= Single-layer greenhouse model
A <single-layer greenhouse model> replaces the vertical structure of an <exoplanet atmosphere> by one layer transparent to incoming stellar light and having thermal <emissivity> $\epsilon$. Let the surface be a uniform <blackbody> receiving absorbed stellar <irradiance> $Q$. In <radiative equilibrium>, the layer absorbs $\epsilon\sigma T_s^4$ and emits $\epsilon\sigma T_a^4$ in each direction, so for $\epsilon>0$,
$$
T_a^4=T_s^4/2,\qquad T_s^4=\frac{Q}{\sigma(1-\epsilon/2)}.
$$
The second equation follows from surface balance, $\sigma T_s^4=Q+\epsilon\sigma T_a^4$. It demonstrates how absorption and downward reemission warm the surface at fixed absorbed power. The limiting $\epsilon=1$ surface is $2^{1/4}$ times the <planetary equilibrium temperature>. The model omits <convection>, spectral variation and a continuous <atmospheric pressure-temperature profile>, so the fitted <emissivity> is not a unique atmospheric composition.
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