3. Radiative transfer

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
1
Energy flux conservation
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:
4 r F (r )  const .  L
2
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.
2
Transport of energy
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
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
3
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)
dEν = Iν (cos θ) cos θ dA dω dν dt
([Iν ]: erg cm−2 sr−1 Hz−1 s−1 )
Iλ =
c
I
λ2 ν
(from Iλ dλ = Iν dν and ν = c/λ)
Intensity depends on location in
space, direction and frequency
4
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 dA dω dν dt
dEν0 = Iν0 (cos θ0 ) cos θ0 dA0 dω 0 dν dt
5
Invariance of the specific intensity
energy is conserved:
dEν = dEν0
and
dω =
projected area
distance2
dω 0 =
and
dEν = Iν (cos θ) cos θ dA dω dν dt
dEν0 = Iν0 (cos θ 0 ) cos θ0 dA0 dω 0 dν dt
Iν = Iν0
=
dA0 cos θ0
r2
dA cos θ
r2
Specific intensity is constant
along rays - as long as there
is no absorption and emission
of matter between emitter and
receiver
In TE: I = B
6
Spherical coordinate system and solid angle dω
solid angle : dω =
dA
r2
4πr 2
Total solid angle = 2 = 4π
r
dA = (r dθ)(r sin θ dφ)
→ dω = sin θ dθ dφ
define μ = cos θ
dμ = − sin θ dθ
dω = sin θ dθ dφ = −dμ dφ
7
Radiative flux
How much energy flows through surface element dA?
dE ~ I cosθ dω
 integrate over the whole solid angle (4):
πFν =
Z
4π
“astrophysical flux”
Iν (cos θ) cos θ dω =
Z
0
2π
Z
π
Iν (cos θ) cos θ sin θdθdφ
0
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.
8
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):
πFν =
“astrophysical flux”
Z
Iν (cos θ) cos θ dω =
Z
0
4π
2π
Z
π
Iν (cos θ) cos θ sin θdθdφ
0
We distinguish between the outward direction (0 <  < /2)
and the inward direction (/2 < so that the net flux is
=
Z
0
2π
Z
0
πFν = πFν+ − πFν− =
π/2
Iν (cos θ) cos θ sin θdθdφ +
Z
0
2π
Z
π
Iν (cos θ) cos θ sin θdθdφ
π/2
Note: for /2 < cosθ < 0  second term negative !!
9
Total radiative flux
Integral over frequencies 
Z
0
∞
πFν dν = 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.
10
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 ) =
πFν+ (Ro )
=
Z
2π
0
Z
π/2
Iν (cos θ) cos θ sin θdθdφ
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
4πRo2
·
Z
0
∞
4πRo2 · πFν (Ro ) = Lν
πFν+ (Ro )dν = L (Luminosity)
11
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
Z
Z
Eν =
dE = ∆ω ∆ν ∆t
1/2 sphere
Iν (cos θ) cos θ dA
1/2 sphere
in spherical symmetry: dA = Ro2 sin θ dθ dφ
→ Eν =
∆ω ∆ν ∆t Ro2
Z
o
2π
Z
π/2
Iν (cos θ) cos θ sin θ dθ dφ
0
πFν+
 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 !!
