stellar Atmospheres: Solution Strategies

1

Solution Strategies


2

All equations

Radiation Transport Iv(z), Jv(z), Hv(z), Kv(z)

Energy Balance T(z)

Hydrostatic Equilibrium ne(z)

Saha-Boltzmann / Statistical Equilibrium nijk(z)

Huge system with coupling over depth (RT) and frequency (SE)

Complete Linearisation (Auer Mihalas 1969)

Separate in sub-problems

2


3

RT: Short characteristic method

Olson & Kunasz, 1987, JQSRT 38, 325

Solution along rays passing through whole plane-parallel slab

max

maxmax

0

( , , ) ( , , )exp ( )exp

( , , ) (0, , ) exp ( ) exp

Solution on a discrete depth grid , 1, with boundi

dI v I v S

dI v I v S

i ND

τ

τ

τ

τ τ τ τ ττ µ τ µ τ

µ µ µ

τ τ τ ττ µ µ τ

µ µ µ

τ

+ +

− −

′ ′ − −′= − + −

′ ′−

′= − + −

=

∫

∫

1

max

ary conditions:

( , ) (0, , )

( , ) ( , , )ND

I v I v

I v I v

µ µ

µ τ µ

− −

+ +

=

=


4

Short characteristic method

Rewrite with previous depth point as boundary condition for

the next interval:

( )

( )

( )

1

1 1

1

1

( , , ) ( , , ) exp ( , , )

( , , ) ( , , ) exp ( , , )

with

using a linear interpolation for the spatial variation of

the intergrals can be evaluated

i i i i

i i i i

i i

i

i

I v I v I S v

I v I v I S v

S

I

τ µ τ µ τ µ

τ µ τ µ τ µ

τ ττ

µ

+ + ++

− − −− −

−

−

±

= −∆ + ∆

= −∆ +∆

−∆ =

∆

1 1

as

i i i i i i iI S S Sα β γ± ± ± ±− +∆ = + +

3


5


Out-going rays:

( )

1 1

1

/ / / /

/ / / /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 11

1

i i

i i

i ii

ii i

a a b a

i a

a b b a

i b

d dI S v S S

x g x x a b

ew e e e e

w e e e e

τ τ

τ τ

µ µ µ µ

µ µ µ

τ τ ττ τ τµ

µ µ µ µ µ

τττ τ

µ µ

β

γ

+ +

+

+

−∆+ − − −

+ − − −

′ ′ ′ ′ −∆ = − = −

′ ∆

= = − = = ∆ =

− ⇒ = = + − = + ∆ ∆

⇒ = = − − −∆

∫ ∫

( ) 1eeµ

−∆−∆ −

= − − ∆


6


In-going rays:

( )

11

11

/ / / /

/ /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 1

1

i i

i i

i ii

ii i

b a b a

i a

b b b

i b

d dI S v S S

x g x x a b

ew e e e e e

w e e e

τ τ

τ τ

µ µ µ µ

µ µ

τ τ ττ τ τµ

µ µ µ µ µ

τττ τ

µ µ

α

β

−

−

−

−−

−∆−− −∆

−−

′ ′ ′ ′−∆ = − = −

′ ∆

= = = = ∆ =

− ⇒ = = − + − = − + ∆ ∆

⇒ = = −∆

∫ ∫

( )/ / 11

a ee

µ µ−∆−

− = − ∆

4


7


Also possible: Parabolic instead of linear interpolation

Problem: Scattering

Requires iteration

1

1

1 , ( )

2e e e e e e

n J I dκ σ η κ κ µ µ−

= = = ∫


8

Solution as boundary-value problem

Feautrier scheme

Radiation transfer equation along a ray:

Two differential equations for inbound and outbound rays

Definitions by Feautrier (1964):

( )( ) ( )

pp:

sp:

vv v

dII S

d

dtd

d

d dZ

ττ τ

τ

τ κµ

τ κ

±±± = −

=

= −

( )

( )

symmetric, intensity-like

antisymmetric, flux-l

1

2

k1

i e2

+ −

+ −

= +

= −

u I I

v I I

5


9

Feautrier scheme

Addition and subtraction of both DEQs:

One DEQ of second order instead of two DEQ of first order

2

2

( )( ) ( )

( )( )

(1)

(2

( )( )

)

( )

= −

=

⇒ = −

v

v

dvu S

d

duv

d

d uu S

d

ττ τ

ττ

ττ

ττ τ

τ


10

Feautrier scheme

Boundary conditions (pp-case)

Outer boundary ... with irradiation

Inner boundary


0

(2

(

)

0) 0 ( 0) ( 0)

( ) ( 0)

−

=

= = → = = =

⇒ = =

I u v

duu

d τ

τ τ τ

ττ

τ

max max

max

max

max max max

max

( ) ( ) ( )

(( )

) )2 (

+ + +

+

=

= = → + =

⇒ = −

I I u v I

duI u

d

τ τ

τ

τ τ

τ τ τ τ

ττ

τ

0

0

0

( 0)

( )( 0)

I I u v I

duu I

d τ

τ

ττ

τ

− − −

−

=

= = → − =

⇒ = = −

6


11

Finite differencesApproximation of the derivatives by finite differences:

( )

1 2 1 2

2

2

1 2 1

1 1

1 1

discretization on a scale

at interfirst derivative

second

mediate points:

1

2

( ) ( ) and

(

derivative

)

:

)

(

+ −

+ +

+ −

+ −

− = −

= +

− −≈ ≈

− −

=

i i

i

i i i

i i i i

i i i i

d uu S

d

u u u udu du

d d

du

dd du

d d

τ τ

τ

ττ

τ τ τ

τ τ

τ τ τ τ τ τ

τ

τ

τ τ

( )

1 2 1 2

1 2 1 2

1 1

2

1 1

2

1 1

( )

( )

1

2

+ −

+ −

+ −

+ −

+ −

−

−

− −−

− −≈

−

i i

i

i i

i i i i

i i i i

i i

du

d

u u u u

du

d

τ τ

τ

τ

τ τ

τ τ

τ τ τ ττ

τ τ τ


12

Finite differences

Approximation of the derivatives by finite differences:

( )

( )( )

( ) ( )

2

2

1 1

1 1

1 1

1 1

1

1 1 1

1

1 1 1

discretisation on a scale

, 2 11

2

, 2 1

1

2

1

2

1

i i i i

i i i ii i

i i

i i i i i i i

i i i i i

i i i i i

i

d uu S

d

u u u u

u S i ND

Au B u C u S i ND

A

C

B

ττ

τ τ τ τ

τ τ

τ τ τ τ

τ τ τ τ

+ −

+ −

+ −

− +

−

− + −

−

+ + −

− = −

− −−

− −⇒ − = = −

−

⇒ − + − = = −

= − −

= − − =

L

L

i iA C+ +

7


13

Linear system of equations

Linear system for ui

Use Gauss-Jordan elimination for solution

11 1 1 1

2 2 2 2 2

1 1 1

2

1

*

0

0

=

ND ND

ND

ND ND

ND ND ND ND

B C u W

A B C u W

C u

S

S

S

S

W

A B u W

−− − −

− − − = −

−


14

Upper diagonal matrix

1st step:

( ) ( )

1 11

2 22

1 11

1 1

1 1 1 1 1 1

1 1

1 1

1

1 0

0

1

1

2

ND NDND

ND ND

i i i i i i i i i

u WC

u WC

u WC

u W

i C B C W B W

i ND C B AC C W B AC

− −−

− −

− −

− −

−

− = −

= = =

= = − = −

%%

%%

%%%

% %

% % %%L ( )1i i iW AW −+ %

8


15

Back-substitution

2nd step:

Solution fulfils differential equation as well as both boundary

conditions

Remark: for later generalization the matrix elements are

treated as matrices (non-commutative)

11 1

ND ND

i i i i

i ND u W

i ND u W C u +

= =

= − = +

%

%%L


16

Complete Linearization

Auer & Mihalas 1969

Newton-Raphson method in 5n

Solution according to Feautrier scheme

Unknown variables:

Equations:

System of the form:

[ ]1 , 1 , , , ,Ti

i i ND

i

Ji ND

nψ ψ ψ ψ ψ

= = =

rr r r r

L L Lr

, 1, , , , 1, , ( ) 0 NF transfer equations

( ) 0 NL equations for SE

− +− + − − =

− =

r

rr ri k i k i k i k i k i k i k i

i i i

A J B J C J S n

P J n b

, ( ) 0 , 1if NF NLα ψ α= = +L

9


17


Start approximation:

Now looking for a correction so that

Taylor series:

Linear system of equations for ND(NF+NL) unknowns

Converges towards correct solution

Many coefficients vanish

0

, ( ) 0if α ψ ≠

0

, ( ) 0 , if iα ψ δψ α+ = ∀

0

0

, ,

, ,0

, , ,

1 1 1, ,

0 ( ) ( )

( )

i i

ND NF NLi i

i i k i l

i k li k i l

f f

f ff J n

J n

α α

α αα

ψ

ψ ψ δψ

ψ δ δ= = =

= = +

∂ ∂ = + + +

∂ ∂ ∑ ∑ ∑ L

, , , i k i lJ nδ δ


18

Complete Linearization - structureOnly neighbouring depth points (2nd order transfer equation

with tri-diagonal depth structure and diagonal statistical

equations):

Results in tri-diagonal block scheme (like Feautrier)

, , 1 1( ) ( , , )i i i i if fα αψ ψ ψ ψ− +=r r r

1 1

, 1 ,

1

,

0 0

0

0 0

0 0

0

0

0

0 0

i i i i i i i

i k i i k i

i i

i k

A B C L

A J B J

n n

C

δψ δψ δψ

δ δ

δ δ

− +

−

−

− + − =

− +

−

rr r r

O O r r

O O

r r

O

O1

0

,

1

( )

i

i

i

J

f

n

α

δ

ψ

δ

+

+

= −

r

r

10


19

Complete Linearization - structure

Transfer equations: coupling of Ji-1,k , Ji,k , and Ji+1,k at the

same frequency point:

→Upper left quadrants of Ai, Bi, Ci describe 2nd derivative

Source function is local:

→ Upper right quadrants of Ai, Ci vanish

Statistical equations are local

→ Lower right and lower left quadrants of Ai, Ci vanish

2

2

d J

dτ


20

Complete Linearization - structureMatrix Bi:

( )( )

,

,

,

,

, ,1 ,

0

0

i k

i k

i l

i

NLi l m

i m i l lm i k

SB

n

B

Pn P

J

′

′= ′

∂ − ∂

= ∂

∂

∑

O M

L L

O M

M M

L L L L

M M

1 ... NF 1 ... NL

1 ... N

F1 ... N

L

11


21


Alternative (recommended by Mihalas): solve SE first and

linearize afterwards:

Newton-Raphson method:

• Converges towards correct solution

• Limited convergence radius

• In principle quadratic convergence, however, not achieved

because variable Eddington factors and τ-scale are fixed

during iteration step

• CPU~ND (NF+NL)3 → simple model atoms only

– Rybicki scheme is no relief since statistical equilibrium not as

simple as scattering integral

1( ) 0 ( )i i i i i iP J n b n P J b−− = → =r rr rr r


22

Energy Balance

→ Including radiative equilibrium into solution of radiative

transfer → Complete Linearization for model atmospheres

→Separate solution via temperature correction

+ Quite simple implementation

+ Application within an iteration scheme allows completely linear

system → next chapter

− No direct coupling

− Moderate convergence properties

12


23

Temperature correction – basic scheme

0. start approximation for

1. formal solution

2. correction

3. convergence?

