Projected-area factor in planetary energy balance ($⟨F_{\mathrm{absorbed}}⟩=(1-A)F_{\ast }/4$)

A sphere intercepts a parallel stellar beam over projected area $\pi R_p^2$ and radiates over surface area $4\pi R_p^2$. Dividing total absorbed power by total surface area gives the factor one-quarter, independent of planet radius. Under uniform-temperature reradiation this implies $T=T_{\ast }\sqrt{R_{\ast }/(2a)}(1-A{)}^{1/4}$. Without heat redistribution it constrains the area mean of $T^4$, not the arithmetic mean of $T$.