The topology of transit orbits in multibody systems with applications to dynamical astronomy and mission design
Keywords: Aeronautics and Astronautics.

In recent years there has been a growing interest in missions to multibody systems. In particular, Jupiter and its Galilean moons have become targets of a great deal of scientific investigation, and are now a primary focus of space exploration efforts because they are believed to have the potential to harbor life. Several missions to explore these icy moons are planned for the upcoming decades, including the Jupiter Icy Moon Explorer (JUICE), Europa Clipper, and Europa Lander. These missions have complex architectures that demand innovative trajectory solutions leveraging the dynamics of multibody systems, as well as the ability to perform rapid and well-informed design iteration. The qualitative exploration of multibody trajectory solutions requires knowledge of the underlying dynamical structures, invariant manifolds - tube-like structures formed by the set of trajectories asymptotic to a given periodic orbit - which form the basis of multibody analysis and design. Although our understanding of multibody dynamics has advanced greatly in recent decades, there is still much to learn. In particular, the range of energies over which invariant manifolds govern transit orbits in the three-body problem is unknown. In the four-body problem there exists no strong theorems on this subject at all. In order to meet mission requirements, a deeper understanding of the dynamical structures present in multibody systems is needed. To address these gaps in knowledge, we investigate the role of invariant manifolds in controlling transit orbits of the planar three-body and four-body problems. This objective is achieved through a study of the topology of the equilibrium regions, the spaces surrounding Lagrange points, of the three-body and four-body problems. By visualizing trajectories directly in the energy surface, we will make clear the governing role of invariant manifolds at the most basic, physical level. With this insight we build upon Conley's foundational theorems and show that invariant manifolds of the three-body problem govern transit orbits over a much larger range of energies than previously proven. Our understanding of the topology of the neck region, the bottleneck shaped region of allowable motion near the first and second Lagrange points, will allow us to derive a method to approximate the four-body problem as an effective three-body problem at a perturbed energy level, and show that the invariant manifolds of the effective three-body problem control transit orbits of the four-body problem. This effectively reduces the four-body analysis and design challenge to a three-body analysis and design challenge. We then apply these theories to practical examples from dynamical astronomy and mission design. Through a study of Lyapunov orbits, halo orbits, and their associated invariant manifolds, we show that periodic orbits act as gateways to the region surrounding Jupiter, and that invariant manifolds controlled the capture of comet Shoemaker-Levy 9 (SL9), ultimately leading to its impact with Jupiter in 1994. The study of small solar system bodies like SL9 is vital for assessing the Earth impact risk, but also informs mission design. Using insights from dynamical astronomy we design initial capture, landing, and end-of-life trajectories for the Europa Lander mission. We demonstrate that dynamical structures in multibody systems provide a means to rapidly generate fuel efficient solutions. Our invariant manifold-informed trajectory designs are shown to significantly outperform two-body baseline designs, enabling mission architectures that would otherwise be infeasible.