Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source

TL;DR

This paper derives analytical formulas for calculating the solid angle of a circular detector as viewed from a point source with a cosine angular distribution, considering different coordinate systems.

Contribution

It provides new analytical expressions for the solid angle subtended by a circular disc at a cosine-distributed point source, including multiple coordinate system references.

Findings

Explicit formulas for the solid angle are derived.

Sample plots illustrate the analytical expressions.

The approach simplifies calculations in radiation detection scenarios.

Abstract

We derive analytical expressions for the solid angle subtended by a circular disc at a point source with cosine angular distribution ($f(\mu)=\mu/\pi$)under the sole condition that the disc lies in the half-space illuminated by the source ($\mu \geq 0$). The expressions are given with reference to two alternative coordinate systems (S and S'), S being such that the z axis is parallel to the symmetry axis of the disc and S' such that the $z'$ axis is aligned with the source direction. Sample plots of the expressions are presented.