= Solution
{c}
It is convenient to use the <optical depth> \b[above] a level,
$$
t(p)=\tau_\infty-\tau(p)=\tau_\infty(p/p_s)^2.
$$
Thermal <radiation> escapes mainly from levels with $t$ of order unity: emission much deeper is attenuated, while the column far above contains little emitting material. Hence, in an optically thick <atmosphere>,
$$
\boxed{p_{\rm rad}\sim\frac{p_s}{\sqrt{\tau_\infty}}
=\sqrt{\frac{2gp_s}{\kappa_s}},\qquad
T_{\rm rad}\sim T_s\tau_\infty^{-a/2}.}
$$
For a particular viewing direction, $t\sim\mu$ instead gives $p_{\rm rad}\sim p_s\sqrt{\mu/\tau_\infty}$. These are characteristic levels, not sharp emitting surfaces; the contribution function depends on the <temperature> exponent $a$. If $\tau_\infty\lesssim1$, the formal level lies below the surface and the surface itself contributes strongly. A useful estimate then clips $p_{\rm rad}$ at $p_s$.
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