The Restricted Six-Body Problem with Stable Equilibrium Points and a Rhomboidal Configuration

We explore the central conguration of the rhomboidal restricted six-body problem in Newtonian gravity, which has four primariesmi (where i 1, . . . 4) at the vertices of the rhombus (a, 0), (− a, 0), (0, b), and (0, − b), respectively, and a fthmassm0 is at the point of intersection of the diagonals of the rhombus, which is placed at the center of the coordinate system (i.e., at the origin (0, 0)). ­e primaries at the rhombus’s opposite vertices are assumed to be equal, that is, m1 m2 m and m3 m4 m̃. After writing equations of motion, we expressm0, m, and m̃ in terms of mass parameters a and b. Finally, we nd the bounds on a and b for positive masses. In the second part of this article, we investigate the motion and dierent features of a test particle (sixth body m5) with innitesimal mass that moves under the gravitational eect of the ve primaries in the rhomboidal conguration. All four cases have 16, 12, 20, and 12 equilibrium points, with case-I, case-II, and case-III having stable equilibrium points. A signicant shift in the position and the number of equilibrium points was found in four cases with the variations of mass parameters a and b. ­e regions for the possible motion of test particles have been discovered. It has also been observed that as the Jacobian constant C increases, the permissible region of motion expands. We also have numerically veried the linear stability analysis for dierent cases, which shows the presence of stable equilibrium points.


