1

Grey Atmosphere(Mihalas 3)

Eddington Approximation SolutionTemperature Stratification

Limb Darkening LawΛ-iteration, Unsőld iteration

Method of Discrete Ordinates

2

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

3

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

4

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

5

Constant from Surface Flux

6

Grey E.A. Limb Darkening Law

7

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

8

Λ Iteration

Further iterations possible, but convergence is slow since operator important only over photon free path.

9

Unsőld Iteration

J F H H

3

4

2

33

2

32 0 ,

*

*

10

Unsőld Iteration

11

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

12

Discrete Ordinates: Use S=J

13

Trial Solution & Substitution

14

Roots of Characteristic Function

T k 2 k 2

1

12

1

22

1

32

15

Roots of Characteristic Function

16

Linear term & Full Solution

17

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

18

Final Solution

•

• Good even with n small (better than 1% for n=3)

19

Exact Solution

J F q 3

4

20

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