Stationary motions in a restricted problem of three rigid bodies

Abstract

The present paper is a continuation of papers by Shinkaric (1972), Vidyakin (1976), Vidyakin (1977), and Duboshin (1978), in which the existence of particular solutions, analogues to the classic solutions of Lagrange and Euler in the circular restricted problem of three points were proved. These solutions are stationary motions in which the centres of mass of the bodies of the definite structures always form either an equilateral triangle (Lagrangian solutions) or always remain on a straight line (Eulerian solutions) The orientation of the bodies depends on the structure of the bodies.

In this paper the usage of the small-parameter method proved that in the general case the centre of mass of an axisymmetric body of infinitesimal mass does not belong to the orbital plane of the attracting bodies and is not situated in the libration points, corresponding to the classical case. Its deviation from them is proportional to the small parameter. The body turns uniformly around the axis of symmetry. In this paper a new type of stationary motion is found, in which the axis of symmetry makes an angle, proportional to the small parameter, with the plane created by the radius-vector and by the normal to the orbital plane of the attracting bodies. The earlier solutions-Shinkaric (1971) and Vidyakin (1976)-are also elaborated, and stability of the stationary motions is discussed.