Closed form expressions for gravitational multipole moments of elementary solids

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.