Radiative equilibrium of a single grey cloud layer ($T_g^4=Q/[\sigma (1-e/2)]$)

A black surface receiving absorbed stellar irradiance $Q$, beneath a single non-solar-absorbing cloud with thermal emissivity $e>0$, satisfies $T_c^4=T_g^4/2$ when both bodies are in pure radiative equilibrium. The surface balance is $\sigma T_g^4=Q+e\sigma T_c^4$, giving the displayed result. If instead convection pins $T_c=q{T}_g$, the whole-column balance gives $T_g^4=Q/[\sigma (1-e+e{q}^4)]$; the nonradiative energy budget cannot then be discarded.