Celestial Mechanics and Dynamical Astronomy

, Volume 121, Issue 1, pp 17–38

# Analytical solution from vector potentials for the gravitational field of a general polyhedron

Original Article

## Abstract

Closed form solutions in terms of elementary functions are given for the Newtonian gravitational field and potential of a general constant density polyhedron, using a gravitational vector potential formulation. The solution constants are given in terms of scalar and vector products involving the position vectors of the field point and the polyhedron’s vertices, giving one analytical expression for each edge of the polyhedron. It is shown that in this vector potential formulation, the gravitational problem is related to the point-in-polygon problem of computational geometry. The solution is derived using a new vector theorem giving the integral over a flat surface of a scalar potential as a line integral of a vector potential around its boundary. The method is also valid in the interior of the polyhedron and can be extended to polyhedra with linear spatial variation of the density. Numerical results are compared with results in the literature and detailed results are given for the five regular Platonic polyhedra.

### Keywords

Gravitational field Vector potential Polyhedra  Analytical solution  Surface integrals Potential theorem

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