= Grey-atmosphere convection onset with eighth-power temperature opacity
{title2=$\tau_c=32/45$}
Take an <ideal gas> <grey atmosphere> with $T^4=T_e^4(3\tau+2)/4$, constant <surface gravity of a star> $g$, vanishing external pressure and $\kappa=\kappa_0P^{1/2}T^8$. Integrating <hydrostatic equilibrium in optical depth> gives $P^{3/2}=8g[1/2-(3\tau+2)^{-1}]/(\kappa_0T_e^8)$. Its logarithmic temperature-pressure gradient is $9\tau/16$, so the <Schwarzschild criterion> for a monatomic gas is crossed at $\tau_c=32/45$. The radiative continuation beyond that depth is unstable and must be replaced by convective transport.
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