Outline

Radiation in the limit of diffusion and high extinction

μdI_ν(r,μ) /dr = − χ_ν I_ν(r,μ) + η_ν.

Divide by χ_ν and define
S_ν ≡ η_ν /χ_ν (the source function)
dτ_ν ≡ −χ_ν dr (the differential optical depth).
Then
μdI_ν(τ_ν,μ) /dτ_ν = I_ν(τ_ν,μ) − S_ν.

The formal solution of this equation (dropping the ν subscript for now) is

I(τ,μ) = I(τ_o,μ) e^{−(τ_o − τ) /μ} + ∫_τ^τ_o S(t) e^{−(t − τ) /μ} dt /μ.

Now, assume we are at such great depth that any boundary τ_o differs greatly from τ, so we may neglect any incoming radiation from the boundary (the first term on the right). Also assume that χ is high (short mean free path 1/χ) so we may expand the (isotropic) source function as:
S(t) = S(&tau) + (t − τ) ∂S /∂τ .
Then the formal solution becomes

I((τ,μ) = S(τ) + μ∂S /∂τ .

With such a linear source function and intensity, K = J /3.

Homework: Show that the linear source function results in the given μ-dependence of I, and that as a result K = J /3.

Notice also that J(τ) = S(τ) in this circumstance. Therefore, the mean radiation field is very local. We may assume we are in a thermodynamic oven with walls at nearly constant temperature, so J_ν(τ) = S_ν(τ) = B_ν(τ).