Numerical Investigation of Periodic Motion in the Three-Dimensional Ring Problem of N Bodies

Abstract

We study numerically the three-dimensional periodic motion of a massless particle in an annular configuration of N bodies. This ring model can approximate the dynamics of celestial systems like clusters and planetary rings. After calculating numerically the families of x-symmetric periodic orbits in the planar case for a certain N, we investigate the families' vertical stability and locate those members that present critical instability in the eigenvalues corresponding to the z-axis. These orbits are the points of bifurcation of the planar families of periodic orbits to three-dimensional families. By means of differentially correcting the vertically critical orbits we extend our numerical investigation in the three-dimensional space and locate several families of three-dimensional periodic orbits symmetric to the xz-plane. The evolution of the families and the significant features of their members are subsequently presented.

The regular polygon problem of (N+2) bodies: a numerical investigation of the equilibrium states of the minor bodies

“…The regular polygon problem of (N + 2) bodies we deal with is a combination of two theoretical models that have independently been presented in the past; the 2 + 2 body problem proposed by Whipple (1984) and the regular polygon problem of (N + 1) bodies (Scheeres 1992;Scheeres and Vinh 1993;Kalvouridis 1998Kalvouridis , 2001Kalvouridis , 2008Hadjifotinou and Kalvouridis 2005;Pinotsis 2005; Barrio et al M.N. Croustalloudi · T.J. Kalvouridis (B) National Technical University of Athens, Athens, Greece e-mail: tkalvouridis@gmail.com M.N.…”

Symmetric planar central configurations of five bodies: Euler plus two

“…The ring problem studies the motion of (n + 1)-bodies where n bodies of equal masses are located at the vertices of a regular polygon centered at the remaining body, thus forming a central configuration. It was proposed by Maxwell in [17] as a model for the motion of the particles surrounding Saturn, and used more recently to model systems like planetary rings, asteroid belts, planets around a star, certain stellar formations, stars with accretion ring, planetary nebula, motion of an artificial satellite about a ring, (see [23,25,24,19,20,12,13,3,9,4,5]). We remark that the ring problem with four equal masses on the ring and a fifth mass at the center of the ring considered in [23] coincides with the special case found in Theorem 1.1 (a); such a configuration has been found to be locally unstable.…”

Homographic solutions in the planar n + 1 body problem with quasi-homogeneous potentials

“…Albeit the problem may be considered as a classical one, it attracted the interest of researchers (Scheeres 1992;Stavinschi 1998, 1999;Kalvouridis 1999Kalvouridis , 2001Hadjifotinou and Kalvouridis 2005;Arribas and Elipe 2004) in the last years because of the possibility of considering this kind of configuration to model some dynamical systems like planetary rings, asteroid belts, planets around a star, certain stellar formations, stars with accretion ring, planetary nebula, motion of an artificial satellite about a ring to mention a few.…”