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

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 m 1 = m 4 at positions (∓0.5, r B ) and m 2 = m 3 at positions (∓α/2, r A ). 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.


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, r =   for all  = 1, 2, . . ., .A central configuration can also be expressed as a critical point for the function  2 , 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  ≥ 0. 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 ( 0 , 1, , 1, ), where  0 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  1 =  4 at positions (∓0.5,   ),  2 =  3 at positions (∓/2,   ), and  0 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.

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  0 ,  1 ,  2 ,  3 , and  4 .The mass  0 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  2 and  3 , and  is the c.o.m of the masses  1 and  4 .Because of the symmetry about -axis, the symmetric masses will be equal.Therefore, we take  2 =  3 =  and  1 =  4 = .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  2 and  3 and   is the distance from the c.o.m of the system to the c.o.m of  1 and  4 .Without loss of generality, it is assumed that  23 = − 41 ,   = |  −   | =  41 .Using these assumptions with (1) we obtain the following equations of motion: Advances in Astronomy where As  0 is taken to be stationary at the c.o.m, therefore r 0 = 0.The c.o.m is at the origin; therefore,  0 = 0. Hence, the CC equation r 0 = − 0 will simply be 0 = 0 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   ,  = 1, 2, 3, 4, , and  41 : This clearly shows that it is enough to study the equations for  = ((+)/2)( 2 + 3 ) and  41 as   ,  = 1, 2, 3, 4, are linear combination of  and  41 : 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.

Four Equal Masses with a Varying Central Mass
Let the four bodies on the vertices of the trapezoid have equal masses; that is,   = ,  = 1, 2, 3, 4 and  = 1.The equations of central configuration, given in (10) and (11), become where The above two equations give  0 and  as functions of  and : where To find regions of central configurations in the -plane where (,  0 ) > (0, 0), the sign analysis of ( − ), (, ), and ℎ(, ) is needed which is given below.
(ii) For sign analysis of ℎ(, ), we obtain the following region where ℎ(, ) < 0: 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 ℎ(, ) < 0 in terms of  or .
Therefore, we approximate it by a polynomial of order 2 in : where  = √ 2 + 0.25.The equation ℎ(, ) app = 0 gives  as a function  which provides a boundary between ℎ < 0 and ℎ > 0: Therefore, region  1 can now be rewritten as below: It is numerically verified that ℎ(, ) = 0 and ℎ(, ) app = 0 have almost identical graphs for all values of  and  (see Figure 2).Region  1 is given in Figure 2.  (iii) For sign analysis of (, ), we obtain the following region where (, ) < 0: To find the region (, ) < 0 which is explicitly defined by a function of  or  we approximate (, ) by a polynomial of order 2 in : The real valued function  app (, ) = 0 gives  as a function  which provides a boundary between  < 0 and  > 0: In the same way as in the case of ℎ(, ), it is numerically verified that (, ) = 0 and (, ) app = 0 have almost identical graphs for all values of  and  (see Figure 2).Therefore, region  2 can now be rewritten as below: Numerically, region  2 is given in Figure 2.
As the numerator of  is negative for all values of  and , therefore  2 defines a central configuration region in the plane where  > 0.
defines the CC region for this particular setup of the 5body problem where all the masses are positive.Numerically, regions  1 ,  2 , and  3 are shown in Figure 2. The Central Configuration regions where  0 > 0 and  > 0 are given in Figure 3.

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  2 and  3 , and  is the center of mass of where To obtain regions of central configurations in the trapezoidal five-body problem we simultaneously solve the above equation for  0 and : The above equations representing  and  0 give regions of central configurations for the trapezoidal five-body problem in the -plane.For example, when (1)  = 0.2,  = 0.8, and  = 0.3, (,  0 ) = (0.26, 0.04), and when (2)  = 0.4,  = 0.7, and  = 0.5, (,  0 ) = (−0.5,0.4).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)  * −  * > 0 : it is straightforward to show that  * −  * > 0 in where Region  4 is given in Figure 4(a): (2) The analysis of   0 (, , ) is similar in nature to ℎ(, ) but it is comparatively easier to write a closed form solution of   0 (, , ) = 0 as (, ) = ( +  − 2)/(( − )).Therefore,   0 (, , ) > 0 in the following region: (31) Like in the four equal masses case we can write a polynomial approximation of   (, , ) as follows: Equation   0app (, , ) = 0 will give  as a function of  and  which will provide a boundary between the regions where   0app < 0 and   0app > 0: Region  5 can now be rewritten as It is worth mentioning here that the approximation error between   0 and   0app is almost zero.Region  5 is given in Figure 4(c).The polynomial approximation of   (, , ) is given as below: app (, , ) where As above   app (, , ) = 0 will give  as a function of  and  which will provide a boundary between the regions where   app < 0 and   app > 0: Region  6 can now be rewritten as Region  6 is given in Figure 4(b).
From the above analysis we conclude that the CC region where  > 0 is Similarly, the CC region where  0 > 0 is This gives the CC region for this particular setup of the fivebody problem as Region  9 is given in Figure 5.In the complement of this region no central configurations are possible.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  7 and  8 and is given by  9 =  7 ∩  8 .Numerically, region  9 is shown by the colored part of Figure 5.To aid the understanding of the central configuration region  9 its cross sections are given in Figures 6 and 7 for various values of .
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 configurations are possible for  < 0.42.For  > 0.42 there exists at least one  such that both  0 and  are positive and form a 4 + 1-body trapezoidal central configuration.

Conclusions
In this paper, we model isosceles trapezoidal four-and fivebody 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 (  = ,  = 1, . . ., 4) and form expressions for  0 and  as functions of  and  which gives central configurations in the trapezoidal 5-body problems.CC regions are identified in the -plane where both  0 and  are positive.In the same way we form expressions for  0 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

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

Figure 7 :
Figure 7: Cross sections of the central configurations region in the general case when ,  0 > 0 and (a)  = 0.7, (b)  = 0.8, (c)  = 0.9, and (d)  = 1 give the case with four equal masses and a varying central mass.