Second-species dynamics in the restricted planar circular three body problem: chaos, final motions and periodic orbits

TL;DR

This paper explores complex dynamical behaviors in the restricted planar circular three-body problem, constructing orbits with various final motions, including oscillations, ejections, collisions, and large periodic orbits, using advanced mathematical techniques.

Contribution

It introduces a novel combination of regularization and perturbation methods to explicitly construct a wide range of orbit types in the three-body problem with small mass ratios.

Findings

Construction of orbits with any combination of past and future motions.

Existence of arbitrarily large ejection-collision orbits.

Presence of large periodic orbits near Jupiter.

Abstract

Consider the Restricted Planar Circular Three Body Problem (RPC3BP), which models the motion of a massless particle (Asteroid) under the gravitational influence of two massive bodies (the primaries) moving on circular orbits. By considering the ratio between the masses of the primaries to be arbitrarily small, we construct orbits with close encounters with the smaller primary (Jupiter) that realize any combination of past and future final motions (in the sense of Chazy's), including oscillatory motions. We also obtain arbitrarily large ejection-collision orbits with Jupiter and ejection-collision orbits between the two primaries (Sun and Jupiter), as well as arbitrarily large periodic orbits that pass arbitrarily close to Jupiter. Our approach combines singular perturbation theory and Levi-Civita regularization near Jupiter, and McGehee regularization near infinity and near the Sun, together with a global analysis that leads to transverse intersections of invariant manifolds.