Computer programs for spherical cap harmonic analysis of potential and general fields

Abstract

Spherical cap harmonic analysis is an analytical technique for modeling either a potential function and its spatial derivatives over and above a spherical cap, or a general function and its surface derivatives on a spherical cap surface. In either situation, it gives a uniformly convergent series expansion, the basis functions of which involve associated Legendre functions of integral order but not necessarily of integral degree. Temporal variations are accommodated in the usual way by expressing the coefficients as polynomials in time. For both the potential and the general function, and for any of their derivatives not involving colatitude, one set of mutually orthogonal basis functions is sufficient. For differentiation of either the potential or the general function with respect to colatitude, a second set of mutually orthogonal basis functions is required. For a potential function, and its spatial derivatives, the expansion involves coefficients resulting from both internal and external sources. For a general function on the spherical cap surfaces, and its surface derivatives, there is no distinction between internal and external sources and the expansion in this instance is the same as the potential expansion at the spherical cap surface with internal and external sources combined. The practical importance of spherical cap harmonic analysis in modeling potential fields already has been demonstrated, and the computer programs and subprograms presented here, with accompanying discussion, will allow the technique to be applied easily to both potential and general surface fields by anyone with access to a computer.