Solid angle subtended by a cylindrical detector at a point source in terms of elliptic integrals

TL;DR

This paper derives formulas for calculating the solid angle of a cylindrical detector at a point source using elliptic integrals, enabling precise computation for arbitrary source positions.

Contribution

The paper introduces new elliptic integral-based expressions for the solid angle of a cylinder from an arbitrary point source, improving accuracy and computational efficiency.

Findings

Derived expressions for cylindrical surface solid angle using elliptic integrals.

Provided formulas for end circle contributions in terms of elliptic integrals.

Enabled calculation of total solid angle for any source position relative to the cylinder.

Abstract

The solid angle subtended by a right circular cylinder at a point source located at an arbitrary position generally consists of a sum of two terms: that defined by the cylindrical surface ($\Omega_{cyl}$) and the other by either of the end circles ($\Omega_{circ}$). We derive an expression for $\Omega_{cyl}$ in terms of elliptic integrals of the first and third kinds and give similar expressions for $\Omega_{circ}$ using integrals of the first and second kinds. These latter can be used alternatively to an expression also in terms of elliptic integrals, due to Philip A. Macklin and included as a footnote in Masket (Rev. Sci. Instr., 28 (3), 191-197, 1957). The solid angle subtended by the whole cylinder when the source is located at an arbitrary location can then be calculated using elliptic integrals.