A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane

Abstract

Methods for calculating the solid angles subtended at a point by closed contours of some given objects are required in many areas of optical and nuclear physics to estimate the fluxes or particle beams of radiation. It is therefore of interest to derive solid angle equations for various closed contours and the location of such contours with respect to a single point or some points. In this article two methods are described for calculating the solid angles for objects having the shape of a circular disk. In the first method the formula for calculating the solid angle at a point subtended by a circular disk was presented in a more general form than the expressions reported by S. Tryka ͓Opt. Commun. 137, 317 ͑1997͔͒. In the second method this formula was used to derive another formula for calculating the average solid angle subtended by a circular disk from uniformly distributed multiple points lying on the planar circular surface coaxial to the disk. Both of these formulas were represented by superpositions of simple elementary functions with the complete Legendre-Jacobi elliptic integrals. These superpositions were given as dependencies on the radii of the disk and of the circular surface and on the distance between the disk and the surface. The influences of these radii and distance on the solid angle value were illustrated graphically by some three-dimensional surface plots. Some representative results were calculated to eight decimal places and given in tabular form. In addition, a similar table of results computed for the solid angle at a point subtended by a circular disk was included. It was shown that the formulas obtained are simple and directly applicable in some high-language programs. As an example of such an application all plots and calculations in this article were performed using MATHEMATICA 2.2.3 software.

Calculating the solid angle for different source shapes and orientations as viewed by a detector with a cylindrical collimator

“…Several approaches to the problem have been published [1][2][3][4][5][6][7][8][9][10][11][12], mostly devoted to the commonly encountered disk-shaped radiation source and un-collimated detectors. This paper presents a method for calculating the solid angle from either circular, elliptical, or rectangular sources oriented at any angle relative to a collimated detector.…”