On evolution of libration points similar to eulerian in the model problem of the binary-asteroids dynamics

The binary asteroids are of current interest in the modern dynamics as there have been up to 50 discoveries of binaries. Estimates are that about 20% of near-Earth asteroids may be binary asteroids. Nevertheless the known asteroids pairs are rather rare objects in the Solar System. There are a number of papers studying the various aspects of asteroid pair dynamics. In this paper we study some stationary motions in the system of binary asteroid. Using the model for the first time suggested in [1], we approximate the bigger asteroid by the dumbbell-shaped rigid body. Moreover we assume that the smaller asteroid has mass close to zero. It was shown in [1] that the motion equations for the considered system have the stationary solutions corresponding to the smaller asteroid's equilibria relative to the axis of the regular precession and the dumbbell rod These equilibria similar to the libration points in the Restricted Circular Problem of Three Bodies (RCP3B). There are two types of such equilibria. The equal distances from the dumbbell endpoints characterize the equilibria of the first type, therefore we shall name them ‘the libration points similar to Lagrangian’ or ‘the triangular libration points’ (TLPs) by analogy to a classical problem. At difference with RCP3B there are two or one TLPs or they do not exist. Equilibria of the second type in something are similar to the Eulerian libration points. They belong to the plane containing the bigger asteroid's angular momentum and the dumbbell rod. Moreover, these equilibria belong to the strip bounded by straight lines crossing the dumbbell endpoints and being perpendicular to angular momentum. Therefore we shall name them ‘the coplanar libration points’(CLP). The CLPs coordinates are computed by following procedure. Two algebraic equations are deduced. One of these equations determines the curve containing all CLPs. We say that this curve is ‘the Geometrical locus of CLPs’ (GL). The second equation allows to locate CLPs in GL. Studying evolution of CLPs in GL it can be proved that the number of CLPs varies from 3 up to 7, but if the dumbbell consist of equal spheres then only the odd number of CLPs is possible. However, if the dumbbell is asymmetric then the number of CLPs can be equal 4 or 6 for some rare situations