3. radiative transfer

1

3. Transport of energy: radiation

specific intensity, radiative

flux

optical depth

absorption & emission

equation of transfer, source function

formal solution, limb darkening

temperature distribution

grey atmosphere, mean opacities

2

No sinks and sources of energy in the atmosphere

all energy produced in stellar interior is transported through the atmosphere

at any given radius r

in the atmosphere:

F is the energy flux per unit surface and per unit time. Dimensions: [erg/cm2/sec]

The energy transport is sustained by the temperature gradient.

The steepness of this gradient is dependent on the effectiveness

of the energy transport through the different atmospheric layers.

24 ( ) .r F r const L

Energy flux conservation

3

Mechanisms of energy transport

a.

radiation: Frad

(most important)

b.

convection: Fconv

(important especially in cool stars)

c.

heat production: e.g. in the transition between solar cromosphere

and corona

d.

radial flow of matter: corona and stellar wind

e.

sound waves: cromosphere

and corona

Transport of energy

We will be mostly concerned with the first 2 mechanisms: F(r)=Frad

(r) + Fconv

(r). In the outer layers, we always have Frad

>> Fconv

4

The specific intensity

Measures of energy flow: Specific

Intensity

and Flux

The amount of energy dEν

transported

through a surface area dA

is proportional to dt

(length of time), dν

(frequency width), dω

(solid angle) and the projected unit surface area cos

θ

dA.

The proportionality factor is the specific Intensity Iν

(cosθ)

Intensity depends on location in space, direction and frequency

dEν = Iν(cos θ) cos θ dAdω dν dt([Iν ]: erg cm

−2 sr−1 Hz−1 s−1)

Iλ =cλ2Iν

(from Iλdλ = Iνdν and ν = c/λ)

5

Invariance of the specific intensity

The area element dA

emits radiation towards dA’. In the absence of any matter between emitter and receiver (no absorption and emission on the light paths between the surface elements) the amount of energy emitted and received through each surface elements is:

dEν = Iν(cos θ) cos θ dAdω dν dtdE0ν = I 0ν(cos θ0) cos θ0 dA0 dω0 dν dt

6

Invariance of the specific intensity

energy is conserved: dEν = dE0ν and dω = projected area

distance2 = dA0 cos θ0r2

dω0 = dA cos θr2

Iν = I0ν

In TE: I

= B

Specific intensity is constant along rays -

as long as there is no absorption and emission of matter between emitter and receiver

dEν = Iν(cos θ) cos θ dAdω dν dtdE0ν = I 0ν(cos θ0) cos θ0 dA0 dω0 dν dt

and

7

solid angle : dω =dA

r2

Total solid angle =4πr2

r2= 4π

dA = (r dθ)(r sin θ dφ)

→ dω = sin θ dθ dφ

define μ = cos θ

dμ = − sin θ dθdω = sin θ dθ dφ = −dμ dφ

Spherical coordinate system and solid angle dω

8

Radiative

flux

How much energy flows through surface element dA?

dE

~ I

cosθ

dω

integrate

over the whole solid angle (4):

πFν =

Z4π

Iν(cos θ) cos θ dω =

Z 2π

0

Z π

0

Iν(cos θ) cos θ sin θdθdφ

“astrophysical flux”

Fν

is the monochromatic radiative

flux. The factor π

in the definition is historical.

Fν

can also be interpreted as the net rate of energy flow through a surface element.

9

Radiative

flux

The monochromatic radiative

flux

at frequency

gives the net rate of energy flow through a surface element.

dE

~ I

cosθ

dω

integrate

over the whole solid angle (4):

We distinguish between the outward direction (0 <

< /2) and the inward direction (/2 < so that the net flux is

πFν = πF+ν − πF−ν =

=

Z 2π

0

Z π/2

0

Iν(cos θ) cos θ sin θdθdφ+

Z 2π

0

Z π

π/2


πFν =

Z4π

Iν(cos θ) cos θ dω =

Z 2π

0

Z π

0


Note: for /2 < cosθ

< 0 second term negative !!

“astrophysical flux”

10

Total radiative

flux

Integral over frequencies

Z ∞

0

πFνdν = Frad

Frad

is the total radiative

flux.

It is the total net amount of energy going through the surface element per unit time and unit surface.

