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

We give analytic expressions for the gravitational inner spherical multipole moments, qlm with l ≤ 5, for 11 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