Dynamical aspects of an equilateral restricted four-body problem.(Research Article)(Report)

1. Introduction

Dynamical systems with few bodies (three) have been extensively studied in the past, and various models have been proposed for research aiming to approximate the behavior of real celestial systems. There are many reasons for studying the four-body problem besides the historical ones, since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems. Around one-fifth of these is a part of triple systems, while a rough estimate suggests that a further one-fifth of these triples belongs to quadruple or higher systems, which can be modeled by the four-body problem. Among these models, the configuration used by Maranhao [1] and Maranhao and Llibre [2], where three point masses form at any time a collinear central configuration (Euler configuration, see [3]),is of particular interest not only for its simplicity but mainly because in the last 10 years, an increasing number of extrasolar systems have been detected, most of them consisting of a "sun" and a planet or of a "sun" and two planets.

We study the motion of a mass point of negligible mass under the Newtonian gravitational attraction of three mass points of masses [m.sub.1], [m.sub.2], and [m.sub.3] (called primaries) moving in circular periodic orbits around their center of mass fixed at the origin of the coordinate system. At any instant of time, the primaries form an equilateral equilibrium configuration of the three-body problem which is a particular solution of the three-body problem given by Lagrange (see [4] or [3]). Two of these primaries have equal masses and are located symmetrically with respect to the third primary.

We choose the unity of mass in such a way that [m.sub.1] = 1 - 2y and [m.sub.2] = [m.sub.3] = y are the masses of the primaries, where [mu] [member of] (0,1/2). Units of length and time are chosen in such a way that the distance between the primaries is one.

For studying the position of the infinitesimal mass, [m.sub.4], in the plane of motion of the primaries, we use either the sideral system of coordinates, or the synodical system of coordinates (see [5] for details). In the synodical coordinates, the three point masses [m.sub.1], [m.sub.2], and [m.sub.3] are fixed at ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) respectively. In this paper, the equilateral restricted four-body problem (shortly, ERFBP) consists in describing the motion of the infinitesimal mass, [m.sub.4], under the gravitational attraction of the three primaries [m.sub.1], [m.sub.2], and [m.sub.3]. Maranhao's PhD thesis [1] and the paper [2] by Maranhao and Llibre studied a restricted four body problem, where three primaries rotating in a fixed circular orbit define a collinear central configuration.

In the ERFBP, the equations of motion of [m.sub.4] in synodical coordinates (x, y, z) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.2)

We remark that the ERFBP becomes the central force problem when [mu] = 0, and [m.sub.1] = 1 is situated in the origin of the system, while [mu] = 1/2 results in the restricted three-body problem with the bodies [m.sub.2] and [m.sub.3] of mass 1/2.

Mathematical Problems in Engineering

Our paper is organized as follows: Section 2 is devoted to describing the most important dynamical phenomena that governs the evolution of asteroid movement and states the problem under consideration in the present study. In Section 3 reductions of the problem are discussed and a comprehensive treatment of streamline analogies is given. Section 4 is devoted to the principal qualitative aspect of the restricted problem--the surfaces and curves of zero velocity, several uses of which are discussed. The regions of allowed motion and the location and properties of the equilibrium points are established. We describe the Hill region. The description of the number of equilibrium points is given in Section 5, and in the symmetrical case (i.e., [mu] = 1/3), we describe the kind of stability of each equilibrium. In Section 6, the planar case is considered. There, we prove the existence of periodic solutions as a continuation of periodic Keplerian orbits, and also when the parameter [mu] is small and when it is close to 1/2. Finally, in Section 8 we present the conclusions of the present work.

Next, we will enunciate some four-body problem that has been considered in the literature. Cronin et al. in [6, 7] considered the models of four bodies where two massive bodies move in circular orbits about their center of mass or barycenter. In addition, this barycenter moves in a circular orbit about the center of mass of a system consisting of these two bodies and a third massive body. It is assumed that this third body lies in the same plane as the orbits of the first two bodies. The authors studied the motion of a fourth body of small mass which moves under the combined attractions of these three massive bodies. This model is called bicircular four-body problem. Considering this restricted four-body problem consisting of Earth, Moon, Sun, and a massless particle, this problem can be used as a model for the motion of a space vehicle in the Sun-Earth-Moon system. Several other authors have considered the study of this problem, for example, [8-11] and references therein. The quasibicircular problem is a restricted four body problem where three masses, …