Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

Abstract

Abstract. We theoretically investigate the existence of families of periodic orbits in the planar N-body ring problem and we give a qualitative picture of the motion of a small particle. This study yields four families of periodic orbits which we also found numerically: two families of periodic orbits around the central body and two families around all the peripherals. These results are valid for different values of N and β. Also we investigate the evolution of simple periodic motions as well as their stability. We found stable and unstable orbits around the central body, around all the peripherals and around one or more peripherals which form rings of stability. Some families present other types of bifurcations, such as bifurcations of families of non-symmetric periodic orbits of the same period and period-doubling bifurcations.

Periodic motions of a small body in the Newtonian field of a regular polygonal configuration of ν+1 bodies

“…However, as ν increases the variety of different kind of orbits increases too and the evolution of the families and particularly those that belong to classes 1 and 4, changes with ν. These differences become more obvious between configurations characterized by an odd number of peripheral primaries (for example ν = 7 which is the case studied by Pinotsis 2005) or an even one (for example ν = 8 which is our case). At this point we would like to remind that the regular polygon configuration has ν axes of symmetry that form 2π ν angles between them.…”

“…Stavinschi (1998, 1999) used post-Newtonian fields such as Manev-type or Schwarzshild-type ones and recently, Arribas et al (2006) extended this study to any field which depends on the inverse of the mutual distances of the primary bodies. Maxwell's model has been used in the last fifteen years as a force field generator by many investigators who mainly intended to study the motion of a small body in the combined force field created by this formation (Goudas 1991;Scheeres 1992;Grebenikov 1997;Gadomski 1998;Kalvouridis 1999aKalvouridis , 1999bKalvouridis , 1999cKalvouridis , 2001aKalvouridis , 2001bKalvouridis , 2003Kalvouridis , 2004Kalvouridis and Tsogas 2002;Elipe 2003, 2004;Kazazakis and Kalvouridis 2004;Bang and Elmabsout 2004;Psarros and Kalvouridis 2005;Pinotsis 2005;Hadjifotinou and Kalvouridis 2005;Croustalloudi 2006;Hadjifotinou et al 2006;Elipe et al 2007;Croustalloudi and Kalvouridis 2007;etc.). In the gravitational version of this problem, the system is characterized by two parameters: the mass parameter, which is the ratio of the central mass to a peripheral one (β = m 0 /m) and the number ν of the peripheral primaries.…”

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.…”

Linear stability of ring systems with generalized central forces

“…Another body of mass m 0 is placed at the center of the n-gon. More recently, the problem has been revisited and extended in several aspects, like the computing of periodic orbits of an infinitesimal mass attracted by the ring under Newtonian forces (Scheeres 1992;Kalvouridis 1999Kalvouridis , 2001Kalvouridis , 2003Pinotsis 2005), or considering more general forces (Arribas & Elipe 2004;Elipe et al 2007). Arribas et al (2007) have proved that the ring configuration was central for quasihomogeneous potentials.…”