Based on the concept of orbital stability introduced by G. W. Hill, a method is presented to facilitate the determination of the orbital stability of solutions to the planar elliptic restricted problem of three bodies. The invariant relation introduced by Szebehely and Giacaglia (1964) contains an integral which is expanded here about a Keplerian solution to the problem. If the expansion converges, it can be used to determine the conditions for Hill stability. With it one can also define stability in a periodic sense.

This is a generalization of the paper by Bhatnagar and Beena Gupta ‘Resonance in the restricted problem of three bodies with short-periodic perturbations’. The motion of an asteroid moving in the gravitational field of Jupiter is considered. In the original paper it was assumed that Jupiter is moving in a circular orbit around the Sun. In the present paper we consider the orbit to be elliptic. The series occurring in the problem are expanded in powers of a small parameter ε, which represents the ratio of the mass of Jupiter to that of the Sun. The perturbations in the osculating elements are obtained up to 0(ε).

The existence of periodic orbits of the first kind using Delaunay’s canonical variables is established through analytic continuation in the planar restricted three-body problem when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion.

The family of trajectories near a triple collision solution in the planar problem of three bodies is investigated by means of linearization in the neighborhood of the parabolic free fall. The local topological structure of this family is found to be that of a saddle point in the R8. The corresponding stable manifold is the set of all triple collision solutions, whereas the instable manifold is formed by the parabolic solutions.

In important cases of nonzero total energy the family of close encounters is quantitatively described in terms of hypergeometric functions. By means of homothetic transformations the close triple encounters are then related to three-body motion with zero energy and zero angular momentum. In this way almost all close encounters near a homothetic solution can be treated by using a small number of particular solutions of the three-body problem that may be calculated once for all.

In practical examples the first order theory presented here predicts the escape velocity after a close triple encounter with a relative accuracy comparable to the closeness of the encounter.

The notion of the velocity ellipsoid for the planar three body problem is given. Using the sufficient conditions for escape of one member of a triple system, given by Standish (1971), a region is found on the velocity ellipsoid for which escape is guaranteed.