Fractal structures for the Jacobi Hamiltonian of restricted three-body problem

TL;DR

This paper investigates the fractal structures and chaotic dynamics in the planar circular restricted three-body problem, revealing how fractal dimensions relate to system parameters and how particle survival probabilities decay over time.

Contribution

It introduces a detailed analysis of fractal dimensions of non-escaping orbits and links the spiral fractal structure to particle distribution and decay behaviors in the system.

Findings

Fractal dimension varies with mass ratio and Jacobi integral.

Spiral fractal structures lead to spiral density distributions.

Survival probability exhibits exponential then algebraic decay.

Abstract

We study the dynamical chaos and integrable motion in the planar circular restricted three-body problem and determine the fractal dimension of the spiral strange repeller set of non-escaping orbits at different values of mass ratio of binary bodies and of Jacobi integral of motion. We find that the spiral fractal structure of the Poincar\'e section leads to a spiral density distribution of particles remaining in the system. We also show that the initial exponential drop of survival probability with time is followed by the algebraic decay related to the universal algebraic statistics of Poincar\'e recurrences in generic symplectic maps.