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

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