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Grey Atmosphere(Mihalas 3)
Eddington Approximation SolutionTemperature Stratification
Limb Darkening LawΛ-iteration, Unsőld iteration
Method of Discrete Ordinates
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Grey or Constant Opacity Case
• Simplifying assumption Χν = Χ independent of wavelength
• OK in some cases (H- in Sun; Thomson scattering in hot stars)
• Good starting point for iterative solutions
• Use some kind of mean opacity
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Mean Opacities
• Flux weighted mean(radiation pressure)
• Rosseland mean(good at depth; low opacity weighted)
• Planck mean(good near surface;near rad. equil.)
F H d H d
/0 0
R T
B
Td
13
1
04
P B d B d
/0 0
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Frequency Integrated Form of TE
• TE
• Radiative equilibrium
• Recall moments of TE:
• Apply Eddington approximation K/J = 1/3
I
I S
J S B T
d H
dJ S
d K
dH K
Fconst
0
4.
J S F const
F q
3
43
4
.
H=F/4=conserved quantity
q= Hopf function in general
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Constant from Surface Flux
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Grey E.A. Limb Darkening Law
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Improvements by Iteration
• All based on K/J=1/3 which is too small close to the surface
• Flux is not rigorously conserved (close)
• Two improvement schemes used to revise the grey solution and bring in closer to an exact solution: Lambda Λ and Unsőld iteration methods
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Λ Iteration
Further iterations possible, but convergence is slow since operator important only over photon free path.
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Unsőld Iteration
J F H H
3
4
2
33
2
32 0 ,
*
*
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Unsőld Iteration
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Unsőld Iteration
• Initial estimate
• Work out ΔH and ΔB
• Next estimate
• Converges at all depths
H F F
1
4
1
4
3
4
2
3
H B B 1
4 0
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Discrete Ordinates: Use S=J
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Trial Solution & Substitution
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Roots of Characteristic Function
T k 2 k 2
1
12
1
22
1
32
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Roots of Characteristic Function
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Linear term & Full Solution
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Boundary Conditions
• Lower limit on semi-infinite atmosphere
• No incident radiation from space at top(n equations, n unknowns for Q, Lα)
• Set b according to flux
lim
I e L0 0
011
1
QL
kii
n
b F3
4
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Final Solution
•
• Good even with n small (better than 1% for n=3)
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Exact Solution
J F q 3
4
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Next steps …
• Grey atmosphere shows general trends
• But need to account for real opacities that are frequency dependent
• Need to check if temperature gradient is altered by convection, another way stars find to transport flux outwards