Triangular libration points and their stability in the restricted, circular, plane three-body problem

The effect of the non-sphericity of a body of specific form on the type of relative equilibrium of a passively gravitating body and on the conditions for their stability in a restricted, circular plane three-body problem is studied (it is assumed that the mass of one of the bodies is negligibly small compared with the mass of the other two bodies and has no effect on their motion). Rectilinear libration points of a passively gravitating point in the field of attraction of a sphere with a spherical distribution of masses and a homogeneous rod, were studied in /1/, where it was assumed that the centres of the rod and the sphere were at a constant distance from each other, that the sphere and the rod rotated with constant angular velocity about the common centre of mass and that the rod was collinear with the radius vector. In this paper we investigate the triangular libration points of a passively gravitating point in the field of attraction of a sphere and a rod, under the same assumptions concerning the motions of the sphere and rod. It is shown that taking into account the size of the rod leads to displacement of the classical triangular libration points corresponding to a rod of zero length, towards the rod in both directions (i.e. towards the straight line connecting the centres of mass of the sphere and rod, as well as towards the straight line orthogonal to it and passing through the centre of the rod).