 Effective emission pressure

For a source linear in downward effective optical depth $t\in [0,D]$, two-stream approximation weighting $e^{-t}$ has normalized mean $t_{\mathrm{eff}}=1-D/(e^D-1)$. Constant mass opacity gives $p_{\mathrm{rad}}=p_s{t}_{\mathrm{eff}}/D$, tending to $p_s/2$ for small $D$ and $p_s/D$ for large $D$. The atmospheric flux is $(1-e^{-D})S(t_{\mathrm{eff}})$, 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.