 Convective stability of a constant-opacity irradiated atmosphere

A deep irradiated layer can remain stable against convection when its thermal opacity and outward radiative flux are constant. Specify a plane-parallel atmosphere in hydrostatic equilibrium, with constant gravity $g>0$, positive constant Rosseland mean opacity ${\kappa }_R$ and upward radiative flux $F>0$. At depths where incident stellar light has already been absorbed, radiative diffusion givesConsequentlywhere $T_0>0$ is the extrapolated boundary temperature of this simplified diffusion solution. The radiative temperature gradient obeysFor a homogeneous diatomic ideal gas with fixed rotational degrees of freedom, $\gamma =7/5$ gives adiabatic temperature gradient ${\nabla }_{\mathrm{ad}}=2/7>1/4$. The Schwarzschild criterion therefore predicts stability throughout this model. Variable opacity, changing thermodynamics or varying radiative flux can change that conclusion. The diffusion approximation also needs large optical depth; it does not establish stability of the entire optically thin irradiated atmosphere.