= Solution
{c}
At fixed $T_s$ and $a>0$, the outgoing <radiant flux> scales as $F\propto\tau_\infty^{-2a}$. Increasing the <optical depth> shifts the effective emitting region to lower <pressure> and lower <temperature>. A surface <temperature> cannot be inferred from emission alone: specify the absorbed stellar <radiant flux> $F_0$ and impose <radiative equilibrium>, $F=F_0$. If the no-<atmosphere> <blackbody> has $T_0=(F_0/\sigma)^{1/4}$, then
$$
\boxed{\frac{T_s}{T_0}\sim
\left[\frac{a+1}{\Gamma(2a+1)}\right]^{1/4}\tau_\infty^{a/2}.}
$$
The vertical-ray approximation omits $a+1$ from this prefactor. For fixed $a>0$ and sufficiently large <optical depth>, the surface is warmer than $T_0$: this is the thermal <greenhouse effect> within the prescribed <temperature> profile. It does not by itself establish a self-consistent vertical heating or <convection> balance.
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