Spherical Harmonic Expansions for the Gravitational Field of a Polyhedral Body with Polynomial Density Contrast

We provide the spherical harmonic solutions to evaluate the external gravitational field of a general polyhedral body with arbitrary polynomial density contrast, including the gravitational potential and its arbitrary-order derivatives. The linear recursive algorithm for computation of the spherical harmonic coefficients of the potential is derived by using the Gauss divergence theorem and the Stokes theorem, and the computations are performed on the basis of the division of general polygonal pyramid of the polyhedron instead of the division of tetrahedron. The algorithm of this paper can handle the density contrast in both horizontal and vertical directions and the polynomial function of the density at arbitrary degree. Both the conversion relations of the density function and the arbitrary-order derivatives of the spherical harmonic potential between the initial and rotated reference frames are given in tensor product forms, which assist the calculations. The space-domain method for evaluating the gravity field of the polyhedral body may leads to numerical problems at a remote observation point. However, the spherical harmonic method is numerically stable at arbitrary observation points outside the smallest enclosed sphere. The numerical experiments for three actual and synthetic polyhedral models including a right rectangular prism with cubic density varying with depth, the asteroid 433 EROS with cubic polynomial density and a right rectangular prism with quartic polynomial density, are implemented to test the accuracy, convergence, and numerical stability of the spherical harmonic algorithm.