11

Stellar luminosity

At the outer boundary of atmosphere

(r = Ro

)

there is no incident radiation

Integral interval over θ

reduces from [0,π] to [0,π/2].

πFν(Ro) = πF+ν (Ro) =

Z 2π

0

Z π/2

0


This is the monochromatic energy that each surface element of the star radiates in all directions

If we multiply by the total stellar surface 4πR02

monochromatic stellar luminosity

at frequency

and integrating over

total stellar luminosity

4πR2o · πFν(Ro)

4πR2o ·Z ∞

0

πF+ν (Ro)dν = L (Luminosity)

= Lν

12

Observed flux

What radiative

flux is measured by an observer at distance d? integrate specific intensity Iν

towards observer over

all surface elements note that only half sphere contributes

πF+ν

Eν =

Z1/2 sphere

dE =∆ω∆ν∆t

Z1/2 sphere

Iν(cos θ) cos θ dA

in spherical symmetry: dA = R2o sinθ dθ dφ

→ Eν =∆ω∆ν∆tR2o

Z 2π

o

Z π/2

0

Iν(cos θ) cos θ sinθ dθ dφ

because of spherical symmetry

the integral of intensity towards the observer over the stellar surface is proportional to F

+, the flux emitted into all directions by one surface element !!

13

Observed flux

Solid angle of telescope at distance d:

Fobsν =radiative energy

area · frequency · time =R2od2

πF+ν (Ro)

This, and not I

, is the quantity generally measured for stars. For the Sun, whose disk is resolved, we can also measure I

(the variation of I

over the solar disk is called the limb darkening)

unlike I

, F

decreases with increasing distance

∆ω = ∆A/d2

+

Eν = ∆ω∆ν ∆t R2o πF+ν (Ro)

flux received = flux emitted x (R/r)2

R∞0F obsν,¯ dν = 1.36 KW/m

2

14

Mean intensity, energy density & radiation pressure

Integrating over the solid angle and dividing by 4:

Jν =1

4π

Z4π

Iν dω mean intensity

energy density

radiation pressure (important in hot stars)

uν =radiation energy

volume=1

c

Z4π

Iν dω =4π

cJν

pν =1

c

Z4π

Iν cos2 θ dω

pressure =force

area=d momentum(= E/c)

dt

1

area

15

Moments of the specific intensity

0th

moment

1st

moment (Eddington

flux)

2nd

moment

Jν =1

4π

ZIν dω =

1

4π

Z 2π

0

dφ

Z 1

−1Iν dμ =

1

2

Z 1

−1Iν dμ

Hν =1

4π

ZIν cos θ dω =

1

2

Z 1

−1Iν μ dμ =

Fν4

Kν =1

4π

ZIν cos

2 θ dω =1

2

Z 1

−1Iν μ

2 dμ =c

4πpν

for azimuthal

symmetry

16Convention:

= 0 at the outer edge of the atmosphere, increasing inwards

Interactions between photons and matterInteractions between photons and matter

absorption of radiation

I

ds

Io I

(s)s

Over a distance s:Iν (s) = I

oν e

−sR0

κν ds

τν :=

sZ0

κν ds

optical depth(dimensionless)

or:

d

=

ds

loss of intensity in the beam (true absorption/scattering)microscopical

view:

n

dIν = −κνIνds

κν : absorption coefficient

[κν ] = cm−1

17

optical depth

Iν(s) = Ioν e

−τν

if τν = 1→ Iν =Ioνe' 0.37 Ioν

The quantity

has a geometrical interpretation in terms of mean free path of photons :̄s

photons travel on average for a length before absorption

s̄

We can see through atmosphere until

~ 1

optically thick

(thin) medium:

>

(<) 1

The optical thickness of a layer determines the fraction of the

intensity passing through the layer

τν = 1 =

s̄Zo

κν ds

18

photon mean free path

What is the average distance over which photons travel?

