Advances in Astronomy

Volume 2016 (2016), Article ID 9897681, 11 pages

http://dx.doi.org/10.1155/2016/9897681

## Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

^{1}Department of Mathematics, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia^{2}Abu Dhabi Men’s College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, UAE^{3}Applied Mathematics and Sciences, Khalifa University of Science Technology and Research, P.O. Box 127788, Abu Dhabi, UAE

Received 2 November 2015; Revised 11 January 2016; Accepted 18 January 2016

Academic Editor: Elmetwally Elabbasy

Copyright © 2016 Muhammad Shoaib et al. 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

We study central configuration of a set of symmetric planar five-body problems where the five masses are arranged in such a way that , , and are collinear and , , and are collinear; the two sets of collinear masses form a triangle with at the intersection of the two sets of collinear masses; four of the bodies are on the vertices of an isosceles trapezoid and the fifth body can take various positions on the axis of symmetry both outside and inside the trapezoid. We form expressions for mass ratios and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We also show that the triangular configuration is not possible.

#### 1. Introduction

The equations of motion for positive masses subject to Newtonian Gravitation is given by where is the Newtonian potential, is the position vector of the th body, is the mass of the th body, and is the universal constant of gravitation. The central configuration of an -body system is obtained, if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration; that is,where the constant .

Using the Laura/Andoyer/Dziobek equations [1], the condition that form a planar, noncollinear, central configuration is where and .

The study of the central configurations plays a key role in understanding the Gravitational -body problems (cf. [2–4]). Central configurations are useful in providing explicit homographic solutions of the equations of motion and families of periodic solutions [5]. They are also useful in understanding the nature of solutions near collisions and the energy level sets that hold the central configuration determine the topology of the integral manifolds. Understanding the four- and five-body problem is very important as it is known that approximately two-thirds of stars in our Galaxy are part of multistellar systems. There is a growing interest in studying spatial central configurations of Newtonian five-body problem with various symmetrical restrictions. Several authors have recently studied central configuration of restricted four-body and five-body problems [6–9]. According to [10], finiteness problem of -body central configurations is an open problem for the 21st century. Hampton and Moeckel [11] proved that finite central configurations are possible for the Newtonian three-body and four-body problems with positive masses. Finiteness of central configurations of planar five-body problem is recently established by Albouy and Kaloshin in [12]. Hampton and Jensen [13] showed that number of spatial central configurations are finite for positive masses in the Newtonian five-body problem.

Several authors studied the inverse problem of central configurations by finding the mass vectors which makes the configuration central for a given configuration of bodies. Albouy and Moeckel [14] show that two-parameter family of masses exists for collinear central configurations with . Xie [15] proved the existence of a singular curve where it is possible to have permutational admissible set of mass vectors in the collinear central configuration four-body problem. According to [16], families of stacked central configurations can be formed by adding extra bodies to known central configurations of three bodies. Gidea and Llibre [1] studied the stacked symmetric planar central configuration of five bodies with some special symmetries. They have shown that central configuration is possible in rhomboidal arrangement where four masses are kept at the vertices and a fifth mass in the center and a trapezoidal arrangement where four masses are at the vertices and a fifth mass at the midpoint of one of the parallel sides. Shoaib et al. [17] have studied central configuration of the rhomboidal 5-body problem and identified CC regions using similar approaches.

Xie [18] studied central configuration of the planar Newtonian four-body problems, where two equal mass pairs are kept at adjacent vertices of a trapezoid and possible central configuration mass ratios are expressed in terms of the size of quadrilateral. While much is known about trapezoid central configurations in the restricted four-body problem, there is less known about trapezoid central configurations in the restricted five-body problem. Shoaib [19] recently investigated the inverse problem of central configuration in a symmetric -body problem and derived regions of central configuration.

In this work, we study the central configuration of the isosceles trapezoidal five-body problem and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We are motivated by the work of [1] and follow similar ideas to study planar symmetric five-body problems. The problems we investigate include two types of trapezoidal five-body problems with four masses on its vertices and a fifth mass on the axis of symmetry which can be both inside and outside the trapezoid. The third type of problem which is investigated here is a triangular problem with two pairs of masses and a fifth mass on the perpendicular bisector of the triangle which is not necessarily equal to any of the four masses. In Section 2, we present the main theorems related to the three five-body problems and their proofs are given in Sections 3, 4, and 5, respectively.

#### 2. Main Results

Theorem 1. *Consider a 5-body noncollinear configuration and a positive mass vector , where Assume that and are placed symmetrically on the vertices of an isosceles trapezoid and the fifth mass is placed on the axis of symmetry of the trapezoid.*(a)*When , there is a continuous family of central configurations determined by the region and the function , given in Figure 2. There are no central configurations when .*(b)*When and there exists a continuous family of central configurations for . There are no central configurations when or when .*

*Theorem 2. Consider a 5-body noncollinear configuration with a positive mass vector , where are given by (5). The five masses are arranged in such a way that , , and are collinear and , , and are collinear. The two sets of collinear masses form a triangle with at the intersection of the two sets of collinear masses. Then there does not exist any such type of central configuration.*

*3. Proof of Theorem 1*

*3.1. Proof of Theorem 1(a)*

*Consider five bodies of masses , , , , and placed at , , , , and : Four of the masses make a trapezoid (see Figure 1) while the mass is on the line of symmetry. Using the inherent symmetries of the trapezoidal 5-body model we obtain the following system of four equations from (4) which define the central configurations for the model described above: where Write (7) to (10) as a linear homogeneous system in given by the matrix *