Generalized Solutions for the Gravitational Potential of Constant-Density Polyhedra

Finding mathematical solutions for the gravitational potential, V, inside, on, and outside three-dimensional bodies of constant density has been an active research area for more than a century. Existing solutions for constant-density polyhedra, including cuboids and tetrahedra, are valid exterior to the body, and, in some cases, extant solutions are also valid on the surface of the body. But they are not valid inside the body. This is because these solutions produce numerical singularities in the interior of body. However, these singularities are purely numerical, and therefore removable, using special case handling of the arctangents and logarithmic terms in which the problem arises. We present generalized, singularity-free analytical expressions for the gravitational potential of a homogeneous solid cuboid, tetrahedron, and (arbitrary) polyhedron, which are valid at any point in space. To validate the solutions, we introduce the Laplacian Test, a robust method that numerically evaluates second-order derivatives of the potential using high-precision Julia’s BigFloat arithmetic. This test confirms that our solutions satisfy Poisson’s equation within the body and Laplace’s equation outside of it. Since the solutions to Laplace’s equation are unique, this validates our algorithms and codes. We also show that vector field of gravitational acceleration is continuous in space, including on and across the surface. We apply this framework to each geometric primitive. For the cuboid, the formulation systematically arranges logarithmic and arctangent functions to maintain numerical stability. The tetrahedron, as a fundamental element of polyhedral structures, is shown to admit a singularity-free potential under this unified approach. Extending this to arbitrary polyhedra, we demonstrate a scalable method for modeling complex, uniform mass distributions. The solutions for V due to these constant-density mass elements, will be useful in planetary science, exploration geophysics, and the precise characterization of irregular celestial bodies such as asteroids and comets.