Several possibilities for step 2 based on radiative equilibrium

or flux conservation

Generalization to non-LTE not straightforward

With additional equations towards full model atmospheres:


• Statistical equilibrium

0( ) ( )T Tτ τ←

( )v v v

J S T= Λ

( ) ( ) ( )T T Tτ τ τ← +∆


24

LTE

Strict LTE

Scattering

Simple correction from radiative equilibrium:

( ) ( ( ))v vS B Tτ τ=

( ) (1 ) ( ( )) ( )v e v e vS B T Jτ β τ β τ= − +

( )

[ ]( )

( )

0

0

( )0

( )0 0

( , ) ( , ) ( ( ), ) 0

( , ) ( , ) ) 0

0

( )(

∞

=

∞

∆

=

∞

==

∞ ∞

== =

− ≠

→ − + =

∂⇒ − − =

∆

∆ ∂

∂⇒ = −

∂∆

∫

∫

∫

∫ ∫

v v

v

v vT

v

v

v v

T Tv

vv v

T Tv v

v J v B T v dv

v J v B T dv

BJ B dv

T

BJ B dv dv

T

T

T

T

τ

τ

κ τ τ τ

κ τ τ τ

κ

κ

τ

κ

13


25

LTE

Problem:

Gray opacity (κ independent of frequency):

deviation from constant flux provides temperature correction

( )( )0 0

independent of the temperature 0

vv v

T Tv v

v v

BT J B dv dv

T

J B T

τ

τ

κ κ∞ ∞

== =

→∞

∂∆ = −

∂

→ ⇒ ∆ →

∫ ∫

( )0

0.Moment equation

( ) ( )

( ) 0

( )

v v

v

v J B dv J B

J B B

J B B

dHB

dt

κ κ

κ

κ κ

κ

∞

=

− → −

→ − −∆ =

→ − = ∆

→ = ∆

∫


26

Unsöld-Lucy correction

Unsöld (1955) for gray LTE atmospheres, generalized by

Lucy (1964) for non-gray LTE atmospheres

( )

0 0

0

0-th moment

1st m

:

, , ,

:

,

now new quantities , , fulfill

omen

ing ra

t

diat

∞ ∞

= =

∞

=

= −

→ = − = = =

=

→ = =

′ ′ ′

∫ ∫ ∫

∫ ∫

L

L

vv v v

JB v v J v v B

B v v

vv v

HH v v

B v

dHJ B

dt

dHdv J B B B dv J J dv d dt

d

dKH

dt

dKdv H H H dv

d

J H K

κ

κκ κ κ κ τ κ

τ κ

κ

κκ κ

τ κ

4

eff

radiative equi

ive equilibrium (local) and

flux conservation (non local)

: librium

flux conservation

0

: 4

′′ ′= − =

′′= =

J

B

H H

B B

dHJ B

d

dKH T

d

κ

τ κ

κ κ σ

τ κ κ π

14


27


Now corrections to obtain new quantities:

0

0 0 0 0

0

integrate (0)

, (0) (0) (0)

(0)

(0) (0)

(0) (0

) 1

′=

∞ ∞ ∞ ∞

′=

′∆ = −

′→ ∆ = ∆ + ∆

= = = = = =

∆′→ ∆ = + ∆ =

∆→ ∆ = − + +

∆= ∆

∆= −∆

∆

∆∆

∫

∫ ∫ ∫ ∫

∫

HH

B

J

B

v v v v v v

H

B

B

J

B

X X X

K K Hd

K K dv f J dv fJ H H dv h J dv hJ

f H

d H

d

K Hd fh

f HB

fh

d KH

d

d H

fB

J

Jd

τ

τ

τ

τ

τ

κτ

κ

κ

κ

τ κ

τ

τκ

κ

κ

κ

κ

κ

0

3

0

3

0

4 (0) (0) 1

(0) (0) 1

4

′=

′=

′=

′∆

∆

′∆ = ∆ = − + + ∆

∆′∆ = + + ∆

′+

−

∫

∫

∫J

B

H

B

J H

B B

J H

B B

Hd

T dH dH

d d

f HB T Hd

fh f

f HT HdB

fh fJ

T

τ

τ

τ

τ

τ

τ

τκ

κ κστ

π κ κ

κ κπτ

κ κ

κ

κσ

τ τ


28


„Radiative equilibrium“ part good at small optical depths but

poor at large optical depths

„Flux conservation“ part good at large optical depths but poor

at small optical depths

Unsöld-Lucy scheme typically requires damping

Still problems with strong resonance lines, i.e. radiative

equilibrium term is dominated by few optically thick

frequencies

3

0

(0) (0) 1

4

J J H

B B B

f HT J B Hd

T fh f

τ

τ

κ κ κπτ

σ κ κ κ′=

∆′∆ = − + + ∆

∫

J B→

0dH

dτ→

15


29

NLTE Model Atmospheres

Radiation Transport and Sattistical Equilibrium are very

closely coupled

Simple separation (Lamda Iteration) does not work


Accelerated Lambda Iteration


30

Lambda Iteration

Split RT and SE+RE:

• Good: SE is linear (if a separate T-correction scheme is used)

• Bad: SE contain old values of n,T (in rate matrix A)

Disadvantage: not converging, this is a Lambda iteration!

( )0

( , )

( , )

( , , ) ( , , ) 0

new old

new

v v

J S n T

A J T n b

v n T J S v n T dvκ∞

= Λ

=

− =∫

RT formal solution

SE

RE

16


31

Accelerated Lambda Iteration (ALI)

Again: split RT and SE+RE but now use ALI

• Good: SE contains new quantities n, T

• Bad: Non-Linear equations → linearization (but without RT)

Basic advantage over Lambda Iteration: ALI converges!

( )

* *

0

( , ) ( , ) ( , )

( , )

( , , ) ( , , ) 0

new old old old new new new old old old

newnew new

new new new new new

v v

J S n T S n T S n T

A J T n b


= Λ +Λ −Λ

=

− =∫

RT

SE

RE


32

Example: ALI working on Thomson scattering problem

amplification factor

( )

( )( ) ( )( )

( ) ( )( )( )

* *

* *

*

*

:=formal solution on

solve for

1

1

1 1

⇒ = Λ −Λ +Λ

= Λ −Λ +Λ

= −

= −

− +

− −Λ − =

= − Λ −

⇒

+

− +

new

new new

e e

new new new

e e

e e

old old

old old old

new old

FS

FS

FS

e

old new new

e e

new

e

S B J

S S

B J B

S

B J

B J

J S

J

J

J J J J

J

B

β β

β β

β

β

β

β

β

β

( )

( )

1* *

1*

1 subtract on both sides

1

−

−

= − Λ −Λ

⇒ − = − Λ −

FS old old

e e

new old FS old

e

J J J

J J J J

β β

β

Interpretation: iteration is driven by difference (JFS-Jold) but: this difference

is amplified, hence, iteration is accelerated.

Example: βe=0.99; at large optical depth Λ* almost 1 ĺ strong amplifaction

source function with scattering, problem: J unknownĺiterate

17


33

What is a good ȁ*?

The choice of ȁ* is in principle irrelevant but in practice it decides about the success/failure of the iteration scheme.

First (useful) ȁ* (Werner & Husfeld 1985):

A few other, more elaborate suggestions until Olson & Kunasz (1987): Best ȁ* is the diagonal of the ȁ-matrix(ȁ-matrix is the numerical representation of the integral operator ȁ)

We therefore need an efficient method to calculate the elements of the ȁ-matrix (are essentially functions of ττττνννν ).Could compute directly elements representing the ȁ-integral operator, but too expensive (E1 functions). Instead: use solution method for transfer

equation in differential (not integral) form: short characteristics method

( )*( )

, ' ( ')0

>Λ =

≤v

v v

SS

τ τ γτ τ τ

τ γ


34

Towards a linear scheme

ȁ* acts on S, which makes the equations non-linear in the

occupation numbers

• Idea of Rybicki & Hummer (1992): use J=ǻJ+Ȍ*Șnew instead

• Modify the rate equations slightly:

( )*

0 0

*3

2

0

*3

*

2

0

4 4 ( )

24

24 ( )

ij ij

ij i i v

ijiji j j v

j

i

i

j

ji

j

R n n J dv J dvhv hv

n hvR n n J dv

n hv c

n hvJ dv

n

n n

nhv

nc

σ σπ π η

σπ

σπ η

∞ ∞

∞

∞

= = Ψ +∆

= +

= Ψ +∆ +

∫ ∫

∫

∫

18


35

Stellar Atmospheres

This was the contents of our lecture:

Radiation field

Radiation transfer

Emission and absorption

Energy balance and Radiative equilibrium

Hydrostatic equilibrium

Solution Strategies for Stellar atmosphere models


36

Stellar Atmospheres


Radiation field

Radiation transfer


Radiative equilibrium


Stellar atmosphere models

The End

19


37

Stellar Atmospheres


Radiation field

Radiation transfer





The End

Thank you for

listening !

1

Stellar Atmospheres: Non-LTE Stellar Atmospheres

1

Stellar Atmospheres in Non-LTE


2

Stellar Atmospheres in Non-LTE

radiation transfer Iv or Jv

radiative equilibrium solve consistently (∗) T

hydrostatic equation N

Parameters: Teff , log g, yk

LTE: strict LTE

including scattering

Population numbers by Saha-Boltzmann equations

NLTE: rate equations

(1 )

v v

v v v

S B

S J Bρ ρ

=

= + −

0idnAn b

dt= → =

2


3

Solution Methods

(∗) is a non-linear system of equations, we look for the solution vector:

Solution principle: Newton-Raphson iteration

Solution methods:

1. Complete Linearization (Auer & Mihalas 1969)

2. Multi-frequency/multi-gray (Anderson 1985)

3. ALI method (Werner, Husfeld 1985, 1986)

All methods have in common: linearization and iteration

For that, it is necessary to invert matrices (Jacobi matrix) with rank = number of equations

Numerical limit: matrix inversion limits rank to the order of

~ 100

( )( ) ( )

11, , , , , , , ,

with

=

=

L LNF

dd

NL e v v

d d d

n n N T n J J

M c

ψ

ψ ψ ψ

d=depth index


4


Linearizes all equations

Enabled break-through for first calculation of NLTE models,

quite robust method

Depth coupling by radiation transfer

⇒ Feautrier scheme

Disadvantage: Capacity limit quickly reached

e.g. model atom with hydrogen and helium:

20 NLTE levels

80 frequency points

(5 for each spectral line, 2 for each bound-free edge)

Only rather rudimentary representation of plasma

⇒ Number of equations must be reduced

100 equations

3


5

Anderson‘s method

Does not linearize the transfer equation with respect to all

frequency points. First: grouping of frequency points in

energy blocks. Then: linearization of these quantities.

Number of blocks determines the dimension of the system of

equations.

In some sense related to multi-grid methods.

Very clever method, BUT: requires physical motivation for

grouping of frequencies. Must be done manually, quite

cumbersome, much experience and physical insight by

user necessary. Was essentially used by inventor himself,

is not used any more.


