Hyperbolic and Bi-hyperbolic solutions in the planar restricted $(N+1)$-body problem

TL;DR

This paper proves the existence of hyperbolic and bi-hyperbolic solutions in the planar restricted (N+1)-body problem with specific asymptotic directions and velocities, for any positive energy and collision-free primaries.

Contribution

It establishes the existence of hyperbolic and bi-hyperbolic solutions with prescribed asymptotic directions and velocities in the planar restricted (N+1)-body problem, extending previous results.

Findings

Existence of hyperbolic solutions with given asymptotic directions.

Existence of bi-hyperbolic solutions with prescribed asymptotic behavior.

Results hold for any positive energy and collision-free primaries.

Abstract

Consider the planar restricted $(N+1)$-body problem with trajectories of the $N(\ge 2)$ primaries forming a collision-free periodic solution of the $N$-body problem, for any positive energy $h$ and directions $\theta_{\pm} \in [0, 2\pi)$, we prove that starting from any initial position $x$ at any initial time $t_x$, there are hyperbolic solutions $\gamma^{\pm}|_{[t_x, \pm \infty)}$ satisfying $\gamma^{\pm}(t_x) =x$ and $$ \lim_{t \to \pm \infty} \gamma^{\pm}(t) / |\gamma^{\pm}(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}^{\pm}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$ Moreover we also prove the existence of a bi-hyperbolic solution $\gamma|_{\mathbb{R}}$ satisfying $$ \lim_{t \to \pm \infty} \gamma(t) / |\gamma(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$