Transport geometry of the restricted three-body problem

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Virginia Tech


This dissertation expands across three topics the geometric theory of phase space transit in the circular restricted three-body problem (CR3BP) and its generalizations. The first topic generalizes the low energy transport theory that relies on linearizing the Lagrange points in the CR3BP to time-periodic perturbations of the CR3BP, such as the bicircular problem (BCP) and the elliptic restricted three-body problem (ER3BP). The Lagrange points are no longer invariant under perturbation and are replaced by periodic orbits, which we call Lagrange periodic orbits. Calculating the monodromy matrix of the Lagrange periodic orbit and transforming into eigenbasis coordinates reveals that the transport geometry is a discrete analogue of the continuous transport geometry in the unperturbed problem. The second topic extends the theory of low energy phase space transit in periodically perturbed models using a nonlinear analysis of the geometry. This nonlinear analysis relies on calculating the monodromy tensors, which generalize monodromy matrices in order to encode higher order behavior, about the Lagrange periodic orbit. A nonlinear approximate map can be obtained which can be used to iterate initial conditions within the linear eigenbasis, providing a computationally efficient means of distinguishing transit and nontransit orbits that improves upon the predictions of the linear framework. The third topic demonstrates that the recently-discovered "arches of chaos" that stretch through the solar system, causing substantial phase space divergence for high energy particles, may be identified with the stable and unstable manifolds to the singularities of the CR3BP. We also study the arches in terms of particle orbital elements and demonstrate that the arches correspond to gravity assists in the two-body limit.



Three-body problem, Phase space transit, Astrodynamics, Invariant manifolds