Weak capture and the weak stability boundary

Abstract

Weak capture in the restricted problem of three bodies refers to transitory motion between the primaries. Such behavior has interesting applications in space mission design, notably, in finding optimal transfer trajectories that surpass the performance of traditional Hohmann transfers. The Weak Stability Boundary (WSB) is a mathematical set that describes the collection of points in phase space that support such motion. Two definitions exist for the WSB region: a numerical algorithmic definition and an analytical definition. The numerical algorithmic definition is the most accurate description of the WSB region; however it cumbersome to use in practice and does not provide insight into the underlying dynamics of the 3-Body Problem (3BP). An analytical definition has recently been developed to circumvent the need to integrate trajectories to establish the WSB region in phase space. This definition consists of intersections of a number of different mathematical sets. Though this definition can be implemented easily, it is inaccurate in describing the WSB region: it overestimates the region in phase space and includes regions consisting of invariant tori around the two primaries that do not permit transit. The aim of this study is to investigate a possible improvement of the analytical definition and to visualize the internal structure of the WSB region. The internal structure of the WSB region is probed by using extended-Poincarections, which are Poincarections with an additional dimension indicating orbital stability with respect to one of the primaries. The WSB region is investigated in the Planar Circular Restricted 3-Body Problem (PCR3BP), which is a simplified 3BP model. The existence of the WSB region and its internal structure are sketched for a number of different Jacobi energy values. From these results, a number of poignant conclusions can be drawn about the underlying dynamics of the PCR3BP. The methodology presented to visualize the internal structure of the WSB region is proposed as a robust framework to study other dynamical systems in astrodynamics, such as the three-dimensional Circular Restricted 3-Body Problem (CR3BP) and the Elliptic Restricted 3-Body Problem (ER3BP), which are interesting for space mission design.