= Grey-atmosphere convection onset with density-linear opacity
{title2=$\tau_b=\frac23\left[\left(\frac{4\beta+1}{5}\right)^{1/(\beta-1)}-1\right]$}
Assume an ideal-gas <grey atmosphere> with $P(0)=0$, constant <surface gravity of a star> $g$, $T^4=T_e^4(1+3\tau/2)/2$ and $\kappa=\kappa_0\rho T^{4\beta+1}$ for $\beta>1$. Using $\rho=\mu P/(\mathcal RT)$ in $dP/d\tau=g/\kappa$ and integrating gives
$$
P^2=\frac{2^{\beta+2}\mathcal Rg}{3\kappa_0(\beta-1)\mu T_e^{4\beta}}\left[1-(1+3\tau/2)^{1-\beta}\right].
$$
Differentiation yields $d\log T/d\log P=[(1+3\tau/2)^{\beta-1}-1]/[2(\beta-1)]$. Applying the <Schwarzschild criterion> with monatomic adiabatic gradient $2/5$ gives $\tau_b=\frac23[((4\beta+1)/5)^{1/(\beta-1)}-1]$. Since $\tau_b$ depends only on $\beta$, matching to $P=K T^{5/2}$ fixes $K\propto g^{1/2}T_e^{-2\beta-5/2}$.
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