 Solution

The formula assumes spherical geometry, an effective uniform emission temperature, steady balance, specified bolometric emissivity, no important internal heating and appropriate redistribution or averaging. An atmosphere is not mathematically required: for any surface-temperature pattern the same global balance holds with $T_{eq}^4=⟨T^4⟩$ if emissivity is uniform. Without redistribution on an ideal airless surface, local dayside equilibrium instead gives $T(\zeta )=[F_S(1-A)\cos \zeta /(ϵ\sigma R_{AU}^2){]}^{1/4}$ and the instantaneous night side would approach zero in the no-storage limit. Atmospheric absence alone does not justify replacing the factor four by one for a global temperature.