Uncertainties in the gravity spherical harmonics coefficients arising from a stochastic polyhedral shape

The increasing interest towards exploring small bodies in the Solar system has paved the way for the investigation of different gravity field models allowing robust orbit design and dynamical environment characterization. Among these, the spherical harmonics representation and the polyhedral gravity model are the most employed and studied—the former for historical reasons and the latter due to the fact that it is an exact and closed-form representation of the gravity field arising from a constant-density polyhedron. The exact algorithm computing the spherical harmonics coefficients from a given polyhedron is available in the literature. Unfortunately, little to no insight into the uncertainty in the spherical harmonics coefficients is provided alongside their computed value, as the polyhedron is customarily considered as a deterministically known solid. During orbit design, spacecraft operations and small body characterization, it is crucial to account for the uncertainty in the spherical harmonics coefficients and how it influences the overall mission architecture. This paper provides the analytical derivation of the partial derivatives of the transformation between the constant density polyhedron and the spherical harmonics coefficients with respect to the vertices of the polyhedron. This derivation allows for the quantification of the spherical harmonics coefficients’ uncertainties when a stochastic polyhedral shape is considered, i.e., a shape whose vertices do not have a deterministic position. As a result, the analytical expressions of the uncertainty in the gravity potential and acceleration are assembled to rigorously quantify the influence of a stochastic polyhedron on the surrounding dynamical environment. A brief explanation of the implementation procedure and the polyhedron vertices covariance matrix definition is provided to the reader. The final result of the paper is a set of numerical simulations validating the proposed model and demonstrating its capability to provide the uncertainties in the spherical harmonics coefficients, the gravity potential and acceleration arising from a stochastic polyhedral shape.