Determination of the potential of homogeneous polyhedral bodies using line integrals

Abstract.

For the determination of the potential of irregular inhomogeneous bodies they can be decomposed into (polyhedral) parts of homogeneous density. Efficient formulas for the computation of the gravitational potential (and its first and second derivatives) of homogeneous polyhedral bodies are presented. They are obtained using a transformation of the volume integral into line integrals.

The most important property of the solution is that all ten quantities under consideration (potential, 3 components of the gravitation vector, 6 components of the tensor of the second derivatives) can be represented by using only two different line integrals. Furthermore, all coordinate transformations needed in the evaluation are chosen in such a way that they do not appear in the final result.

The consequence, favorable for efficient programming, is that the same transcendental expressions along each edge of the polyhedron are needed for all ten quantities; even the same linear combinations of them for individual surfaces are appearing in different formulas. The expressions obtained are probably the simplest possible, which is also reflected in the fact that for the special case of a right rectangular prism they may easily be specialized to the usual well-known formulas.