6

ALI method


Eliminates the explicit inclusion of the transfer equation into

the linearization scheme by using instead an implicit

approximate solution for Jv:

Lambda iteration:

ALI:

( )( )

( )

(

( 1)1

()

(

)

) ii

v

i

v

i

v

i

J n

n

S

A bJ

− −= Λ

=

( ) ( ) ( )( )

( )

( 1) (( 1) * ( ) * ( 1)( )

( )

( )

( )

(

1)

)

* ( ) ( 1)

( )

i i i

v v v

i i

i

v

i

v

i i

i

v

i

i

i

v

v

n nnS S S

S

J

bJA

n JJ

n

− −− −

−

= Λ + Λ − Λ

= Λ + ∆

=

i=iteration counter

4


7

ALI method

Advantage: number of frequency points no longer appears in

dimension of equation system to be linearized (but

calculation of derivatives of ην,κν w.r.t. source function)

No explicit depth coupling, i.e. local linearized equations for

every depth point

Starting solution

Calculate correction

from linearized equation

Improved solution

( )

( )

( )

1

1

1

, , , , ,

, , , , ,

dd

NL e

dd

NL e

d d d

d d d

d d d

n n N T n

n n N T n

M c

M c

ψ

δψ δ δ δ δ δ

δψ

δψ

ψ δψ ψ

−

=

=

=

=

+ →

L

L

1

Stellar Atmospheres: Radiation Transport as Boundary-Value Problem

1

Radiation Transport as Boundary-Value

Problem of Differential Equations


2

Solution with given source function

• Formal Solution, applications:

– Strict LTE,

– Step within iterative method

• Numerical integration, short characteristics method

• Algebraic equation, Feautrier method

( )v vS B T=

2


3

Solution by numerical integration

Emergent intensity, plane-parallel geometry

Depth grid, ND depth points

• Geometrical depth

• Optical depth

z

ϑϑϑϑ

( 0, ), cosvI τ µ µ ϑ= =

10t =

Lt

maxNDt t=

1,L L NDt = L

0

1 ,, ( ) ( , )Lt

L L

t

L ND v v t dtτ τ κ′=

= ′ ′= ∫L

F


4

Emergent intensity

Numerical integration:

Trapezoidal rule,

naive approach:

Not very smart, systematic

summation of approximation

errors

max

maxmax

0

( 0, ) ( ) exp ( )expd

I S I

τ

ν ν ν

τ

ττ ττ µ τ τ

µ µ µ+ +

′=

′ ′ ′= = − + −

∫ F

1

( 0, )ND

L L

L

I S wν τ µ+

=

= =∑

Laτ

µ= 1Lb

τ

µ+=

( )2

a b

a b

b aI S e S e− − −

= +

3


5

Proper integration weights

Including weight function in integration weights is smart if:

( )

Problem: ( ) ( )

Interval: ,

Integrand: ( )

Weight function: ( )

b

a

I f x g x dx

a b

f x

g x

= ∫

( ) is less curved than ( ) ( )

( ) and ( ) have handy antiderivatives

f x f x g x

g x x g x


6

Determination of proper weights

( )f x

x

r1− 0 1

a m b

∆

( )2

2

2

( ) ( ) ( ) ( )( ) , even and odd

2 2

b a

b am

x m r

dx dr

f b f a f b f af x r

∆ = −

+=

∆= +

∆=

+ −= +

( )( ) ( )

( ) ( )( )

( ) ( )( )

1

1

( ) ( ) ( ) ( ) ( ) ( )24

like wise:

( ) ( ) ( )

1( )

2 22

1( )

2 22

b

a r

e o

e

o

I f x g x dx f b f a r f b f a g m r dr

g x g r g r

g r g m r g m r

g r g m r g m r

=−

∆ ∆= = + + − +

= +

∆ ∆= + + −

∆ ∆= + − −

∫ ∫

4


7

Determination of proper weights

Integration over symmetric interval leaves only even

integrands:

( ) ( )

( ) ( )

( ) ( )

1 1

1 1

1 1

0 0

( ) ( ) ( ) ( ) ( ) ( )4

( ) ( ) ( ) ( ) ( ) ( )2

( ) ( ) G ( ) ( ) ( ) ( )2

( )2

( )2

e o

r r

e o

r r

a b

a

b

I f b f a g r dr f b f a rg r dr

f b f a g r dr f b f a rg r dr

f b f a f b f a H w f a w f b

w G H

w G H

=− =−

= =

∆= + + −

∆

= + + −

∆= + + − = +

∆⇒ = −

∆⇒ = +

∫ ∫

∫ ∫


8

Examples

,6

2 6

6

( )

2

a

b

g x

G m H

w

w m

x

m

∆= =

∆ ∆ = −

∆ ∆+

=

=

2

22

22

22

,12 3

2 12 3

2 12 3

( )

a

b

g

G m H m

w

x

m

w m m

x

m

∆ ∆= + =

∆ ∆ ∆= + −

∆ ∆ ∆

+

=

= +

( )

( )

( )

( )

1

1 2

(

1

1

)

a b

a b a b

a a b

a

b a

b

x

b

G e e

H e e

g x e

e e

w e e e

w e e e

− −

− − − −

−

− − −

− − −

= −∆

= − − + −

∆ ∆

= −

−∆

=

−∆

= + −

( )

( )

( )

( )

1

(

2

)

1

1

1

a b

a b a b

a a b

a

b a b

x

b

G e e

H e e e e

w e e e

w e e

g e

e

x

−

= − +∆

= + + −

∆ ∆

= − − −∆

= + −∆

=

5


9


Olson & Kunasz, 1987, JQSRT 38, 325

Solution along rays passing through whole plane-parallel slab

max

maxmax

0

( , , ) ( , , )exp ( )exp

( , , ) (0, , ) exp ( ) exp

Solution on a discrete depth grid , 1, with boundi

dI v I v S

dI v I v S

i ND

τ

τ

τ


µ µ µ


µ µ µ

τ

+ +

− −

′ ′ − −′= − + −

′ ′−

′= − + −

=

∫

∫

1

max

ary conditions:

( , ) (0, , )

( , ) ( , , )ND

I v I v

I v I v

µ µ

µ τ µ

− −

+ +

=

=


10


Rewrite with previous depth point as boundary condition for

the next interval:

( )

( )

( )

1

1 1

1

1

( , , ) ( , , ) exp ( , , )

( , , ) ( , , ) exp ( , , )

with


the intergrals can be evaluated

i i i i

i i i i

i i

i

i

I v I v I S v

I v I v I S v

S

I

τ µ τ µ τ µ

τ µ τ µ τ µ

τ ττ

µ

+ + ++

− − −− −

−

−

±

= −∆ + ∆

= −∆ + ∆

−∆ =

∆

1 1

as

i i i i i i iI S S Sα β γ± ± ± ±− +∆ = + +

6


11


Out-going rays:

( )

1 1

1

/ / / /

/ / / /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 11

1

i i

i i

i ii

ii i

a a b a

i a

a b b a

i b

d dI S v S S

x g x x a b

ew e e e e

w e e e e

τ τ

τ τ

µ µ µ µ

µ µ µ

τ τ ττ τ τµ

µ µ µ µ µ

τττ τ

µ µ

β

γ

+ +

+

+

−∆+ − − −

+ − − −

′ ′ ′ ′ −∆ = − = −

′ ∆

= = − = = ∆ =

− ⇒ = = + − = + ∆ ∆

⇒ = = − − −∆

∫ ∫

( ) 1eeµ

−∆−∆ −

= − − ∆


12


In-going rays:

( )

11

11

/ / / /

/ /

( , , ) exp exp exp

, ( ) exp( ) , , ,

1 1

1

i i

i i

i ii

ii i

b a b a

i a

b b b

i b

d dI S v S S

x g x x a b

ew e e e e e

w e e e

τ τ

τ τ

µ µ µ µ

µ µ

τ τ ττ τ τµ

µ µ µ µ µ

τττ τ

µ µ

α

β

−

−

−

−−

−∆−− −∆

−−

′ ′ ′ ′−∆ = − = −

′ ∆

= = = = ∆ =

− ⇒ = = − + − = − + ∆ ∆

⇒ = = −∆

∫ ∫

( )/ / 11

a ee

µ µ−∆−

− = − ∆

7


13


Also possible: Parabolic instead of linear interpolation

Problem: Scattering

Requires iteration

1

1

1 , ( )

2e e e e e e

n J I dκ σ η κ κ µ µ−

= = = ∫


14

Determination of mean intensity and flux

Discrete angular points

Solution along each ray

Angular integration

Gauss integration with 3 points sufficient for pp-RT

Alternative: numerical integration of moment equation

Problem: numerical approximation of

10t =

Lt

maxNDt t=

, 1j

j NAµ = L

1

weights depend on ND

j L Lj j

L

I S w µ±

=

=∑

1 1

1 1 , H

2 2

NA NA

j j j j j

j j

J I w I wµ= =

= =∑ ∑

F2 ( )E τ

8


15

Spherical geometry

Observer

Z

P

core

r=1

Pj Zi

rl

Z1

r=Rmax

1

1 max

, 1 ,

0, , inters

Impact parameter

ecting the core

1

s

, ,+

= = +

=

= =

L

L

L

j

NC

NC NP

P j NP NP ND NC

P P

P P R

1 max

, 1

, 1

Radii

l

ND

r l ND

r R r

=

= =

L

( )

2

i

2

2 2

2 2

, 1 intersecting t

Z

he core

, 1 1- ,

, 1 1- , 2 1

points

= + − =

= + − = + =

= − − = + = + − −

L

L

L

i l j

i l j

i l j

Z r P i ND

Z r P l NP j i l

Z r P l NP j i NP j l


16

Spherical geometry

Numerical integration on this grid, e.g.:

Optical depth

Emergent intensity

Emergent flux

( )1

2 2iZ

i i j

Z

r Z P dZτ κ= = +∫

maxmax

01

( )i

j l i

i

I S e d S wτ

τ

ττ τ′+ −

′==

′ ′= =∑∫

max

01

1 1( )

2 2

NPR

j jP

j

H I P PdP I w+ + +

==

= = ∑∫ F

9


17

Solution as boundary-value problem

Feautrier scheme

Radiation transfer equation along a ray:

Two differential equations for inbound and outbound rays

Definitions by Feautrier (1964):

( )( ) ( )

pp:

sp:

vv v

dII S

d

dtd

d

d dZ

ττ τ

τ

τ κµ

τ κ

±±± = −

=

= −

( )

( )

symmetric, intensity-like

antisymmetric, flux-l

1

2

k1

i e2

+ −

+ −

= +

= −

u I I

v I I


18

Feautrier scheme

Addition and subtraction of both DEQs:

One DEQ of second order instead of two DEQ of first order

2

2

( )( ) ( )

( )( )

(1)

(2

( )( )

)

( )

= −

=

⇒ = −

v

v

dvu S

d

duv

d

d uu S

d

ττ τ

ττ

ττ

ττ τ

τ

10


19

Feautrier scheme

Boundary conditions (pp-case)

Outer boundary ... with irradiation

Inner boundary


0

(2

(

)

0) 0 ( 0) ( 0)

( ) ( 0)

−

=

= = → = = =

⇒ = =

I u v

duu

d τ

τ τ τ

ττ

τ

max max

max

max

max max max

max

( ) ( ) ( )

(( )

) )2 (

+ + +

+

=

= = → + =

⇒ = −

I I u v I

duI u

d

τ τ

τ

τ τ

τ τ τ τ

ττ

τ

0

0

0

( 0)

( )( 0)

I I u v I

duu I

d τ

τ

ττ

τ

− − −

−

=

= = → − =

⇒ = = −


20

Feautrier scheme

Boundary conditions (spherical)

Core rays:

Like pp-case

Non-core rays:

