Spherical harmonic representation of the gravitational potential of discrete spherical mass elements

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We present expressions in a spherical harmonic framework for the gravitational potential of discrete point, surface, and volume mass elements located at any depth within a sphere. Through analysis of the spherical harmonic spectrum, insight is gained into the properties of the potentials arising from a variety of mass distributions. A point mass at the surface of a sphere displays the richest harmonic spectrum in all degrees; spectra become increasingly reddened as the source mass is distributed through larger elements of area or volume, or is located at greater depths below the surface of the reference sphere. The spectra of dipolar distributions, useful in representing compensated masses, are depressed, especially in the low harmonic degrees, relative to the spectra of monopole elements.