Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on a planar circular surface coaxial to the disc

Abstract

Average solid angles subtended by an external contour of a given body from several points are required for the calculation of radiation fluxes or particle beams that are incident on the body from multiple-point isotropic emitters of analogous kinds. To consider the refraction effects for fluxes of light propagating through various optical media, knowledge of the angular distributions of such average solid angles is necessary. In this paper the formula describing angular dependence of the average solid angle subtended by a circular disc from uniformly distributed points within the circular surface of a parallel and coaxial disc was derived analytically and used for the calculation of some representative results. The solution has been made in the cylindrical coordinate system. The final and some intermediate formulas were expressed as functions of the polar angle, of the radii of both discs and of the distance between their planes. These formulae were represented by superpositions of simple elementary functions, single integrals of, these superpositions and by incomplete and complete Legendre—Jacobi elliptic integrals of all three kinds. Mathematica 2.2.3 software was used to illustrate graphically the relationships between some computed data. These data indicate that the derived formula is directly applicable in any computer programs calculating the Legrendre—Jacobi elliptic integrals to estimate the fluxes of optical radiation and particle beams propagated within a non-absorbing homogeneous medium. The expressions obtained may also be used to calculate the fluxes of optical radiation propagated through various homogeneous media.