Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

Hadjifotinou, K. G.; Kalvouridis, T. J.; Gousidou-Koutita, M.

doi:10.1016/j.mechrescom.2006.03.002

Cited by 6 publications

(2 citation statements)

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“…peripheral primaries and we have found a large number of periodic orbits of various multiplicity (simple and multiple) and types; planar and three-dimensional, planetary, satellite, interplanetary, simple and multiple symmetric, direct and retrograde etc. (see [25]- [27], [31], [35]- [37], [42]- [43], [47]). In Figures 4 and 5 we present a few samples of them.…”

Section: Planar and Three Dimensional Periodic Orbitsmentioning

confidence: 99%

The ring problem of (N+1) bodies: ten years of research (1999–2009)

Kalvouridis

2009

SPIE Proceedings

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One of the most popular N-body models that has been under investigation in the last ten years, is the so-called ring problem of (N+1) bodies, otherwise known as the regular polygon problem of (N+1) bodies. Here we present an overview of the scientific work that has been done throughout these years as well as the major results obtained so far. An extended bibliography is displayed at the end of this article, aiming to become a source of information for further analytical study. 1． INTRODUCTIONThe ring problem of (N+1) bodies describes the motion of a small body S which moves in the force field created by N major bodies called the primaries. The ν=N-1 of them, have equal masses and are located at the vertices of an imaginary ν-gon. The N-th body has a different mass and is located at the center of mass of this formation (Figure 1). Figure 1. The general configuration of the ring problem of (N+1) bodiesThe problem is characterized by two parameters: the number ν of the peripheral primaries and the mass parameter β which is the ratio of the central mass m 0 to a peripheral one named m. The regular polygon formation of the big bodies with another one at their center of mass was proposed by Maxwell in 1865 [1] in order to explain the rings of Saturn. Since then, this configuration has often been found to be the center of special scientific interest and many papers have been written aiming to prove its central character and to find homographic solutions, relative equilibria, conditions for stability etc. not only for the simple gravitational case but also for cases that include post-Newtonian potentials ([2]-[21]). However, the ring problem as a new dynamical system under the description made at the beginning of this paragraph, first appeared in Scheeres' PhD Thesis ([22]) and a little later in a joint paper with Vinh ([23]). The problem was considered anew by the author of the present articles and his collaborators, who devoted ten years (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009) of scientific research in investigating many of its aspects ([24] -[48]). He reformulated the whole problem and he improved the original model by considering not only gravitational forces but also forces coming either from radiation or from post-Newtonian potentials. He also examined some cases where the small body is a tri-axial body or a gyrostat. The configuration is quite simple and, apart from other benefits, can be reduced to some already known problems-models of

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Section: Planar and Three Dimensional Periodic Orbitsmentioning

confidence: 99%

The ring problem of (N+1) bodies: ten years of research (1999–2009)

Kalvouridis

2009

SPIE Proceedings

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Croustalloudi

Kalvouridis

2007

Astrophys Space Sci

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We study the simple periodic orbits of a particle that is subject to the gravitational action of the much bigger primary bodies which form a regular polygonal configuration of (ν + 1) bodies when ν = 8. We investigate the distribution of the characteristic curves of the families and their evolution in the phase space of the initial conditions, we describe various types of simple periodic orbits and we study their linear stability. Plots and tables illustrate the obtained material and reveal many interesting aspects regarding particle dynamics in such a multi-body system.

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The Ring Problem of (N+1) Bodies: An Overview

Kalvouridis

2011

Dynamical Systems and Methods

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Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

Cited by 6 publications

References 5 publications

The ring problem of (N+1) bodies: ten years of research (1999–2009)

The ring problem of (N+1) bodies: ten years of research (1999–2009)

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

The Ring Problem of (N+1) Bodies: An Overview

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