Gravity Degree-Depth Relationship Using Point Mass Spherical Harmonics

Relationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree-depth relationship. Our relationship resembles an earlier one derived by Bowin (1983), with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. (2022) over-estimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to d = (1 - e-1/n+1)R, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin (1983) affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modeling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.