Restrict to one quadrant (symmetry) inner boundary at Z=0:

max

max max max( ) ( ) ( ) 0

( )0

I I v

du

d τ τ

τ τ τ τ τ

τ

τ

+ −

=

= = = → =

⇒ =

max 0τ τ τ= =

max 0τ τ τ= =

11


21

Finite differencesApproximation of the derivatives by finite differences:

( )

1 2 1 2

2

2

1 2 1

1 1

1 1

discretization on a scale

at interfirst derivative

second

mediate points:

1

2

( ) ( ) and

(

derivative

)

:

)

(

+ −

+ +

+ −

+ −

− = −

= +

− −≈ ≈

− −

=

i i

i

i i i

i i i i

i i i i

d uu S

d

u u u udu du

d d

du

dd du

d d

τ τ

τ

ττ

τ τ τ

τ τ

τ τ τ τ τ τ

τ

τ

τ τ

( )

1 2 1 2

1 2 1 2

1 1

2

1 1

2

1 1

( )

( )

1

2

+ −

+ −

+ −

+ −

+ −

−

−

− −−

− −≈

−

i i

i

i i

i i i i

i i i i

i i

du

d

u u u u

du

d

τ τ

τ

τ

τ τ

τ τ

τ τ τ ττ

τ τ τ


22

Finite differences

Approximation of the derivatives by finite differences:

( )

( )( )

( ) ( )

2

2

1 1

1 1

1 1

1 1

1

1 1 1

1

1 1 1

discretisation on a scale

, 2 11

2

, 2 1

1

2

1

2

1

i i i i

i i i ii i

i i

i i i i i i i

i i i i i

i i i i i

i

d uu S

d

u u u u

u S i ND

Au B u C u S i ND

A

C

B

ττ

τ τ τ τ

τ τ

τ τ τ τ

τ τ τ τ

+ −

+ −

+ −

− +

−

− + −

−

+ + −

− = −

− −−

− −⇒ − = = −

−

⇒ − + − = = −

= − −

= − − =

L

L

i iA C+ +

12


23

Discrete boundary conditions

Outer boundary: first order

[ ]

1

2 11 1

2 1

2 11

2 1

1 1 1 2

1

1 2 1 1 1

0

0

, 1

u uduu u

d

u uu

B u C u

C B C

ττ τ τ

τ τ

τ τ−

−= → =

−

−→ − =

−

⇒ − =

= − = +


24


Numerically better is a second-order condition:

Taylor expansion of u(τ) around τ1:

( ) ( )

( ) ( ) ( )

( )

[ ] [ ] [ ]

1 1

2

2 1 2 1 2 1

2

2 1 2 1 2 1

1 1 21 1 1 1 1 2 12

2 1 2 1

2 1 2

1 2 1 1 2 1 2 1

2

2

1 11

1

2

1

2

boundary condit

2 2

2 ,

ion

1 2

D

2

EQ

− − −

= + − + −

= + − + −

−+ + = ⇒ − =

− −

= − = + − + −

−

u u

u u

u u

d

uu S

d u

d

u S

B u C u S

u

B

d

C

u

τ τ

τ τ τ τ

τ τ τ τ

τ τ τ τ

τ τ τ

τ

τ τ

τ

τ

13


25


Inner boundary: first order

[ ]

max

1

1

1

1

*

1

1

1

pp or core rays( )

0 non-core rays 0

0 0

1 ,

ND NDND ND

ND ND

ND ND ND

ND ND

ND ND ND ND ND

ND

ND ND ND ND

ND

I u I uu udu

d

u Iu u

A u B u S

AA B

A

τ τ

τ

τ τ τ

τ τ

τ τ

+ +

−

= −

+

−

−

−

−

−

− −−= → =

−

−→ + =

− ⇒ − + =

+= − = * ,

0ND

IS

+=


26

Discrete boundary conditionsOuter boundary: second order

( ) ( )

( )

2

1 1 1

1 1

2

2

boundary condition

1

2

0 non-core

c

DE

r

Q

o e

− − −

+

− −

= + − + −

= − −

+

−

ND ND

ND ND ND ND ND ND

ND ND ND

D

D

N

N

du d u

d

u uI u

du u

τ τ

τ τ τ ττ τ

τ τ

( ) ( )

( )

[ ] [ ] [ ]

[ ]

2

1

* *112

1

2 2

1

1

1 1

1*

1

1

1

2

2

2 , 1 2

2

2

2

−

−−

−

− −

− −

− +

+

−

−

−

−

− −

−+ = ⇒ − =

−−

−

= − = +

−

+ − −

= + −

ND ND ND ND

ND NDND ND ND ND ND ND ND

ND ND

ND ND ND ND ND ND

ND ND ND

ND

N

ND ND

ND N

D

D

u S

u uu S B u A u S

A B

S S I

I u

τ τ

τ τ

τ τ τ τ

τ

τ

τ τ

τ

τ

14


27




11 1 1 1

2 2 2 2 2

1 1 1

2

1

*

0

0

=

ND ND

ND

ND ND

ND ND ND ND

B C u W

A B C u W

C u

S

S

S

S

W

A B u W

−− − −

− − − = −

−


28

Upper diagonal matrix

1st step:

( ) ( )

1 11

2 22

1 11

1 1

1 1 1 1 1 1

1 1

1 1

1

1 0

0

1

1

2

ND NDND

ND ND

i i i i i i i i i

u WC

u WC

u WC

u W

i C B C W B W

i ND C B AC C W B AC

− −−

− −

− −

− −

−

− = −

= = =

= = − = −

%%

%%

%%%

% %

% % %%L ( )1i i iW AW −+ %

15


29

Back-substitution

2nd step:

Solution fulfils differential equation as well as both boundary

conditions

Remark: for later generalization the matrix elements are

treated as matrices (non-commutative)

11 1

ND ND

i i i i

i ND u W

i ND u W C u +

= =

= − = +

%

%%L


30

Solution with linear dependent source function

Coherent scattering:

General form (complete redistribution)

Thomson scattering

Results in coupling of equations for all directions

1 1

1 0

1 , ( ) ( )

2v v v v vS J J I d u d

µ µ

α β µ µ µ µ=− =

= + = =∫ ∫

( )

,

1 ,

e e e e e

tot e e e e e e

e e e e e e

tot ee e e

e

n J

S S J

S S J

κ σ η κ

η η κ η κ κ κ κκ η

κ κ κ κ κ κ κ κ κ κ κ κ

κβ β β

κ κ

= =

+ + −⇒ = = + = +

+ + + + +

= − + =+

( ) ( )( )

[ ]21

2

0

,, ( ) , 1 ( )e e

d uu u d S

dµ

τ µτ µ β τ τ µ µ β τ

τ=

− − = −∫

16


31

Discretization

Generalization of Feautrier scheme to a block-matrix scheme:

Angular discretization:

Depth discretization as previously:

( )1

10

,NA

ji j

j

u d u wµ

τ µ µ==

≈∑∫

1 1

equations for vector

− +− + − =i i i i i i i

i

Au B u C u W

u


32

Discretization

( ) ( )( )

[ ]21

2

0

1 1

1

,, ( ) , 1 ( )

equations for vector

0 0

, , C

0 0

0

0

=

− +

− − = −

→ − + − =

= = =

=

∫

e e

i i i i i i i

i

i i

i j i i i i

NA i i

i

i i

i

d uu u d S

d

Au B u C u W

u

u A C

u u A A C

u A C

B

B B

B

µ

τ µτ µ β τ τ µ µ β τ

τ

( )

( )

( )

1

1

1 ( )

( ) , 1 ( )

1 ( )

− − = −

−

NA e i

e i e i

NA e i

w w i S

i W i S

w w i S

β

β β

β

Identical lines

pp-case

identical in all

depths

17


33

1;1,1 1;1, 1

1; ,1 1; , 1

2 2;1,1 2;1, 2

2 2; ,1 2; , 2

1;1,1 1;1, 1

1; ,1 1; , 1

;1,1 ;1,

; ,1 ; ,

NA

NA NA NA

NA

NA NA NA

ND ND NA ND

ND NA ND NA NA ND

ND ND ND NA

ND ND NA ND NA NA

B B C

B B C

A B B C

A B B C

B B C

B B C

A B B

A B B

− − −

− − −

− − − −

− −

−

−

−

−

1;1 1;1

1; 1;

2;1 2;1

2; 2;

1;1 1;1

1; 1;

;1 ;1

; ;

NA NA

NA NA

ND ND

ND NA ND NA

ND ND

ND NA ND NA

u W

u W

u W

u W

u W

u W

u W

u W

− −

− −

=

( )

( ) ( )

;

1

; 1

1 ( ) , 1 1

1 ( ) 2

i j e i

ND j e ND ND ND j

W i S i ND

W ND S I

β

β τ τ− +

−

= − = −

= − + −

L

Block matrix


34

Feautrier scheme

Thermal source function

– Decoupled equation for each direction

– Separate Feautrier scheme for each ray

– ND•5 multiplications

Thermal source function + coherent scattering (e.g. Thomson)

– Coupled equations for all directions

– Feautrier scheme in block matrix form

– ND•NA3 multiplications (matrix inversions)

Spherical geometry

– Radius dependent angular integration

– Block matrix size depends on radius

– ~ND4 multiplications

12 2

, 21 , 1 ( ) , ( ) 1-

j

j l

l

Pj JMAX l JMAX l NP l

rµ

= − = = +

L

18


35

Solution with line scattering

Two-level atom, complete redistribution

Photon conservation within a spectral line

Approximately realized for resonance lines

The frequency independent line source function ( ) is a

weighted mean of the mean intensity

ground state

1st excited state

( ) ( ) , ( ) ( )

photon conservation:

( ) ( ) ( )

( )

( ) (generali scattering intergralzed)

= =

=

=

= = +

∫ ∫

∫

∫

L L

v

Line Line

L L v v

Line

L v v L L

Line

v v v v

v dv v J v dv

J v dv

S J v dv S J

κ κ ϕ η η ϕ

η κ

η κ ϕ

ϕ α β

F


36


(non-coherent scattering)

Transfer equation:

Each one equation for each angular and frequency point

Each equation contains intensities from all other points

⇒coupling of all transfer equations to an integro-differential-

equation

This system is linear with respect to the intensity

⇒solution with Gauss-Jordan elimination

1

1

( , , )( , , ) ( )

( )

( , , ) 1( , , ) ( ) ( , , )

( ) 2

vv L

vv v

Line

dI vI v S

d v

dI vI v v I v d d

d v µ

µ τµ µ τ τ

τ

µ τµ µ τ ϕ µ τ µ τ

τ

+

=−

= −

= − ∫ ∫

19


37


Discretization:

Analogous solution as with coherent scattering

Scattering integral:

With appropriate weights including the line profile function

(e.g. Gauss function)

1 2

2

1

( , , )( , , ) ( ) ( ) ( , , ) ( )

( )Line

d u vu v v u v d dv

d vµ

µ τµ τ α τ ϕ µ τ µ β τ

τ

+

=−

− − =∫ ∫

0

01

( ) ( )

v v NF

L k k

kv

J J v v dv J wν

ϕ+∆

=−∆

′= →∑∫

1 1

NA NF

L jk j k

j k

J u w w= =

′→∑∑


38




1 1 1 1

2 2 2 2 2

1 1 1

0

0 ND ND ND

ND ND ND ND

B C u W

A B C u W

C u W

A B u W

− − −

− − − = −

−

20


39

Block matrix

,11

,

,

1 1 1

0 0

= , , C

0 0

0

( )

