Lambertian sphere at full phase ($F=\frac{2A}{3}{F}_{\star }(R/d{)}^2$)

A spherical body of radius R observed from distance $d\gg R$ is illuminated and viewed from the same direction, with incident irradiance $F_{\star }$. For uniform Lambertian reflectance A, a patch at angle $\theta$ has outgoing radiance $A{F}_{\star }\cos \theta /\pi$. The observer receives this radiance times projected area divided by $d^2$. Since ${\int }_{\mathrm{hemisphere}}{\cos }^2\theta dS=2\pi R^2/3$, integration proves the displayed full-phase irradiance. Its geometric albedo is $2A/3$. Total reflected power alone does not specify the brightness in one direction for a non-Lambertian surface.