= 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 $T_{eq}^4=\langle T^4\rangle$ if emissivity is uniform. Without redistribution on an ideal airless surface, local dayside <equilibrium> instead gives $T(\zeta)=[F_S(1-A)\cos\zeta/(\epsilon\sigma R_{AU}^2)]^{1/4}$ 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 \i[global] <temperature>.
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