= Solution
{c}
Write $\tau_0=\tau_\infty$, $n=2a$, and $t=\tau_0(p/p_s)^2$. The integrated <Planck function> is $B(T)=\sigma T^4/\pi$ by the <Stefan–Boltzmann law>, and
$$
B(t)=\frac{\sigma T_s^4}{\pi}\left(\frac t{\tau_0}\right)^n.
$$
The formal <radiative transfer equation> solution for upward <radiance> at the top is
$$
I(0,\mu)=\frac{\sigma T_s^4}{\pi}e^{-\tau_0/\mu}
+\int_0^{\tau_0}B(t)e^{-t/\mu}\frac{dt}{\mu},\qquad0<\mu\leq1.
$$
The first term is transmitted surface emission; the second is emission from the <atmosphere>. Integrating <radiance> over the upward hemisphere gives the outgoing <radiant flux> per unit area,
$$
\boxed{F=2\sigma T_s^4\left[E_3(\tau_0)+\tau_0^{-n}
\int_0^{\tau_0}t^nE_2(t)\,dt\right],\qquad
E_j(t)=\int_0^1\mu^{j-2}e^{-t/\mu}\,d\mu.}
$$
Here $E_j$ is an <exponential integral>. For fixed $a$ and $\tau_0\to\infty$, surface transmission is negligible and the upper limit can be extended to infinity. Positivity permits interchanging the <integrals>; putting $t=\mu x$ gives
$$
\int_0^\infty t^nE_2(t)\,dt
=\int_0^1\mu^{n+1}\,d\mu\int_0^\infty x^ne^{-x}\,dx
=\frac{\Gamma(n+1)}{n+2}.
$$
Consequently the exact hemispheric optically thick result is
$$
\boxed{F\sim\sigma T_s^4\tau_0^{-2a}\,
\frac{\Gamma(2a+1)}{a+1}.}
$$
This is a dimensional scale times a dimensionless definite <integral> depending only on $a$. In the commonly used vertical-ray approximation, define $F_{\rm v}=\pi I(0,1)$. Then
$$
F_{\rm v}=\sigma T_s^4\left[e^{-\tau_0}+\tau_0^{-2a}\int_0^{\tau_0}t^{2a}e^{-t}\,dt\right]
\sim\sigma T_s^4\tau_0^{-2a}\Gamma(2a+1).
$$
A vertical-ray approximation therefore has the same scaling but omits the angular factor $1/(a+1)$. Stating which quantity is used prevents a purely angular normalization discrepancy.
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