Convection onset in a logarithmic-pressure grey atmosphere

For a grey atmosphere with $T^4=\frac{3}{4}{T}_{\mathrm{eff}}^4(\tau +2/3)$ and $P^2=P_0^2\log (1+3\tau /2)$, logarithmic differentiation givesA monatomic perfect gas of uniform composition has adiabatic temperature gradient $2/5$. The Schwarzschild criterion therefore predicts the first marginally unstable layer at ${\tau }_c=\frac{2}{3}(e^{4/5}-1)$, and convection for larger optical depth. Beyond that onset, the assumed purely radiative profile is no longer a self-consistent transport solution.