2.7

Question 2

Grey Atmosphere Goals

Answer 2

Get radiation quantities as a function of optical depth (τ), including I(τ), S(τ), U(τ), P(τ), and I(0, μ) at surface τ=0.

Question 3

Grey Atmosphere Assumption

Answer 3

Optical depth (τ) is frequency-independent
-> U, F, P not depending on frequency.

Question 4

Radiative Transfer Equation

(in grey atmosphere approximation)

Answer 4

μ ∂I(τ, μ) / ∂τ = I - S,

where I is intensity and S is source function.

Question 5

Average Specific Intensity (J)

Answer 5

J = (1/2) ∫ from -1 to +1 I dμ = U · c / (4π), linking intensity to energy density.

Question 6

Radiative Equilibrium

what is the condition?

Answer 6

• J = S
• average intensity = source function

implying radiative equilibrium similar to Iν = Sν for optically thick objects.

Question 7

Radiation Pressure (P)

in grey atmosphere aproximation

Answer 7

P = F · c · (τ + q),

where F is flux, c is speed of light, and q is an integration constant.

τ is optical depth

Question 8

Eddington Approximation

Formula

Answer 8

Assumes P ≈ U / 3 below surface, leading to:

S = J = (3c/4π)P = 3/(4π) · F · (τ + q)

Question 9

Specific Intensity at Surface (I(0, μ))

Answer 9

I(0, μ) = (3F/4π)(q + μ),

derived from the surface intensity equation.

Question 10

Constant of Integration (q)

Answer 10

Derived as q = 2/3, from the surface flux equation F = 2π∫ from 0 to 1 I(0, μ)μ dμ.

Question 11

Final Results: Source Function (S(τ))

Answer 11

Key result of the grey atmosphere model:

S(τ) = (3F/4π)(τ + 2/3)

Question 12

Final Results: Energy Density (U(τ))

Answer 12

c · U(τ) = 3F(τ + 2/3),
linking energy density to flux and optical depth.

Question 13

Final Results: Temperature Structure (T(τ))

Answer 13

T⁴(τ) = (3/4)T_eff⁴(τ + 2/3),

showing how temperature varies with optical depth.

Question 14

Limb Darkening Law

Answer 14

I(0, μ)/I(0,1) = (3/5)(μ + 2/3), describing how intensity varies with angle.

just seems random to me, idk how relevant, maybe good to know that its linear