0

j ijk ijk

j k ijk ijki j j i i

j NF ijk ijkNA

ijk NF

ijki

ijk

u A Cu

u A Cu u u A

u A Cu

B w w w w

BB i

B

α

= = =

′ ′ = −

1

1 1 1 1

( )

, ( )

( )

1

NA NA NF

i

NF NA NA NF

ijk jk ijk

w w w w i

W i

w w w w w w w w i

B A C

β

β

β

′ ′ = ′ ′ ′ ′

= + +

L

no longer identical, depend on k but not on j

Identical lines

( )

1 1

1 1

1 11 1

1

2

i i i i

NA NFi i i i

i i jk j k i

j ki i

u u u u

u u w wτ τ τ τ

α βτ τ

+ −

+ −

= =+ −

− −−

− −′− − =

−∑∑


40


The modified Feautrier scheme is not very efficient

pp: inversion of matrices of rank NA•NF ~ 3 •5

ND•NA3 •NF3 multiplications

sp: inversion of matrices of rank NA•NF ~ 70 •5 \

ND•ND3 •NF3 multiplications

Repeated calculation of identical scattering integrals at each

frequency point

21


41

Rybicki scheme

Rybicki (1971)

Coherent scattering, e.g. Thomson scattering

Again: Feautrier transfer equation:

Discretization:

But now: additional variable

With additional equations

Use alternative grouping

( )1tot

v e v e vS S Jβ β= − +

( )( )

[ ]2

2

,, ( ) ( ) 1 ( ) ( )e e

d uu J S

d

τ µτ µ β τ τ β τ τ

τ− − = −

, 1, , , , 1,1

1 (pp) , 1 (sp)

(1),2 1, ( )

, , like in usual Feautrier scheme

e e

i j i j i j i j i j i j i i i i

i i i

A u B u C u J S

j NA j JMAX

i ND ND

A B C

β β− + − + − − = −

= =

= −

L L

L

FiJ

, ,

j

=i i j i j

J u w∑


42

Rybicki scheme( )

( )

( )

1 11

1

1

e

e

j i i

e

ND ND

S

K S

S

β

β

β

−

= − −

r

,

,

,

0

0

i j

i jj

ND j

w

wW

w

=

1,

,

,

j

i jj

ND j

u

uu

u

=

r

11 1 1

1

1

0

0

1 0 0

0 1 0

jj j j

NANA NA NA

j NA

ND

uT U K

uT U K

uT U K

J

W W W

J

=

− −

rr

rr

rr

1, 1, 1

, , ,

, ,

2

2

0 0

0 0

represents , J describes for the ray J

Scattering integral is repre

− − − −= = −

− −

− − → + =

r r rr r

ej j

ei j i j i jj j i

eND j ND j ND

j j j e j j j j

B C

A B CT U

A B

d uT u u U J j T u U K

d

β

β

β

βτ

1

sented by last block line 0=

− =∑rrNA

j j

j

W u J

22


43

Rybicki scheme

1st step: diagonal elements to unit matrixes

Corresponds to the solution of NA tri-diagonal systems of

equations of rank ND → cpu-time ~ NA•ND2

111

1

1

1 1

1

0

1

0

1

1 00

0 10

,

jj j

NANANA

j NA

ND

j j j j j j

KuU

uU K

uUK

J

W W W

J

U T U K T K− −

= − −

= =

rr %%

rr% %

r r%

%

r r% %


44

Rybicki scheme

2nd step: Wj to zero

Corresponds to cpu-time ~ NA•ND2

111

1 1

1

0

1

0

1

0 0 0

1 ,

jj j

NANANA

NA NA

j j j j

j j

KuU

uU K

uU KJW

Q

W W U Q W K= =

=

= − − = −∑ ∑

rr %%

rr% %

r r% %r

r

rr% %

23


45

Rybicki scheme

3rd step: solve for J

Corresponds to cpu-time ~ ND3

Finished if only J is required

4th step: solve for Feautrier variables u:

1WJ Q J W Q−= → =r rr r

1 1

j j j j ju T K T U J− −= −

r r rr


46

Comparison Rybicki vs. Feautrier

Thomson scattering

Few angular points: take Feautrier

Many angular points: take Rybicki

C1 NP ND2+ C 2ND3C ND4Spherical

C1 NA ND2+ C 2ND3C NA3 NDPlane-parallel

RybickiFeautrier

24


47



Two-level atom or Compton scattering

Each one block line

describes transfer equation for each ray j,k

(direction and frequency dependent!)

→ Huge system of equations

L LS Jα β= +

Jjk jk jk jkT u U K+ =r rr

11 1 1

1

1

0

0

1 0 0

0 1 0

jkjk jk jk

NANFNANF NANF NANF

jk NANF

ND

uT U K

uT U K

uT U K

J

W W W

J

=

− −

rr

rr

rr


48



1

jk i

ND

K

β

β

β

=

r

1,

,

j

ijjk

ND j

u

uu

u

=

r

,

,

,

0

0

i j k

i j kjk

ND j k

w w

w wW

w w

′ ′= ′

1

0

0

jk i

ND

U

α

α

α

− = − −

( ) ( )i v iJ J v dvτ ϕ= ∫

11 1 1

1

1

0

0

1 0 0

0 1 0

jkjk jk jk

NANFNANF NANF NANF

jk NANF

ND

uT U K

uT U K

uT U K

J

W W W

J

=

− −

rr

rr

rr

1, 1,

, , ,

, ,

0

0

jk jk

i jk i jk i jkjk

ND jk ND jk

B C

A B CT

A B

− − −= −

25


49

Comparison Rybicki vs. Feautrier

Line scattering or non-coherent scattering, e.g. Compton

scattering

Few frequency points: take Feautrier or Rybicki

Many frequency points: take Rybicki

Spherical symmetry: take Rybicki

C1 NP NF ND2+ C 2ND3C NF3 ND4Spherical

C1 NA NF ND2+ C 2ND3C NA3 NF3 NDPlane-parallel

RybickiFeautrier


50

Variable Eddington factors

0-th moment

1st moment

2nd moment

Eddington factor:

0-th moment of RT (pp)

1st moment of RT (pp)

1 1

1 0

1( ) ( )

2J I d u d

µ µ

µ µ µ µ+

=− =

= =∫ ∫1 1

1 0

1( ) ( )

2H I d v d

µ µ

µ µ µ µ µ µ+

=− =

= =∫ ∫1 1

2 2

1 0

1( ) ( )

2K I d u d

µ µ

µ µ µ µ µ µ+

=− =

= =∫ ∫

/f K J= F

( , )( , ) ( , )

( )

vv v

dH vJ v S v

d v

ττ τ

τ= −

( , )( , )

( )

vv

dK vH v

d v

ττ

τ=

26


51


Plane-parallel

With variable Eddington factor

With given f and S 2nd-order DEQ for J

Outer boundary:

2

2

( , )( , ) ( , )

( )

vv v

d K vJ v S v

d v

ττ τ

τ= −

[ ]2

2

( , ) ( , )( , ) ( , )

( )

v

v v

d f v J vJ v S v

d v

τ ττ τ

τ= −

[ ]

1 1

0 0

0

( 0) ( 0) / ( 0)

( 0) ( 0) ( 0)

( 0) ( ) ( )

( , ) ( , )( 0) ( , 0)

( )

v

v

h H J

I u v

h u d u d

d f v J vh J v

d v

µ µ

τ

τ τ τ

τ τ τ

τ µ µ µ µ µ

τ ττ τ

τ

−

= =

=

= = = =

= → = = =

→ = =

= = =

∫ ∫


52


Inner boundary:

2nd-order boundary conditions from Taylor series

[ ]

max

max max max

1 1

max

0 0

!

1 1 1

max

0 0 0

max max max

( ) ( ) / ( )

( ) ( ) ( )

: ( ) ( ) ( )

( , ) ( , )( ) ( ) ( , )

( )

H J

v

v v

h H J

h u d u d

H v d I d u d H hJ

d f v J vH h J v

d v

µ µ

µ µ µ

τ τ

τ τ τ τ τ τ

τ τ µ µ µ µ µ

τ τ µ µ µ µ µ µ µ µ µ

τ ττ τ τ τ τ τ

τ

= =

≠ =

+ +

= = =

+

=

= = = =

= =

= = = − = −

= = − = =

∫ ∫

∫ ∫ ∫

1442443 14243

F

27


53

Solution

Discretization → algebraic equation

Tri-diagonal system, solution analogous to Feautrier scheme

Thomson scattering:

Possible without extra costs

[ ]2

2

1 1

( , ) ( , )( , ) ( , )

( )

v

v v

i i i i i i i

d f v J vJ v S v

d v

A J B J C J S

τ ττ τ

τ

− +

− =

→ − + − =

F

[ ]2

2

1 1

(1 )

( , ) ( , )(1 ) ( , ) (1 ) ( , )

( )

( ) (1 )

tot

e e

v

e v e v

e e

i i i i i i i i i

S S J

d f v J vJ v S v

d v

A J B J C J S

β β

τ τβ τ β τ

τ

β β− +

= − +

→ − − = −

→ − + − − = −


54


But the Eddington factors are unknown → iteration

1. Formal solution u with S=B

2. Start value

3. Solution of the moment equation for J→S tot

4. Formal solution u with given S → K

5. New Eddington factors f

6. Converged?

1 3if ≡

, ,

1 1

2

contains , contains

NA NA

i i j j i j j

j j

j j

f u w u w

w wµ µ

= =

′=

′

∑ ∑

28


55


0-th moment

1st moment

2nd moment

0-th moment of RT (sp)

1st moment of RT (sp)

1 1

2

1 0

1( ) ( ) ,

2J I d u d J r J

µ µ

µ µ µ µ+

=− =

= = =∫ ∫ %

1 1

2

1 0

1( ) ( ) ,

2H I d v d H r H

µ µ

µ µ µ µ µ µ+

=− =

= = =∫ ∫ %

1 1

2 2 2

1 0

1( ) ( ) ,

2K I d u d K r K

µ µ

µ µ µ µ µ µ+

=− =

= = =∫ ∫ %

2

( , )( , ) ( , ) ( , )v

v v

r S

dH v rv r S v r J v r

drκ

=

= −

%% %123

[ ]( , ) 1

3 ( , ) ( , ) ( , ) ( , )vv v v

dK v rK v r J v r v r H v r

dr rκ+ − = −

F

F


56


1st moment of RT (sp)

Eddington factor

Introduction of the sphericality factor (Auer 1971),

which corresponds to the integrating factor for the DEQ

[ ]( , ) 1

3 ( , ) ( , ) ( , ) ( , )vv v v

dK v rK v r J v r v r H v r

dr rκ+ − = −

[ ]( )

3 1d fJ J

f Hdr r

κ+ − = −

F

( )2

2 2

1

( )3 ( ) 1 3 ( ) 1( ) exp , ( )

( ) ( )

r d r q rf r f rr q r dr r q r

r f r dr rf r

′ − −′= =

′ ′ ∫

( )( ) ( ) , ( ) exp ( )y f x y g x M x f x dx′ + = = ∫

29


57


1st moment equation:

( ) ( )

( )

2 2

2

2 2

( ) ( )( )

3 1 ( )( ) ( )

d r q r K d r q r dKK r q r

dr dr dr

f d fJr q r fJ r q r H

rf drκ

= +

−= + = −

( ) ( )

( )

( )( )

( )

2

2

2

2

( )

( )

