Closed form expressions for gravitational multipole moments of elementary solids

TL;DR

This paper derives comprehensive closed form expressions for gravitational multipole moments of elementary solids, enabling precise modeling of complex structures in gravitational experiments without numerical integration.

Contribution

It provides the first complete set of closed form formulas for all degrees and orders of multipole moments for elementary solids.

Findings

Closed form expressions for all multipole moments derived

Facilitates accurate gravitational modeling of complex structures

Eliminates the need for numerical integration in multipole calculations

Abstract

Perhaps the most powerful method for deriving the Newtonian gravitational interaction between two masses is the multipole expansion. Once inner multipoles are calculated for a particular shape this shape can be rotated, translated, and even converted to an outer multipole with well established methods. The most difficult stage of the multipole expansion is generating the initial inner multipole moments without resorting to three dimensional numerical integration of complex functions. Previous work has produced expressions for the low degree inner multipoles for certain elementary solids. This work goes further by presenting closed form expressions for all degrees and orders. A combination of these solids, combined with the aforementioned multipole transformations, can be used to model the complex structures often used in precision gravitation experiments.