Numerical investigation on the Hill's type lunar problem with homogeneous potential

TL;DR

This paper numerically analyzes the planar Hill's lunar problem with a homogeneous potential, classifying trajectory types, examining the no-return property at L2, and exploring basin boundary structures across different potential regimes.

Contribution

It provides a detailed numerical classification of trajectories and reveals fractal basin boundaries in the subcritical case, advancing understanding of the system's dynamical complexity.

Findings

Escaping trajectories are scattered exponentially.

In the supercritical case, basin boundaries are smooth.

In the subcritical case, basin boundaries exhibit fractal properties.

Abstract

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $\alpha \geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $\alpha < 2$ the boundaries between the different types of basins exhibit fractal properties.