Restricted Concave Kite Five-Body Problem

. Te restricted concave kite fve-body problem is a problem in which four positive masses, called the primaries, rotate in the concave kite confguration with a mass at the center of the triangle formed by three of the primaries. Te ffth body has negligible mass and does not infuence the motion of the four primaries. It is assumed that the ffth mass is in the same plane of the primaries and that the masses of the primaries are m 1 , m 2 , m 3 , and m 4 , respectively. Tree diferent types of concave kite confgurations are considered based on the masses of the primaries. In case I, one pair of primaries has equal masses; in case II, two pairs of primaries have equal masses; in case III, three of the primaries have equal masses. For all three cases, the regions of central confguration are obtained using both analytical and numerical techniques. Te existence and uniqueness of equilibrium positions of the in-fnitesimal mass are investigated in the gravitational feld of the four primaries. It is numerically confrmed that none of the equilibrium points are linearly stable. Te Jacobian constant C is used to investigate the regions of possible motion of the infnitesimal mass.


Introduction
Dynamical systems with few bodies (three) have been extensively studied in the past, and various models have been proposed for research aiming to approximate the behavior of real celestial systems. Tere are many reasons for studying the fve-body problem besides the historical ones since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems. In this manuscript, the motion of a body of negligible mass under the Newtonian gravitational attraction of four bodies (called primaries) moving each one in circular periodic orbits around their center of mass is considered. At any instant of time, the primaries form a kite confguration (central confguration) which is a particular solution to the general fve-body problem. Central confgurations (CC) play a vital role in the understanding of the n-body problem of celestial mechanics. It can be used to fnd simple or special solutions to the n-body problem since the geometry formed by the arrangement of the primaries remains constant for all time (cf. Saari [1]; Moeckel [2]; Farantos [3]; Deng and Zhang [4]; MacMillan and Bartky [5]; Sim [6]; Llibre and Mello [7]; Papadakis and Kanavos [8]).
Te restricted fve-body problem mainly takes into consideration a ffth body generally referred to as the test particle with negligible mass and does not infuence the motion of the four primaries. Kulesza et al. [9] discussed restricted rhomboidal fve-body problem, and they found that 11, 13, or 15 equilibrium solutions are all unstable. Siddique et al. [10] did a stability analysis of the rhomboidal restricted six-body problem. Ansari and Alhussain [11] investigated the fve-body problem with kite confguration where four bodies are placed at the vertices of a kite, and the ffth infnitesimal body is moving in the space under the infuences of these four primaries but not infuencing them gravitationally. After evaluating the equations of motion of the infnitesimal body, they investigated the location of Lagrange points and their linear stability, zero velocity curves, region of motion for the infnitesimal body, and Poincare surfaces of section. Shahbaz Ullah et al. [12] investigated the series solutions of the Sitnikov kite confguration by the methods given by Lindstedt-Poincaré, using Green's function and MacMillan. Bengochea et al. [13] discussed the planar four-body problem emanating from a kite confguration with three equal masses by using analytical and numerical tools and showed the existence of families of quasiperiodic orbits. Tey introduced a new coordinate system that measures (in the planar four-body problem) how far an arbitrary confguration from a kite confguration is. Hampton [14] studied the fniteness of planar relative equilibria of the Newtonian fve-body problem and the fve-vortex problem in the case that confgurations form a symmetric kite. Tey proved that the equivalence classes of such relative equilibria are fnite with some possible exceptional cases. Tese exceptional cases are given explicitly as polynomials in the masses (or vorticities in the vortex problem).Álvarez-Ramírez and Llibre [15] using mutual distances as coordinates showed that any fourbody central confguration forming a Hjelmslev quadrilateral must be a right kite confguration. A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.
In this work, we study the motion of a body of negligible mass under the Newtonian gravitational attraction of four bodies of masses m 1 , m 2 , m 3 , and m 4 (called primaries) moving each one in circular periodic orbits around their center of mass. At any instant of time, the primaries form a confguration (central confguration) of a concave kite which is a particular solution to the fve-body problem. Here, we study the position of the infnitesimal mass, m 5 , in the plane of motion of the primaries, and we use either the sideral system of coordinates or the synodic system of coordinates (see [4] or [8] for details). r 1 � (− 1, − α), r 2 � (0, β), r 3 � (1, − α), r 4 � (0, 0) where α and β are positive numbers. We call this as restricted concave kite fvebody problem (RK5BP) (see Figure 1). Te rest of the paper is organized as follows: in Section 2, we derive the general equations for (RK5BP) and list the main results. Tree specifc cases of CC's for four primaries are investigated. In Section 3, the equations of motion for m 5 are set up in synodic system of coordinates. We explore the hill region and possible region of motion of m 5 according to the Jacobian constant. In Section 4, we discuss, analytically, the equilibrium points along the coordinate axes and numerically of the axes for diferent cases of CCs. In Section 5, the linear stability of equilibrium points is discussed, and conclusions are drawn in the last section.

