Stable Orbits in the Small-Body Problem

Abstract

Over the past few years, space missions to minor celestial bodies have gained increased attention from the space community. A prime example is the selection of Psyche as NASA’s fourteenth Discovery Program mission. Orbital dynamics in the vicinity of small irregular bodies pose great challenges for trajectory design. In particular, science orbits close to the surface of small bodies are often strongly perturbed by the irregular gravity field. These irregularities can potentially perturb the trajectory of a spacecraft in such a way that it becomes uncontrollable, resulting in impact on the surface of the central body or escape from the system. A less dramatic consequence would be the need for costly orbit maintenance maneuvers to counteract these instabilities, which is undesirable. For uniformly rotating bodies such as the Psyche asteroid, mean motion resonances with the asteroid rotation amplify instabilities even further, warranting the need for a systematic study of orbital stability in such systems.

To increase our understanding of these dynamically challenging environments, we carry out an extensive numerical characterization of stability in the uniformly rotating second degree and order gravity field by uniformly sampling the initial condition space and the gravity field harmonic coefficients. In doing so, we consider two conceptually different notions of stability: Bounded-Input, Bounded-Output (BIBO) stability and the regularity/chaoticity of spacecraft trajectories, quantified by the Fast Lyapunov Indicator (FLI). Through this study we have been able to generate stability plots that can be used to quickly inform mission designers on stable and unstable regions in the phase space. Our results show that FLI maps are an effective complementary mission design tool that flag orbit instabilities sooner when compared to BIBO stability constraints. Finally, our numerical results are complemented by analytically derived equations of the semi-major axis and the eccentricity in the uniformly rotating second degree and order gravity field. These equations clearly show the origin and impact of resonant instabilities. This initial work can be developed further into a complementary mission design tool for future small-body missions.