Periodic orbits in the gravity field of a fixed homogeneous cube

TL;DR

This study investigates the existence and stability of periodic orbits around a fixed homogeneous cube, demonstrating the effectiveness of the homotopy method and revealing extensive orbit presence in symmetry planes, with implications for non-spherical celestial bodies.

Contribution

It introduces the homotopy method for finding periodic orbits around a cube, extending orbit dynamics understanding to complex-shaped celestial bodies.

Findings

Periodic orbits are prevalent in symmetry planes of the cube.

The homotopy method effectively finds periodic orbits in the cube's gravity field.

Stable orbits are identified through eigenvalue analysis of the monodromy matrix.

Abstract

In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincar\'e surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.