Four-Body Concave Kite Central Configurationś
Erdi and Czirják [16] studied the planar central confgurations of four bodies when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry and gave a complete solution for a symmetric case. Te derived formulae represent exact analytical solutions of the four-body problem. Tey also discussed one convex and two concave cases of central confguration in detail. Benhammouda et al. [17] studied the central confguration of the kite four-body problems. Tey considered two diferent types of symmetrical confgurations. In both cases, the existence of a continuous family of central confgurations for positive masses is shown. Tey also provided numerical explorations via Poincaré cross sections, to show the existence of periodic and quasiperiodic solutions for the four-body problem. Hampton [18] discussed the existence of a new family of planar fve-body central confgurations. Tis family is unusual in that it is a stacked central confguration, i.e., a subset of the points also forms a central confguration. Mello and Fernandes [19] studied the existence of kite central confgurations in the planar four-body problem which lies on a common circle. Tey also proved the existence of kite central confgurations in the spatial fve-body problem which lies on a common sphere. Perez-Chavela and Santoprete [20] proved that there is a unique convex noncollinear central confguration of the planar four-body problem when two equal masses are located at opposite vertices of a quadrilateral, and at most, only one of the remaining masses is larger than the equal masses. Such a central confguration possesses a symmetry line, and it is a kite-shaped quadrilateral. Tey also showed that there is exactly one convex noncollinear central confguration when the opposite masses are equal. Such a central confguration also possesses a symmetry line, and it is a rhombus. Corbera et al. [21] prove that any four-body convex central confguration with perpendicular diagonals must be a kite confguration. Tey extended the result to general power-law potential functions, including the planar four-vortex problem. Ansari et al. [22] presented a numerical investigation of some characteristics and parameters related to the motion of an infnitesimal body with variable mass in a fve-body problem. Te whole system forms a cyclic kite confguration. Tey also determined the positions of Lagrangian points and basins of attraction for the infnitesimal body. Tey also investigated the linear stability of the Lagrangian points and found that Lagrangian points are unstable. We consider, initially, four primaries with masses m 1 � m 3 � m, m 2 , and m 4 in a concave kite confguration. As it is a classical approach, in such cases, the system will be treated in a synodical system of coordinates in order to eliminate the time dependence. Te whole system rotates with constant angular velocity, the centrifugal force compensates for the Newtonian attraction, and the fve bodies are in equilibrium in such a rotating system, the so-called "relative equilibria solutions." Te central confguration of this particular was derived in [17], and we will give a brief review of their results with minor improvement. If we denote by r j the position of the body with mass m j , and by r ij � ‖r j − r j ‖, the distance between the body with mass m j and the body with mass m i , then the algebraic system of equations that must be satisfed for the bodies to be in noncollinear central confguration is where R ij � r − 3 ij and ∆ ijk � | (r i − r j )∧(r i − r k )| represent the area of the triangle determined by the sides ‖r i − r j ‖ and ‖r i − r k ‖. After eliminating redundant equations due to the symmetries of the problem, i.e., f 13 � f 24 ≡ 0, f 12 � f 23 , f 14 � f 34 , and m 1 � m 3 , [17], we get the following two equations: We will consider three special cases based on the relationship between the masses: Ten, the simultaneous solution of equations (2) and (3) gives With some elementary computations, the region of central confgurations for positive masses is derived as follows: Te values of the mass ratios μ 1 and μ 2 are depicted in Figure 2.  (2) and (3) give where (3/2) . Te function f(α, β) � 0 is a necessary condition for the central confgurations to exist. It is not possible to analytically solve f(α, β) � 0 for either α or β, and therefore, we use interpolation to write β � φ(α), where φ(α) � 2349.87α 12 − 16627.1α 11 + 52450.α 10 − 97275.4α 9 + 117771.α 8 − 97711.8α 7 + 56737.3α 6 − 23129.5α 5 + 6542.32α 4 − 1250.54α 3 + 155.539α 2 − 12.1183α + 1.66263.

(11)
Tis allows us to write the mass ratio μ as a function of α only. Te function μ(α) is an increasing function and attains its maximum at α � 1. Te values of the mass ratios μ are depicted in Figure 4.

Setting Up of the Problem and Preliminary Results
We consider the motion of an infnitesimal mass m 5 in the gravitational feld of the concave kite confguration described in Section 2. Te equations of motion of the body m 5 are where is the efective potential. Defne a frst Jacobi-type integral by It is trivial to show that C(ζ, η) is the frst integral of motion of the system (12) by proving that _ C(ζ, η) � 0.

