A Fast Quadrature-Based Gravity Model for the Homogeneous Polyhedron

To date several probes have been sent to explore the Solar system's asteroids and comets. These bodies are often irregular in shape and to safely navigate probes in their vicinity accurate gravity models are required. For an arbitrarily shaped constant-density body, the gravitational field can be determined from the surface topology and bulk properties. This is achieved by replacing the body's true geometry with a polyhedron that closely resembles it and for which analytic equations for the gravitational field exist. For some applications however, these equations are too computationally expensive and it can be beneficial to replace them with numerically amenable approximations. In this work, a numerical-quadrature-based model for the gravitational field of a polyhedron consisting of triangular facets is derived. The proposed approximate model is found to be faster than its analytic counterpart. The error of the approximation is found to be negligible for the potential and Laplacian calculations. The approximate model introduces singularities to the surface of the acceleration calculation degrading the solution at altitudes less than the average edge length of the polyhedron.