12
Observed flux
Solid angle of telescope at distance d:
∆ω = ∆A/d2
+
Eν = ∆ω ∆ν ∆t Ro2 πFν+ (Ro )
flux received = flux emitted x (R/r)2
Fνobs
radiative energy
Ro2
=
= 2 πFν+ (Ro )
area · frequency · time
d
unlike I, F decreases with
increasing distance
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)
R∞
0
obs
F ν,¯
dν = 1.36 K W/m 2
13
Mean intensity, energy density & radiation pressure
Integrating over the solid angle and dividing by 4:
1
Jν =
4π
Z
mean intensity
Iν dω
4π
radiation energy
1
uν =
=
volume
c
Z
Iν dω =
4π
Jν
c
energy density
4π
1
pν =
c
Z
Iν cos2 θ dω
4π
pressure =
radiation pressure
(important in hot stars)
force
d momentum(= E/c) 1
=
area
dt
area
14
Moments of the specific intensity
1
Jν =
4π
Z
1
Iν dω =
4π
Z
2π
0
Z
1
1
dφ
Iν dμ =
2
−1
for azimuthal symmetry
1
Hν =
4π
Z
1
Iν cos θ dω =
2
Z
1
Kν =
4π
Z
1
Iν cos2 θ dω =
2
1
Iν μ dμ =
−1
Z
1
−1
Z
1
Iν dμ
−1
Fν
4
Iν μ2 dμ =
0th moment
c
pν
4π
1st moment
(Eddington flux)
2nd moment
15
Interactions between photons and matter
absorption of radiation
ds
dIν = −κν Iν ds
κν : absorption coefficient
I
[κν ] = cm−1
microscopical view: n 
loss of intensity in the beam (true absorption/scattering)
Over a distance s:
s
o
I
I(s)
Convention:  = 0 at the outer edge
of the atmosphere, increasing
inwards
Iν (s) =
Iνo
τν :=
Zs
0
e
−
Rs
κ ν ds
0
optical depth
κν ds
or:
(dimensionless)
d =  ds
16
optical depth
Iν (s) =
Iνo
e
The optical thickness of a layer
determines the fraction of the
intensity passing through the layer
−τν
Iνo
' 0.37 Iνo
if τν = 1 → Iν =
e
We can see through
atmosphere until  ~ 1
optically thick (thin) medium:  > (<) 1
The quantity  has a geometrical interpretation in terms of mean
free path of photons s̄:
τν = 1 =
Zs̄
κν ds
o
photons travel on average
s̄
for a length
before absorption
17
photon mean free path
What is the average distance over which photons travel?
expectation value
< τν >=
Z∞
τν p(τν ) dτν
0
probability of absorption in interval [+d]
= probability of non-absorption between 0 and  and absorption in d
- probability that photon is absorbed: p(0, τν ) =
∆I(τ )
Io − I(τν )
I(τν )
=
= 1−
Io
Io
Io
- probability that photon is not absorbed: 1 − p(0, τν ) =
I(τν )
= e−τν
Io
- probability that photon is absorbed in [τν , τν + dτν ] : p(τν , τν + dτν ) =
total probability: e−τν dτν
dIν
I(τν )
= dτν
18
photon mean free path
< τν >=
Z∞
τν p(τν ) dτν =
0
Z∞
τν e
−τν
Z
dτν = 1
xe−x dx = −(1 + x) e−x
0
mean free path corresponds to <>=1
if κν (s) = const :
(homogeneous
material)
∆τν = κν ∆s → ∆s = s̄ =
1
κν
19
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)
20
radiative acceleration
In the absorption process photons release
momentum E/c to the atoms, and the
corresponding force is:
force =dfphot =
momentum(=E/c)
dt
The infinitesimal energy absorbed is:
dEνabs = dIν cos θ dA dω dt dν = κν Iν cosθ dA dω dt dν ds
The total energy absorbed is (assuming that κν does not depend on ω):