Te Hill Sphere and Region of Motion for m 5 .
Te zero velocity curves in case I for μ 1 � 0.502, μ 2 � 3.579 and μ 1 � 0.991, μ 2 � 0.704 are given in Figure 5. Te Hill spheres are the circular regions surrounding the four primaries masses shown in Figure 5. It is clear from equation (14) that C + Ω ≥ 0. Terefore, Ω � − C will defne a boundary between the region of permitted and prohibited motions. Te region of possible motion of m 5 in case I for μ 1 � 0.502, μ 2 � 3.579 and μ 1 � 0.991, μ 2 � 0.704 is shown in Figure 6for six different values of Jacobian constant C. Te shaded regions represent the permitted regions of motion for the infnitesimal mass m 5 . It is numerically confrmed that the permitted regions of motion are connected when C ≥ − 4.15.
For decreasing values of C, the permitted regions of motion begin to disconnect. Tey completely disconnect at C � − 5.52. It can be seen from Figure 6 that the disconnection occurs in six stages, and hence, the infnitesimal mass m 5 will be completely trapped in the shaded region when C ≤ − 5.52. Similar restrictions exist for all combinations of μ 1 and μ 2 . However, the disconnection doesn't always occur in six stages. For small values of μ 2 , the disconnection occurs in three stages. For example, when μ 1 � 0.965281 and μ 2 � 0.0880651, the complete disconnection is achieved in three stages at C � − 3.85, C � − 3.87, and C � − 3.95, and when μ 1 � 1.38 and μ 2 � 0.025, the complete disconnection is achieved in two stages at C � − 3.853 and C � − 3.865. Te zero velocity curves in case II for μ � 1.73878 and μ � 1 are given in Figure 7. Te Hill spheres are the circular regions surrounding the four primaries' masses as shown in Figure 7. In this case of four equal masses, the regions of permitted motion are connected when C � − 4.4, partially connected when C � − 4.9 and completely disconnected when C � − 5. Te regions of motion are given in Figure 8, and when μ � 1.73878, the complete disconnection is achieved in four stages as shown in Figure 9. We have shown the graph of zero-velocity curves and efective potential for case III for μ � 3 and μ � 0.01 in Figure 10, respectively. Similar to case I and case II, the complete disconnection of permitted regions is achieved in multiple stages as presented in two Figures 11 and 12. In the frst instance, when μ � 0.01, the complete disconnection is achieved in three stages while for the higher mass ratio of μ � 3, it is achieved in four stages.

Advances in Astronomy
In the following sections, we will study the equilibrium solutions of all three cases introduced in Section 2. Initially, we study the existence and number of equilibrium solutions on the axes and then of the coordinate axes.
Tis proves the existence of unique equilibrium solutions inside the concave kite on the η− axis. Now consider η > β and rewrite Φ 1 (η) as When η ⟶ β + , the term − μ 1 /(β − η) 2 will increase indefnitely and will dominate the remaining terms therefore
Terefore, we have shown that there are three equilibrium solutions on the η− axis. One of the solutions is inside the kite when α < 0.8. Tese solutions are shown in Figure 14.

Case III.
Consider m 1 � m 2 � m 3 � 1, μ � m 4 , and ζ � 0, then equilibria on the η− axis will be the solution of Te method and procedure to prove the existence and uniqueness of the equilibrium solutions are similar to case I and case II; therefore, we leave the proof of existence and uniqueness to the interested reader and provide only numerical evidence. On the η− axis, there are three equilibrium solutions one of which is inside the Kite. Te inside solution is always between m 2 and m 4 , and the outside solutions are on either side of m 2 and m 4 outside the triangle formed by m 1 , m 2 , and m 3 . Tese solutions are shown in Figure 15 Similarly to get equilibrium solutions on ζ− axis, we substitute η � 0 in Ω ζ (ζ, η) � 0 and Ω η (ζ, η) � 0 which gives Φ 6a (ζ) � 0 and Φ 6b (ζ) � 0, where In this case, there are only two equilibrium solutions as shown in Figure 15(b). Te solutions are at (α, β, ζ) � (0.164903, 1.42885, ± 0.265228). In this case, the central mass (μ � 0.0907689) is much smaller than the masses at the vertices of the triangle.

