= Pressure-linear grey opacity
{title2=$\kappa\propto p,\quad t\propto p^2$}
A <grey atmosphere> model with <mass absorption coefficient> proportional to <pressure>. Under <hydrostatic balance>, $\kappa=\kappa_sp/p_s$ gives downward <optical depth> $t=\kappa_sp^2/(2gp_s)$. For $T=T_s(p/p_s)^a$, a non-scattering optically thick column emits hemispheric <radiant flux> $F\sim\sigma T_s^4t_s^{-2a}\Gamma(2a+1)/(a+1)$. The angular factor follows from integrating ray attenuation over the upward hemisphere; a vertical-ray approximation omits it. The extension of the emitting-depth <integral> to infinity requires fixed $a$ as $t_s\to\infty$.
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