3. Transport of energy: radiation

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 Lπ = =

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: Iν = Bν

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?

dEν ~ Iν cosθ dω

integrate over the whole solid angle (Ω = 4π):

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

dEν ~ Iν cosθ dω integrate over the whole solid angle (Ω = 4π):

We distinguish between the outward direction (0 < θ < π/2) and the inward direction (π/2 < θ < π), 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 π/2 < θ < π −> cosθ < 0 second term negative !!

“astrophysical flux”

10

Total radiative flux

Integral over frequencies ν

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 ν

and integrating over ν

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) cosθ dA

in spherical symmetry: dA = R2o sinθ dθ dφ

→ Eν =∆ω∆ν∆tR2o

Z 2π

o

Z π/2

0

Iν(cos θ) cosθ sinθ dθ dφ

because of spherical symmetry the integral of intensity towards the observer over the stellar surface is proportional to πFν

+, 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 Iν, is the quantity generally measured for stars. For the Sun, whose disk is resolved, we can also measure Iν(the variation of Iν over the solar disk is called the limb darkening)

unlike Iν, Fν 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 4π:

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

Interactions between photons and matterInteractions between photons and matter

absorption of radiation

Iν

ds

Iνo Iν(s)s

Over a distance s:Iν (s) = I

oν e−

sR0

κν ds

τν :=

sZ0

κν ds

optical depth(dimensionless)

or: dτν = κν ds

loss of intensity in the beam (true absorption/scattering)microscopical view: κν=n σν

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 τν = 1 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 τν ~ 1

optically thick (thin) medium: τν > (<) 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 [τν,τν+dτν]

= probability of non-absorption between 0 and τν and absorption in dτν

- 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 <τν>=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 τν ≈ 1

In the continuum κν 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):

π Fν

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

dωdA

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


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


Spherical symmetry

angle θ 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


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 Iν(cos θ,τν= 0) depends on μ = cos θ , τν(Ri) and Sν

The physics of Sν is crucial for radiative transfer κν =dτνds ≈ ∆τν

∆s ≈ 1s̄

Sν =²νκν≈ ²ν · s̄

τ = 1 corresponds to free mean path of photons

source function Sν corresponds to intensity emitted over the free mean path of photons

source function

dim [Sν] = [Iν]

−κν dx = dτν

τν = −xZ

Ro

κν dx

28


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(τ) solution of transfer equation Iν(μ,τν)

independent of radiation field

29



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

the absorption process is characterized by the absorption coefficient σν, analogous to κν:

dIν = −σνIνds

ν = ν’

and at each frequency ν: 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



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-τν/μ and integrate between τ1 (outside) and τ2 (> τ1, inside)

d

dτν(Iν e

−τν/μ) = −Sν e−τν/μ

μ

linear 1st order differential equation

μdIνdτν

= Iν − Sν

hIν e− τν

μ

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 τ2 decreased by exponential factor to τ1 contribution to the intensity by

emission along the path from τ2 to τ1(at each point decreased by the exponential factor)

integral form of equation of radiation transfer

Formal solution! actual solution can be complex, since Sν can depend on Iν

hIν e− τν

μ

iτ2τ1= −

τ2Zτ1

Sν e− τν

μdtνμ

Iν(τ1, μ) = Iν(τ2,μ) e− τ2−τ1μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

33

Boundary conditionsBoundary conditions

a. incoming radiation: μ < 0 at τ2 = 0

usually we can neglect irradiation from outside: Iν(τ2 = 0, μ < 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−τ1μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

34

Boundary conditionsBoundary conditions

b. outgoing radiation: μ > 0 at τ2 = τmax ∞

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)Iν increases less rapidly than the exponential

Ioutν (τν ,μ) =

∞Zτν


μdt

μ

Iν(τν) = Ioutν (τν) + I

inν (τν)

and at a given position τν in the atmosphere:

Iν(τ1,μ) = Iν(τ2,μ) e− τ2−τ1μ +

τ2Zτ1

Sν(t) e− t−τ1

μdt

μ

35

Emergent intensityEmergent intensity

from the latter emergent intensity

τν = 0, μ > 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 τ/μ < 1.

