The Restricted Three-Body Problem

Abstract

The determination of the motion of N point-like masses under their mutual gravitational forces is the basic problem in celestial mechanics, with important applications to fundamental astrophysical problems, like the dynamical structure of planetary and satellite systems and the evolution of multiple stellar systems, ranging from multiple stars to stellar clusters and galaxies. However, even for N as small as 3 it is impossible to find general solutions of this problem in terms of simple analytical functions (like in the case of the two-body problem). A great variety of orbits is possible and one can show that only very particular choices of the initial conditions give rise to a periodic behaviour. Even the perturbative techniques worked out in Ch. 11 can be applied only in the particular case of hierarchical systems (see Sec. 15.1). As a consequence, the gravitational N-body problem has been studied by two other methods which can yield only partial results, but are in some way complementary. The first method is the numerical integration of the orbits, starting from a given set of initial conditions. In this way the motion can be determined in detail, but only for a limited span of time (due to limitations in computer time and accumulating numerical errors; see Sec. 15.1); and often it is not possible to generalize some property of the chosen orbits to significant regions of the phase space.