= Projected-area factor in planetary energy balance
{title2=$\langle F_{\rm absorbed}\rangle=(1-A)F_*/4$}
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 $T=T_*\sqrt{R_*/(2a)}(1-A)^{1/4}$. Without heat redistribution it constrains the area mean of $T^4$, not the arithmetic mean of $T$.
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