Post on 29-Jan-2016
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Analytical Relations for the Transfer Equation (Mihalas 2)
Formal Solutions for I, J, H, KMoments of the TE w.r.t. Angle
Diffusion Approximation
2
Schwarzschild – Milne Equations
•
• Formal solution
I
J I d
H I d
K I d pcKR
1
2
1
2
1
2
4
1
1
1
1
2
1
1
Specific intensity
Mean intensity
Eddington flux
Pressure term
3
Semi-infinite Atmosphere Case
• Outgoing radiation, μ>0
• Incoming radiation, μ<0
z
Τ=0
4
Mean Field J (Schwarzschild eq.)
5
6
F, K (Milne equations)
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Operator Short Forms
J = Λ
F = Φ
K = ¼ Χ
f(t) = S(t) Source function
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Properties of Exponential Integrals
E x E x E E x dx
E x e xE x n
E x dx e n E
En
E
E xe
x
n n n n
nx
n
n n
n n
x b ign
x
1 1
1
1 1
1
1
01
10
( )
/
( )
lim
9
Linear Source FunctionS=a+bτ
• J =
• For large τ
• At surface τ = 0
J a b bE aE
a b S
1
2 3 2
J a b a a b S
1
2
1
2
1
1
1
2
1
2
1
21 2 /
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Linear Source FunctionS=a+bτ
• H = ¼Φ(a+bτ)
• For large τ, H=b/3 (gradient of S)
• At surface τ = 0
H b a b
a bS
1
4
4
32
1
2
1
3
4 6
1
42 3 /
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Linear Source FunctionS=a+bτ
• K = ¼ Χ(a+bτ)
• Formal solutions are artificial because we imagine S is known
• If scattering is important then S will depend on the field for example
• Coupled integral equations
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Angular Moments of the Transfer Equation
• Zeroth moment and one-D case:
II S
LH SId I d
H
RHS I S d J S
HJ S
1
2
1
2
1
2
1
1
1
1
1
1
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Radiative Equilibrium
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Next Angular Moment: Momentum Equation
• First moment and one-D case:
II S
LH SId I d
K
RHS I S d H
KH
p
z cH dR
1
2
1
2
1
2
4
2
1
1
2
1
1
1
1
0
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Next Angular Moment: Momentum Equation
• Radiation force (per unit volume) = gradient of radiation pressure
• Further moments don’t help …need closure to solve equations
• Ahead will use variable Eddington factorf = K / J
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Diffusion Approximation (for solution deep in star)
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Diffusion Approximation Terms
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Only Need Leading Terms
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Results
• K / J = 1/3
• Flux = diffusion coefficient x T gradient
• Anisotropic term small at depth