< τν >=

∞Z0

τν p(τν ) dτνexpectation value

probability of absorption in interval [

+d

]

= probability of non-absorption between 0 and

and absorption in d

- probability that photon is not absorbed: 1 − p(0, τν) = I(τν)

Io= e−τν

- probability that photon is absorbed: p(0, τν) =∆I(τ )

Io=Io − I(τν)

Io= 1− I(τν)

Io

total probability: e−τν dτν

- probability that photon is absorbed in [τν , τν + dτν ] : p(τν , τν + dτν) =dIνI(τν )

= dτν

19

photon mean free path

< τν >=

∞Z0

τν p(τν) dτν =

∞Z0

τν e−τν dτν = 1

mean free path corresponds to <

>=1

if κν (s) = const : ∆τν = κν ∆s→ ∆s = s̄ =1

κν(homogeneous

material)

Zxe−x dx = −(1 + x) e−x

20

Principle of line formation

observer sees through the atmospheric layers up to

1

In the continuum

is smaller than in the line see deeper into the atmosphere

T(cont) > T(line)

21

radiative

acceleration

In the absorption process photons release momentum E/c to the atoms, and the corresponding force is:

The infinitesimal energy absorbed is:

The total energy absorbed is (assuming that κν

does not depend on ω):

F

force =dfphot =momentum(=E/c)

dt

dEabsν = dIν cos θ dA dω dt dν = κν Iν cosθ dA dω dt dν ds

Eabs =

∞Z0

κν

Z4π

Iν cos θ dω dν dA dt ds = π

∞Z0

κν Fν dν dA dt ds

22

radiative

acceleration

dfphot =π

c

∞R0

κν Fν dν

dtdA dt ds = grad dm (dm = ρ dA ds)

grad =π

cρ

∞Z0

κν Fν dν

23

emission of radiation

ds

dA d

dV=dA

ds

energy added by emission processes within dV

dEemν = ²ν dV dω dν dt

²ν : emission coefficient

[²ν ] = erg cm−3 sr−1Hz−1 s−1

24

The equation of The equation of radiativeradiative

tranfertranfer

If we combine absorption and emission together:

dEabsν = dIabsν dA cos θ dω dν dt = −κν Iν dA cos θ dω dt dν ds

dEabsν +dEemν = (dIabsν +dIemν ) dA cos θ dω dν dt = (−κν Iν+²ν) dA cos θ dω dν dt ds

dEemν = dIemν dA cos θ dω dν dt = ²ν dA cos θ dω dν dt ds

dIν = dIabsν + dIemν = (−κν Iν + ²ν) ds

dIνds = −κν Iν + ²ν

differential equation describing the flow of radiation through matter

25


tranfertranfer

Plane-parallel symmetry

dx = cos θ ds = μ ds

d

ds= μ

d

dx

μ dIν(μ,x)dx = −κν Iν(μ, x) + ²ν

26


tranfertranfer

Spherical symmetry

angle

between ray and radial direction is not constant

d

ds=dr

ds

∂

∂r+dθ

ds

∂

∂θ

∂

∂θ=

∂μ

∂θ

∂

∂μ= − sin θ ∂

∂μ

=⇒ d

ds= μ

∂

∂r+sin2 θ

r

∂

∂μ= μ

∂

∂r+1 − μ2

r

∂

∂μ

μ ∂∂r Iν(μ, r) +

1−μ2r

∂∂μ Iν(μ, r) = −κν Iν(μ, r) + ²ν

−r dθ = sin θ ds (dθ < 0)→ dθ

ds= − sin θ

r

dr = ds cos θ → dr

ds= cos θ (as in plane−parallel)

27


tranfertranfer

Optical depth and source function

In plane-parallel symmetry:optical depth increasing

towards interior:

μdIν(μ,x)

dx = −κν(x) Iν(μ, x) + ²ν(x)

μ dIν(μ,τν)dτν

= Iν(μ, τν) − Sν(τν)

Sν =²νκν

Observed emerging intensity I

(cos

,

= 0) depends on = cos

,

(Ri

) and

S

The physics of S

is crucial for radiative

transfer κν =dτνds ≈ ∆τν

∆s ≈ 1s̄

Sν =²νκν≈ ²ν · s̄

= 1 corresponds to free mean path of photons

source function S

corresponds to intensity emitted over the free mean path of photons

source function

dim [S

] = [I

]

−κν dx = dτν

τν = −xZ

Ro

κν dx

28


tranfertranfer

Source function: simple cases

a. LTE (thermal absorption/emission)

