### Grey Atmosphere (Mihalas 3)

```Grey Atmosphere
(Mihalas 3)
Eddington Approximation Solution
Temperature Stratification
Limb Darkening Law
Λ-iteration, Unsőld iteration
Method of Discrete Ordinates
1
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
2
Mean Opacities
• Flux weighted mean
(radiation pressure)


0
0
 F    H d /  H d
 R1


1  B


d
3  
T
4 T 0
• Rosseland mean
(good at depth;
low opacity weighted)


• Planck mean
 P     B d /  B d
0
0
(good near surface;
near rad. equil.)
3
Frequency Integrated Form of TE
• TE
I

 I S

• Radiative equilibrium
J  S  B T 
• Recall moments of TE:
H=F/4=conserved quantity
• Apply Eddington
approximation K/J = 1/3
q= Hopf function in general


dH
 JS 0
d
dK
F
 H  K    const .
d
4
3
J    S    F  const .
4
3
 F   q 
4
4
Constant from Surface Flux
5
Grey E.A. Limb Darkening Law
6
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
7
Λ Iteration
Further iterations possible, but convergence is slow
since operator important only over photon free path.
8
Unsőld Iteration
*
*
J
3 
2
2

F      3H      2 H ,   0

4 
3
3
9
Unsőld Iteration
10
Unsőld Iteration
• Initial estimate
1
1 3 
H  F    F  
4
4 4 
• Work out ΔH and ΔB
• Next estimate
1
H   B0   B
4

2 

3 

• Converges at all depths
11
Discrete Ordinates: Use S=J
12
Trial Solution & Substitution
13
Roots of Characteristic Function
 
T k
2
k2
1
1
1
 12
 22
 32
14
Roots of Characteristic Function
15
Linear term & Full Solution
16
Boundary Conditions
• Lower limit on semi-infinite atmosphere
lim I  e    0  L  0
 
• No incident radiation from space at top
(n equations, n unknowns for Q, Lα)
n 1
L
0  Q  i  
 1 1  k  i
• Set b according to flux
3
b F
4
17
Final Solution
•
• Good even with n small (better than 1% for n=3)
18
Exact Solution
3
J  F   q 
4
19
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
20
```