E abs =
Z∞
0
κν
Z
Iν cos θ dω dν dA dt ds = π
4π
Z∞
κν Fν dν dA dt ds
0
 F
21
radiative acceleration
dfphot
π
=
c
R∞
κν Fν dν
0
dt
dA dt ds = grad dm
grad =
π
cρ
Z∞
(dm = ρ dA ds)
κν Fν dν
0
22
emission of radiation
ds
d
dA
dV=dA ds
energy added by emission processes within dV
dEνem = ²ν dV dω dν dt
²ν : emission coefficient
[²ν ] = erg cm−3 sr−1 Hz−1 s−1
23
The equation of radiative tranfer
If we combine absorption and emission together:
dEνabs = dIνabs dA cos θ dω dν dt = −κν Iν dA cos θ dω dt dν ds
dEνem = dIνem dA cos θ dω dν dt = ²ν dA cos θ dω dν dt ds
dEνabs +dEνem = (dIνabs +dIνem ) dA cos θ dω dν dt = (−κν Iν +²ν ) dA cos θ dω dν dt ds
dIν = dIνabs + dIνem = (−κν Iν + ²ν ) ds
dIν
ds
= −κν Iν + ²ν
differential equation
describing the flow of
radiation through
matter
24
The equation of radiative tranfer
Plane-parallel symmetry
dx = cos θ ds = μ ds
d
d
=μ
ds
dx
(μ,x)
μ dIνdx
= −κν Iν (μ, x) + ²ν
25
The equation of radiative tranfer
Spherical symmetry
d
dr ∂
dθ ∂
=
+
ds
ds ∂r
ds ∂θ
dr = ds cos θ →
dr
= cos θ
ds
−r dθ = sin θ ds (dθ < 0) →
(as in plane−parallel)
dθ
sin θ
=−
ds
r
∂
∂μ ∂
∂
=
= − sin θ
∂θ
∂θ ∂μ
∂μ
d
∂
sin2 θ ∂
∂
1 − μ2 ∂
=⇒
=μ
+
=μ
+
ds
∂r
r ∂μ
∂r
r ∂μ
angle  between ray and radial direction
is not constant
μ
∂
∂r Iν (μ, r)
+
1−μ2 ∂
r ∂μ Iν (μ, r)
= −κν Iν (μ, r) + ²ν
26
The equation of radiative tranfer
Optical depth and source function
In plane-parallel symmetry:
μ
dIν (μ,x)
dx
= −κν (x) Iν (μ, x) + ²ν (x)
optical depth increasing
towards interior:
−κν dx = dτν
Zx
τν = − κν dx
Ro
μ
dIν (μ,τν )
dτν
Sν =
= Iν (μ, τν ) − Sν (τν )
²ν
κν
source function
dim [S] = [I]
κν =
Sν =
dτν
∆τν
1
ds ≈ ∆s ≈ s̄
²ν
κν ≈ ²ν · s̄
= 1 corresponds to free mean path of
photons
source function S corresponds to intensity
emitted over the free mean path of photons
Observed emerging intensity I(cos ,= 0)
depends on = cos  , (Ri) and S
The physics of S is crucial for radiative transfer
27
The equation of radiative tranfer
Source function: simple cases
a. LTE (thermal absorption/emission)
Sν =
²ν
κν
= Bν (T )
independent of radiation field
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(,)
28
The equation of radiative tranfer
Source function: simple cases
b. coherent isotropic scattering (e.g. Thomson scattering)
the absorption process is characterized by the
scattering coefficient , analogous to 
dEνem
=
Z
²sc
ν dω
4π
dEνabs =
Z
4π
σν Iν dω
dIν = −σν Iν ds
 = ’
incident = scattered
and at each frequency  dEνem = dEνabs
Z
4π
²sc
ν
²sc
ν dω =
Z
Z
σν Iν dω
4π
dω = σν
4π
Z
4π
Iν dω
²sc
1
ν
=
σν
4π
Z
Iν dω
4π
completely dependent on radiation field
Sν = Jν
not dependent on temperature T
29
The equation of radiative tranfer
Source function: simple cases
c. mixed case
²ν + ²sc
κν
σν ²sc
²ν
ν
ν
Sν =
=
+
κν + σν
κν + σν κν
κν + σν σν
²ν + ²sc
κν
σν
ν
Sν =
=
Bν +
Jν
κν + σν
κν + σν
κν + σν
30
Formal solution of the equation of radiative tranfer