Sν =²νκν= Bν(T ) Kirchhoff’s law

photons are absorbed and re-emitted at the local temperature T

Knowledge of T stratification T=T(x) or T() solution of transfer equation I

(,

)

independent of radiation field

29


tranfertranfer


b. coherent isotropic scattering (e.g. Thomson scattering)

the absorption process is characterized by the scattering coefficient

, analogous to

dIν = −σνIνds

= ’

and at each frequency dEemν = dEabsν

incident = scattered

Z4π

²scν dω =

Z4π

σν Iνdω

²scν

Z4π

dω = σν

Z4π

Iνdω

dEemν =

Z4π

²scν dω

dEabsν =

Z4π

σν Iνdω

²scνσν=1

4π

Z4π

Iνdω

Sν = Jν

completely dependent on radiation field

not dependent on temperature T

30


tranfertranfer


c. mixed case

Sν =²ν + ²

scν

κν + σν=

κνκν + σν

²νκν+

σνκν + σν

²scνσν

Sν =²ν + ²

scν

κν + σν=

κνκν + σν

Bν +σν

κν + σνJν

31

Formal solution of the equation of Formal solution of the equation of radiativeradiative

tranfertranfer

we want to solve the equation of RT with a known source function and in plane-parallel geometry

multiply by e-

and integrate between 1

(outside) and 2

(> 1

, inside)

d

dτν(Iν e

−τν/μ) = −Sν e−τν/μ

μ

linear 1st

order differential equation

μdIνdτν

= Iν − Sν

hIν e

− τνμ

iτ2τ1= −

τ2Zτ1

Sν e− τν

μdtνμ

check, whether this really yields transfer equation above

32

Formal solution of the equation of Formal solution of the equation of radiativeradiative

tranfertranfer

intensity originating at 2

decreased by exponential factor to 1 contribution to the intensity by

emission along the path from 2

to 1

(at each point decreased by the exponential factor)

integral form of equation of radiation transfer

Formal solution! actual solution can be complex, since S

can depend on I

hIν e

− τνμ

iτ2τ1= −

τ2Zτ1

Sν e− τν

μdtνμ

Iν(τ1, μ) = Iν(τ2,μ) e− τ2−τ1

μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

33

Boundary conditionsBoundary conditions

a. incoming radiation:

< 0 at

0

usually we can neglect irradiation from outside: I

(

0,

< 0) = 0

solution of RT equation requires boundary conditions, which are different for incoming and outgoing radiation

I inν (τν ,μ) =

0Zτν

Sν(t) e− t−τν

μdt

μ

Iν(τ1,μ) = Iν(τ2,μ) e− τ2−τ1

μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

34

Boundary conditionsBoundary conditions

b. outgoing radiation:

> 0 at

= max

∞

We have either

or

Iν(τmax,μ) = I+ν (μ)

finite slab or shell

limτ→∞ Iν(τ,μ) e

−τ/μ = 0 semi-infinite case (planar or spherical)I

increases less rapidly than the exponential

Ioutν (τν ,μ) =

∞Zτν


μdt

μ

Iν(τν) = Ioutν (τν) + I

inν (τν)

and at a given position

in the atmosphere:

Iν(τ1,μ) = Iν(τ2,μ) e− τ2−τ1

μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

35

Emergent intensityEmergent intensity

from the latter emergent intensity

= 0,

> 0

Iν(0,μ) =

∞Z0

Sν(t) e− tμdt

μ

intensity observed is a weighted average of the source function along the line of sight. The contribution to the emerging intensity comes mostly from each depths with

< 1.

36


suppose that S

is linear in

(Taylor expansion around

= 0):

Eddington-Barbier

relation

emergent intensityIν(0,μ) =

∞Z0

(S0ν + S1νt)e− tμdt

μ= S0ν + S1νμ

Iν(0,μ) = Sν(τν = μ)

we see the source function at location

the emergent intensity corresponds to the source function at

= 1 along the line of sight

Sν(τν) = S0ν + S1ντν Zxe−x dx = −(1 + x) e−x

37


= 1 (normal direction):

Iν(0, 1) = Sν(τν = 1)

= 0.5 (slanted direction):

Iν(0, 0.5) = Sν(τν = 0.5)

in both cases:

1

spectral lines:

compared to continuum

= 1 is reached at higher layer in the atmosphere

Sline

< Scont

a dip is created in the spectrum

38

Line formationLine formation

simplify:

= 1, 1

=0 (emergent intensity), 2

=

S

independent of location

Iν(0) = Iν(τν) e−τν + Sν

τνZ0

e−t dt = Iν(τν) e−τν + Sν (1 − e−τν )

39

Line formationLine formation

Optically thick object: Iν(0) = Iν(τν) e−τν + Sν (1 − e−τν ) = Sν

Optically thin object: Iν(0) = Iν(τν) + [Sν − Iν(τν)] τν

∞

exp(-

)

1 -

40

independent of

, no line (e.g. black body B

)

I

=

S

=

ds

S

e.g. HII region, solar corona

enhanced

Iν(0) = Iν(τν) + [Sν − Iν(τν)] τν

e.g. stellar absorption spectrum (temperature decreasing outwards)

e.g. stellar spectrum with temperature increasing outwards (e.g.

