Grey-atmosphere convection onset with density-linear opacity (${\tau }_b=\frac{2}{3}\left[{\left(\frac{4\beta +1}{5}\right)}^{1/(\beta -1)}-1\right]$)

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 $\rho =\mu P/(\mathcal{R}T)$ in $dP/d\tau =g/\kappa$ and integrating givesDifferentiation 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 ${\tau }_b=\frac{2}{3}[((4\beta +1)/5{)}^{1/(\beta -1)}-1]$. Since ${\tau }_b$ depends only on $\beta$, matching to $P=K{T}^{5/2}$ fixes $K\propto g^{1/2}{T}_e^{-2\beta -5/2}$.