Spherical harmonic representation of the gravitational potential of a point mass, a spherical cap, and a spherical rectangle

The gravitational potentials of a point mass, a spherical cap, and a spherical rectangle are presented in a spherical harmonic framework. The point mass displays the richest harmonic spectrum at all degrees. Equal mass distributed as a spherical cap or rectangle shows loss of representation at high degrees; the spectral weakening moves to progressively lower degrees as the mass is distributed more widely. The spectrum of a cap is less smooth than that of a similar rectangle with the same area. The characteristics of these three mass configurations are compared with those of other mass distributions and modeling techniques.