1

1

dx q dr

d qfJd r q r KH H

r q r dr q dr

dHJ S

d qfJdx qJ S

dx qd qfJH

dx

κ

κκ

=−

= − → = −

→

= −

= −

=

%%

%%%

%%%

%%


58

Solution

Iteration of Eddington factors like in pp case

Additional integration of sphericality factor

f=1/3 is a bad starting point, f→1 for r →∞

Computational demand much smaller than for formal solution

2 3 1exp , include weights for

NDl

i i l l

l i l

f drq r w w

f r

−

=

−=

∑

30


59

Non-coherent scattering and moment equation

Two-level atom or Compton scattering

For each frequency point one moment equation of 2nd order

for mean intensity and Eddington factor

Coupled by frequency integral

pp

sp

LS Jα β= +

( ) , ( , )v k ik i k

J v f vτ

1

( )NF

v k k

k

J J v dv J J wϕ=

= → =∑∫

( )2

2

( , ) ( , )( , ) ( ) ( )

( )

v

v vv

d f v J vJ v J v dv

d v

τ ττ α ϕ β τ

τ− − =∫

( )2

2

( , ) ( , ) ( , )( , ) ( , ) ( )

( )

( )

v

v vv

d q r v f r v J r vJ r v q r v J v dv

dx v

r

α ϕ

β

− −

=

∫%

% %

%


60

Non-coherent scattering and moment equation

Feautrier:

cpu-time~ND•NF3

Rybicki:

cpu-time~NF•ND2+ND3

[ ]1

1 1

2

21

, , , , , 1

( )

T

i i ik iNF

i i i i i i i

NF

i ik k

k

J J J J i ND

AJ B J C J

d fJJ J w

d

β

τ

− +

=

= =

− + − =

=∑

rL L L

rr r r

[ ]1

1

1

, , , , , 1

, , , ,

, 0

T

k k ik NDk

T

i ND

NF

k k k k k k

k

J J J J k NF

J J J J

T J U J K W J J=

= =

=

+ = − =∑

rL L L

L L

r rr r

31


61

Multi-level atom

Atmospheric structure assumed to be given

(or accounted for by iteration)

Two sets of equations:

Radiative transfer equation for mean intensity

Statistical equilibrium

Both equations are coupled via radiative rates in matrix P

( )2

2

sum over all bb-, bf-, and ff-transitions

( , ) ( , )( , ) ( , )

( )

( , ) ( )

− =

=∑ ∑

v

v v

lu lu

v v

lu lu

d f v J vJ v S v

d v

S v v

τ ττ τ

τ

τ η κ

[ ]1( ) , T

v NLP J n b n n n= =rr r

L


62

Multi-level atomCoupling of J twofold:

• Over frequency via SE

• Over depth via RT

→ Simultaneous solution

→ non-linear in J

Lambda Iteration:

0. start approximation for n

1. formal solution with given S

2. solution of statistical equilibrium with given J

3. converged?

Not convergent for large optical depths F

32


63


Auer & Mihalas 1969

Newton-Raphson method in 5n

Solution according to Feautrier scheme

Unknown variables:

Equations:

System of the form:

[ ]1 , 1 , , , ,Ti

i i ND

i

Ji ND

nψ ψ ψ ψ ψ

= = =

rr r r r

L L Lr

, 1, , , , 1, , ( ) 0 NF transfer equations

( ) 0 NL equations for SE

− +− + − − =

− =

r

rr ri k i k i k i k i k i k i k i

i i i

A J B J C J S n

P J n b

, ( ) 0 , 1if NF NLα ψ α= = +L


64


Start approximation:

Now looking for a correction so that

Taylor series:

Linear system of equations for ND(NF+NL) unknowns

Converges towards correct solution

Many coefficients vanish

0

, ( ) 0if α ψ ≠

0

, ( ) 0 , if iα ψ δψ α+ = ∀

0

0

, ,

, ,0

, , ,

1 1 1, ,

0 ( ) ( )

( )

i i

ND NF NLi i

i i k i l

i k li k i l

f f

f ff J n

J n

α α

α αα

ψ

ψ ψ δψ

ψ δ δ= = =

= = +

∂ ∂ = + + +

∂ ∂ ∑ ∑ ∑ L

, , , i k i lJ nδ δ

33


65

Complete Linearization - structureOnly neighbouring depth points (2nd order transfer equation

with tri-diagonal depth structure and diagonal statistical

equations):

Results in tri-diagonal block scheme (like Feautrier)

, , 1 1( ) ( , , )i i i i if fα αψ ψ ψ ψ− +=r r r

1 1

, 1 ,

1

,

0 0

0

0 0

0 0

0

0

0

0 0

i i i i i i i

i k i i k i

i i

i k

A B C L

A J B J

n n

C

δψ δψ δψ

δ δ

δ δ

− +

−

−

− + − =

− +

−

rr r r

O O r r

O O

r r

O

O1

0

,

1

( )

i

i

i

J

f

n

α

δ

ψ

δ

+

+

= −

r

r


66

Complete Linearization - structure

Transfer equations: coupling of Ji-1,k , Ji,k , and Ji+1,k at the

same frequency point:

→ Upper left quadrants of Ai, Bi, Ci describe 2nd derivative

Source function is local:

→ Upper right quadrants of Ai, Ci vanish

Statistical equations are local

→ Lower right and lower left quadrants of Ai, Ci vanish

2

2

d J

dτ

34


67

Complete Linearization - structureMatrix Bi:

( )( )

,

,

,

,

, ,1 ,

0

0

i k

i k

i l

i

NLi l m

i m i l lm i k

SB

n

B

Pn P

J

′

′= ′

∂ − ∂

= ∂

∂

∑

O M

L L

O M

M M

L L L L

M M

1 ... NF 1 ... NL1 ... N

F1 ... N

L


68


Alternative (recommended by Mihalas): solve SE first and

linearize afterwards:

Newton-Raphson method:

• Converges towards correct solution

• Limited convergence radius

• In principle quadratic convergence, however, not achieved

because variable Eddington factors and τ-scale are fixed

during iteration step

• CPU~ND (NF+NL)3 → simple model atoms only

– Rybicki scheme is no relief since statistical equilibrium not as

simple as scattering integral

1( ) 0 ( )i i i i i iP J n b n P J b

−− = → =r rr rr r

1

Stellar Atmospheres: Temperature Correction Schemes

1

Temperature Correction Schemes


2

Motivation

Up to now: Radiation transfer in a given atmospheric

structure

No coupling between radiation field and temperature included

→ Including radiative equilibrium into solution of radiative

transfer → Complete Linearization for model atmospheres

(next chapter)

→ Separate solution via temperature correction

+ Quite simple implementation

+ Application within an iteration scheme allows completely linear

system → next chapter

− No direct coupling

− Moderate convergence properties

2


3

Temperature correction – basic scheme

0. start approximation for

1. formal solution

2. correction

3. convergence?

Several possibilities for step 2 based on radiative equilibrium

or flux conservation

Generalization to non-LTE not straightforward

With additional equations towards full model atmospheres:


• Statistical equilibrium

0( ) ( )T Tτ τ←

( )v v vJ S T= Λ

( ) ( ) ( )T T Tτ τ τ← + ∆


4

LTE

Strict LTE

Scattering


( ) ( ( ))v vS B Tτ τ=

( ) (1 ) ( ( )) ( )v e v e vS B T Jτ β τ β τ= − +

( )

[ ]( )

( ) ( )

0

0

( )0

( )0 0 0 0

( , ) ( , ) ( ( ), ) 0

( , ) ( , ) ( ) 0( )

0

∞

=

∞

∆

=

∞

==

∞ ∞ ∞ ∞

== = = =

− ≠

→ − + =

∂⇒ − − = ∂

∂⇒ = − = − ∆

∂

∆

∆

∆

∫

∫

∫

∫ ∫ ∫ ∫

v v

v

v vT

v

v

v v

T Tv

vv v v v v

T Tv v v v

v J v B T v dv

v J v B T dv

BJ B dv

T

BJ B dv

T

T

T dv J B dv B dvT

τ

τ

κ τ τ τ

κ τ τ τ

κ κ

τ

κ

κ κ

3


5

LTE

Strict LTE

Scattering


( ) ( ( ))v vS B Tτ τ=

( ) (1 ) ( ( )) ( )v e v e vS B T Jτ β τ β τ= − +

( )

[ ]( )

( )

0

0

( )0

( )0 0

( , ) ( , ) ( ( ), ) 0

( , ) ( , ) ) 0

0

( )(

∞

=

∞

∆

=

∞

==

∞ ∞

== =

− ≠

→ − + =

∂⇒ − − =

∆

∆ ∂

∂⇒ = −

∂∆

∫

∫

∫

∫ ∫

v v

v

v vT

v

v

v v

T Tv

vv v

T Tv v

v J v B T v dv

v J v B T dv

BJ B dv

T

BJ B dv dv

T

T

T

T

τ

τ

κ τ τ τ

κ τ τ τ

κ

κ

τ

κ


6

LTE

Problem:

Gray opacity (κ independent of frequency):

deviation from constant flux provides temperature correction

( )( )0 0

independent of the temperature 0

vv v

T Tv v

v v

BT J B dv dv

T

J B T

τ

τ

κ κ∞ ∞

== =

→∞

∂∆ = −

∂

→ ⇒ ∆ →

∫ ∫

( )0

0.Moment equation

( ) ( )

( ) 0

( )

v v

v

v J B dv J B

J B B

J B B

dHB

dt

κ κ

κ

κ κ

κ

∞

=

− → −

→ − − ∆ =

→ − = ∆

→ = ∆

∫

4


7


Unsöld (1955) for gray LTE atmospheres, generalized by

Lucy (1964) for non-gray LTE atmospheres

( )

0 0

0

0-th moment

1st m

:

, , ,

:

,

now new quantities , , fulfill

omen

ing ra

t

diat

∞ ∞

= =

∞

=

= −

→ = − = = =

=

→ = =

′ ′ ′

∫ ∫ ∫

∫ ∫

L

L

vv v v

JB v v J v v B

B v v

vv v

HH v v

B v

dHJ B

dt

dHdv J B B B dv J J dv d dt

d

dKH

dt

dKdv H H H dv

d

J H K

κ

κκ κ κ κ τ κ

τ κ

κ

κκ κ

τ κ

4

eff

radiative equi

ive equilibrium (local) and

flux conservation (non local)

: librium

flux conservation

0

: 4

′′ ′= − =

′′= =

J

B

H H

B B

dHJ B

d

dKH T

d

κ

τ κ

κ κ σ

τ κ κ π


8


Now corrections to obtain new quantities:

0

0 0 0 0

0

integrate (0)

, (0) (0) (0)

(0)

(0) (0)