Introduction
e system of point masses is considered to be under motion due to the mutual gravitational force as explained by Newton's gravitational law. Dynamical systems with n point masses have been extensively studied, and various models have been proposed for research aiming at approximating the behaviour of real celestial systems.
ere are many reasons for studying the n-body problem since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems.
In celestial mechanics, the central con guration (CC) plays an important role in the investigation of n-body problems. It is the con guration that can be used to nd simple or special solutions of the n-body problem since the shape of the gure formed by the arrangement of the bodies remains constant for all time. at is why the CCs lead to homographic solutions. e rst three collinear homographic solutions for n 3 were found by Euler in 1767, and two equilateral triangular homographic solutions were found by Lagrange [1] in 1772 for n 3. e central con guration for n > 3 has been investigated a lot. Xia [2] used the analytical continuation approach to nd the exact number of CCs for an open set of n positive masses in 1991. Llibre and Mello [3] investigated 7-body problem for the existence of families of central con gurations. In 2017, Deng et al. [4] investigated the central con guration of four-body problem with equal masses and showed that for the planar Newtonian four-body problem having adjacent equal masses, that is, m 1 m 2 ≠ m 3 m 4 , and equal lengths for the two diagonals, any convex noncollinear central con guration must have a symmetry and must be an isosceles trapezoid. ey also showed that when the length between m 1 and m 4 equals the length between m 2 and m 3 , the central con guration is also an isosceles trapezoid. In the same year, Marchesin [5] explored the stability of rhomboidal configurations with a mass in the center. He considered a system with five bodies at r 0 � (0, 0), r 1 � (a, 0), r 2 � (− a, 0), r 3 � (0, b), and r 4 � (0, − b) with masses m 0 , m 1 � m 2 � m and m 3 � m 4 � m that are moving in a plane. ey investigated the effect of altering the mass of the central body on the stability of such a configuration for several values of mass parameter λ � m/m. Saari [6] studied the role and properties of central configuration for n-body problem, and the other work on central configuration is given in Refs. [7][8][9][10][11][12][13][14][15] (for latest publications on CCs, cf. Suraj et al. [16], Zotos, and Papadakis [17], and Cornelio et al. [18]). e three-body problem has been a great challenge for the scientists as it needed some special assumptions for the third body. e ever first problem solved using restriction for the infinitesimal body is "restricted three-body problem" (RTBP). e RTBP originally arose from the work of Newton, but unfortunately, Newton was unable to solve the problem throughout his life. Alexis Clairaut [19,20] in 1747 solved the problem with some approximations, and in 1757, he calculated the return of "Halley's Comet." Ollongren [21] in 1988 was the first to introduce the restricted five-body problem. Ollongren investigated the motion of the fifth infinitesimal mass that moved under the gravitational attraction of the other four massive bodies. He discovered nine equilibrium points, three of which were stable, and the rest were linearly unstable. In 2007, Papadakis and Kavanos [22] extended the work of Ollongren and investigated the restricted five-body problem numerically, and they showed that the counts of equilibrium points depend on the radiation factors as well as on the mass parameter. Marchesin and Vidal [23] in 2013 studied the stability of five-body problem by assuming that the central body makes a generalized force on the other four bodies. In 2018, Aggarwal et al. [24] investigated the effect of small perturbation in Coriolis and centrifugal forces on the presence of equilibrium points in the restricted four-body problem. e three primaries were placed at the vertices of equilateral triangle and the fourth infinitesimal mass moves under the gravitational effect of the primaries. ey showed the existence of 8 to 10 equilibrium points and found that all the liberation points are unstable. Mello et al. [25], in 2009, introduced the stacked CC for the spatial six-body problem, and they illustrated three novel families of stacked spatial central configurations for the six-body problem with a regular tetrahedron model. Recently, in 2020, Idrisi and Ullah [26] investigated the existence and linear stability of equilibrium points in a six-body problem with a square CC and a central mass, and discovered that there are twelve equilibrium points, four of which are stable for a given value of the mass parameter μ c . Furthermore, in 2020 [27], Ansari et al. studied six body CC by placing the primary at the vertices of a square and a fifth mass at the center of a circle, which is thought to be the orbit of motion for the rest of the primaries. Using the Jeans law and the Mesheherskii space time transformation, they discovered twelve equilibrium points for the sixth small body of variable mass, all of which are unstable. Kulesza et al. [28] in the restricted rhomboidal fivebody problem found that the number of libration points depends on the semidiagonal ratio λ and they showed that there can be eleven, thirteen, and even fifteen libration points. In 2013, Marchesin and Vidal [29] studied the restricted rhomboidal five-body problem and found that no chaos exists and the behaviour of the fifth mass is quite predictable. ey also showed the existence of the periodic solutions of infinitesimal mass and also studied numerically their linear horizontal stability. e rhomboidal six-body CC was presented by Siddique et al. in 2021 [30] by putting the four primaries at the corners of the rhombus and the fifth mass at the intersection of the two diagonals of rhombus.
ey computed the region of probable motion of the sixth mass using the first integral of motion and the values of the Jacobian constant C for various energy intervals and found the limitation on the region of motion for infinitesimal mass. ey demonstrated the existence and uniqueness of equilibrium solutions on and off the axes using semi-analytic approaches, and established that there are always twelve unstable equilibrium points when b ∈ (1/ � 3 √ , 1.1394282249562009) and a � 1. Recently, Ref. [31] examined the basins of convergence by deploying the well-known Newton-Raphson iterative scheme, associated with the libration points (indeed, act as attractors), in the restricted rhomboidal six-body problem.
In this study, we focus on the rhomboidal restricted sixbody problem, which has four primaries at the corners of the rhombus and a fifth primary where the diagonals of the rhombus cross, which is in the center of the coordinate system, that is, (0, 0). ere is a sixth test mass m 5 that moves under the gravitational effect of the five primaries. e primaries at the opposite vertices of rhombus are assumed to be equal, that is, m 1 � m 2 � m and m 3 � m 4 � m. In section 2, we investigate the continuous families of five primaries central configurations for different values of the mass parameters a and b. In section 3, we derived the equation of motion for infinitesimal mass and discovered the Jacobian constant C. In section 4, we explore the Hill's and permissible regions of motion of the m 5 . Sections 5 and 6 analyze the positions and stability of the equilibrium points for the sixth body, m 5 . Concluding remarks are given in section 7.

