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Advances in Astronomy
Volume 2015, Article ID 649352, 11 pages
http://dx.doi.org/10.1155/2015/649352
Research Article

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 .

Figure 1: Trapezoidal four- and five-body configurations.

Without loss of generality, it is assumed that , . Using these assumptions with (1) we obtain the following equations of motion: whereAs is taken to be stationary at the c.o.m, therefore . The c.o.m is at the origin; therefore, . Hence, the CC equation will simply be which is why we do not have a fifth equation. Let   (Figure 1) and then using the geometry of our proposed problem we arrive at the following relationships between , ,  , and :This clearly shows that it is enough to study the equations for and as , , are linear combination of and :Using (3) in conjunction with (9) and taking the center of mass at the origin, we obtain the following equations of central configurations for the trapezoidal five-body problem:

To obtain regions of central configurations in the trapezoidal five-body problems we will need to simultaneously solve (10) and (11) for and . This will give values of the two masses which define the central configurations for the trapezoidal five-body problem.

3. Four Equal Masses with a Varying Central Mass

Let the four bodies on the vertices of the trapezoid have equal masses; that is, ,   and . The equations of central configuration, given in (10) and (11), becomewhere The above two equations give and as functions of and :where To find regions of central configurations in the -plane where , the sign analysis of ,  , and is needed which is given below.(i)As ,  ; therefore, for all values of and which means .(ii)For sign analysis of , we obtain the following region where :Ideally we would like to find a region which is explicitly defined by a function of or . Because of the involvement of radicals it is impossible to find a closed form solution of in terms of or . Therefore, we approximate it by a polynomial of order 2 in :where . The equation gives as a function which provides a boundary between and :Therefore, region can now be rewritten as below:It is numerically verified that and have almost identical graphs for all values of and (see Figure 2). Region is given in Figure 2.(iii)For sign analysis of , we obtain the following region where :To find the region which is explicitly defined by a function of or we approximate by a polynomial of order 2 in :The real valued function gives as a function which provides a boundary between and :In the same way as in the case of , it is numerically verified that and have almost identical graphs for all values of and (see Figure 2). Therefore, region can now be rewritten as below:Numerically, region is given in Figure 2.

Figure 2: CC regions in trapezoidal 5-BP with four equal masses and a varying central mass.

As the numerator of is negative for all values of and , therefore defines a central configuration region in the -plane where . Similarly in . As and , thereforedefines the CC region for this particular setup of the 5-body problem where all the masses are positive. Numerically, regions , , and are shown in Figure 2. The Central Configuration regions where and are given in Figure 3.

Figure 3: (a) (colored) (b) (colored).

4. Two Pairs of Equal Masses and a Central Mass

As stated earlier the geometry in Figure 1 is taken to be symmetric about the line , where is the center of mass of and , and is the center of mass of and . Let and . We further assume that , , and , where . Therefore, (9) give uswhereTo obtain regions of central configurations in the trapezoidal five-body problem we simultaneously solve the above equation for and :whereThe above equations representing and give regions of central configurations for the trapezoidal five-body problem in the -plane. For example, when (1) , , and , , and when (2) , , and , . As can be seen in number (2), one of the masses has become negative for specific values of and , which is impractical. Therefore, we would like to identify compact regions in the central configuration space where none of the masses can become negative. The sign analysis of , , and is needed which is given below.(1) it is straightforward to show that inwhereRegion is given in Figure 4(a):(2)The analysis of is similar in nature to but it is comparatively easier to write a closed form solution of as . Therefore, in the following region:Like in the four equal masses case we can write a polynomial approximation of as follows:Equation will give as a function of and which will provide a boundary between the regions where and :Region can now be rewritten asIt is worth mentioning here that the approximation error between and is almost zero. Region is given in Figure 4(c).(3)It is similarly easy to write as a function and from . Therefore, is positive in the following region:The polynomial approximation of is given as below:whereAs above will give as a function of and which will provide a boundary between the regions where and :Region can now be rewritten asRegion is given in Figure 4(b).

Figure 4: (a) (colored). (b) (colored). (c) (colored).

From the above analysis we conclude that the CC region where isSimilarly, the CC region where is This gives the CC region for this particular setup of the five-body problem asRegion is given in Figure 5. In the complement of this region no central configurations are possible.

Figure 5: Central configurations region in the isosceles trapezoid 4 + 1-body problem.

Hence, the central configuration region for the isosceles trapezoidal five-body problem with two pairs of masses and a stationary central mass is determined by and and is given by . Numerically, region is shown by the colored part of Figure 5. To aid the understanding of the central configuration region its cross sections are given in Figures 6 and 7 for various values of .

Figure 6: Cross sections of the central configurations region in the general case when and (a) , (b) , and (c) .
Figure 7: Cross sections of the central configurations region in the general case when and (a) , (b) , (c) , and (d) give the case with four equal masses and a varying central mass.

To complete the analysis of the 4 + 1-body case with two pairs of masses we study a special case where . This reduces the number of parameters from three to two while we still maintain the general setup with two pairs of masses. The CC regions are given only numerically in Figure 8 as it is easy to derive the analytical expressions from (25) by replacing with the following new values: Using , and , CC regions are given for in Figure 8. It is obvious from Figure 8(c) that no central configurations are possible for . For there exists at least one such that both and are positive and form a 4 + 1-body trapezoidal central configuration.

Figure 8: Central configurations region in the trapezoidal 5-BP when (a) , (b) , and (c) and .

5. Conclusions

In this paper, we model isosceles trapezoidal four- and five-body problems where four of the masses are placed at the vertices of an isosceles trapezoid and the fifth mass is placed at the center of mass of the system. To make use of the symmetries of the problem chosen we rewrite the position vectors of all the point masses as a linear combination of two vectors. This helped us halve the dimensions of the problem to a manageable level. Initially, we study a special case (, and form expressions for and as functions of and which gives central configurations in the trapezoidal 5-body problems. CC regions are identified in the -plane where both and are positive. In the same way we form expressions for and as functions of , , and in the 4 + 1-body cases which give central configuration regions in the trapezoidal 4 + 1-body problems. In this case there are two pairs of masses and a varying central mass. CC regions are identified both analytically and numerically. We also identify regions in the configuration space where no central configurations are possible for ,  , and .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author thanks the Deanship of Research at the University of Hail, Saudi Arabia, for funding this work under Grant no. SM14014.

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