(0) (0

) 1

′=

∞ ∞ ∞ ∞

′=

′∆ = −

′→ ∆ = ∆ + ∆

= = = = = =

∆′→ ∆ = + ∆ =

∆→ ∆ = − + +

∆= ∆

∆= − ∆

∆

∆∆

∫

∫ ∫ ∫ ∫

∫

HH

B

J

B

v v v v v v

H

B

B

J

B

X X X

K K Hd

K K dv f J dv fJ H H dv h J dv hJ

f H

d H

d

K Hd fh

f HB

fh

d KH

d

d H

fB

J

Jd

τ

τ

τ

τ

τ

κτ

κ

κ

κ

τ κ

τ

τκ

κ

κ

κ

κ

κ

0

3

0

3

0

4 (0) (0) 1

(0) (0) 1

4

′=

′=

′=

′∆

∆

′∆ = ∆ = − + + ∆

∆′∆ = + + ∆

′+

−

∫

∫

∫J

B

H

B

J H

B B

J H

B B

Hd

T dH dH

d d

f HB T Hd

fh f

f HT HdB

fh fJ

T

τ

τ

τ

τ

τ

τ

τκ

κ κστ

π κ κ

κ κπτ

κ κ

κ

κσ

τ τ

5


9


„Radiative equilibrium“ part good at small optical depths but

poor at large optical depths

„Flux conservation“ part good at large optical depths but poor

at small optical depths

Unsöld-Lucy scheme typically requires damping

Still problems with strong resonance lines, i.e. radiative

equilibrium term is dominated by few optically thick

frequencies

3

0

(0) (0) 1

4

J J H

B B B

f HT J B Hd

T fh f

τ

τ

κ κ κπτ

σ κ κ κ′=

∆′∆ = − + + ∆

∫

J B→

0dH

dτ→


10


Generalization for scattering

All the rest is the same

Difficulties for scattering dominated regions: weak coupling

between radiation field and temperature

( ) ( )

0 0

:

(1 ) , (1 ) ,

(0-th moment

1 )

∞ ∞

= =

= =

→ = −

= − = −

− − − −

=

∫

∫ ∫

L

M

v

tvv v v v

J

B

B e v v J e v v

o

B

e v e

v

v

v

tdHJ J

dt

dHdv J B

d

B B dv J J dv d d

S B J

t

κ κ

κ

τ κ

κ β κ κ β κ τ κ

β β

0

1 0

0

B

e J

B

J

d H

d

κ

β κ

τ

∆ →→ ⇒ ∆ →

∆ →

6


11


Generalization to non-LTE (Werner & Dreizler 1998, Dreizler 2003)

All the rest is the same

should contain only terms which couple directly to the

temperature, i.e. bf and ff transitions

Depth dependent damping (need to play with parameters c):

( )

0 0

0 :

, ( ) ,

-th

moment

∞ ∞

= =

= =

→ = −

= = − =

− − −

∫

∫ ∫

%

M

%

L

v v v v v

v

vv

v

v v v

J

B

B v J v v B

v v

dHJ J

dt

dHdv J B

d

B B dv J J dv d dt

S B Jκ κ

κ

τ κ

κ

κ

τ

γ

κκ γκ κ

vκ%

0

0 0

3

0

// / 3

1 2

(1 )(1

()

0) (0)

4

−−

=

−

′

∆′∆ = − + + ∆

−− ∫J J H

B B B

f HT J B Hd

c ec e c

he

T f f

τ ττ τ

τ

τ

τ τκ κ κπτ

σ κ κ κ


12

Stellar Atmospheres


Radiation field

Radiation transfer





7


13

Stellar Atmospheres


Radiation field

Radiation transfer





The End


14

Stellar Atmospheres


Radiation field

Radiation transfer





The End

Thank you for

listening !

8


15

Avrett-Krook methodIn case that flux conservation and radiative equilibrium is not

fulfilled, Unsöld-Lucy can only change the temperature

Change of other quantities, e.g. opacity, is not accounted for

→ Avrett & Krook (1963)

strict LTE assumed, generalization straightforward

Current quantities:

Does not fulfill flux conservation and radiative equilibrium

New quantities:

( )

0

0 00 0 0 0 0

0 0( ( )) with some kind of mean opacity

v

v vv v

dII B T

dχ

κµ τ κ

τ κ=

= −

( )( ( )) with mean opacity

v

v vv v

dII B T

dχ

κµ τ κ

τ κ=

= −


16

Avrett-Krook methodLinear Taylor expansion of the new quantities from old ones:

Radiative transfer equation:

10 1 0 1 0 1 0 1 4

eff0 0

0 1 0 1 0 1 0 1

0 0

1 ( ) 4v v v v v

v vv v v v v v v v

d dT T T I I I H H H dv T

d d

d dBB B B B T

d dT

τ ττ τ τ σ π

τ τ

χχ χ χ χ τ

τ

= + → = + = + = + = + =

= + = + = + = +

∫

( )

( )( )

( ) ( )( )

( ) ( )

( )

0 10 1 0 1 0 1

0 0 0

1 10 0 0 0 1 0 1 0 1

0 0

1 10 0 0 0 1 0 0 1 0 1

0 0

10 1 1

0

( ( ))

1

vv v v

v vv v v v v v

vv v v v v v v v v

vv v v v v v v v v v

vv v v

dII B T

d

dI dI dI I B B

d d d

dI dI B I I B B

d d

dI dI B I I B B

d d

dII B

d

µ χ ττ

τµ µ χ χ

τ τ τ

τχ µ χ χ

τ τ

τχ µ χ χ χ

τ τ

µ χ χτ

= −

+ = + + − −

− + = + + + − −

− + = + + + − −

= − + ( )1

1 0 0 0

0v v v v

dI B

d

τχ

τ

+ −

9


17

Avrett-Krook method

1st moment:

( ) ( )1 1

0 1 1 1 0 0 0

0 0

1 1 10 1 1 0 0 0 1 1 0 0

0 0 0 0

010 1 1

0 0 0

010 1 0

ef0 0 0

0

: 0

4

vv v v v v v v

v vv v v v v v v v v

v vv v

v

v v

v

dI dI B I B d

d d

dK dd dH H H H

d d d d

d HdH H dv

d d

d HdH dv H T

d d

τµ χ χ χ µ µ

τ τ

χτ τχ χ χ χ τ χ

τ τ τ τ

χττ

τ τ χ

χτ στ

τ τ χ π

∞

= − + + −

→ = + + = + + =

→ + = −

→ + = −

∫

∫

∫

L

L

0

4

f

4 01 eff

0 0 0 0 0

0 0 0

, linear DEQ of first order

1 1( ) 1 , ( ) exp

( ) 4 ( )

x

v v

v

T d HM x dx M x dy dv

M H x d H

τ σ χτ

τ π χ τ

∞ ⇒ = − =

∫ ∫ ∫

F


18

Avrett-Krook method

Outer boundary:1 1

4 41 4 0 0 0eff eff

eff 0 0

0

40 1eff

0

0 0

4 01 eff

0

11

0

(0) (0)

(0) (0) 1 (0) 1 (0)4 4 (0) 4 (0)

1 (0) (0)4 (0)

(0)(0) 1

4 (0)

: 0 (

v v v

v

v v v

vv

v

vv v

H h J dv

T TH T H H H dv

H H

TH dv h J dv

H

T HJ

H h

dKf J

d

σ σσ

π π π

σ

π

σ

π

τ

∞

∞ ∞

=

= − = − = −

→ − =

→ = −

= ⇒

∫

∫

∫ ∫

L

4 00 1 eff

0

4 01 0 eff

0

(0)) const (0) 1

4 (0)

(0)( ) 1

4 (0)

vv v

v

vv

v v

T Hf J

H h

T HJ

H f h

στ

π

στ

π

= = = −

⇒ = −

10


19

Avrett-Krook method0-th moment:

( ) ( )

( ) ( )

( )

1 10 1 1 1 0 0 0

0 0

1 10 1 1 1 0 0 0

0 0

1 10 1 1 1 0 0 0

0 0 0 0

1 00 1 1 0

0 0

0

vv v v v v v v

vv v v v v v v

v v vv v v v v

vv v v

dI dI B I B d

d d

dH dJ B J B

d d

dH dB d dJ T J B dv

d dT d d

dBdH dHJ dv T

d d dT

τµ χ χ χ µ

τ τ

τχ χ χ

τ τ

χ τχ τ χ

τ τ τ

χ χτ τ

∞

= − + + −

→ = − + + −

→ = − + + −

→ = − = −

∫

∫

∫

L

L

( )

( )

1 01 0 0

0 0 0 0

0 0

1 0

0

0

4 01 01 0 0 0eff

0 0 0 0

0 0

1

(0)1 1

4 (0)

vv v

vv

v vv v v

v v

d d dHdv J B dv

d d d

dBT dv

dT

d T Hd dHJ B dv dv

d d d H f h

χ ττ

τ τ τ

χ

χ σττ χ

τ τ τ π

∞ ∞

∞

∞ ∞

+ − +

⇒ =

− + + + −

∫ ∫

∫

∫ ∫


20

Radiative equilibrium and Complete Linearization

(LTE)Simultaneous solution of RT and RE radiation transfer:

( )

2

2

1, , , 1

0 0 0 0 0 0

0 0

( , )( , ) ( , ( )) 0

( )

( , ) 0 , , , , ( , , , , )

, ( , ) 0 correction , ( , ) 0

Taylor expansion: ( , )

vv v

ik k k k i k ND k i ND

k ik k k ik k k

ik k k i

d J vJ v B v T

d v

f J T J J J J T T T T

J T f J T J T f J J T T

f J J T T

ττ τ

τ

δ δ δ δ

δ δ

− − =

→ = = =

≠ ⇒ → + + =

+ +

=

r r r rL L L L

r r r r r r r r r r

r r

( )

( )

0 0

1 1

1 1

0 0

0 ( , )

: tri-diagonal with usual , ,

( , ): diagonal

( , )

ik ik ik ikik k i k ik i k i

i k ik i k i

k k k k

k ik ik ik

v k ik k ii

i

k ik ki

f f f ff J T J J J T

J J J T

T J U T K

T A B C

B v TU U

T

K f J T

δ δ δ δ

δ δ

− +

− +

∂ ∂ ∂ ∂= + + + +

∂ ∂ ∂ ∂

→ + =

− −

∂= −

∂

= −

r r

r r r

r r r

F

11


21


(LTE)Simultaneous solution of RT and RE radiative equilibrium:

( )

( ) ( )

0

, 1 ,1 , ,

1

( , ) ( , ) ( , ) 0

, , , , , ( , ) 0

i v i v i

v

NF

i NF i i k i NF i k ik v k i

k

v J v B v T dv

f J J J T w J B v T

κ τ τ∞

=

+=

− =

→ = − =

∫

∑L L

( )

( )

( )

( )

0 0

, 11,

NF, 1 , 10 0

, 1

1 k=1

1

0

Taylor expansion: ,

0 ( , )

: diagonal

( , ): diagonal

i NF k k ik NF

NFi NF i NF

i NF k ik i k k

k ik i

k k kii

NFv k i

kiik i

k iki

f J J T T

f ff J T J T W J D T L

J T

W W w

B v TD D w

T

L w J B

δ δ

δ δ δ δ

+=

+ +

+=

′′

′=

+ +

∂ ∂= = + + → + =

∂ ∂

=

∂= −

∂

= − −

∑ ∑

∑

r r

r r r r r

r( )0

1

( , )NF

v k i

k

v T=∑


22


(LTE)Together: Rybicki scheme:

RE takes the part of the scattering integral

Instead of solve for temperature corrections

Non-linear → iteration

During the iteration: new opacities, Eddington factors

1 1 1 1

1

0

0

k k k k

NF NF NF NF

k NF

T U J K

T U J K

T U J K

W W W D T L

δ

δ

δ

δ

=

r r

r r

r r

r r

J

12


23


(NLTE)Direct generalization at least problematic due to weak

coupling of NLTE source function to the temperature

Take into account the change of the population numbers

→ Add RE to the Complete Linearization scheme

(Auer & Mihalas 1969)

Radiative equilibrium in NLTE:

Linearization:

F

( )

( ) ( )

