Abstract
The restricted concave kite five-body problem is a problem in which four positive masses, called the primaries, rotate in the concave kite configuration with a mass at the center of the triangle formed by three of the primaries. The fifth body has negligible mass and does not influence the motion of the four primaries. It is assumed that the fifth mass is in the same plane of the primaries and that the masses of the primaries are , , , and , respectively. Three different types of concave kite configurations 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 configuration are obtained using both analytical and numerical techniques. The existence and uniqueness of equilibrium positions of the infinitesimal mass are investigated in the gravitational field of the four primaries. It is numerically confirmed that none of the equilibrium points are linearly stable. The Jacobian constant is used to investigate the regions of possible motion of the infinitesimal mass.
1. 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. There are many reasons for studying the five-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 configuration (central configuration) which is a particular solution to the general five-body problem. Central configurations (CC) play a vital role in the understanding of the n-body problem of celestial mechanics. It can be used to find 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]).
The restricted five-body problem mainly takes into consideration a fifth body generally referred to as the test particle with negligible mass and does not influence the motion of the four primaries. Kulesza et al. [9] discussed restricted rhomboidal five-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 five-body problem with kite configuration where four bodies are placed at the vertices of a kite, and the fifth infinitesimal body is moving in the space under the influences of these four primaries but not influencing them gravitationally. After evaluating the equations of motion of the infinitesimal body, they investigated the location of Lagrange points and their linear stability, zero velocity curves, region of motion for the infinitesimal body, and Poincare surfaces of section. Shahbaz Ullah et al. [12] investigated the series solutions of the Sitnikov kite configuration 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 configuration with three equal masses by using analytical and numerical tools and showed the existence of families of quasiperiodic orbits. They introduced a new coordinate system that measures (in the planar four-body problem) how far an arbitrary configuration from a kite configuration is. Hampton [14] studied the finiteness of planar relative equilibria of the Newtonian five-body problem and the five-vortex problem in the case that configurations form a symmetric kite. They proved that the equivalence classes of such relative equilibria are finite with some possible exceptional cases. These 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 four-body central configuration forming a Hjelmslev quadrilateral must be a right kite configuration. 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 and (called primaries) moving each one in circular periodic orbits around their center of mass. At any instant of time, the primaries form a configuration (central configuration) of a concave kite which is a particular solution to the five-body problem. Here, we study the position of the infinitesimal mass, , 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). where and are positive numbers. We call this as restricted concave kite five-body problem (RK5BP) (see Figure 1). The rest of the paper is organized as follows: in Section 2, we derive the general equations for (RK5BP) and list the main results. Three specific cases of CC’s for four primaries are investigated. In Section 3, the equations of motion for are set up in synodic system of coordinates. We explore the hill region and possible region of motion of according to the Jacobian constant. In Section 4, we discuss, analytically, the equilibrium points along the coordinate axes and numerically off the axes for different cases of CCs. In Section 5, the linear stability of equilibrium points is discussed, and conclusions are drawn in the last section.
2. Four-Body Concave Kite Central Configurations
Érdi and Czirják [16] studied the planar central configurations 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. The derived formulae represent exact analytical solutions of the four-body problem. They also discussed one convex and two concave cases of central configuration in detail. Benhammouda et al. [17] studied the central configuration of the kite four-body problems. They considered two different types of symmetrical configurations. In both cases, the existence of a continuous family of central configurations for positive masses is shown. They 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 five-body central configurations. This family is unusual in that it is a stacked central configuration, i.e., a subset of the points also forms a central configuration. Mello and Fernandes [19] studied the existence of kite central configurations in the planar four-body problem which lies on a common circle. They also proved the existence of kite central configurations in the spatial five-body problem which lies on a common sphere. Perez-Chavela and Santoprete [20] proved that there is a unique convex noncollinear central configuration 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 configuration possesses a symmetry line, and it is a kite-shaped quadrilateral. They also showed that there is exactly one convex noncollinear central configuration when the opposite masses are equal. Such a central configuration also possesses a symmetry line, and it is a rhombus. Corbera et al. [21] prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. They 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 infinitesimal body with variable mass in a five-body problem. The whole system forms a cyclic kite configuration. They also determined the positions of Lagrangian points and basins of attraction for the infinitesimal body. They also investigated the linear stability of the Lagrangian points and found that Lagrangian points are unstable.
