= Solution
{c}
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 therefore
$$
\boxed{\mathcal R_c=e\sigma(T_g^4-2T_c^4)=e\sigma T_g^4(1-2q^4).}
$$
For $e>0$, 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> requires
$$
\boxed{T_{c,\mathrm{rad}}=2^{-1/4}T_g.}
$$
This equals the ambient dry-<adiabatic process> <temperature> precisely when
$$
q^4=\frac12,\qquad \boxed{\frac{p_c}{p_s}=2^{-1/(4\kappa)}}.
$$
For ordinary dry air, $\kappa\simeq2/7$, 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>.
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