Sun in the UV)

From Rutten’s

web notes

41

Line formation example: solar coronaLine formation example: solar corona

I

=

S

42

The diffusion approximationThe diffusion approximation

At large optical depth in stellar atmosphere photons are local: S

B

Expand S

B

as a power-series:

Sν(t) =

∞Xn=0

dnBν

dτnν(t − τν)

n/n!

In the diffusion approximation (

>>1) we retain only first order terms:

Bν(t) = Bν(τν) +dBν

dτν(t− τν)

Ioutν (τν ,μ) =

∞Zτν

[Bν(τν) +dBν

dτν(t − τν)]e

−(t−τν)/μ dtμ

43

The diffusion approximationThe diffusion approximation

Substituting:

At

= 0 we obtain the Eddington-Barbier

relation for the observed emergent intensity.

It is given by the Planck-function and its gradient at

= 0.

It depends linearly on μ

= cos

θ.

∞Z0

uke−udu = k!

I inν (τν , μ) = −τν/μZ0

[Bν(τν) +dBν

dτνμu]e−u du

Ioutν (τν ,μ) =

∞Z0

[Bν(τν) +dBν

dτνμu]e−u du = Bν(τν) + μ

dBν

dτν

Ioutν (τν ,μ) =

∞Zτν

[Bν(τν) +dBν

dτν(t − τν)]e

−(t−τν)/μ dtμ

t→ u =t − τνμ

→ dt = μ du

44

Solar limb darkeningSolar limb darkening

Iν(0,μ)Iν(0,1)

=Bν(0)+

dBνdτν

μ

Bν(0)+dBνdτν

from the intensity measurements B

, dB

/d

Bν(t) = Bν(0) +dBν

dτνt = a+ b · t = 2hν3

c21

ehν/kT(t)−1

T(t): empirical temperature stratification of solar photosphere

center-to-limb variation of intensity

diffusion approximation:

45

Solar limb darkeningSolar limb darkening

...and also giant planets

46

Solar limb darkening: temperature stratificationSolar limb darkening: temperature stratification

Iν(0,μ) =

∞Z0

Sν(t) e− tμdt

μ

exponential extinction varies as -

/cos

From S

= a + b

Iν(0,μ) = Sν(τν = μ) S

Iν(0, cos θ) = aν + bν cos θ

R. Rutten,web notes

Unsoeld, 68

47

EddingtonEddington

approximationapproximation

In the diffusion approximation we had:

Bν(t) = Bν(τν) +dBν

dτν(t− τν)

0 <

< 1

-1 <

< 0

>> 1

I inν (τν ,μ) = −τν/μZ0

[Bν(τν) +dBν

dτνμu]e−u du

Ioutν (τν ,μ) = Bν(τν) + μdBν

dτν

I−ν (τν ,μ) = Bν(τν) + μdBν

dτν

we want to obtain an approximation for the radiation field –

both inward and outward radiation -

at large optical depth

stellar interior, inner boundary

of atmosphere

48

EddingtonEddington

approximationapproximation

Jν =1

2

Z 1

−1Iν dμ = Bν(τν)

Hν =Fν4=1

2

Z 1

−1μ Iν dμ =

1

3

dBν

dτν

Kν =1

2

Z 1

−1μ2 Iν dμ =

1

3Bν(τν)

= −13

1

κν

dBν

dx= − 1

3κν

dBν

dT

dT

dx

flux F

~ dT/dxdiffusion: flux ~ gradient (e.g. heat conduction)

Kν =13 Jν Eddington

approximation

With this approximation for Iν

we can calculate the angle averaged momenta

of the intensity

simple approximation for photon flux and a relationship between mean intensity Jν

and Kν

very important for analytical estimates

49

Jν =1

2

1Z−1

Iν dμ =1

2

1Z0

Ioutν dμ+1

2

0Z−1

I inν dμ

Jν =1

2

⎡⎣ 1Z0

∞Zτν

Sν(t)e−(t−τν)/μ dt

μdμ−

0Z−1

τνZ0

Sν (t)e−(t−τν )/μ dt

μdμ

⎤⎦

substitute w = 1μ ⇒ dw

w = − 1μdμ w = − 1

μ ⇒ dww = − 1

μdμ

< 0

> 0

After the previous approximations, we now want to calculate exact solutions for tha

radiative

momenta

Jν

, Hν

, Kν

.

