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Advances in Astronomy

Volume 2015, Article ID 649352, 11 pages

http://dx.doi.org/10.1155/2015/649352

## Regions of Central Configurations in a Symmetric 4 + 1-Body Problem

Department of Mathematics, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia

Received 30 September 2014; Revised 12 January 2015; Accepted 28 January 2015

Academic Editor: Dean Hines

Copyright © 2015 Muhammad Shoaib. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The inverse problem of central configuration of the trapezoidal 5-body problems is investigated. In this 5-body setup, one of the masses is chosen to be stationary at the center of mass of the system and four-point masses are placed on the vertices of an isosceles trapezoid with two equal masses at positions and at positions . The regions of central configurations where it is possible to choose positive masses are derived both analytically and numerically. It is also shown that in the complement of these regions no central configurations are possible.

#### 1. Introduction

To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of a central configuration. For a system to be in central configuration, the acceleration of the th mass must be proportional to its position (relative to the center of mass of the system); thus, for all . A central configuration can also be expressed as a critical point for the function , where is the moment of inertia. Central configuration is one of the most important and fundamental topics in the study of the few-body problem. Therefore, few-body problem in general and central configurations in particular has attracted a lot of attention over the years (see, e.g., Albouy and Llibre [1] and Shoaib and Faye [2]). The straight line solutions of the -body problem were first published by Moulton [3]. Moulton arranged masses on a straight line so that they always remained collinear and then solved the problem of the values of the masses at arbitrary collinear points. Palmore [4, 5] presented several theorems on the classification of equilibrium points in the planar -body problem.

To overcome the complexities of higher dimensions of the general -body problem, various restriction methods have been used. The two most common restriction methods used are to neglect the mass of one of the bodies and introduce some kind of symmetries. Papadakis and Kanavos [6] studied the photogravitational restricted five-body problem. They study the motion of a massless particle under the gravitational attraction of four equidistant particles on a circle. More recently, Kulesza et al. [7] studied a restricted rhomboidal five-body problem. They arrange the primaries on the vertices of a rhombus and the fifth massless particle in the same plane as the primaries. Ollongren [8] studied a restricted five-body problem having three bodies of equal mass placed on the vertices of the equilateral triangle; they revolve in the plane of the triangle around their gravitational center in circular orbits under the influence of their mutual gravitational attraction; at the center a mass of is present where . A fifth body of negligible mass compared to moves in the plane under the gravitational attraction of the other bodies. Other noteworthy studies on the restricted five-body problem include Kalvouridis [9] and Markellos et al. [10].

Another method of restriction used to study the five-body problem is the introduction of some kind of symmetries. For example, Roberts [11] discussed relative equilibria for a special case of the five-body problem. He considered a configuration which consists of four bodies at the vertices of a rhombus. The fifth body is located at the center. Mioc and Blaga [12] discussed the same problem but in the post-Newtonian field of Manev. They prove the existence of monoparametric families of relative equilibria for the masses , where is the central mass, and prove that the Manev square five-body problem admits relative equilibria regardless of the value of the mass of the central body. Albouy and Llibre [1] dealt with the central configurations of the 1 + 4-body problem. They considered four equal masses on a sphere whose center is a bigger fifth mass. More recent studies on the symmetrically restricted five-body problem include Shoaib et al. [13, 14], Lee and Santoprete [15], Gidea and Llibre[16], and Marchesin and Vidal [17].

So far, in the noncollinear general four- and five-body problems the main focus has been on the common question: for a given set of masses and a fixed arrangement of bodies does a unique central configuration exist? In this paper, we ask the inverse of the question, that is, given a four- or five-body configuration, if possible, find positive masses for which it is a central configuration. Similar question has been answered by Ouyang and Xie [18] for a collinear four-body problem and by Mello and Fernandes [19] for a rhomboidal four- and five-body problems.

We consider four-point masses on the vertices of an isosceles trapezoid with two equal masses at positions , at positions , and at the center of mass (c.o.m). We derive, both analytically and numerically, regions of central configurations in the phase space where it is possible to choose positive masses. The rest of the paper is organized as follows. In Section 2, the equations of motion for the trapezoidal four- and five-body problems are derived. In Section 3, using both analytical and numerical techniques, the regions of central configurations for a special case of the trapezoidal five-body problem where four of the masses on the vertices of the trapezoid are taken to be equal are studied. In Section 4, the isosceles trapezoidal five-body problem in its most general form is investigated for the regions of central configurations. The regions of central configurations are given both numerically and analytically. Conclusions are given in Section 5.

#### 2. Equations of Motion

The classical equation of motion for the -body problem has the form where the units are chosen so that the gravitational constant is equal to one, is the location vector of the th body, is the self-potential, and is the mass of the th body.

A* central configuration* is a particular configuration of the -bodies where the acceleration vector of each body is proportional to its position vector, and the constant of proportionality is the same for the -bodies; therefore,where

Let us consider five bodies of masses , and . The mass is stationary at the c.o.m of the system. The remaining four bodies are placed at the vertices of an isosceles trapezoid shown in Figure 1. The geometry of the system is taken to be symmetric about the -axis. As shown in Figure 1, is the center of mass of and , and is the c.o.m of the masses and . Because of the symmetry about -axis, the symmetric masses will be equal. Therefore, we take and . We choose the coordinates for the five bodies as follows: where is the distance from the c.o.m of the system to the center of mass of and and is the distance from the c.o.m of the system to the c.o.m of and .