Three-dimensional periodic motion in the vicinity of the equilibrium points of an asteroid

Abstract

Abstract. A way is described to find the initial conditions for the simplest three-dimensional periodic motions (3DPMs) in the vicinity of the equilibrium points (EPs) of an elongated asteroid whose shape is approximated by a triaxial ellipsoid (TE). A condition for the existence of these 3DPMs is formulated which may or may not be satisfied, depending on the spin period of the asteroid. A closed integral form of the gravitational potential of a TE is used. The general formulae for the computation of the initial conditions and some parameters of a 3DPM are derived from analysing the variational equations of motion. These initial conditions are used to integrate numerically the non-linear equations of motion in the vicinity of the EPs of a specific model of a triaxial ellipsoid based on the asteroid 243 Ida for a number of different rotation periods. We found that the period and some parameters of a 3DPM obtained by the numerical integration (with initial conditions predicted from the variational equations) have approximately the values computed from the variational equations. We show that the equatorial projections of all 3DPMs constructed in the current work are ellipses with sizes dependent on one free parameter. General formulae are presented for the computation of the ratio of the semi-axes of the ellipse and the period of motion in it. A special study is made for a boundary case where the EPs lying along the greatest ellipsoid axis (strongly unstable) are touching the asteroid surface. This critical value of the spin period of the asteroid (at which the loss of loose regoliths or destruction of the asteroid can start) is shown to be the same for the asteroids with similar densities and axial ratios. Some comments are given on the condition under which a 3-D periodic motion can exist in the vicinity of the EPs of the Jacobi ellipsoid.

Dynamics of Equilibrium Points in a Uniformly Rotating Second-Order and Degree Gravitational Field

“…This work is devoted to the dynamics around EPs in a rotating 2ODgravitational field, with asteroids as the research targets. Compared with previous studies (Scheeres 1994;Vasilkova 2005;Jiang et al 2014), which mainly focus on the linear stability of the EPs and the motions in their vicinity, theglobal dynamics around theEPs are studied in this work by computing periodic orbits (POs) and the invariant manifolds associated with them, with special focus on the genealogy and stability of the periodic families. Since theorbits around the EPs are also resonance orbits, which are in 1:1 resonance with the asteroid's rotation in the inertial framerather than theEPs themselves, this work can be treated as a study ofthe 1:1 resonance orbitsbut carried out in a rotating frame with the tools of dynamical systems.…”

Detection by MEGNO of the gravitational resonances between a rotating ellipsoid and a point mass satellite

“…Periodic orbits for a few resonances are considered in the first paper. Conditions for existence of 3-D periodic motions near the equilibrium points of an elongated triaxial ellipsoid are studied (with application to an Ida-like ellipsoid) in Vasilkova (2005). In Scheeres (2002aScheeres ( ,b, 2006, an analytical study of binary asteroids with equivalent masses is based on the full two-body problem.…”

Coplanar libration points of the generalized restricted circular problem of three bodies for conjugate complex masses of attracting centers

“…В настоящей статье изучается существование и проводится классификация равновесий второго типа, названных ранее компланарными точками либрации (КТЛ) [13] и являющихся аналогами эйлеровых точек либрации классической ООКЗ3Т [2,4] (о точках либрации гравитирующих тел см. также [5][6][7]), в случае комплексно-сопряженных масс притягивающих центров, «лежащих» на оси динамической симметрии и имеющих на этой оси чисто мнимые координаты. (Сумма потенциалов таких центров, очевидно, будет действительной, но обладающей целым отрезком особых точек.)…”