Orbits and manifolds near the equilibrium points around a rotating asteroid

Abstract

Abstract. We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than four, and we find at least 1 family of periodic orbits near the resonant equilibrium points. Besides, this theory is 2 applied to asteroids 216 Kleopatra,

The dynamical environment of asteroid 21 Lutetia according to different internal models

“…As reported in many previous studies (Szebehely 1967;Scheeres 1994;Murray & Dermott 1999;Hu & Scheeres 2008;Yu & Baoyin 2012;Jiang et al 2014;Wang, Jiang, & Gong 2014), we can also examine the stability of the equilibria determined above. The linearized state equations in the neighborhood of the equilibrium points are summarized as…”

“…For more information, we recommend that interested readers review equation 14 in Jiang et al (2014). The linearization method is applied using the classical polyhedral model and the Mascon 8 approach with uniform density and Mascon 8 gravity model considering the two multiple layers structures.…”

“…The eigenvalues of the linearized system are listed in table 4. The classification of the equilibrium points defined in Jiang et al (2014); Wang, Jiang, & Gong (2014) shows that E1 and E2 belong to Case 2 (two pairs of imaginary eigenvalues and one pair of real eigenvalues). As a consequence, the saddle equilibrium points are unstable, whereas E3 and E4 belong to Case 1 (purely imaginary eigenvalues), that leads to a linear stability of center equilibrium points.…”

Stabilization of coupled orbit–attitude dynamics about an asteroid utilizing Hamiltonian structure

“…The precision of traditional spacecraft dynamics, in which orbital and attitude motions are modeled separately, will degenerate in the close proximity of small asteroids, due to the significant orbit-attitude coupling [1,2]. The orbital dynamics about asteroids, in which the spacecraft is considered as a point mass, has been studied broadly [3][4][5][6][7][8][9][10][11]. This point mass model can be improved by considering the orbit-attitude coupling, especially in the case of a large-sized spacecraft.…”

Periodic orbits in the Chermnykh problem

“…Certain periodic solutions associated with the local manifolds can be obtained from the linearized solutions around equilibrium points of the system [4]. However, obtaining large-scale periodic orbits is usually not possible.…”