Astrophys Space Sci (2009) 323: 261–272 DOI 10.1007/s10509-009-0063-1 ORIGINAL ARTICLE Asymptotic orbits in the (N + 1)-body ring problem K.E. Papadakis Received: 22 April 2009 / Accepted: 19 June 2009 / Published online: 1 July 2009 © Springer Science+Business Media B.V. 2009 Abstract In this paper we study the asymptotic solutions of the (N + 1)-body ring planar problem, N of which are finite and ν = N − 1 are moving in circular orbits around their center of masses, while the N th + 1 body is infinitesimal. ν of the primaries have equal masses m and the N th most- massive primary, with m 0 = βm, is located at the origin of the system. We found the invariant unstable and stable man- ifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L 1 and L 2 . We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymp- totic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x - axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β = 2, are found and presented. The eighteen (more geo- metrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is com- puted and also presented. Keywords (N + 1)-body ring problem · Asymptotic orbits · Homoclinic orbits · Heteroclinic orbits K.E. Papadakis ( ) Department of Engineering Sciences, Division of Applied Mathematics and Mechanics, University of Patras, 26504 Patras, Greece e-mail: k.papadakis@des.upatras.gr 1 Introduction and equations of motion Poincaré (1957), while studying the so-called Poincaré map associated with an unstable periodic orbit, defined a ho- moclinic point as a point whose orbit is asymptotic to the hyperbolic fixed point in both directions. Also, Strömgren in 1935 calculated heteroclinic asymptotic orbits connect- ing the two triangular critical points in the classical re- stricted three-body problem. After that, many papers have been written about stable and unstable invariant manifolds and homoclinic, heteroclinic, asymptotic orbits associated with an equilibrium point or with a Lyapunov periodic or- bit (among others Deprit and Henrard 1965; Conley 1968; McGehee 1969; Llibre and Simó 1980; Llibre et al. 1985; Gómez et al. 1988). The interest in these concepts has been revived recently due to the fact that stable and unstable manifold tubes associated with bounded orbits around the collinear libration points L 1 and L 2 , provide a framework for understanding dynamical phenomena such as the con- duction particles to and from the smaller primary body and between primaries for separate three-body systems (for de- tails see Koon et al. 1999, 2001b, 2001a; Gómez et al. 2004). Our goal in this paper is to study the asymptotic solutions around periodic orbits and around the equilibrium points of the planar (N + 1)-body ring problem. We will calculate the invariant stable and unstable manifolds and the correspond- ing homoclinic asymptotic orbits around the Lyapunov pe- riodic solutions as well as the homo- and heteroclinic orbits associated with the equilibrium configurations of this prob- lem. The planar (N + 1)-body ring problem is a two degrees of freedom problem and describes the motion of an infini- tesimal particle attracted by the gravitational field of ν + 1 primary bodies (N = ν + 1). We consider a central primary body of mass m 0 = βm which is located at the center of