linear 1st order differential equation
we want to solve the equation of RT
with a known source function and in
plane-parallel geometry
μ
dIν
= Iν − Sν
dτν
multiply by e- and integrate between 1 (outside)
and 2 (> 1, inside)
Sν e−τν /μ
d
−τν /μ
(Iν e
)=−
dτν
μ
h
Iν e
− τμν
iτ 2
τ1
=−
Zτ2
τ1
S ν e−
τν
μ
check, whether this really yields transfer
equation above
dtν
μ
31
h
Iν e
− τμν
iτ 2
τ1
=−
Zτ2
S ν e−
τν
μ
τ1
dtν
μ
Formal solution of the equation of radiative tranfer
integral form of equation
of radiation transfer
Iν (τ1 , μ) = Iν (τ2 , μ) e
τ −τ
− 2μ 1
+
Zτ2
Sν (t) e
−
t−τ1
μ
dt
μ
τ1
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)
Formal solution! actual solution
can be complex, since S can
depend on I
32
Iν (τ1 , μ) = Iν (τ2 , μ) e
τ −τ
− 2μ 1
+
Zτ2
Sν (t) e
−
t−τ1
μ
dt
μ
τ1
Boundary conditions
solution of RT equation requires
boundary conditions, which are
different for incoming and outgoing
radiation
a. incoming radiation:  < 0 at   0
usually we can neglect irradiation from outside: I(  0,  < 0) = 0
Iνin (τν , μ)
=
Z0
Sν (t) e
ν
− t−τ
μ
dt
μ
τν
33
Iν (τ1 , μ) = Iν (τ2 , μ) e
τ −τ
− 2μ 1
+
Zτ2
Sν (t) e−
t−τ1
μ
dt
μ
τ1
Boundary conditions
b. outgoing radiation:  > 0 at  = max  ∞
We have either
Iν (τmax , μ) = Iν+ (μ)
or
lim Iν (τ, μ) e−τ /μ = 0
finite slab or
shell
semi-infinite
case (planar
or spherical)
τ →∞
I increases less rapidly than the exponential
Iνout (τν , μ) =
Z∞
τν
Sν (t) e−
t−τν
μ
dt
μ
and at a given position  in
the atmosphere:
Iν (τν ) = Iνout (τν ) + Iνin (τν )
34
Emergent intensity
from the latter  emergent intensity
 = 0,  > 0
Iν (0, μ) =
Z∞
t
Sν (t) e− μ
dt
μ
0
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.
35
Emergent intensity
suppose that S is linear in  (Taylor expansion around  = 0):
Sν (τν ) = S0ν + S1ν τν
emergent intensity
Z
Z∞
t dt
= S0ν + S1ν μ
Iν (0, μ) = (S0ν + S1ν t)e− μ
μ
xe−x dx = −(1 + x) e−x
0
Iν (0, μ) = Sν (τν = μ)
we see the source function
at location  
Eddington-Barbier relation
the emergent intensity corresponds to
the source function at  = 1 along the
line of sight
36
Emergent intensity
 = 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
Sline < Scont
 a dip is created in the spectrum
37
Line formation
simplify:  = 1, 1=0 (emergent intensity), 2 = 
S independent of location
Iν (0) = Iν (τν ) e−τν + Sν
Zτν
e−t dt = Iν (τν ) e−τν + Sν (1 − e−τν )
0
38
Line formation
Optically thick object:
∞
Optically thin object:
Iν (0) = Iν (τν ) e−τν + Sν (1 − e−τν ) = Sν
Iν (0) = Iν (τν ) + [Sν − Iν (τν )] τν
exp(-)  1 - 
39
I =  S =  ds S
e.g. HII region, solar corona
enhanced 
independent of , no line
(e.g. black body B)
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)
40
From Rutten’s web notes
Line formation example: solar corona
I =  S
41
The diffusion approximation
At large optical depth in stellar atmosphere photons are local: S  B
Expand S B as a power-series:
∞
X
dn B ν
Sν (t) =
(t − τν )n /n!
