On rejection to infinity and exterior motion in the restricted problem of three bodies

Abstract

The cases of the restricted problem of three bodies which have hitherto been considered in detail deal with motion of the particle in the interior of a closed oval of zero velocity, or in the neighborhood of points of equilibrium or periodic orbits. In none of these cases does the question of rejection of the infinitesimal body to infinity play any r6le. It is to the investigation of this phenomenon, and of types of motion in which its possibility is an essential feature, that the present paper is devoted. Attention is restricted to the case of motion in a plane. Chapter I consists of a qualitative study of the individual half orbits which are described by the particle as it goes to or comes from infinity. The methods are altogether elementary. The first salient result is the proof of the existence of such half orbits, with a practical criterion for finding them. The second is the theorem that every half orbit which extends to infinity resembles a hyperbolic or parabolic orbit in this: that it has an asymptote (in the fixed space), and the particle has a limiting velocity. In a special case, resembling the parabolic, this velocity is zero and the asymptote is at infinity (projectively speaking). In Chapter II, we take up the investigation of infinity from the point of view of a singularity of the differential equations of motion. It is shown that by means of a rational substitution this singularity can be reduced to a well known type: a point, at which the right-hand members of a first-order system are analytic, but vanish simultaneously. The power expansions of these functions about the point start with terms of the third degree. When all further terms are omitted, a system is obtained whose solution is wholly known; unfortunately, no method is at hand for extending its properties to the given system. We finish the chapter with a discussion of the totality of half orbits which reach to infinity, regarded as an ensemble.