36


suppose that Sν is linear in τν (Taylor expansion around τν = 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 τν = μ

the emergent intensity corresponds to the source function at τν = 1 along the line of sight

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

37


μ = 1 (normal direction):

Iν(0, 1) = Sν(τν = 1)

μ = 0.5 (slanted direction):

Iν(0, 0.5) = Sν(τν = 0.5)

in both cases: Δτ/μ ≈ 1

spectral lines: compared to continuum τν/μ = 1 is reached at higher layer in the atmosphere

Sνline < Sν

cont

a dip is created in the spectrum

38

Line formationLine formation

simplify: μ = 1, τ1=0 (emergent intensity), τ2 = τ

Sν 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

τ ∞

exp(-τν) ≈ 1 - τν

40

independent of κν, no line (e.g. black body Bν)

Iν = τν Sν = κν dsν Sν

e.g. HII region, solar corona

enhanced κν

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

Iν = τν Sν

42

The diffusion approximationThe diffusion approximation

At large optical depth in stellar atmosphere photons are local: Sν Bν

Expand Sν (= Bν) as a power-series:

Sν(t) =

∞Xn=0

dnBνdτnν

(t − τν)n/n!

In the diffusion approximation (τν >>1) we retain only first order terms:

Bν(t) = Bν(τν) +dBνdτν

(t− τν)

Ioutν (τν ,μ) =

∞Zτν

[Bν(τν) +dBνdτν

(t − τν)]e−(t−τν)/μ dt

μ

43

The diffusion approximationThe diffusion approximation

Substituting:

At τν = 0 we obtain the Eddington-Barbier relation for the observed emergent intensity.

It is given by the Planck-function and its gradient at τν = 0.

It depends linearly on μ = cos θ.

∞Z0

uke−udu = k!

I inν (τν , μ) = −τν/μZ0


μu]e−u du

Ioutν (τν ,μ) =

∞Z0


μu]e−u du = Bν(τν) + μdBνdτν

Ioutν (τν ,μ) =

∞Zτν


(t − τν)]e−(t−τν)/μ dt

μ

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 Bν(0), dBν/dτν

Bν(t) = Bν(0) +dBνdτνt = a+ b · t = 2hν3

c21

ehν/kT(t)−1

T(t): empirical temperature stratification of solar photosphere

center-to-limb variation of intensity

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 -τν /cosθ

From Sν = a + bτν:

Iν(0,μ) = Sν(τν = μ) Sν

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) +dBνdτν

(t− τν)

0 < μ < 1

-1 < μ < 0

τ >> 1

I inν (τν ,μ) = −τν/μZ0

[Bν(τν) +dBν

dτνμu]e−u du

Ioutν (τν ,μ) = Bν(τν) + μdBνdτν

I−ν (τν ,μ) = Bν(τν) + μdBνdτν

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

dBνdτν

Kν =1

2

Z 1

−1μ2 Iν dμ =

1

3Bν(τν)

= −13

1

κν

dBνdx

= − 1

3κν

dBνdT

dT

dx

flux Fν ~ 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

μ < 0μ > 0

After the previous approximations, we now want to calculate exact solutions for tharadiative momenta Jν, Hν, Kν. Those areimportant 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 Λ operator:

Jν(τν) = Λτν [Sν(t)]

Similarly for the other 2 moments of Intensity:

Hν (τν) =1

2

∞Z0

Sν(t)E2(|t− τν |)dt = Φτν [Sν(t)]

Λτν [f(t)] =1

2

∞Z0

f(t)E1(|t− τν |) dt

Milne’s equations

Kν(τν) =1

2

∞Z0

Sν (t)E3(|t− τν |)dt = Xτν [Sν (t)]