Those are

important to calculate spectra and atmospheric structureSchwarzschildSchwarzschild--Milne equationsMilne equations

Ioutν (τν ,μ) =

∞Zτν


μdt

μ

I inν (τν ,μ) =

0Zτν


μdt

μ

Jν =1

2

⎡⎣ ∞Z1

∞Zτν

Sν(t)e−(t−τν)wdt

dw

w+

∞Z1

τνZ0

Sν(t)e−(τν−t)wdt

dw

w

⎤⎦

50

SchwarzschildSchwarzschild--Milne equationsMilne equations

Jν =1

2

⎡⎣ ∞Zτν

Sν(t)

∞Z1

e−(t−τν)wdw

wdt +

τνZ0

Sν(t)

∞Z1

e−(τν−t)wdw

wdt

⎤⎦> 0 > 0

Jν =1

2

∞Z0

Sν (t)

∞Z1

e−w|t−τν|dw

wdt =

1

2

∞Z0

Sν(t)E1(|t− τν |)dt

Schwarzschild’s

equation

51


where

E1(t) =

∞Z1

e−txdx

x=

∞Zt

e−x

xdx

is the first exponential integral (singularity at t=0)

Exponential integrals

En(t) = tn−1

∞Zt

x−ne−xdx

En(0) = 1/(n− 1),En(t→∞) = e−t/t→ 0

dEndt

= −En−1,ZEn(t) = −En+1(t)

E1(0) =∞ E2(0) = 1 E3(0) = 1/2 En(∞) = 0

Gray, 92

52


Introducing the

operator:

Jν(τν) = Λτν [Sν(t)]

Similarly for the other 2 moments of Intensity:

Λτν [f(t)] =1

2

∞Z0

f(t)E1(|t− τν |) dt

Milne’s equations

J

, H

and K

are all depth-weighted means of S

53


the 3 moments of Intensity:

J

, H

and K

are all depth-weighted means of S

the strongest contribution comes from the depth, where the argument of the exponential integrals is zero, i.e. t=

Gray, 92

54

The temperatureThe temperature--optical depth relationoptical depth relation

RadiativeRadiative

equilibriumequilibrium

The condition of radiative

equilibrium (expressing conservation of energy) requires that the flux at any given depth remains constant:

4πr2F(r) = 4πr2 · 4π∞Z0

Hν dν = const = L

In plane-parallel geometry r

R = const 4π

∞Z0

Hν dν = const

and in analogy to the black body radiation, from the Stefan-Boltzmann law we define the effective temperature:

4π

∞Z0

Hν dν = σT 4eff

F(r) = πF =

∞Z0

Z4π

Iν cos θ dω dν = π

∞Z0

Fν dν = 4π

∞Z0

Hν dν

55

The The effective temperatureeffective temperature

The effective temperature is defined by:

It characterizes the total radiative

flux transported through the atmosphere.

It can be regarded as an average of the temperature over depth in the atmosphere.

A blackbody radiating the same amount of total energy would have a temperature T = Teff

.

4π

∞Z0

Hν dν = σT 4eff

56

RadiativeRadiative


Let us now combine the condition of radiative

equilibrium with the equation of radiative

transfer in plane-parallel geometry:

μdIνdx

= −(κν + σν) (Iν − Sν)

1

2

1Z−1

μdIνdx

dμ = −12

1Z−1(κν + σν) (Iν − Sν) dμ

H

d

dx

⎡⎣12

1Z−1

μ Iν dμ

⎤⎦ = −(κν + σν) (Jν − Sν)

57

RadiativeRadiative


Integrate over frequency:

d

dx

∞Z0

Hν dν = −∞Z0

(κν + σν) (Jν − Sν) dν

const

∞Z0

(κν + σν) (Jν − Sν) dν = 0 su b s t itu t e S ν =κ ν

κ ν + σ ν

B ν +σ ν

κ ν + σ ν

J ν

∞Z0

κν [Jν − Bν(T )] dν = 0

4π

∞Z0

Hν dν = σT 4effin addition:

