1

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)

7

Operator Short Forms

J = Λ

F = Φ

K = ¼ Χ

f(t) = S(t) Source function

8

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 /

11

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

12

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

13

Radiative Equilibrium

14

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

16

Diffusion Approximation (for solution deep in star)

17

Diffusion Approximation Terms

18

Only Need Leading Terms

19

Results

• K / J = 1/3

• Flux = diffusion coefficient x T gradient

• Anisotropic term small at depth