= Grey-atmosphere convection onset with thirteenth-power opacity
{title2=$T_b=(13/20)^{1/8}T_e$}
In a thin ideal-gas <grey atmosphere> with $T^4=T_e^4(1/2+3\tau/4)$, constant enclosed <mass> $M$ and <luminosity> $L$, and $\kappa=\kappa_0\rho T^{13}$, divide the <stellar hydrostatic equation> by <radiative diffusion in a star>. This gives $P\,dP/dT=16\pi acGM\mathcal R\,T^{-9}/(3\kappa_0L\mu)$. Integrating from $P=0$, $T^4=T_e^4/2$, yields
$$
P^2=\frac{4\pi acGM\mathcal R}{3\kappa_0L\mu T_e^8}\left(4-\frac{T_e^8}{T^8}\right).
$$
<Logarithmic derivative> gives $\nabla=T^8/T_e^8-1/4$. For a <monatomic gas> gas, the <Schwarzschild criterion> reaches equality at $\nabla=2/5$, hence $T/T_e=(13/20)^{1/8}$. The purely radiative profile cannot be continued past this point as a convectively stable model.
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