Recursive computation of gravitational field of a right rectangular parallelepiped with density varying vertically by following an arbitrary degree polynomial

Abstract

We present a recursive formulation to compute the potential, the acceleration vector and the force gradient tensor of the gravitational field of a right rectangular parallelepiped or a prism when its volume mass density varies vertically by following an arbitrary degree polynomial. First, the potential of the parallelepiped is expressed as the triple difference of an indefinite volume integral. Next, for the density described by an arbitrary degree polynomial of the vertical coordinate, the integral is expressed as a finite sum of several elementary functions and three groups of auxiliary functions, the latter of which are recursively computable. Then, the acceleration vector and the force gradient tensor are obtained by analytically differentiating the potential. As a result, all the field quantities turn out to be linear combinations of seven basic functions: three arc tangents, three logarithms and a square root. Also, the employed recurrence formulae are as simple as of a leapfrog form. Thus, the execution of the new formulation is fast. For example, in the computation of the vertical gravity component only, it runs 30-2000 times faster than the existing formulation (Zhang & Jiang) when the maximum degree of polynomial increases from 2 to 40. However, the analytical expressions suffer from the catastrophic cancellation in the far region although they are accurate inside and near the prism.