Dynamical Evaluation of Gravity Spherical Harmonic Coefficients due to Generally Shaped Polyhedra

Abstract

The gravitational potential uncertainty process arising from the stochastic consideration of generally shaped polyhedra is outlined and tested on the real shape model of asteroid Psyche. The examined method is based on the computation of partial derivatives of spherical harmonic coefficients as implied by corresponding coordinate changes of the polyhedron’s vertices, while the derived results are compared with gravity signal differences induced by the shape’s variations using the line integral analytical approach. For the numerical tests, 3 regular grids of points with dimensions 600 km2 were considered. The differences of the obtained results between the two approaches range from 85 m2/s2 to 300 m2/s2 for the gravitational potential uncertainties and from 2% to 2.4% for the normalized gravitational potential uncertainties. Additional tests were carried out on different points with increasing distance from the asteroid’s surface to correlate the computed uncertainties with the spherical harmonic coefficients’ maximum degree of expansion. As seen, inside the uncertainty region defined by the boundary of Brillouin sphere, the computed normalized gravitational potential uncertainties differ at the level of 0.04% for solutions of maximum degree of expansion {5, 10, 15, 20} while outside they gradually become identical. Therefore, the position of the computation points as well as the morphology of the examined mass distribution that defines the Brillouin sphere seem to strongly affect the derived results.