Analytical shape uncertainties in the polyhedron gravity model

The exploration of small bodies in the Solar system and the ability to perform remote or in situ science is tied to the understanding of the dynamical environment of such objects. As such, the evaluation of the gravity field arising from small bodies is key to this understanding. However, remote observations can only produce shape estimates, from which only uncertain gravity fields can be computed. The current disconnect in the literature between the uncertainty in the shape and that of the gravity field properties is detrimental to small body science and robust mission design. In particular, the literature does not provide any quantitative means to capture this link in the polyhedron gravity model, one of the main gravity model representations. With this in mind, this paper derives the expressions of the first variations and partial derivatives in the potential, acceleration and slopes computed from the polyhedron gravity model with respect to the vertices of the underlying body. These formulae are then combined with a Gaussian description of the uncertainty in the vertex coordinates so as to obtain analytical predictions of the potential, slope variances as well as the covariance in the acceleration at arbitrary locations around the body, treated as a stochastic shape. This linearized analytical approach was able to capture the statistical variation in the dynamical environment about asteroid 25143 Itokawa and 16 Psyche under the assumption of stochastic errors in the bodies’ shape models, at a lower computational cost than Monte–Carlo simulations. These methods should be of benefit to planetary scientists and mission designers seeking for more insight into the dynamical environment of uncertain small body shapes.