Grey and Eddington Approximations ***

Problem: Determine the values of I_in and I_out as functions of vertical optical depth (τ), using the Eddington Approximation for a plane-parallel atmosphere. Then find the optical depth at which the radiation field is isotropic to within 5%.

Hints:

Begin by equating the first two equations below and then the second two equations below, and substituting.

Find an expression for I_out.

Find an expression for I_in.

Data:

Useful Equations:

Solution:

Add to find an expression for I_out.

Take half of this expression and subtract from the sum of I_in and I_out to find an expression for I_in.

For the radiation field to be isotropic to within 5%, the following condition must hold:

Solve for τ.

At optical depth τ = 26, the radiation is isotropic to within 5%.