Oscillatory motions, parabolic orbits and collision orbits in the planar circular restricted three-body problem

TL;DR

This paper investigates complex orbit behaviors in the planar circular restricted three-body problem, including oscillatory motions, collision orbits, and the construction of various dynamic trajectories near the primaries.

Contribution

It introduces new methods to construct orbits with specific asymptotic behaviors, including arbitrarily large ejection-collision and oscillatory motions, in the PCRTBP with small mass ratios.

Findings

Existence of orbits with any combination of past and future final motions.

Construction of arbitrarily large ejection-collision orbits.

Establishment of oscillatory motions with unbounded position and velocity.

Abstract

In this paper we consider the planar circular restricted three body problem (PCRTBP), which models the motion of a massless body under the attraction of other two bodies, the primaries, which describe circular orbits around their common center of mass. In a suitable system of coordinates, this is a two degrees of freedom Hamiltonian system. The orbits of this system are either defined for all (future or past) time or eventually go to collision with one of the primaries. For orbits defined for all time, Chazy provided a classification of all possible asymptotic behaviors, usually called final motions. By considering a sufficiently small mass ratio between the primaries, we analyze the interplay between collision orbits and various final motions and construct several types of dynamics. In particular, we show that orbits corresponding to any combination of past and future final motions can be created to pass arbitrarily close to the massive primary. Additionally, we construct arbitrarily large ejection-collision orbits (orbits which experience collision in both past and future times) and periodic orbits that are arbitrarily large and get arbitrarily close to the massive primary. Furthermore, we also establish oscillatory motions in both position and velocity, meaning that as time tends to infinity, the superior limit of the position or velocity is infinity while the inferior limit remains a real number.