Rhomboidal Central Configurations
e classical equation of motion for the n-body problem has the following form: where the units are chosen so that the gravitational constant is equal to one. A central configuration is a particular configuration of the n-bodies where the acceleration vector of each body is proportional to its position vector, and the constant of proportionality is the same for the n-bodies. erefore, a central configuration is a configuration that satisfies the following equation: 2 Advances in Astronomy where ω is an angular speed and is the center of mass on n-bodies. We choose the coordinates (for positions, see Figure 1) of five primaries m j , where j � 0 − 4 as Expanding equation (2) for (i � 0 − 4) and using equation (3) and letting m 1 � m 2 � m and m 3 � m 4 � m, we get the following equation for central configurations for the masses m 1 and m 2 : and for the masses m 3 and m 4 , the equation of motion is e mass m 0 is stationery so its equation of motion ends up zero. We take sum of all primaries is equal to unity that is Taking ω � 1 and solving equations (5)- (7) give where Proof. Let us define the following: To prove D(a, b) < 0, we need to prove P(α) < 1. For this, differentiate P(α) with respect to α and find the critical points of ). One can easily see that P(α) is monotonically increasing function because (dP(α)/dα) < 0 in (0, 1) and P(α) is monotonically decreasing function because (dP(α)/dα) > 0 in (1, ∞) (see Figure 2). When α ⟶ 0 or ∞, P(α) ⟶ (15/64). So P(α) < 1 for α ∈ (0, ∞), and hence, D(a, b) < 0 for a > 0 and b > 0.
To prove m(a, b) and m(a, b) positive, we need to prove N m and N m must be negative for a > 0 and b > 0. Because N m and N m are nonlinear algebraic functions of a and b, so it is difficult to solve these inequalities. For this, we draw the region (shaded region of Figure 3) where both N m and N m are negative. From the graph, we can easily find the approximate bounds for a > 0 and b > 0 where N m and N m are negative.
Using equation (7) to equation (9), one can easily see that m(a, b) � m(b, a) and 0 < m(a, b) < 0.5 , 0 < m(a, b) < 0.5 and m 0 (a, b) > 0 for 0.5 < a < 1 and 0.5 < b < 1. In Figure4 we show the region of central configuration for which m, m, m 0 > 0 are positive. Because Figure 4 is symmetric about the line b � a, we divide here Figure 4 into two parts: upper region (R u ) and lower region (L u ). Here, we discuss only the central configuration for the region of R u . e upper CC region is approximately surrounded by the following three interpolating polynomials:

Equation of Motion of Infinitesimal Body
In this section, we derive the equation of motion of the infinitesimal body, m 5 , that moves under the gravitational attraction of the five primaries. e sixth body has a significantly smaller mass as compared to the masses of the primaries, that is, (m 5 ≪ m i ) where i � 0 − 4. Due to this fact, the sixth body acts as an infinitesimal test particle, and therefore, it does not have any influence on the motion of the five primaries. In the above scenario, the equation of motion of the test particle m 5 is e dot represents the derivative with reference to time. We discuss here the dynamics of the infinitesimal mass with respect to the five primaries. e equations of motion of the m 5 in synodical coordinates ξ and η [32] are where is the effective potential, where the mutual distances are e effective potential is shown in Figure 6 (right) for the four different cases of rhomboidal restricted six-body problem. We define the first Jacobian-type integral by By demonstrating that _ C(ξ, η) � 0, it is now straightforward to establish that C(ξ, η) is the first integral of motion of system (18).
Equation (18) shows that C + Ω ≥ 0. en, Ω � − C will establish a boundary between both the allowed and forbidden regions and Ω � − C presents the zero velocity curves for various values of C.

The Spheres of Influence
Spheres of influence or gravitational spheres of influence are areas surrounding celestial objects where other celestial objects experience the greatest pull and can become satellites of the huge celestial object relative to its mass; these regions are also known as Hill's regions, after George William Hill. e zero velocity curves (ZVC) are the contours of the Jacobian constant C as mentioned above, and they are available in Figure 6 on the left, with their corresponding effective potentials on the right. e Hill's regions are tightly packed circular regions surrounding primaries; in Figure 6 (i), for case-I when b � 0.58 and a � 0.68, the Hill's region for m 1 stretches between L 2 and L 4 , whereas for m 3 , it extends between L 5 and L 7 . Figure 6 (i) also shows that the spheres of influence shrink as the particle mass decreases, since m 0 has a relatively tiny Hill's region as compared to the other primaries. e pink and green lines in Figures 6 (i), (iii), (v), and (vii) on the left depict the contours of equations (19) and (20). e black dots indicate masses and are represented by the symbol m i ; i � 0, 1, 2, 3, 4, the orange dots

Equilibrium Solutions
All rates of change should be zero for equilibrium solutions; hence, the right-hand side of the system in (15) can be set to zero, that is, Ω ξ � 0 and Ω η � 0, and the solution of resulting equations will give the problem's equilibrium solutions. Ω ξ and Ω η are

Advances in Astronomy
We considered four cases with different values of a and b that show a significant change in the number and location of equilibrium points, namely, a � 0.68 and b � 0.58 for case-I; a � 0.68 and b � 0.60 for case-II, a � 0.78 and b � 0.67 for case-III; and a � 0.62 and b � 0.80 for case-IV. e corresponding equilibrium points can be seen in subfigures (i), (iii), (v), and (vii) of Figure 6.

