In this paper: Planet temperatures with surface cooling parameterized it states in the "radiation model" the following:
The Eddington–Milne approximation relates $T_0$ and $T_e$ through the gray IR optical depth $\tau$:
I looked this approximation up on google, and found it was more commonly known as the Milne-Eddington Approximation. However, this approximation seems to (mainly) be applied to stars, and I can't exactly see the connection between it and the temperature of a planet. (I also don't completely understand what an "optical depth" is.)
What is this approximation and why can it be applied to approximate the temperature of a planet with an atmosphere?