Case I.
To fnd a relationship between the masses and the equilibrium points, we defne a new mass ratio μ 3 � μ 1 /μ 2 : 12 Advances in Astronomy  For all the masses to remain positive, we choose α and β from R μ 1 μ 2 given in equation (6). Based on the values of α, the region R μ 1 μ 2 is divided into two parts, i.e., when 0 < α < 1/ � 3 √ , μ 3 is a decreasing function of β for fxed value of α and is an increasing function when α > 1/ � 3 √ as shown in Figure 16. Depending on the distance between the central mass m 4 and m 1 or m 3 , the number of equilibrium points changes from seven to thirteen. Tis efect is best explained by Figure 17. When α � 0.1, the equilibrium points are distributed as follows which shows a clear dependence on the ratio between m 2 and m 4 . Te equilibrium solutions in case I are shown in Table 1 for diferent values of α ranging between 0.1 and 1.

Case II and Case III.
In the case when m 1 � m 3 and m 2 � m 4 (case II), there are a total of 9 equilibrium points. Tese include the three equilibriums which were shown to exist on the η− axis. Te remaining 6 equilibrium points are of the axes. Tere is no change in the number of equilibrium points for the varying values of any of the masses are parameters. Te parameter α or the mass relation μ only afects the position of the equilibrium points. Tis efect is best explained in Figure 18.
In the case when m 1 � m 2 � m 3 and m 4 � μ(α), the number of equilibrium points depends on α and hence μ(α). For α � 0.11, there are only four equilibrium points as shown in Figure 19(a). Two of the four equilibrium points are on the η− axis. For α ∈ (0.11, 0.18), there are seven equilibrium points with three inside the kite and four outside the kite, as shown in Figure 19(b). For α ∈ [0.18, 0.2, there are nine equilibrium points with fve inside the kite and four outside the kite, shown in Figure 19(c). For α ∈ [0.2, 0.23), there are eleven equilibrium points with fve inside the kite and six outside the kite, shown in Figure 19(d). For α ∈ [0.23, 0.26), the number of equilibrium points increases to 13 with fve inside and 8 outside the kite; however, for α ≥ 0.26, the number of equilibrium points reduces to nine with three inside the kite and six outside the kite. Tere is a clear trend of dependence of equilibrium points on α when α < 0.26; however, the trend is broken when α ≥ 0.26, and therefore, we cannot conclude that the number of equilibrium either depends on the parameter α or the masses.

Stability Analysis of Equilibrium Points
To study the stability of the equilibrium points, the standard procedure of linearization is followed. Let the location of an equilibrium point in the RK5BP be denoted by (ζ, η) and consider a small displacement (x, y) to the new position (ζ + x, η + y). Using Taylor's series expansion, a new set of second-order linear diferential equations is obtained.
Te matrix form of the linearized equations is where For ζ and η to be a stable solution, all the eigenvalues of A must be pure imaginary. To fnd these eigenvalues, write the characteristic polynomial for A as where F 1 � 4 − Ω ζζ − Ω ηη and F 2 � Ω ζζ Ω ηη − Ω 2 ζη . For λ to be pure imaginary, we must either F 1 > 0 and 0 < F 2 ≤ F 2 1 /4 or F 1 > 0 and F 2 1 � 4F 2 . We will numerically identify regions where at least one of the above conditions is satisfed. In case I, when m 1 � m 3 , the stability regions for two values of α are given in Figure 20. Upon inspection, it is seen that none of the corresponding equilibrium points are in the stability region, and hence, the equilibrium points are unstable. We  Figure 21, and however, it has been numerically confrmed that none of the equilibrium points intersects the stability regions, and hence, all the equilibrium points are unstable. Examples are given in Tables 2 to 4.

Conclusion
In this paper, we studied the motion of a body of negligible mass under the Newtonian gravitational attraction of four bodies (primaries). Te primaries move in circular periodic orbits around their center of mass and maintain their geometric arrangement at all times. We have considered here three cases of central confguration based on the masses of the four primaries placed at the vertices of a concave kite. Using analytical techniques, regions of central confgurations are derived for all three cases. To complement the analytical results, these regions are also explored numerically. Te ffth mass (negligible mass with respect to the primaries) is moving in the gravitational feld of four primaries. We have obtained equations of motion of the infnitesimal mass moving in the plane of motion of the primaries in synodical coordinates to get rid of the time dependency of the equation of motion of m 5 . Te equations of motion of m 5 are nonlinear ordinary diferential equations. We did a qualitative analysis of the equation of motion of m 5 and found equilibrium solutions for the infnitesimal mass for all three diferent cases of the central confguration of the four primaries. We investigated the uniqueness and existence of equilibrium points on the coordinate axes and numerically of the coordinate axes. We can confrm that the number of equilibrium points is between 7 and 13 for three cases of central confguration for diferent values of mass parameters. Te linear stability analysis revealed that none of the equilibrium points are stable. Additionally, we have obtained the permissible region of motions for m 5 in the feld of four primaries in kite confguration according to the Jacobian constant. Te Hill sphere and the zero velocity curves of primaries are also discussed.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.