What is the Milne-Eddington Approximation? In this paper: Planet temperatures with surface cooling parameterized it states in the &quot;radiation model&quot; the following:

The Eddington–Milne approximation relates $T_0$ and $T_e$ through the
gray IR optical depth $\tau$:
$T_0=T_e(1+\frac{3}{4}\tau)^{0.25}$

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 &quot;optical depth&quot; is.)
What is this approximation and why can it be applied to approximate the temperature of a planet with an atmosphere?
I don't have time to write up a full answer with derivation, but you might want to have a look into the standard radiation transport literature (e.g. the book by Mihalas&amp;Mihalas) or the radiation transport in stellar interiors (e.g. Kippenhahn) or literature for irradiated exoplanets (e.g. Guillot (2010)). The ME-approximation is a standard result for grey radiation transport at optical depths &gt;1, when the radiation flux is diffusive, i.e. $\rm F_{rad} = c/(3\rho \kappa_R)\nabla E_{rad}$.
Then you can couple the equations for $\rm E_{rad}$ and $\rm E_{gas}$, integrate and solve for $\rm E_{gas}$, which is $\rm \rho c_vT$. This gives you the ME relation, which is valid for the diffusive radiative transport in gases, and precludes interesting things like the greenhouse-effect, temperature inversions, etc.