n
dτν
n=0
In the diffusion approximation ( >>1) we retain only
first order terms:
Bν (t) = Bν (τν ) +
dBν
(t − τν )
dτν
Z∞
dBν
dt
Iνout (τν , μ) = [Bν (τν ) +
(t − τν )]e−(t−τν )/μ
dτν
μ
τν
42
Z∞
dBν
dt
Iνout (τν , μ) = [Bν (τν ) +
(t − τν )]e−(t−τν )/μ
dτν
μ
τν
The diffusion approximation
Substituting:
t→u=
t − τν
→ dt = μ du
μ
Z∞
uk e−u du = k!
0
Iνout (τν , μ) =
Z∞
[Bν (τν ) +
0
Iνin (τν , μ) = −
dBν
dBν
μu]e−u du = Bν (τν ) + μ
dτν
dτν
τZν /μ
[Bν (τν ) +
0
dBν
μu]e−u du
dτν
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.
43
diffusion approximation:
Solar limb darkening
Iν (0,μ)
Iν (0,1)
=
ν μ
Bν (0)+ dB
dτν
center-to-limb variation of
intensity
ν
Bν (0)+ dB
dτν
from the intensity measurements
 B, dB/d
Bν (t) = Bν (0) +
dBν
dτν t
= a+b·t =
2hν 3
1
2
hν/kT
(t) −1
c
e
T(t): empirical temperature stratification of solar
photosphere
44
Solar limb darkening
...and also giant planets
45
Solar limb darkening: temperature stratification
Iν (0, μ) =
Z∞
t
Sν (t) e− μ
dt
μ
0
exponential extinction varies as - /cos
From S = a + b
Iν (0, cos θ) = aν + bν cos θ
Iν (0, μ) = Sν (τν = μ)
S
R. Rutten,
web notes
Unsoeld, 68
46
we want to obtain an approximation for the
radiation field – both inward and outward
radiation - at large optical depth
Eddington approximation
 stellar interior, inner boundary of
atmosphere
In the diffusion approximation we had:
dBν
(t − τν )
dτν
Bν (t) = Bν (τν ) +
Iνout (τν , μ) = Bν (τν ) + μ
Iνin (τν , μ) = −
τZν /μ
dBν
dτν
[Bν (τν ) +
0
 >> 1
0<<1
dB ν
μu]e−u du
dτν
Iν− (τν , μ) = Bν (τν ) + μ
-1 <  < 0
dBν
dτν
47
With this approximation for Iν we can calculate
the angle averaged momenta of the intensity
Eddington approximation
1
Jν =
2
Z
 very important for analytical estimates
1
Iν dμ = Bν (τν )
−1
1
Fν
=
Hν =
4
2
1
Kν =
2
simple approximation for photon flux and a
relationship between mean intensity Jν and Kν
Z
1
Z
1
1 dBν
μ Iν dμ =
3 dτν
−1
1
μ Iν dμ = Bν (τν )
3
−1
2
Kν =
1
3
Jν
=−
1 dBν dT
1 1 dBν
=−
3 κν dx
3κν dT dx
flux F ~ dT/dx
diffusion: flux ~ gradient (e.g.
