Abstract

The study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades. Some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body. They replaced each contour by a horizontal irregular n-sided polygonal lamina and chose the combination of the analytical solution for the attraction of an arbitrary polygonal lamina (Jung, 1961; Talwani, 1973) with a numerical integration along the z-direction for the computation of the body's gravity field (Talwani and Ewing, 1960; Hellinger, 1983). Others devoted their efforts to the derivation of purely analytical expressions, based on the technique of reducing the volume Newtonian integral to a summation of line integrals along the segments of each facet, applying Gauss' divergence theorem twice: for the conversion first to surface and then to line integrals. The pioneering work in this field came from Paul (1974), Plouff (1976), and Barnett (1976). Analytical formulas for the potential due to a homogeneous polyhedral body of arbitrary shape and its derivatives up to second order were presented in a compact manner, suitable for efficient computer programming, by Petrović (1996) and Werner and Scheeres (1997). Optimized analytical formulas for the gravity field of a homogeneous polyhedral body, appropriate for efficient calculations, have been presented by Pohanka (1988). An error analysis and further optimization of these expressions were undertaken by Holstein and Ketteridge (1996) and Holstein et al. (1999). Recently, the research in this field was extended to the nonhomogeneous case (Pohanka, 1998; Hansen, 1999). The goal of this research note is to draw attention to the singularities appearing in the numerical evaluation of the closed expressions for the gravitational field of a homogeneous polyhedron. These appear in the transition from surface to line integrals when the orthogonal projection of the computation point onto the plane defined by a facet of the polyhedron lies inside the polygon defining the facet or falls on the line segment describing the edge. Indeed, certain singularities arise when one of these cases occur. We will show, however, that the gravity field is nevertheless defined, if one treats the singularities in the usual way of potential field theory, excluding a small sphere or circle around the singular point. Thus, certain correction terms arise, which should be taken into account when the respective singularities occur.

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