0

, 1 ,1 , ,1 ,

1

( , ) ( , ) ( , ) 0

, , , , , , ( , ) ( , ) 0

i v i v i

v

NF

i NF NL i i NF i i NL i k k i ik v k i

k

v J v S v dv

f J J n n T w v J v

κ τ τ τ

κ τ η τ

∞

=

+ +=

− =

→ = − =

∫

∑L L

0

, 1

, 1 , 1 , 10

, 1

1 1

( )

0 ( )

i NF NL

NF NLi NF NL i NF NL i NF NL

i NF NL ik il i

k lik il i

f

f f ff J n T

J n T

ψ δψ

ψ δ δ δ

+ +

+ + + + + +

+ += =

+ =

∂ ∂ ∂= + + +

∂ ∂ ∂∑ ∑

r r

r

k kk ik

il il

w Jn n

κ η ∂ ∂−

∂ ∂


24


(NLTE)Together with RT and SE: F

0

,

, , ,0

, , ,

1 1 1, ,

1( )

0 ( )

i

ND NF NLi i i

i i k i l i

i k li k i l i

NF NLf

f f ff J n T

J n T

α

α α αα

αψ δψ

ψ δ δ δ= = =

= ++ =

∂ ∂ ∂ = + + +

∂ ∂ ∂ ∑ ∑ ∑

L

1 1

1, ,

1

1

0 0

0

0 0

0 0

0 0

i i i i i i i

i ii k i k

i i

i i

A B C L

J JA B

n n

T T

δψ δψ δψ

δ δ

δ δ

δ δ

− +

−

−

−

− + − =

− +

rr r r

O O r r

O O

r r

1,

0

,

1

0

0

0

( )

0 0

0 0

ii k

i

i

i

JC

f

n

T

α

δ

ψ

δ

δ

+

+

− = −

O r

O

r

13


25


(NLTE)Matrix Bi: F

( )( )

( )

, ,

,

,

, ,

, ,1 ,

1 1 1

, ,

0

0

i k i k

i k

i l i

iNL

i il m l m

i m i il lm i k i

iNF NL iNF NL iNF NL

i k i l i

S SB

n T

BP P

n P TJ T

f f f

J n T

′

′= ′

+ + + + + +

′ ′

∂ ∂ − − ∂ ∂ = ∂ ∂

∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

∑

O M

L L

O M

M M

L L L L

M M

1 ... NF 1 ... NL T1 ... N

F1 ... N

L T

1

Stellar Atmospheres: Accelerated Lambda Iteration

1



2

Motivation

Complete Linearization provides a solution scheme, solving

the radiation transfer, the statistical equilibrium and the

radiative equilibrium simultaneously.

But, the system is coupled over all depths (via RT) and all

frequencies (via SE, RE) → HUGE!

Abbreviations used in this chapter:

RT = Radiation Transfer equations

SE = Statistical Equilibrium equations

RE = Radiative Equilibrium equation

F

2


3

Multi-frequency / multi-gray

Ways around:

Multi-frequency / multi-gray method by Anderson (1985,1989)

• Group all frequency points according to their opacity into

bins (typically 5) and solve the RT with mean opacities of

these bins. → Only 5 RT equations instead of thousands

• Use a Complete Linearization with the reduced set of

equations

• Solve RT alone in between to get all intensities, Eddington-

factors, etc.

• Main disadvantage: in principle depth dependent grouping


4

Lambda Iteration

Split RT and SE+RE:

• Good: SE is linear (if a separate T-correction scheme is used)

• Bad: SE contain old values of n,T (in rate matrix A)

Disadvantage: not converging, this is a Lambda iteration!

( )0

( , )

( , )

( , , ) ( , , ) 0

new old

new

v v

J S n T

A J T n b


= Λ

=

− =∫

RT formal solution

SE

RE

3


5

Accelerated Lambda Iteration (ALI)

Again: split RT and SE+RE but now use ALI

• Good: SE contains new quantities n, T

• Bad: Non-Linear equations → linearization (but without RT)

Basic advantage over Lambda Iteration: ALI converges!

( )

* *

0

( , ) ( , ) ( , )

( , )

( , , ) ( , , ) 0

new old old old new new new old old old

newnew new

new new new new new

v v

J S n T S n T S n T

A J T n b


= Λ + Λ − Λ

=

− =∫

RT

SE

RE


6

Example: ALI working on Thomson scattering problem

amplification factor

( )

( )( ) ( )( )

( ) ( )( )

( )

* *

* *

*

*

:=formal solution on

solve for

1

1

1 1

⇒ = Λ − Λ + Λ

= Λ − Λ + Λ

= −

= −

− +

− −Λ − =

= − Λ −

⇒

+

− +

new

new new

e e

new new new

e e

e e

old old

old old old

new old

FS

FS

FS

e

old new new

e e

new

e

S B J

S S

B J B

S

B J

B J

J S

J

J

J J J J

J

B

β β

β β

β

β

β

β

β

β

( )

( )

1* *

1*

1 subtract on both sides

1

−

−

= − Λ − Λ

⇒ − = − Λ −

FS old old

e e

new old FS old

e

J J J

J J J J

β β

β

Interpretation: iteration is driven by difference (JFS-Jold) but: this difference

is amplified, hence, iteration is accelerated.

Example: βe=0.99; at large optical depth Λ* almost 1 ĺ strong amplifaction

source function with scattering, problem: J unknownĺiterate

4


7

What is a good ȁ*?

The choice of ȁ* is in principle irrelevant but in practice it

decides about the success/failure of the iteration scheme.

First (useful) ȁ* (Werner & Husfeld 1985):

A few other, more elaborate suggestions until Olson &

Kunasz (1987): Best ȁ* is the diagonal of the ȁ-matrix

(ȁ-matrix is the numerical representation of the integral operator ȁ)

We therefore need an efficient method to calculate the

elements of the ȁ-matrix (are essentially functions of ττττνννν ).

Could compute directly elements representing the ȁ-integral operator, but

too expensive (E1 functions). Instead: use solution method for transfer

equation in differential (not integral) form: short characteristics method

( )*( )

, ' ( ')0

>Λ =

≤v

v v

SS

τ τ γτ τ τ

τ γ


8

In the final lecture tomorrow, we will learn two important methods to

obtain numerically the formal solution of the radiation transfer

equation.

1. Solution of the differential equation as a boundary-value problem

(Feautrier method). [can include scattering]

2. Solution employing Schwarzschild equation on local scale (short

characteristics method). [cannot include scattering, must ALI iterate]

The direct numerical evaluation of Schwarzschild equation is much too

cpu-time consuming, but in principle possible.

5


9

Olson-Kunasz ȁ*

Short characteristics with linear approximation of source

functionmax

maxmax

0

( , , ) ( , , ) exp ( )exp

( , , ) (0, , ) exp ( ) exp

dI v I v S

dI v I v S

τ

τ

τ


µ µ µ


µ µ µ

+ +

− −

′ ′ − −′= − + −

′ ′−

′= − + −

∫

∫

( )

( )

( )

1

1 1

1

1

( , , ) ( , , ) exp ( , , )

( , , ) ( , , ) exp ( , , )

with


the intergrals can be evalu

i i i i

i i i i

i i

i

i

I v I v I S v

I v I v I S v

S

I

τ µ τ µ τ µ

τ µ τ µ τ µ

τ ττ

µ

+ + ++

− − −− −

−

−

±

= −∆ + ∆

= −∆ + ∆

−∆ =

∆

1 1

ated as

i i i i i i iI S S Sα β γ± ± ± ±− +∆ = + +


10

Olson-Kunasz ȁ*

Short characteristics with linear approximation of source

function

0

11

1

i

i

i

e

ee

α

β

γ

+

−∆+

−∆+ −∆

=

−= +

∆

−= − −

∆

1

11

0

i

i

i

ee

e

α

β

γ

−∆− −∆

−∆−

−

−= − +

∆

−= −

∆

=

( ) ( )

( ) ( )

1 1

0 0

1 1

2 2

use 0, ,1, ,0 for 0, , , ,0 to project colums of T

J I I d S S d

S i

µ µµ µ+ − + −= + = Λ + Λ

= Λ

∫ ∫L L L L

6


11

Inward

( )

1

1 1

1

2

1

1 1

1 1 1

1 1

0

ˆ

ˆ ˆ0 exp( ) 0

ˆ ˆexp( )

ˆ

0

0 exp( ) 0

exp( ) exp( )

ˆ exp( )

i

i i i

i i i

k i ND

i

i i i

i i i i i

k k

i

I

I I

I I

I

i

I

τ

τ

γ

γ τ β

γ τ β τ α

τ

−

−

− −

− −

− −

+

−

= +

−

−

− −− −

− − −

− − +

−

− −

∆ =

−∆ + ∆ −∆ + ∆

−∆ + −∆ + −∆ +

−∆

L

M

M

M

M

M

M

M

M

1

1 1

0

0

0 0

exp( )

ˆ exp( )

i

i i i

k k

i

I

β

β τ α

τ

−

− −+

−− −

= −∆ +

−∆

M

M

M

M


12

Outward

( )

0 2

1 1

1

1

1 1

1 1 1

1

1

ˆ

ˆ ˆexp( )

ˆ ˆ0 exp( ) 0

ˆ

0

ˆ exp( )

exp( ) exp( )

0 exp( ) 0

0

k i

i i i

i i i

i

k k

i i i i i

i i i

i

i

I

I I

I I

I

i

I

τ

τ

τ

α τ β τ γ

α τ β

α

+

= −

+ +

− −

+ ++

+

+

+

+ −

+ + +

+ − −

+ +

+

+

+

−∆ + ∆

= −∆ + ∆

∆

−∆ −∆ + −∆ +

−∆ +

L

M

M

M

M

M

M

M

M

1 1

1 1

ˆ exp( )

exp( )

0 0

0

0

k k

i i i

i

i

I τ

β τ γ

β

+

+ −

+ +

− −

+

−∆ −∆ +

=

M

M

M

M

7


13

ȁ-Matrix

( )

( )

( )

( )

1

1 1

0

1

1 1

0

1

*,

0

1

1

0

1

1 1

0

1 ˆ exp( )2

1exp( )

2

10 0

2

1exp( )

2

1 ˆ exp( )2

k k

i i i

i i i

i i i

k k

d I

d

d

d

d I

µ τ

µ β τ γ

µ β β

µ β τ α

µ τ

+

+ −

+ +− −

+ −

− −+

−

− −

−∆ −∆ +

Λ = +

−∆ +

−∆

∫

∫

∫

∫

∫

M

M

M

M


14

Towards a linear scheme

ȁ* acts on S, which makes the equations non-linear in the

occupation numbers

• Idea of Rybicki & Hummer (1992): use J=ǻJ+Ȍ*Șnew instead

• Modify the rate equations slightly:

( )*

0 0

*3

2

0

*3

*

2

0

4 4 ( )

24

24 ( )

ij ij

ij i i v

ijiji j j v

j

i

i

j

ji

j

R n n J dv J dvhv hv

n hvR n n J dv

n hv c

n hvJ dv

n

n n

nhvn

c

σ σπ π η

σπ

σπ η

∞ ∞

∞

∞

= = Ψ + ∆

= +

= Ψ + ∆ +

∫ ∫

∫

∫

Documents

Stellar atmospheres lecture (2003) S. Dreizler · 1 Stellar Atmospheres: Motivation 1 Stellar Atmospheres: Literature • Dimitri Mihalas – Stellar Atmospheres, W.H. Freeman, San