Synchronous motion in the Kinoshita problem : Application to satellites and binary asteroids

A Lie-Poisson integrator with Wisdom-Holman type splitting is constructed for the problem of a rigid body and a sphere (the Kinoshita problem). The algorithm propagates not only the position, momentum and angular momentum vector of the system, but also the tangent vector of "infinitesimal displacements". The latter allow to evaluate the maximum Lyapunov exponent or the MEGNO indicator of Cincotta and Simo. Three exemplary cases are studied: the motion of Hyperion, a fictitious binary asteroid with Hyperion as one of the components, and the binary asteroid 90 Antiope. In all cases the attitude instability of the rotation state with spin vector normal to an equatorial orbit influences stability of the system at lower rotation rates. The MEGNO maps with variations restricted to the orbital plane for position and momentum, and to the orbit normal direction for the angular momentum resemble usual Poincare sections. But if no restriction is imposed on the variations, some stable zones turn into highly chaotic regions, often retaining the shape of their boundaries.