= Effective emission pressure
{title2=$p_{\rm rad}$}
For a source linear in downward effective <optical depth> $t\in[0,D]$, <two-stream approximation> weighting $e^{-t}$ has normalized mean $t_{\rm eff}=1-D/(e^D-1)$. Constant mass opacity gives $p_{\rm rad}=p_s t_{\rm eff}/D$, tending to $p_s/2$ for small $D$ and $p_s/D$ for large $D$. The atmospheric flux is $(1-e^{-D})S(t_{\rm eff})$, separate from transmitted ground radiation. Exact angular transfer with literal vertical optical depth instead has weight $2E_2(t)$ and thick-limit mean $2/3$; the convention matters.
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