On a property of zero-velocity curves in N-body ring-type systems

Zero-velocity curves are a useful tool in the investigation of various aspects of a dynamical system. These curves that distinguish the regions where the motion of a particle is permissible from the regions where this motion is not permitted, present some basic properties. In this paper, we prove that in symmetric ring-type systems where a small particle moves under the resultant gravitational field of N coplanar big bodies, of which ν=N−1 are arranged at equal distances among them on the periphery of a circle, a new property concerning these curves, exists. All the zero-velocity curves drawn in the space of the initial conditions (x0,C) and concerning configurations with the same number of peripheral primaries but various mass parameters, pass through two different focal points, the position of which does not depend on the value of the mass parameter.