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# Grey and Eddington Approximations

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Grey and Eddington Approximations ***

Problem: Determine the values of Iin and Iout 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 Iout.

Find an expression for Iin.

Data:

 (1)

Useful Equations:

Solution:

 (2)

 (3)

 (4)

 (5)

 (6)

Add to find an expression for Iout.

 (7)

 (8)

 (9)

Take half of this expression and subtract from the sum of Iin and Iout to find an expression for Iin.

 (10)

 (11)

 (12)

 (13)

 (14)

 (15)

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

 (16)

 (17)

Solve for τ.

 (18)

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