= Grey-atmosphere convection onset with seventeenth-power opacity
{title2=$T_b=(17/40)^{1/12}T_e$}
For a <grey atmosphere> with $T^4=T_e^4(2+3\tau)/4$, an <ideal gas> and $\kappa=\kappa_0\rho T^{17}$, <hydrostatic equilibrium in optical depth> integrates to $P^2=512\mathcal Rg[1/8-(2+3\tau)^{-3}]/(9\mu\kappa_0T_e^{16})$, taking $P(0)=0$. The logarithmic radiative gradient is $[(2+3\tau)^3-8]/48$. Equating it to the monatomic <adiabatic temperature gradient> $2/5$ gives $(2+3\tau_b)^3=136/5$ and the displayed onset temperature. Beyond this point the radiative profile must be replaced by convective transport.
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