Equilibrium Solutions:
On the Coordinate Axes. We shall limit our investigation to the first quadrant, ξ ≥ 0 and η ≥ 0, because the potential given in equation (15) is unchanged under the symmetry (ξ, − η), (− ξ, η), and (− ξ, − η). To determine the presence and number of equilibrium solutions on the y-axis, we set ξ � 0 and then write equations (19) and (20) as   Advances in Astronomy e η axis is subdivided into 0 < η < b and η > b to figure out Ω η � 0 for equilibrium solutions. η � b means collisions of m 5 with m 3 or m 4 from inside. We will not discuss collisions cases here. (21) is rewritten as
en, there are total 16 equilibrium points for case-I and out of which L 5,6,7,8 are along η axis, while L 1,2,3,4 are on ξ axis; similarly for case-II, III, and IV, there are four equilibrium points on each coordinate axis for distinct values of a and b, but the number of equilibrium points is different for other cases, which can be seen from the subfigures (i), (iii), (v), and (vii) of Figure 6.

Stability Analysis
We represent the location of an equilibrium point in our problem by the coordinates (ξ, η) and consider a small displacement from the equilibrium point to be (ξ 0 , η 0 ) in order to linearize around the equilibrium point, and then, the new location of the equilibrium points will be (ξ + ξ 0 , η + η 0 ). e system in equations (14) is subjected to Taylor series expansion, yielding the following set of linearized equations: e linearized equations in the matrix form is Let e equation in (27) can be written as From M, we have where α � 4 − Ω ξξ − Ω ηη and β � Ω ξξ Ω ηη − Ω 2 ξη . Let Ψ 2 � λ, then equation (30) reduces to For an equilibrium point to be linearly stable under a slight disturbance, all four roots of equation (30) must be completely imaginary. As a result, the two roots of equation (31) must be real and negative. For λ ± < 0, we must have (i)α > 0 and 0 < β ≤ α 2 /4, or (ii)α > 0 and α 2 − 4β � 0.
(33) e regions of stability when either condition (i) or (ii) holds true have been determined and presented in Figures 11-14. e stability areas corresponding to the four cases for different values of a and b are shown in Figures 15  and 16, where b is fixed at 0.58, 0.60, 0.67, and 0.80, and the regions are projected across the whole domain of the mass parameter a, that is, 0.5 < a < 1. We investigated the stability of the equilibrium points found in the preceding section for each of the four different cases, and many off-coordinate stable equilibrium points were discovered; the results are reported in Tables 1-4.

Conclusion
Mass parameters a and b are used to examine the restricted rhomboidal six-body problem in depth. A point mass, m 0 , is put at the intersection of two diagonals of the rhombus, and the remaining four point masses are arranged in such a way that they preserve a rhomboidal central configuration during their two-by-two motion. e point masses at the opposite vertices of a rhombus are supposed to be the same, that is, m 1 � m 2 � m and m 3 � m 4 � m. In a synodical coordinate system, the motion of the sixth test mass, m 5 , which has an infinitesimal mass relative to the other five masses, that is, (m 5 ≪ m 0,1,2,3,4 ), is investigated for both the equilibrium points and their linear stability. A significant shift in the position and number of equilibrium points was found in four cases with the variations of mass parameters a and b. Cases-I to IV have 16, 20, 12, and 12 equilibrium points respectively, with case-I and case-II having eight off-thecoordinate-axis stable equilibrium points, case-III having four off-the-coordinate-axis stable equilibrium points, and case-IV having no on or off-the-coordinate-axis stable equilibrium points. e regions for the possible motion of m 5 have been discovered, and it has been observed that as the Jacobian constant C increases, the permissible region of motion expands, and the values of C for which the regions of the possible motion become disconnected or partially disconnected have also been discovered. We also have numerically verified the stability regions for different cases, which shows the presence of stable equilibrium points.

Data Availability
No data were used for the article.