Phase structure of co-orbital motion with Jupiter

Abstract

In this paper, we investigate the dynamics of the inclined co-orbital motion with Jupiter through a torus phase structure in the Sun–Jupiter circular restricted three-body problem. A semi-analytical method to establish the Hamiltonian approximation for the inclined co-orbital motion is proposed. Phase structures of different kinds of co-orbital behaviours are shown in the torus space clearly. Based on numerical computation, we analyse the evolution and the connection of different co-orbital dynamics. Summarizing results and conclusions in this paper, we find two main principles throughout the investigation of the co-orbital motion: (i) the libration amplitude of the resonant angle for the co-orbital motion is bounded by the corresponding Hamiltonian isosurface in the torus space and (ii) the co-orbital behaviour is influenced by collision curves, and with the decrease of the Hamiltonian value, the influence is more significant.

Stability Analysis of Earth Co-orbital Objects

“…The quantities with the subscript "p" refer to the planet host. In the Sunplanet-asteroid CRTBP, the Hamiltonian for the 1:1 prograde MMR can be simplified to that in Qi & de Ruiter (2020a),…”

“…For the given i m , the co-orbital motion in the CRTBP can be approximately described in the (e, ω, j) phase space, where the range of e is from zero to e max , and ranges of ω and j are both from 0°to 360°.  ˜structures can be depicted in a torus space (Qi & de Ruiter 2020a). Figure 1 shows the schematic of the torus space.…”

“…As shown in Figure 1(a), the radius and the phase angle of the point P in the disk are measured by e and ω, respectively. Hence, the Cartesian state of the point P in the torus space can be expressed by (Qi & de Ruiter 2020a). Figure 2(a) shows the  ˜isosurface structure of the Sun-EMB CRTBP with i m = 20°in the torus space, where the colors denote different values of  ˜.…”

“…However, Qi & de Ruiter (2019) proposed a parameter to measure the minimum distance of the Hamiltonian isoline for the planar MMR from the collision curve, and numerical analysis showed that a large distance from the Earth collision curve would provide protection against close encounters with Earth and represent a more stable motion. Meanwhile, according to Qi & de Ruiter (2020a), Hamiltonian isosurfaces with smaller Hamiltonian values are closer to collision curves than those with larger Hamiltonian values, and their co-orbital motions are less stable and more chaotic. The above results inspire us to investigate the stability of ECOs through the distance between Hamiltonian isosurfaces and collision curves, which is the motivation of this paper.…”

“…Their research allowed for the estimation of the size and location of co-orbital stable regions from a few system parameters. In the Sun-Jupiter CRTBP, Qi & de Ruiter (2020a) proposed a semi-analytical approach and studied different co-orbital motions and their evolution through phase structures in the torus space. The semianalytical approach in the torus space has been successfully applied to investigate orbital dynamics of L 4 and L 5 axial orbits (Qi & de Ruiter 2020b) and small co-orbital bodies with Jupiter (Qi & de Ruiter 2021).…”