heat conduction)
Eddington approximation
48
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 structure
Schwarzschild-Milne equations
Iνout (τν , μ) =
Jν =
1
2
Z1
Iν dμ =
1
2
−1
Jν =
Z1
Iνout dμ +
1⎣
2
substitute w =
1
μ
Jν =
Iνin dμ
Iνin (τν , μ) =
Sν (t)e−(t−τν )/μ
dt
dμ −
μ
0 τν
⇒
⎡
1⎣
2
dw
w
= − μ1 dμ
Z∞ Z∞
1 τν
t−τν
μ
dt
μ
Sν (t)e−(t−τν )w dt
Z0 Zτν
Sν (t)e−(t−τν )/μ
−1 0
w = − μ1 ⇒
dw
+
w
Z∞ Zτν
1
0
Z0
Sν (t) e−
t−τν
μ
dt
μ
τν
<0
>0
Z1 Z∞
Sν (t) e−
τν
−1
0
⎡
1
2
Z0
Z∞
dw
w
⎤
dt ⎦
dμ
μ
= − μ1 dμ
Sν (t)e−(τν −t)w dt
⎤
dw ⎦
w
49
Schwarzschild-Milne equations
Jν =
⎡
1⎣
2
Z∞
Sν (t)
τν
1
Jν =
2
Z∞
e−(t−τν )w
1
dw
dt +
w
Zτν
Sν (t)
0
Z∞
1
>0
Z∞
0
Sν (t)
Z∞
1
e−(τν −t)w
>0
dw
1
dt =
e−w|t−τν|
w
2
Z∞
⎤
dw ⎦
dt
w
Sν (t)E1 (|t − τν |)dt
0
Schwarzschild’s equation
50
Schwarzschild-Milne equations
where
E1 (t) =
Z∞
1
e
−tx
dx
=
x
Z∞
e−x
dx
x
t
is the first exponential integral (singularity at t=0)
Exponential integrals
Z∞
En (t) = tn−1 x−n e−x dx
t
En (0) = 1/(n − 1), En (t → ∞) = e−t /t → 0
Z
dEn
= −En−1 , En (t) = −En+1 (t)
dt
E1 (0) = ∞ E2 (0) = 1 E3 (0) = 1/2 En (∞) = 0
Gray, 92
51
Schwarzschild-Milne equations
Introducing the  operator:
Λτν [f (t)] =
1
2
Z∞
0
f (t)E1 (|t − τν |) dt
Jν (τν ) = Λτν [Sν (t)]
Similarly for the other 2 moments of Intensity:
Milne’s equations
J, H and K are all depth-weighted means of S
52
Gray, 92
Schwarzschild-Milne equations
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=
53
The temperature-optical depth relation
Radiative equilibrium
The condition of radiative equilibrium (expressing conservation of
energy) requires that the flux at any given depth remains constant:
F(r) = πF =
Z∞Z
Iν cos θ dω dν = π
0 4π
4πr2 F(r) = 4πr2 · 4π
In plane-parallel geometry r  R = const
Z∞
Fν dν = 4π
0
Z∞
Z∞
Hν dν
0
Hν dν = const = L
0
4π
Z∞
Hν dν = const
0
and in analogy to the black body
radiation, from the Stefan-Boltzmann law
we define the effective temperature:
4π
Z∞
0
4
Hν dν = σTeff
54
The effective temperature
The effective temperature is defined by:
It characterizes the total radiative flux
transported through the atmosphere.
4π
Z∞
4
Hν dν = σTeff
0
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.
55
Radiative equilibrium
Let us now combine the condition of radiative equilibrium with the
equation of radiative transfer in plane-parallel geometry:
μ
1
2
Z1
−1
⎡
dIν
1
μ
dμ = −
dx
2
d ⎣1
dx 2
Z1
−1
⎤
Z1
−1
dIν
= −(κν + σν ) (Iν − Sν )
dx
(κν + σν ) (Iν − Sν ) dμ
μ Iν dμ⎦ = −(κν + σν ) (Jν − Sν )
H
56
Radiative equilibrium
Integrate over frequency:
d
dx
Z∞
0
Z∞
Hν dν = − (κν + σν ) (Jν − Sν ) dν
0
const
Z∞
(κν + σν ) (Jν − Sν ) dν = 0
0
at each depth:
Z∞
⎛
⎝
κν [Jν − Bν (T )] dν = 0
0
in addition:
s u b s t it u t e S ν =
4π
Z∞
0
4
Hν dν = σTeff
Z∞
κν
κν + σν
Bν +
σν
κν + σν
κ ν J ν dν = absorbed energy
0
⎛
⎝
Z∞
κ ν B ν dν = emitted energy
0
T(x) or T()
Jν
⎞
⎠
⎞
⎠
57
Radiative equilibrium
Z∞
κν [Jν − Bν (T )] dν = 0
4π
0
Z∞
4
Hν dν = σTeff
0
T(x) or T()
The temperature T(r) at every depth has to assume the value for which the left
integral over all frequencies becomes zero.
