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

TL;DR

The authors analyze the restricted planar circular three-body problem in the regime of small mass ratio, focusing on second-species dynamics near the smaller primary. They combine Levi-Civita regularization near the Jupiter collision set and McGehee blow-ups near infinity and the Sun to construct two broad families of orbits: large ejection-collision trajectories and ballistic ECOs, and they prove the existence of a hyperbolic invariant set with dynamics conjugate to a full shift, allowing arbitrary past/future combinations of Chazy-type final motions. A key technical achievement is proving transversal intersections of invariant manifolds across near-collision, near-Sun, and infinity regions, enabling a global chaotic structure and the emergence of oscillatory, parabolic, and periodic orbits. The results advance understanding of long-term RPC3BP dynamics, revealing rich chaotic behavior and a mechanism for realizing any sequence of final motions in the Sun-Jupiter system within a rigorous geometric framework.

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.

Figures (3)

• Figure 2.1: Examples of the orbits analyzed in Remark \ref{['remark: parabolic EC orbits with J']}. In the left and right pictures we consider $\check \Theta_0^*=\sqrt2$ and $\check \Theta_0^* = 1$ respectively.
• Figure 7.1: Representation of the "spiraling effect" of the transition map $\hat{\mathcal{P}}$ on the curves $\hat{\gamma}^{-}_{\mathcal{S},\mathcal{J}}\subset \hat{\Sigma}_{\hat{r}_0}^>$ and their transverse intersections with the curves $\hat{\gamma}^{+}_{\mathcal{S},\mathcal{J}}\subset \hat{\Sigma}_{\hat{r}_0}^<$.
• Figure 8.1: Representation of the "ordering of the intersection" given by the angles $A$ and $B$ in \ref{['eqn: angles trasnversality']}. The region shaded in green represents the set $\mathcal{D}$ where the forward and backward return maps to the section $\Sigma_\nu^>$ are well defined. That is, the region where the construction made by Moser in moser2001stable is performed.

Theorems & Definitions (47)

• Theorem 1.1: Chazy, 1922, see also MR2269239
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Remark 2.4