= 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> gives
$$
F=-\frac{16\sigma_{\rm SB}T^3}{3\kappa_R\rho}\frac{dT}{dz},\qquad
\frac{dP}{dz}=-\rho g.
$$
Consequently
$$
\frac{dT^4}{dP}=\frac{3\kappa_RF}{4\sigma_{\rm SB}g},\qquad
T^4=T_0^4+\frac{3\kappa_RF}{4\sigma_{\rm SB}g}P,
$$
where $T_0>0$ is the extrapolated boundary <temperature> of this simplified diffusion solution. The <radiative temperature gradient> obeys
$$
\nabla_{\rm rad}=\frac{d\log T}{d\log P}
=\frac{T^4-T_0^4}{4T^4}<\frac14.
$$
For a homogeneous diatomic <ideal gas> with fixed rotational degrees of freedom, $\gamma=7/5$ gives <adiabatic temperature gradient> $\nabla_{\rm ad}=2/7>1/4$. 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.
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