= Solution
The usual estimate adds a grey, infrared-absorbing <radiative equilibrium> upper atmosphere to the convective lower atmosphere. Consider a very optically thin upper layer: it receives upwelling infrared flux approximately $F_0$ from below, absorbs a small fraction $d\tau$, and emits $\sigma T^4d\tau$ into each hemisphere. Radiative balance gives
$$
2\sigma T_{\rm skin}^4=F_0,\qquad
\boxed{T_{\min}\simeq T_{\rm skin}=\left(\frac{F_0}{2\sigma}\right)^{1/4}\simeq227\,\mathrm K}
$$
for $F_0=300\,\mathrm{W\,m^{-2}}$. The corresponding effective emission <temperature> is $(F_0/\sigma)^{1/4}\simeq270\,\mathrm K$. Without shortwave absorption, the model has no solar-heated stratospheric inversion.
This skin estimate requires the grey absorption/emission and angular approximations just stated. Horizontal mixing and no shortwave absorption alone do not determine an exact minimum for a spectrally selective nitrogen–carbon-dioxide atmosphere: absorption coefficients and spectral windows can change its upper-atmosphere <radiative equilibrium>.
Back to article page