 This determines the local temperature.
58
Iterative method for calculation of a stellar atmosphere:
the major parameters are Teff and g
a. hydrostatic equilibrium
dP
dx
T(x), (x), B[T(x)],P(x), ρ(x)
equation of
transfer
J(x), H(x)
b. equation of radiation
transfer
μ
R∞
κν (Jν − Bν ) dν = 0 ?
R∞
4
?
4π 0 Hν dν = σ Teff
0
= −gρ(x)
dIν
= −(κν + σν ) (Iν − Sν )
dx
c. radiative equilibrium
Z∞
κν [Jν − Bν (T )] dν = 0
0
d. flux conservation
4π
Z∞
4
Hν dν = σTeff
0
T(x), (x), B[T(x)], ρ(x)
e. equation of state
P =
ρkT
μ mH
59
Grey atmosphere - an approximation for the
temperature 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.
Z∞
Z∞
κ̄ [Jν − Bν (T )] dν = 0
κν [Jν − Bν (T )] dν = 0
0
0
With: J =
Z∞
0
Jν dν
H=
Z∞
Hν dν
K=
0
Z∞
0
Kν dν
B=
Z∞
σT 4
Bν dν =
π
0
J =B
4
4πH = σTeff
60
Grey atmosphere
We then assume LTE: S = B.
From
1
Jν (τν ) = Λτν [Sν (t)] =
2
Z∞
Sν (t)E1 (|t − τν |)dt
0
and a similar expression for frequency-integrated
quantities
J(τ¯) = Λτ̄ [S(t)],
dτ̄ = κ̄dx
and with the approximations S = B, B = J:
1
J (τ̄) = Λτ¯[J(t)] =
2
Z∞
0
Milne’s equation
J (t)E 1(|t − τ¯|) dt
!!! this is an integral
equation for J) 
61
The exact solution of the Hopf integral equation
Milne’s equation J() = [J(t)]
 exact solution (see Mihalas, “Stellar Atmospheres”)
J() = const. [  + q()],
with q monotonic
1
√ = 0.577 = q(0) ≤ q(τ̄ ) ≤ q(∞) = 0.710
3
Radiative equilibrium - grey approximation
J() = B() =  T4() = const. [  + q()]
with boundary conditions 
T4() = ¾ T4eff [ + q()]
62
A simple approximation for T
0th moment of equation of transfer (integrate both
sides in d from -1 to 1)
dI
= −κ̄(I − B)
μ
dx
dH
=J −B =0
dτ̄
(J = B)
4
σTeff
H = const =
4π
1st moment of equation of transfer (integrate both
sides in d from -1 to 1)
dI
= −κ̄(I − B)
μ
dx
4
dK
σTeff
=H =
dτ̄
4π
K(τ̄ ) = H τ̄ + constant
From Eddington’s approximation at large depth: K = 1/3 J
63
Grey atmosphere – temperature distribution
T 4 (τ̄ ) =
3πH
(τ̄ + c)
σ
T 4 (τ̄ ) =
3 4
T (τ̄ + c)
4 eff
H=
σ 4
T
4π eff
T4 is linear in 
Estimation of c
Hν (τ̄ = 0) =
⎡
1
3H ⎣
2
Z∞
tE2 (t) dt + c
0
Z∞
0
1/3
⎤
E2 (t) dt⎦
1/2
Z∞
ts E n (t) d t =
s!
