Realizations of the Interior Spherical Bessel Harmonic Coefficients of a Polyhedral Body within the Brillouin sphere

Propagation of satellite orbits about celestial bodies frequently relies on spherical harmonics to model the gravitational field. Yet, the expansion of the gravitational field in terms of spherical harmonics only converges exterior to the Brillouin (circumscribing) sphere of the body, limiting the accuracy of the model for very low orbits or landing trajectories. Interior to this sphere, a complementary expansion in terms of spherical Bessel harmonics provides a rigorous model of the field. To realize this model, past studies have relied on calculating Bessel harmonic coefficients from their spherical harmonic counterparts for the exterior field. This lacks a unique solution. In this article we derive an algorithm for uniquely calculating the Bessel harmonic coefficients directly from the surface integral of an arbitrary shape with homogenous density. We verify the accuracy of such gravity field below the Brillouin sphere of Phobos against the polyhedral gravity field. We also propose a couple methods of allocating the Bessel harmonics from the exterior spherical harmonic coefficients and compare them against the polyhedral gravity field. These methods along with the shape-model integration greatly improve the applicability of the interior spherical Bessel harmonics to problems in spacecraft navigation near small bodies as well as science.