What is the Milne-Eddington Approximation?
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.