⎛⎝ ∞Z0

κν Jν dν = absorbed energy

⎞⎠

⎛⎝ ∞Z0

κν Bν dν = emitted energy

⎞⎠

T(x) or T()

at each depth:

58

RadiativeRadiative


∞Z0

κν [Jν − Bν(T )] dν = 0 4π

∞Z0

Hν dν = σT 4eff

T(x) or T()

The temperature T(r) at every depth has to assume the value for which the leftintegral over all frequencies becomes zero.

This determines the local temperature.

59

Iterative method for calculation of a stellar atmosphere:Iterative method for calculation of a stellar atmosphere:

the major parameters arethe major parameters are

Teff and g

T(x),

(x), B

[T(x)],P(x), ρ(x) J

(x), H

(x)

R∞0

κν(Jν − Bν) dν = 0 ?

4πR∞0Hν dν = σ T 4eff ?

T(x),

(x), B

[T(x)], ρ(x)

equation of transfer

a. hydrostatic equilibrium

b. equation of radiation transfer

dPdx = −gρ(x)

μdIνdx

= −(κν + σν) (Iν − Sν)

c. radiative

equilibrium∞Z0

κν [Jν − Bν (T )] dν = 0

d. flux conservation

4π

∞Z0

Hν dν = σT 4eff

e. equation of stateP = ρ k T

μmH

60

Grey atmosphere Grey atmosphere --

an approximation for the an approximation for the temperature structuretemperature structure

We derive a simple analytical approximation for the temperature structure. We assume that we can approximate the radiative

equilibrium integral by using a frequency-averaged absorption coefficient, which we can put in front of the integral.

∞Z0

κν [Jν − Bν(T )] dν = 0 κ̄

∞Z0

[Jν − Bν(T )] dν = 0

With: J =

∞Z0

Jν dν H =

∞Z0

Hν dν K =

∞Z0

Kν dν B =

∞Z0

Bν dν =σT 4

π

J = B4πH = σT 4eff

61

Grey atmosphereGrey atmosphere

We then assume LTE: S = B.

From

and a similar expression for frequency-integrated quantities

and with the approximations S = B, B = J:

Jν(τν) = Λτν [Sν(t)] =1

2

∞Z0

Sν(t)E1(|t− τν |)dt

Milne’s equation

!!! this is an integral equation for J)

J(τ̄ ) = Λτ̄ [S(t)], d τ̄ = κ̄dx

J (τ̄) = Λτ̄ [J(t)] =1

2

∞Z0

J (t)E1(|t− τ̄ |)dt

62

The exact solution of the The exact solution of the HopfHopf

integral equationintegral equation

Milne’s equation J() =

[J(t)] exact solution

(see Mihalas, “Stellar Atmospheres”)

J() = const. [

+ q()], with q

monotonic

Radiative

equilibrium -

grey approximation

1√3= 0.577 = q(0) ≤ q(τ̄) ≤ q(∞) = 0.710

J() = B() =

T4() =

const. [

+ q()]

with boundary conditions

T4() = ¾

T4eff

[

+ q()]

63

A simple approximation for A simple approximation for TT

0th

moment of equation of transfer (integrate both sides in d

from -1 to 1)

μdI

dx= −κ̄(I −B) dH

dτ̄= J −B = 0 (J = B) H = const =

σT 4eff4π

1st

moment of equation of transfer (integrate both sides in d

from -1 to 1)

dK

dτ̄= H =

σT 4eff4π

K(τ̄) = H τ̄ + constant

From Eddington’s

approximation at large depth: K = 1/3 J

μdI

dx= −κ̄(I −B)

64

Grey atmosphere Grey atmosphere ––

temperature distributiontemperature distribution

T 4(τ̄) =3πH

σ(τ̄ + c) H =

σ

4πT 4eff

T 4(τ̄) =3

4T 4eff (τ̄ + c) T4

is linear in

Estimation of c

∞Z0

tsE n (t) d t =s !

