Orbit classification in the Hill problem: I. The classical case

TL;DR

This study systematically classifies initial conditions of orbits in the classical Hill problem into various categories, analyzing both 2D and 3D cases to understand the complex orbital dynamics and basin structures.

Contribution

It provides a comprehensive numerical classification of orbit types in the Hill problem for both planar and spatial cases, including basin and escape analysis.

Findings

Identification of bounded, escape, and collision basins

Relation of basins to escape and collision times

Complex orbital dynamics revealed by numerical integration

Abstract

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion the Smaller ALingment Index (SALI) method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them with the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.