= Convection onset in a logarithmic-pressure grey atmosphere
For a <grey atmosphere> with $T^4=\tfrac34T_{\rm eff}^4(\tau+2/3)$ and $P^2=P_0^2\log(1+3\tau/2)$, logarithmic differentiation gives
$$
\nabla_{\rm rad}=\frac{d\log T}{d\log P}=\frac12\log(1+3\tau/2).
$$
A monatomic <perfect gas> of uniform composition has <adiabatic temperature gradient> $2/5$. The <Schwarzschild criterion> therefore predicts the first marginally unstable layer at $\tau_c=\tfrac23(e^{4/5}-1)$, and convection for larger <optical depth>. Beyond that onset, the assumed purely radiative profile is no longer a self-consistent transport solution.
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