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Astron. Astrophys. 331, 1108-1112 (1998)

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1. Introduction

A commonly adopted model to study the dynamical evolution of small bodies in the solar system is the planar, circular, Restricted Three-Body Problem (RTBP) (see for instance Szebehely, 1967). In the case of asteroidal motion, the Sun and Jupiter are assumed to be in circular orbits around the center of mass of the system, and an asteroid with negligible mass moves under the gravitational influence of these two main bodies.

Several studies on the main asteroid belt have been made using this model. In general, these studies consisted essentially in numerical integrations of this problem aiming at:

- the search of periodic orbits and the families associated to them as well as the study of their stability with the purpose of characterizing the system orbits through periodic orbits (see for instance Szebehely, 1967, chapters 8 and 9);

- computing the variation of the orbital elements, with the purpose of understanding the evolution of such elements (see for instance Winter and Murray, 1997a);

- obtaining the intersection of the orbits with a transversal surface (Poincaré surface of section method), with the purpose of determining the nature of the orbits, regular or chaotic, and their extent on the phase space (see for instance Winter and Murray, 1994a and 1994b).

However, it is rare to find a method being used to describe in a complete and clear way the orbits in the configuration space, which is the physical space where the orbits actually evolve. In principle, this description could be made in a simple and direct way by just plotting the quasi-periodic orbit in the configuration space. This process is naive and does not work, because even for just one trajectory its path becomes very quickly confused, hiding all the details of the trajectory.

In this work we present a description of some RTBP quasi-periodic orbits. This presentation is made in the configuration space using the caustics of the problem. The caustics are envelopes of the quasi-periodic trajectories, or the contours of the torus that contains each one of these orbits. The construction of the caustics is based on a technique recently developed by Stuchi and Vieira Martins (1995, 1996). The method is built from simple properties of the variational equations solutions, which are fulfilled in the case of not degenerated integrable Hamiltonian systems and, in principle, can be used for almost all quasi-periodic orbits of Hamiltonian systems.

As it will be shown, a relatively small number of caustics characterize the various types of orbits of the RTBP and the shape of these caustics are closely related to the existing resonances.

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998