Analytic expressions for gravitational inner multipole moments of elementary solids and for the force between two rectangular solids

Abstract

We give analytic expressions for the gravitational inner spherical multipole moments, q lm with l ≤ 5, for eleven elementary solid shapes. These moments, in conjunction with their known rotational and translational properties, can be used to calculate precisely the moments of complex objects that may be assembled from the elementary shapes. We also give an analytic expression for the gravitational force between two rectangular solids at arbitrary separations. These expressions are useful for computing the gravitational properties of complex instruments, such as those used in equivalence principle tests, and in the gravitational balancing of drag-free spacecraft.

Closed form expressions for gravitational multipole moments of elementary solids

“…where we note that using the sinc function removes the need for separate cases for m = 0 and m = 0. This equation can be shown to be consistent with the results given in Adelberger et al [11].…”

“…Efficient calculation of inner multipole moments is, as such, of great value. Low-degree (l ≤ 5) inner multipole moments have been calculated individually for each order (m) for a number of elementary solids [11]. For higher degrees, however, either numerical methods must be employed or each order must be calculated explicitly.…”