Stable sets mapping with Taylor differential algebra with application to ballistic capture orbits around Mars

Abstract

Ballistic capture orbits offer safer Mars injection at longer transfer time. However, the search for such an extremely rare event is a computationally-intensive process. Indeed, it requires the propagation of a grid sampling the whole search space. This work proposes a novel ballistic capture search algorithm based on Taylor differential algebra propagation. This algorithm provides a continuous description of the search space compared to classical grid sampling research and focuses on areas where the nonlinearities are the largest. Macroscopic analyses have been carried out to obtain cartography of large sets of solutions. Two criteria, named consistency and quality, are defined to assess this new algorithm and to compare its performances with classical grid sampling of the search space around Mars. Results show that differential algebra mapping works on large search spaces, and automatic domain splitting captures the dynamical variations on the whole domain successfully. The consistency criterion shows that more than 87% of the search space is guaranteed as accurate, with the quality criterion kept over 80%.Ballistic capture (BC) allows a spacecraft to approach a planet and enter a temporary orbit about it without requiring maneuvers in between. As part of the low-energy transfers, it is a valuable alternative to Keplerian approaches. Exploiting BC grants several benefits in terms of both cost reduction (Belbruno and Miller, 1993) and mission versatility (Belbruno and Carrico, 2000;Topputo and Belbruno, 2015), in general at the cost of longer transfer times (Circi and Teofilatto, 2001;Ivashkin, 2002). In the past, the BC mechanism was used to rescue Hiten (Belbruno and Miller, 1990), and to design insertion trajectories in lunar missions like SMART-1 (Racca et al, 2002) and GRAIL (Chung et al, 2010). In the near future, BepiColombo will exploit BC orbits to be weakly captured by Mercury (Benkhoff et al, 2021;Schuster and Jehn, 2014). BC is an event occurring in extremely rare occasions and requires acquiring a proper state (position and velocity) far away from the target planet (Topputo and Belbruno, 2015). In fact, massive numerical simulations are required to find the specific conditions that support capture (Topputo and Belbruno, 2009) and only approximately 1 out of 10 000 states lead to capture (Luo and Topputo, 2015). In a first effort to reduce the computational burden, the variational theory for