Bifurcations and equilibria in the extended N -body ring problem M. Arribas a,b , A. Elipe a, * a Grupo de Mec anica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain b Dpto. Matem atica Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain Received 24 December 2002 Abstract We consider the motion of an infinitesimal particle under the gravitational field of (n þ 1) bodies in ring configu- ration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m 0 ¼ bm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter b). This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the to- pology of the problem is modified. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Bifurcations; Ring configuration; N -body problem 1. Introduction In this paper we consider the N -body ring problem, i.e., we analyze the motion of an infinitesimal satellite attracted by the gravitational field of (n þ 1) primary bodies. These bodies are arranged in a planar ring configuration (Kalvouridis, 1999a; Scheeres, 1992; Scheeres and Vinh, 1993), that consists of n pri- maries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity w. Another primary of mass m 0 ¼ bm with the parameter (b P 0) is placed at the center of the ring. This kind of configuration is considered as a model of observed phenomenon as planetary rings, some stellar formations, asteroids, the motion of small particles close to planetary rings, or in a proto-nebula, etc. The problem may be considered a classic one, although only recently it has attracted the interest of researchers; thus, we find that in 1856 Maxwell (1952) studied the stability of a discrete particle ring to simulate SaturnÕs ring; Tisserand in 1889 reformulated MaxwellÕs analysis and presented a relation between the mass of each ring particle and the number of them in order that the system be linearly stable. Recently, * Corresponding author. Tel.: +34-976-76-1138; fax: +34-976-76-1140. E-mail address: elipe@posta.unizars.es (A. Elipe). 0093-6413/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0093-6413(03)00086-7 Mechanics Research Communications 31 (2004) 1–8 www.elsevier.com/locate/mechrescom MECHANICS RESEARCH COMMUNICATIONS