We consider, initially, four primaries with masses , , and in a concave kite configuration. 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. The whole system rotates with constant angular velocity, the centrifugal force compensates for the Newtonian attraction, and the five bodies are in equilibrium in such a rotating system, the so-called “relative equilibria solutions.” The central configuration of this particular was derived in [17], and we will give a brief review of their results with minor improvement.
If we denote by the position of the body with mass , and by , the distance between the body with mass and the body with mass , then the algebraic system of equations that must be satisfied for the bodies to be in noncollinear central configuration iswhere and represent the area of the triangle determined by the sides and . After eliminating redundant equations due to the symmetries of the problem, i.e., , , , and , [17], we get the following two equations:
We will consider three special cases based on the relationship between the masses: Case I: Case II: and Case III:
2.1. Case I:
Let
Then, the simultaneous solution of equations (2) and (3) gives
With some elementary computations, the region of central configurations for positive masses is derived as follows:
The values of the mass ratios and are depicted in Figure 2.
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2.2. Case II: and
Let and then equations (2) and (3) givewhere and . The function is a necessary condition for the central configurations to exist. It is not possible to analytically solve for either or , and therefore, we use interpolation to write , where
This allows us to write which makes the quantitative analysis of significantly easier. The values of the mass ratio are depicted in Figure 3.
2.3. Case III:
Let and , then equations (2) and (3) givewhere
The function is a necessary condition for the central configurations to exist. It is not possible to analytically solve for either or ; therefore, we use interpolation to write , where
This allows us to write the mass ratio as a function of only. The function is an increasing function and attains its maximum at . The values of the mass ratios are depicted in Figure 4.
3. Setting Up of the Problem and Preliminary Results
We consider the motion of an infinitesimal mass in the gravitational field of the concave kite configuration described in Section 2. The equations of motion of the body arewhereis the effective potential.
Define a first Jacobi-type integral by
It is trivial to show that is the first integral of motion of the system (12) by proving that .
3.1. The Hill Sphere and Region of Motion for
The zero velocity curves in case I for and are given in Figure 5. The Hill spheres are the circular regions surrounding the four primaries masses shown in Figure 5. It is clear from equation (14) that . Therefore, will define a boundary between the region of permitted and prohibited motions. The region of possible motion of in case I for and is shown in Figure 6for six different values of Jacobian constant . The shaded regions represent the permitted regions of motion for the infinitesimal mass . It is numerically confirmed that the permitted regions of motion are connected when . For decreasing values of , the permitted regions of motion begin to disconnect. They completely disconnect at . It can be seen from Figure 6 that the disconnection occurs in six stages, and hence, the infinitesimal mass will be completely trapped in the shaded region when . Similar restrictions exist for all combinations of and . However, the disconnection doesn’t always occur in six stages. For small values of , the disconnection occurs in three stages. For example, when and , the complete disconnection is achieved in three stages at , and , and when and , the complete disconnection is achieved in two stages at and .
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The zero velocity curves in case II for and are given in Figure 7. The 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 , partially connected when and completely disconnected when . The regions of motion are given in Figure 8, and when , the complete disconnection is achieved in four stages as shown in Figure 9. We have shown the graph of zero-velocity curves and effective potential for case III for and 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 first instance, when , the complete disconnection is achieved in three stages while for the higher mass ratio of , it is achieved in four stages.
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4. Equilibrium Solutions
Equilibrium solutions of the restricted problem are the solutions of and . For RK5BP, is given in equation (13). The two first derivatives of are given as follows:
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 off the coordinate axes.
4.1. Equilibrium Solutions on the Coordinates Axes
4.1.1. Case I:
This case is investigated by Gao et al. [23] for equilibrium solutions and zero velocity curves. Our main focus for this case will be on the existence and uniqueness of equilibrium points. Consider in and to investigate the equilibrium solution on the axis. These solutions will be obtained from :
We divide to subintervals and . The restriction and will give the solutions which are outside the kite, and will give equilibrium solutions which are inside the concave kite. Consider , then can be simplified as
At , , and at , , and furthermore, is continuous on , and therefore, by the mean value theorem, there is at least one zero of when and . The derivative of is given by
Lemma 1. positive for and .
Proof. The only negative term of has an absolute maximum value of 2. Consider the first two terms of and simplifying, we get . Solving for and , one can easily see that and or and . Now, we need to show that when and since the absolute maximum value of the negative term is . We will need to show that the positive terms have a combined minimum value greater than 2. Consider the term . The minimum positive value of is 0.22, and the maximum value of is , and therefore, the minimum value of is as 1.24451. It is now clear that the minimum value of the positive terms of is greater than 2. This confirms our claim that .