s + n
0
64
Grey atmosphere – Hopf function
H(0) = H =
3
2
1
H(1 + c) → c =
2
2
3
T 4 (τ̄ ) =
3 4
2
Teff (τ̄ + )
4
3
Remember: More in general J is obtained from
T 4 (τ̄ ) =
based on approximation K/J = 1/3
T = Teff at  = 2/3, T(0) = 0.84 Teff
J(τ¯) = Λτ̄ [S(t)]
3 4
T [τ̄ + q(τ̄ )]
4 eff
q(τ̄ ) : Hopf function
Once Hopf function is specified  solution of the grey atmosphere
(temperature distribution)
1
√ = 0.577 = q(0) ≤ q(τ̄ ) ≤ q(∞) = 0.710
3
65
Selection of the appropriate κν ⇒ κ̄
In the grey case we define a ‘suitable’ mean opacity (absorption coefficient).
κν ⇒ κ̄
I =
Z∞
Iν dν
J =
0
non-grey
Z∞
Jν dν ...
0
grey
dIν
dI
Equation of transfer μ dx = −κν (Iν − Sν ) μ dx = −κ̃ (I − S)
1st moment
2nd moment
dHν = −κ (J − S ) dH
ν ν
ν dx
dx
dKν = −κ H
ν ν
dx
= −κ̂ (J − S)
dK = −κ̄ H
dx
66
Selection of the appropriate κν ⇒ κ̄
non-grey
grey
ν = −κ (I − S ) μ dI = −κ̃ (I − S)
μ dI
ν ν
ν
dx
dx
Equation of transfer
1st
moment
2nd moment
dHν = −κ (J − S )dH
ν ν
ν dx
dx
dKν = −κ H
ν ν
dx
= −κ̂ (J − S)
dK = −κ̄ H
dx
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 
67
Mean opacities: flux-weighted
1st possibility:
Flux-weighted mean
κ̄ =
R∞
κν Hν dν
0
H
allows the preservation of the K-integral
(radiation pressure)
Problem: H not known a priory (requires
iteration of model atmospheres)
68
∞
∞
R 1 dKν
R
κν dx dν = − Hν dν
0
0
dKν = −κ H
ν ν
dx
Mean opacities: Rosseland
2nd possibility: Rosseland mean
to obtain correct integrated energy flux and use local T
Z∞
0
1
=
κ̄
R∞
0
1 dKν
κν dx
1 dKν
1 dK
dν = −H ⇒ (grey) ⇒
= −H
κν dx
κ̄ dx
Kν →
dν
1
Jν , Jν → Bν as τ → ∞
3
dKν
1 dBν
1 dBν dT
→
=
dx
3 dx
3 dT dx
dK
dx
1
κ̄Ross
=
R∞
0
1 dBν (T )
κν
dT
R∞ dBν (T )
0
dT
dν
dν
large weight for low-opacity (more
transparent to radiation) regions
69
Mean opacities: Rosseland
at large  the T structure is accurately given
by
3 4
T 4 = Teff
[τRoss + q(τRoss )]
4
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).
70
T4 vs. 
non-grey
numerical
grey: q() = exact
grey: q= 2/3
71
T vs. log()
non-grey
numerical
grey: q() = exact
grey: q= 2/3
72
Iterative method for calculation of a stellar atmosphere:
the major parameters are Teff and g
a. hydrostatic equilibrium
dP
dx
T(x), (x), B[T(x)],P(x), ρ(x)
equation of
transfer
J(x), H(x)
b. equation of radiation
transfer
μ
R∞
κν (Jν − Bν ) dν = 0 ?
R∞
4
?
4π 0 Hν dν = σ Teff
0
= −gρ(x)
dIν
= −(κν + σν ) (Iν − Sν )
dx
c. radiative equilibrium
Z∞
κν [Jν − Bν (T )] dν = 0
0
d. flux conservation
4π
Z∞
4
Hν dν = σTeff
0
T(x), (x), B[T(x)], ρ(x)
e. equation of state
P =
ρkT
μ mH
73

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