Orbiting below the Brillouin sphere using shifted spherical harmonics

Abstract

Spacecraft trajectories near the south pole of Enceladus violate the Brillouin sphere associated with the convergence radius of spherical harmonics models. In this study, a shifted coordinate frame is demonstrated to ensure a convergent model is available in regions of operational interest. Hypothetical experiments are performed around a simulated celestial body where the truth exterior gravity fields are known exactly. The divergence of the harmonics below the Brillouin sphere of the unshifted models is confirmed, while the shifted harmonics model converges. The method is next applied to the Cassini-derived gravity field for Enceladus, including uncertainties. Using these low-degree and low-order reference models, expected for use in an operational setting, the results show that the shifted and body-centered harmonics models agree to near machine precision for all evaluations at or above the surface, and no divergence is noticed. The results imply that mission designers and navigation engineers can safely use a traditional spherical harmonics field for Enceladus, even in regions that dip below the Brillouin sphere. For low-flying missions to celestial bodies besides Enceladus, the experiments conducted in this study can be repeated. The need for an alternative to the traditional spherical harmonics, such as the presented shifted model, increases for bodies that are increasingly non-spherical and orbits that are deeper inside the Brillouin sphere.