Jν, Hν and Kν are all depth-weighted means of Sν

53


the 3 moments of Intensity:

Hν(τν ) =1

2

∞Z0

Sν(t)E2(|t− τν |)dt = Φτν [Sν (t)]

Kν(τν) =1

2

∞Z0

Sν(t)E3(|t− τν |)dt = Xτν [Sν(t)]

Jν, Hν and Kν are all depth-weighted means of Sν

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

Jν =1

2

∞Z0

Sν (t)

∞Z1

e−w|t−τν|dw

wdt =

1

2

∞Z0


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


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

Hν

d

dx

⎡⎣12

1Z−1

μ Iν dμ

⎤⎦ = −(κν + σν) (Jν − Sν)

57


Integrate over frequency:

d

dx

∞Z0

Hν dν = −∞Z0

(κν + σν) (Jν − Sν) dν

const

∞Z0

(κν + σν) (Jν − Sν) dν = 0 substitute Sν =κν

κν + σνBν +

σνκν + σν

Jν

∞Z0

κν [Jν − Bν(T )] dν = 0

4π

∞Z0

Hν dν = σT 4eff+

⎛⎝ ∞Z0

κν Jν dν = absorbed energy

⎞⎠⎛⎝ ∞Z0

κν Bν dν = emitted energy

⎞⎠

T(x) or T(τ)

58


∞Z0

κν [Jν − Bν(T )] dν = 0 4π

∞Z0

Hν dν = σT 4eff

T(x) or T(τ)

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), κν(x), Bν[T(x)],P(x), ρ(x) Jν(x), Hν(x)

R∞0

κν(Jν − Bν) dν = 0 ?4πR∞0Hν dν = σ T 4eff ?

ΔT(x), Δκν(x), ΔBν[T(x)], Δρ(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


Milne’s equation

!!! this is an integral equation for J(τ) !!!

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(τ) = Λτ[J(t)] exact solution (see Mihalas, “Stellar Atmospheres”)

J(τ) = const. [ τ + q(τ)], with q(τ) monotonic

Radiative equilibrium - grey approximation

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

J(τ) = B(τ) = σ/π T4(τ) = const. [ τ + q(τ)]

with boundary conditions

T4(τ) = ¾ T4eff [τ + q(τ)]

63

A simple approximation for A simple approximation for TT(τ(τ))

0th moment of equation of transfer (integrate both sides in dμ 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 μdμ 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

J(τ̄) = 3H τ̄ + c =σT 4

π

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 τ

Estimation of c

∞Z0

tsEn(t) dt =s!

s + n

1/3 1/2

Hν (τ̄ = 0) =1

2

∞Z0

J(t)E2(t)dt =1

2

∞Z0

[3Ht + c]E2(t)dt

Hν(τ̄ = 0) =1

23H

⎡⎣ ∞Z0

tE2(t) dt+ c

∞Z0

E2(t) dt

⎤⎦

J(τ̄) = 3H τ̄ + c =σT 4

π

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 τ = 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 κκνν

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

67

Selection of the appropriate Selection of the appropriate κκνν

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(τ) = 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

68

Mean opacities: fluxMean opacities: flux--weightedweighted

1st possibility: Flux-weighted mean

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

Problem: Hν 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

dBνdx

=1

3

dBνdT

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

dKνdx 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 τ 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(τ) = exactgrey: q(τ) = 2/3

non-greynumerical

T4 vs. τ

72

T vs. log(τ)

non-greynumerical

grey: q(τ) = exactgrey: q(τ) = 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), κν(x), Bν[T(x)],P(x), ρ(x) Jν(x), Hν(x)

R∞0

κν(Jν − Bν) dν = 0 ?4πR∞0Hν dν = σ T 4eff ?

ΔT(x), Δκν(x), ΔBν[T(x)], Δρ(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

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3. Transport of energy: radiation