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 $ϵ$. Let the surface be a uniform blackbody receiving absorbed stellar irradiance $Q$. In radiative equilibrium, the layer absorbs $ϵ\sigma T_s^4$ and emits $ϵ\sigma T_a^4$ in each direction, so for $ϵ>0$,The second equation follows from surface balance, $\sigma T_s^4=Q+ϵ\sigma T_a^4$. It demonstrates how absorption and downward reemission warm the surface at fixed absorbed power. The limiting $ϵ=1$ 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.

A surface black in the thermal infrared beneath a shortwave-transparent, isothermal grey layer has atmospheric balance ${ϵ}_a\sigma T_s^4=2{ϵ}_a\sigma T_a^4$. For nonzero layer emissivity, $T_a^4=T_s^4/2$. Direct surface transmission plus upward layer emission then gives $F_{\mathrm{out}}=(1-{ϵ}_a/2)\sigma T_s^4$, proving the title. This model only permits $1/2\le {ϵ}_{\mathrm{eff}}\le 1$ for $0\le {ϵ}_a\le 1$. A real vertical opacity/temperature profile is not uniquely determined by this one fitted parameter.