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.
Stirling, J., & Schlamminger, S. (2019). Closed form expressions for gravitational multipole moments of elementary solids. Physical Review D, 100(12), . https://doi.org/10.1103/PhysRevD.100.124053