Periodic orbits in the gravity field of a fixed homogeneous cube

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.Comment: 23 pages, 10 figures, accepted for publication in Astrophysics & Space Scienc

The Gravity Field of a Cube

“…These results were found to have application in the calculation of the gravitational anomalies on the Earth (Mufti, 2006b) and the slowdown of the Earth's rotation rate due to tidal drag (Celnikier, 1990). Further investigations explored the nature of the satellite orbits that would form around a Science Publications PI static cube (Liu et al, 2011a;Werner, 1994), as well as for rotating cubes (Liu et al, 2011b) using simulations and the method of Poincare sections (Liu et al, 2011a;Scheeres et al, 2000). These preliminary investigations of orbits around cubic objects are of significance as they can be considered the first step towards an analysis of orbits around more general shaped bodies.…”

“…A cube has three symmetry planes parallel to its faces, six symmetry planes in the diagonal plane and four asymmetry planes that contain the regular hexagonal cross section. It has been shown (Liu et al, 2011a), that periodic orbits can form in all three cases, although only planar-type orbits form when orbiting around the axis of symmetry, that is parallel to the faces or diagonally across the corners.…”

“…These preliminary investigations of orbits around cubic objects are of significance as they can be considered the first step towards an analysis of orbits around more general shaped bodies. These investigations, for example, have application to future missions to investigate asteroids, where orbits need to be calculated around rotating non-spherical objects (Liu et al, 2011a;Werner and Scheeres, 1996;Michalodimitrakis and Bozis, 1985;Liu et al, 2011b;Scheeres et al, 1998;Yu and Baoyin, 2012a;2012b;2012c).…”

A survey on orbital dynamics and navigation of asteroid missions

“…Scheeres [64] analyzed the dynamics in the vicinity of tri-axial rotating ellipsoids, categorized tri-axial ellipsoids according to the stability of the equilibrium points, and computed the planar periodic orbits of asteroids Vesta and Eros. Liu et al [65] investigated the gravitational field of a homogeneous cube to compute its nearby periodic orbits.…”

Poincaré surfaces of section around a 3D irregular body: the case of asteroid 4179 Toutatis

“…The gravitational potential generated by this segment is two-dimensional, since the nonhomogenity of the segment density is a function that makes the gravitational potential symmetric. Liu et al (2011) used the polyhedron model to compute the gravitational potential of a three-dimensional cube, thus, enabling the mapping by the Poincaré surface of section at the cube vicinity and periodic orbits around the cube were found. All the mentioned works treat two-dimensional or three-dimensional cases.…”