Ray Pierrehumbert explains the Radiative Transfer Equations

In order to elucidate the reasoning behind the differential equation yielding the "grey atmosphere" radiative lapse rate I now take the liberty to present some extracts from Ray Pierrehumbert's first online distribution of the book "Principles of Planetary Climate" dating November 11, 2009. In my view, this is the only understandable derivation that I have found. Take special notice of the formulas (4.6), (4.7), (4.8) and (4.29). Try to reproduce these with your own hand. I believe I have verified that, assuming that the net radiative flux q_T = (I+) - (I-) is independent of altitude, formula (4.29) yields the differential equation appearing first in the Ramanthan/Coakley review paper, except for some numerical factor which probably comes from some angular integral.

A 1978 review paper

Some day ago I came across the following review paper from 1978. Compared to other papers on the subject it appears to be relatively accessible and it contains a lot of references. In particular notice the very first formula describing radiative equilibrium of a "grey" atmosphere. Apparently it originates from a 1958 paper by a guy called Ambartsumyan. That you didn't know, did you?

I am personally convinced that the devil lies somewhere in the equations, and climatologists have indeed made a very good job making these equations as inaccessible as possible, so that virtually nobody can quote them, even less interpret them. But I'll see what information I can extract from it.