s + n

1/3 1/2

Hν (τ̄ = 0) =1

23H

⎡⎣ ∞Z0

tE2(t) dt+ c

∞Z0

E2(t) dt

⎤⎦

65

Grey atmosphere Grey atmosphere ––

HopfHopf

functionfunction

T 4(τ̄) =3

4T 4eff (τ̄ +

2

3)

based on approximation K/J = 1/3T = Teff

at

= 2/3, T(0) = 0.84 Teff

Remember: More in general J is obtained from

T 4(τ̄) =3

4T 4eff [τ̄ + q(τ̄)] q(τ̄) : Hopf function

Once Hopf

function is specified solution of the grey atmosphere (temperature distribution)

1√3= 0.577 = q(0) ≤ q(τ̄) ≤ q(∞) = 0.710

J(τ̄ ) = Λτ̄ [S(t)]

H(0) = H =1

2H(1 +

3

2c)→ c =

2

3

66

Selection of the appropriate Selection of the appropriate

non-grey grey

Equation of transfer

1st

moment

2nd

moment

In the grey case we define a ‘suitable’

mean opacity (absorption coefficient).

I =

∞Z0

Iν dν J =

∞Z0

Jν dν ...

μdIνdx = −κν(Iν − Sν)

κν ⇒ κ̄

μdIdx = −κ̃ (I − S)

dHνdx = −κν(Jν − Sν)

dKνdx = −κνHν

dHdx = −κ̂ (J − S)

dKdx = −κ̄H

κν ⇒ κ̄

67

Selection of the appropriate Selection of the appropriate

non-grey grey

Equation of transfer

1st

moment

2nd

moment

For each equation there is one opacity average that fits “grey equations”, however, all averages are different. Which one to select?

For flux constant models with H() = const. 2nd

moment equation is relevant

μdIνdx = −κν(Iν − Sν)

dHνdx = −κν(Jν − Sν)

μdIdx = −κ̃ (I − S)

dKνdx = −κνHν

dHdx = −κ̂ (J − S)

dKdx = −κ̄H

κν ⇒ κ̄

68

Mean opacities: fluxMean opacities: flux--weightedweighted

1st

possibility: Flux-weighted mean

allows the preservation of the K-integral (radiation pressure)

Problem: H

not known a priory (requires iteration of model atmospheres)

κ̄ =

∞R0

κνHν dν

H

69

Mean opacities: Mean opacities: RosselandRosseland

2nd

possibility:

Rosseland

mean

to obtain correct integrated energy flux and use local T

dKν

dx→ 1

3

dBν

dx=1

3

dBν

dT

dT

dx

Kν → 1

3Jν , Jν → Bν as τ → ∞

1

κ̄Ross=

∞R0

1κν

dBν(T )dT dν

∞R0

dBν(T )dT dν

large weight for low-opacity (more transparent to radiation) regions

1

κ̄=

∞R0

1κν

dKν

dx dν

dKdx

dKνdx = −κνHν

∞R0

1κνdKνdx dν = −

∞R0Hνdν

∞Z0

1

κν

dKν

dxdν = −H ⇒ (grey)⇒ 1

κ̄

dK

dx= −H

70

Mean opacities: Mean opacities: RosselandRosseland

at large

the T structure is accurately given by

T 4 =3

4T 4eff [τRoss + q(τRoss)] Rosseland

opacities used in stellar interiors

For stellar atmospheres Rosseland

opacities allow us to obtain initial approximate values for the Temperature stratification (used for further iterations).

71

grey: q() = exactgrey: q= 2/3

non-greynumerical

T4

vs.

72

T vs. log()

non-greynumerical

grey: q() = exactgrey: q= 2/3

73

Iterative method for calculation of a stellar atmosphere:Iterative method for calculation of a stellar atmosphere:

the major parameters arethe major parameters are

Teff and g

T(x),

(x), B

[T(x)],P(x), ρ(x) J

(x), H

(x)

R∞0

κν(Jν − Bν) dν = 0 ?

4πR∞0Hν dν = σ T 4eff ?

T(x),

(x), B

[T(x)], ρ(x)

equation of transfer

a. hydrostatic equilibrium

b. equation of radiation transfer

dPdx = −gρ(x)

μdIνdx

= −(κν + σν) (Iν − Sν)

c. radiative

equilibrium∞Z0

κν [Jν − Bν (T )] dν = 0

d. flux conservation

4π

∞Z0

Hν dν = σT 4eff

e. equation of stateP = ρ k T

μmH

3. radiative transfer

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