3. radiative transfer

3. Transport of energy: radiation

specific intensity, radiative

optical depth

absorption & emission

equation of transfer, source function

formal solution, limb darkening

temperature distribution

grey atmosphere, mean opacities

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

Mechanisms of energy transport

radiation: Frad

(most important)

convection: Fconv

(important especially in cool stars)

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

and corona

radial flow of matter: corona and stellar wind

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

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

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)

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

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

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

solid angle : dω =dA

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

Radiative

How much energy flows through surface element dA?

integrate

over the whole solid angle (4):

πFν =

Iν(cos θ) cos θ dω =

Iν(cos θ) cos θ sin θdθdφ

“astrophysical flux”

is the monochromatic radiative

flux. The factor π

in the definition is historical.

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

Radiative

The monochromatic radiative

at frequency

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

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 π/2

Iν(cos θ) cos θ sin θdθdφ+

πFν =

Iν(cos θ) cos θ dω =

Note: for /2 < cosθ

< 0 second term negative !!

“astrophysical flux”

Total radiative

Integral over frequencies

πFνdν = Frad

is the total radiative

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

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 π/2

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 ∞

πF+ν (Ro)dν = L (Luminosity)

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

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 π/2

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 !!

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

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

Mean intensity, energy density & radiation pressure

Integrating over the solid angle and dividing by 4:

Jν =1

Iν dω mean intensity

energy density

radiation pressure (important in hot stars)

uν =radiation energy

volume=1

Iν dω =4π

pν =1

Iν cos2 θ dω

pressure =force

area=d momentum(= E/c)

Moments of the specific intensity

moment

moment (Eddington

moment

Jν =1

ZIν dω =

−1Iν dμ =

−1Iν dμ

Hν =1

ZIν cos θ dω =

−1Iν μ dμ =

Kν =1

ZIν cos

2 θ dω =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

Over a distance s:Iν (s) = I

−sR0

κν ds

τν :=

κν ds

optical depth(dimensionless)

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

dIν = −κνIνds

κν : absorption coefficient

[κν ] = cm−1

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

We can see through atmosphere until

optically thick

(thin) medium:

The optical thickness of a layer determines the fraction of the

intensity passing through the layer

τν = 1 =

κν ds

photon mean free path

What is the average distance over which photons travel?

< τν >=

τν p(τν ) dτνexpectation value

probability of absorption in interval [

= 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)

total probability: e−τν dτν

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

= dτν

photon mean free path

< τν >=

τν p(τν) dτν =

τν e−τν dτν = 1

mean free path corresponds to <

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

κν(homogeneous

material)

Zxe−x dx = −(1 + x) e−x

Principle of line formation

observer sees through the atmospheric layers up to

In the continuum

is smaller than in the line see deeper into the atmosphere

T(cont) > T(line)

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):

force =dfphot =momentum(=E/c)

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

Eabs =

Iν cos θ dω dν dA dt ds = π

κν Fν dν dA dt ds

radiative

acceleration

dfphot =π

κν Fν dν

dtdA dt ds = grad dm (dm = ρ dA ds)

grad =π

κν Fν dν

emission of radiation

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

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

tranfertranfer

Plane-parallel symmetry

dx = cos θ ds = μ ds

ds= μ

μ dIν(μ,x)dx = −κν Iν(μ, x) + ²ν

tranfertranfer

Spherical symmetry

between ray and radial direction is not constant

∂r+dθ

∂θ=

∂μ= − sin θ ∂

=⇒ d

ds= μ

∂r+sin2 θ

∂μ= μ

∂r+1 − μ2

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

dr = ds cos θ → dr

ds= cos θ (as in plane−parallel)

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

= 0) depends on = cos

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

κν dx

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

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

²scν dω =

σν Iνdω

²scν

dω = σν

Iνdω

dEemν =

²scν dω

dEabsν =

σν Iνdω

²scνσν=1

Iνdω

Sν = Jν

completely dependent on radiation field

not dependent on temperature T

tranfertranfer

c. mixed case

Sν =²ν + ²

κν + σν=

κνκν + σν

²νκν+

σνκν + σν

²scνσν

Sν =²ν + ²

κν + σν=

κνκν + σν

Bν +σν

κν + σνJν

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

, inside)

dτν(Iν e

−τν/μ) = −Sν e−τν/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

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

(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

τ2Zτ1

Sν(t) e− t−τ1

Boundary conditionsBoundary conditions

a. incoming radiation:

< 0 at

usually we can neglect irradiation from outside: I

< 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

Iν(τ1,μ) = Iν(τ2,μ) e− τ2−τ1

τ2Zτ1

Sν(t) e− t−τ1

Boundary conditionsBoundary conditions

b. outgoing radiation:

> 0 at

We have either

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

Iν(τν) = Ioutν (τν) + I

inν (τν)

and at a given position

in the atmosphere:

Iν(τ1,μ) = Iν(τ2,μ) e− τ2−τ1

τ2Zτ1

Sν(t) e− t−τ1

Emergent intensityEmergent intensity

from the latter emergent intensity

Iν(0,μ) =

Sν(t) e− tμdt

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

suppose that S

is linear in

(Taylor expansion around

Eddington-Barbier

relation

emergent intensityIν(0,μ) =

(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

= 1 (normal direction):

Iν(0, 1) = Sν(τν = 1)

= 0.5 (slanted direction):

Iν(0, 0.5) = Sν(τν = 0.5)

in both cases:

spectral lines:

compared to continuum

= 1 is reached at higher layer in the atmosphere

< Scont

a dip is created in the spectrum

Line formationLine formation

simplify:

= 1, 1

=0 (emergent intensity), 2

independent of location

Iν(0) = Iν(τν) e−τν + Sν

τνZ0

e−t dt = Iν(τν) e−τν + Sν (1 − e−τν )

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

independent of

, no line (e.g. black body B

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

Line formation example: solar coronaLine formation example: solar corona

The diffusion approximationThe diffusion approximation

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

Expand S

as a power-series:

Sν(t) =

∞Xn=0

dτnν(t − τν)

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

The diffusion approximationThe diffusion approximation

Substituting:

= 0 we obtain the Eddington-Barbier

relation for the observed emergent intensity.

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

It depends linearly on μ

uke−udu = k!

I inν (τν , μ) = −τν/μZ0

[Bν(τν) +dBν

dτνμu]e−u du

Ioutν (τν ,μ) =

[Bν(τν) +dBν

dτνμu]e−u du = Bν(τν) + μ

Ioutν (τν ,μ) =

∞Zτν

[Bν(τν) +dBν

dτν(t − τν)]e

−(t−τν)/μ dtμ

t→ u =t − τνμ

→ dt = μ du

Solar limb darkeningSolar limb darkening

Iν(0,μ)Iν(0,1)

=Bν(0)+

dBνdτν

Bν(0)+dBνdτν

from the intensity measurements B

Bν(t) = Bν(0) +dBν

dτνt = a+ b · t = 2hν3

ehν/kT(t)−1

T(t): empirical temperature stratification of solar photosphere

center-to-limb variation of intensity

diffusion approximation:

Solar limb darkeningSolar limb darkening

...and also giant planets

Solar limb darkening: temperature stratificationSolar limb darkening: temperature stratification

Iν(0,μ) =

Sν(t) e− tμdt

exponential extinction varies as -

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

EddingtonEddington

approximationapproximation

In the diffusion approximation we had:

Bν(t) = Bν(τν) +dBν

dτν(t− τν)

I inν (τν ,μ) = −τν/μZ0

[Bν(τν) +dBν

dτνμu]e−u du

Ioutν (τν ,μ) = Bν(τν) + μdBν

I−ν (τν ,μ) = Bν(τν) + μdBν

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

EddingtonEddington

approximationapproximation

Jν =1

−1Iν dμ = Bν(τν)

Hν =Fν4=1

−1μ Iν dμ =

Kν =1

−1μ2 Iν dμ =

3Bν(τν)

= −13

dx= − 1

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

Jν =1

1Z−1

Iν dμ =1

Ioutν dμ+1

0Z−1

I inν dμ

Jν =1

⎡⎣ 1Z0

∞Zτν

Sν(t)e−(t−τν)/μ dt

μdμ−

0Z−1

τνZ0

Sν (t)e−(t−τν )/μ dt

⎤⎦

substitute w = 1μ ⇒ dw

w = − 1μdμ w = − 1

μ ⇒ dww = − 1

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

radiative

momenta

Those are

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

Ioutν (τν ,μ) =

∞Zτν

I inν (τν ,μ) =

0Zτν

Jν =1

⎡⎣ ∞Z1

∞Zτν

Sν(t)e−(t−τν)wdt

τνZ0

Sν(t)e−(τν−t)wdt

⎤⎦

SchwarzschildSchwarzschild--Milne equationsMilne equations

Jν =1

⎡⎣ ∞Zτν

Sν(t)

e−(t−τν)wdw

τνZ0

Sν(t)

e−(τν−t)wdw

⎤⎦> 0 > 0

Jν =1

Sν (t)

e−w|t−τν|dw

Sν(t)E1(|t− τν |)dt

Schwarzschild’s

equation

E1(t) =

e−txdx

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

Exponential integrals

En(t) = tn−1

x−ne−xdx

En(0) = 1/(n− 1),En(t→∞) = e−t/t→ 0

= −En−1,ZEn(t) = −En+1(t)

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

Gray, 92

Introducing the

operator:

Jν(τν) = Λτν [Sν(t)]

Similarly for the other 2 moments of Intensity:

Λτν [f(t)] =1

f(t)E1(|t− τν |) dt

Milne’s equations

are all depth-weighted means of S

the 3 moments of Intensity:

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

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

Hν dν = const

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

Hν dν = σT 4eff

F(r) = πF =

Iν cos θ dω dν = π

Fν dν = 4π

Hν dν

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

Hν dν = σT 4eff

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)

1Z−1

μdIνdx

dμ = −12

1Z−1(κν + σν) (Iν − Sν) dμ

⎡⎣12

1Z−1

μ Iν dμ

⎤⎦ = −(κν + σν) (Jν − Sν)

RadiativeRadiative

Integrate over frequency:

Hν dν = −∞Z0

(κν + σν) (Jν − Sν) dν

(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

κν [Jν − Bν(T )] dν = 0

Hν dν = σT 4effin addition:

⎛⎝ ∞Z0

κν Jν dν = absorbed energy

⎞⎠

⎛⎝ ∞Z0

κν Bν dν = emitted energy

⎞⎠

T(x) or T()

at each depth:

RadiativeRadiative

κν [Jν − Bν(T )] dν = 0 4π

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.

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

(x), B

[T(x)],P(x), ρ(x) J

(x), H

κν(Jν − Bν) dν = 0 ?

4πR∞0Hν dν = σ T 4eff ?

(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

Hν dν = σT 4eff

e. equation of stateP = ρ k T

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.

κν [Jν − Bν(T )] dν = 0 κ̄

[Jν − Bν(T )] dν = 0

With: J =

Jν dν H =

Hν dν K =

Kν dν B =

Bν dν =σT 4

J = B4πH = σT 4eff

Grey atmosphereGrey atmosphere

We then assume LTE: S = B.

and a similar expression for frequency-integrated quantities

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

Jν(τν) = Λτν [Sν(t)] =1

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

J (t)E1(|t− τ̄ |)dt

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

+ q()]

A simple approximation for A simple approximation for TT

moment of equation of transfer (integrate both sides in d

from -1 to 1)

dx= −κ̄(I −B) dH

dτ̄= J −B = 0 (J = B) H = const =

σT 4eff4π

moment of equation of transfer (integrate both sides in d

from -1 to 1)

dτ̄= H =

σT 4eff4π

K(τ̄) = H τ̄ + constant

From Eddington’s

approximation at large depth: K = 1/3 J

dx= −κ̄(I −B)

Grey atmosphere Grey atmosphere ––

temperature distributiontemperature distribution

T 4(τ̄) =3πH

σ(τ̄ + c) H =

4πT 4eff

T 4(τ̄) =3

4T 4eff (τ̄ + c) T4

is linear in

Estimation of c

tsE n (t) d t =s !

1/3 1/2

Hν (τ̄ = 0) =1

⎡⎣ ∞Z0

tE2(t) dt+ c

E2(t) dt

⎤⎦

Grey atmosphere Grey atmosphere ––

HopfHopf

functionfunction

T 4(τ̄) =3

4T 4eff (τ̄ +

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

= 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 +

2c)→ c =

Selection of the appropriate Selection of the appropriate

non-grey grey

Equation of transfer

moment

In the grey case we define a ‘suitable’

mean opacity (absorption coefficient).

Iν dν J =

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

Selection of the appropriate Selection of the appropriate

non-grey grey

Equation of transfer

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

Mean opacities: fluxMean opacities: flux--weightedweighted

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 =

κνHν dν

Mean opacities: Mean opacities: RosselandRosseland

possibility:

Rosseland

to obtain correct integrated energy flux and use local T

dx→ 1

Kν → 1

3Jν , Jν → Bν as τ → ∞

κ̄Ross=

dBν(T )dT dν

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

dx dν

dKνdx = −κνHν

1κνdKνdx dν = −

∞R0Hνdν

dxdν = −H ⇒ (grey)⇒ 1

dx= −H

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).

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

non-greynumerical

T vs. log()

non-greynumerical

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

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

(x), B

[T(x)],P(x), ρ(x) J

(x), H

κν(Jν − Bν) dν = 0 ?

4πR∞0Hν dν = σ T 4eff ?

(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

Hν dν = σT 4eff

e. equation of stateP = ρ k T

3. radiative transfer

Documents

three-dimensional radiative transfer in clouds

radiative transfer: interpreting the observed light ?

radiative transfer – an overview

radiative heat transfer

simple radiative transfer in decomposed domains

3d radiative transfer in cloud resolving models ·...

solar radiation radiative transfer

radiative heat transfer in boiler furnaces

introduction to solar radiative transfer i · introduction...

impact of updated radiative transfer scheme in racmo2.3p3...

the nemesis planetary atmosphere radiative transfer and...

radiative transfer theory

a 3d radiative transfer framework

radiative transfer simulations

part ii radiative transfer

non-linear radiative-conductive heat transfer in a...

section 3. radiative transfer - university of toronto ·...

radiative transfer, canopy radiation and turbulence

arts, the atmospheric radiative transfer simulator ... ·...

radiative transfer in the ocean and applications using the...