The geometrical factor of spherical radiation sources

The geometrical factor of a spherical isotropic source of radiation when viewed from a flat cosine (Lambertian) receiver is $B = \Omega (1 - \frac{\Omega}{4\pi}) \cos \theta_0$, where Ω is the solid angle subtended by the source, while $\theta_0$ is the source's zenith angle. Two different derivations of this result are given in this paper. It holds also for diffuse radiation received from a hemispherical dome. Geometrical factors can also be defined for non-isotropic sources of radiation. Taking into account the limb darkening effect leads to solar temperatures of about 6100 K or 6000 K in a better approximation. These are higher than the value derived assuming the Sun to emit radiation isotropically (5770 K).