3. transport of energy: radiation
TRANSCRIPT
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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
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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
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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
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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)
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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
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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
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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
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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.
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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
Iν(cos θ) cos θ sin θdθdφ
πFν =
Z4π
Iν(cos θ) cos θ dω =
Z 2π
0
Z π
0
Iν(cos θ) cos θ sin θdθdφ
Note: for π/2 < θ < π −> cosθ < 0 second term negative !!
“astrophysical flux”
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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.
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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
Iν(cos θ) cos θ sin θdθdφ
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
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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 !!
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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The equation of The equation of radiativeradiative 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
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The equation of The equation of radiativeradiative tranfertranfer
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)
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The equation of The equation of radiativeradiative 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 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
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The equation of The equation of radiativeradiative 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(τ) solution of transfer equation Iν(μ,τν)
independent of radiation field
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The equation of The equation of radiativeradiative tranfertranfer
Source function: simple cases
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
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The equation of The equation of radiativeradiative tranfertranfer
Source function: simple cases
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
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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
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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
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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
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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
Sν(t) e− t−τν
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
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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.
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Emergent intensityEmergent intensity
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
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Emergent intensityEmergent intensity
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
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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 )
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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
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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
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Line formation example: solar coronaLine formation example: solar corona
Iν = τν Sν
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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
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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
[Bν(τν) +dBνdτν
μu]e−u du
Ioutν (τν ,μ) =
∞Z0
[Bν(τν) +dBνdτν
μu]e−u du = Bν(τν) + μdBνdτν
Ioutν (τν ,μ) =
∞Zτν
[Bν(τν) +dBνdτν
(t − τν)]e−(t−τν)/μ dt
μ
t→ u =t − τνμ→ dt = μ du
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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
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Solar limb darkeningSolar limb darkening
...and also giant planets
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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
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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
Sν(t) e− t−τν
μdt
μ
I inν (τν ,μ) =
0Zτν
Sν(t) e− t−τν
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
SchwarzschildSchwarzschild--Milne equationsMilne equations
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
SchwarzschildSchwarzschild--Milne equationsMilne equations
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
SchwarzschildSchwarzschild--Milne equationsMilne equations
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
Sν(t)E1(|t− τν |)dt
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 equilibriumequilibrium
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
RadiativeRadiative equilibriumequilibrium
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
RadiativeRadiative equilibriumequilibrium
∞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
Sν(t)E1(|t− τν |)dt
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