 Solution

The cloud absorbs $e\sigma T_g^4$ from below and emits $e\sigma T_c^4$ from each face. It absorbs no stellar radiation. Its net radiative heating per unit area is thereforeFor $e>0$, a cloud with $q^4<1/2$ warms in the absence of other energy transfers; one with $q^4>1/2$ cools. For the thin pressure thickness $\delta p$, its initial temperature tendency is approximately $g{\mathcal{R}}_c/(c_p\delta p)$ if its sensible heat capacity dominates. Latent heat and changing condensate modify that conversion from heating to temperature change, but not the sign of the radiative input.

With $T_g$ fixed, pure radiative equilibrium requiresThis equals the ambient dry-adiabatic process temperature precisely whenFor ordinary dry air, $\kappa \simeq 2/7$, so the ratio is about $0.545$. If $e=0$, the cloud exchanges no thermal radiation and its radiation budget does not select any equilibrium temperature.