Lecture 3:

Microphysics of the radiative transfer

Numerical integrationof RT in a simplest case

Local Thermodynamical Equilibrium (LTE, all microprocesses are in detailed balance)

Static (no time dependence)Simple geometry (e.g semi-infinit

medium)One dimension

Equation of radiative transfer (again)

Equation of radiative transfer:

For the case of semi-infinite medium (e.g. stellar atmosphere) boundary condition is set deep (inside a star):

or

or

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

dIk x x I k x x S x

dx

dIx I x S x

dx

dII S

d

( ) ( )I B T

Einstein coefficients• Aul - spontaneous de-excitation.

• Blu - radiative excitation.

• Bul - stimulated de-excitation.

• Einstein relations connect the probabilities:

and that lu transition rate match the ul rate:

Bul

Blu

Aul

3

22

; lu u ul

ul l ul

B g A h

B g B c

l lu u ul u uln B I n A n B I

Absorption/Emission• Energy absorbed:

• Energy emitted:

• Probability profiles are area normalized:

Bul

Blu

Aul

0( )4

bbl lu

hk I n B I

0

0

( )4

( )4

bbu ul

u ul

hj n B I

hn A

0 0 00 0 0

( ) ( ) ( ) 1d d d

Absorption/Emission• Absorption coefficient expressed

through Einstein probabilities:

• Emission coefficient:

• LTE (detailed balance for each frequency) means that probability profiles are the same:

0 0

00

0

( ) ( )4

( )( ) 1

4 ( )

bbl lu u ul

u ll lu

l u

hk n B n B

n ghn B

n g

0( )4

bbu ul

hj n A

0 0 0( ) ( ) ( )

Source function• Source function:

• LTE (detailed balance for each frequency) means that probability profiles are the same:

• Level population in LTE is described by the Boltzmann distribution:

0

0 0

( )

( ) ( )

bbbb u ul

bbl lu u ul

j n AS

n B n Bk

0 0 0( ) ( ) ( )

k k

E hu u uT T

l l l

n g ge e

n g g

Source function and absorption coefficient in LTE

• Source function in LTE is the Planck function:

• Absorption coefficient in LTE:

0

0 0

3

2

( )

( ) ( )

1

2 1( )

1

bb u ul

l lu u ul

u ul ul ul

l lu u ul l u lu ul

h kT

n AS

n B n B

n A A B

n B n B n n B B

hB T

c e

0( ) 14

bb h kTl lu

hk n B e

Continuous opacityo Continuous opacity includes b-f

(photoionization) and f-f transitions.o Hydrogen is often a dominating source

due to its abundance (H, H-).o For b-f transitions only photons with

energy larger than the difference between the ionization energy and the energy of a bound level can be absorbed:

This produces the absorption edges.

2eR 2h h c n m 2v

Lyman, < 912 Å

Balmer, < 3647 Å

Paschen, < 8206 Å

b-f transitions in Hydrogen

n = 1

n = 2

n = 3

0 eV

10.2 eV

12.1 eV

13.6 eV n =

Total opacity in LTEb-f and f-f opacities are described by the

same expression for opacity coefficient as

for b-b. Just Bul and the absorption profiles

are different.The source function is still a Planck

function.Total absorption:

bb bf ffk k k k

Line profileThe last thing left is the absorption

probability profile.Spectral lines are not delta-functions

due to three effects:Damping of radiationPerturbation of atomic energy level system

by neighboring particlesDoppler movements of absorbers/emitters

The convolution the Lorentz and Doppler profile results in Voigt profile:

Gauss

i

an

Lore

ntz

2

2 2H( , )

( )

ya ea dy

y a

vv

Now… we know everythingRT equation:

Boundary condition:

Absorption coefficient:

Absorption profile:

Source function

dIk I k S

dx

( ) ( )I B T

bb bf ffk k k k

H( , )a v

( )B T

Practical implementationMaxwellian velocity distribution:

Boltzmann level population

Saha ionization balance

2

A

1P( ) ; e

xvv

x2kT

v v =mv

u lu u

l l

E E

kTn ge

n g

321 1

2

2 U 21

U

Eii i e kT

i e i

n m kTe

n N h

How good is LTEfor solving RT?

Stellar surfaces

Courtesy of Göran Scharmer/Swedish Solar Vacuum Telescope

Energy transport in starsRadiationConvectionParticle ejectionWaves

Space above surface is not empty!

Courtesy of SOHO

Solar coronaLow density: <107 particle/cm3

Hot: >106 KOptically thinTemperature of radiation is very different from the kinetic temperature

Courtesy of SOHO

Coronal arcades on the Sun

Heating of the outer layers is part of the energy transport

Magnetic fields play a major role

Coherent motions at the photosphere layers dissipate in the corona making it hot

Courtesy of Karel Schrijver/TRACE

How can observe space above solar surface?

At visual spectral range photosphere dominates the total flux

UV lines allow to see chromospheric structures

Going to X-ray is required to observe solar corona

Courtesy of Karel Schrijver/TRACE