This proves the existence of unique equilibrium solutions inside the concave kite on the axis.
Now consider and rewrite asWhen , the term will increase indefinitely and will dominate the remaining terms therefore when . Similarly for large values of , , and , it is, therefore, trivial to see that will dominate the negative terms of and will change the sign from negative to positive. This proves the existence of equilibrium solutions which are outside the concave kite.
Now, consider the case and asIt is immediately clear that when , , and hence, there are no equilibrium solutions. It can be proved in a way similar to the case of that there exists a unique equilibrium solution for each when . Therefore, we have a minimum of three solutions on the axis two of which will be inside the kite and one outside the kite. For example, when and , there are three equilibrium solutions, and when and , there are five equilibrium solutions on the axis.
Now consider in and . The equilibrium solutions on the axis will be obtained from the simultaneous solution of and wherewhere , and . Since and , and therefore, if is an equilibrium solution, then will also be an equilibrium solution. Rewrite asIt is numerically confirmed from Figure 13 that there are two equilibrium solutions on axis.
4.1.2. Case II: ,
Consider , , and in and to find equilibrium solutions on the axis. The equilibrium solutions on the axis will be obtained from
In the case when , the infinitesimal mass will be inside the kite. It will be outside the kite when or . Let and rewrite in simplified form
Because of the term , when . Similarly because of the term , when . Therefore, by the mean value theorem, there is at least one equilibrium solution inside the concave kite. The derivative of is
Since and , therefore, the positive terms of dominate the only negative term, which implies that . This proves the existence of a unique equilibrium solution inside the concave kite with two pairs of equal masses. Now, consider and rewrite as
The derivative of is
By the same argument as in the case of , the existence of a unique equilibrium solution outside the concave kite with two pairs of equal masses is proven. When , becomes
When and , the term dominates the two negative terms and , and hence, ; therefore, there are no equilibrium solutions when . Also implies that implies that , and when , we get
Since , by Descartes rule of sign, the polynomial has only one real root. It is numerically confirmed that this root is at . It is now trivial to show that . When , all the terms of are positive except , and therefore, for a sufficiently smaller value of , will become negative. The derivative of is
Since , , and therefore, only which is clearly dominated by the remaining positive terms, and hence, . This confirms the existence of exactly one equilibrium solution when .
Therefore, we have shown that there are three equilibrium solutions on the axis. One of the solutions is inside the kite when . These solutions are shown in Figure 14.
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To find equilibrium points on the axis, we will need to solve and , which are obtained from and , where
Since and , we will therefore only consider . It is numerically confirmed that has one or three zeros on the positive axis. When , these solutions occur when . It is symmetric counterparts are in the interval . For , there is only one zero when and another one when . These solutions are shown in Figure 14(b). Similarly, has two zeros when and . Therefore, there are no equilibrium solutions on the axis. The curve shaded black corresponds to the solutions of .
4.1.3. Case III
Consider , , and , then equilibria on the axis will be the solution of , where
The 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 axis, there are three equilibrium solutions one of which is inside the Kite. The inside solution is always between and , and the outside solutions are on either side of and outside the triangle formed by , and . These solutions are shown in Figure 15(a), where the black curve represents the solutions that are inside the kite between and .
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Similarly to get equilibrium solutions on axis, we substitute in and which gives and , where
In this case, there are only two equilibrium solutions as shown in Figure 15(b). The solutions are at . In this case, the central mass is much smaller than the masses at the vertices of the triangle.
4.2. Equilibrium Solutions Off the Coordinates Axes
4.2.1. Case I
To find a relationship between the masses and the equilibrium points, we define a new mass ratio :where
For all the masses to remain positive, we choose and from given in equation (6). Based on the values of , the region is divided into two parts, i.e., when and . When , is a decreasing function of for fixed value of and is an increasing function when as shown in Figure 16.
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Depending on the distance between the central mass and or , the number of equilibrium points changes from seven to thirteen. This effect is best explained by Figure 17. When , the equilibrium points are distributed as follows which shows a clear dependence on the ratio between and . The equilibrium solutions in case I are shown in Table 1 for different values of ranging between and .
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4.2.2. Case II and Case III
In the case when and (case II), there are a total of 9 equilibrium points. These include the three equilibriums which were shown to exist on the axis. The remaining 6 equilibrium points are off the axes. There is no change in the number of equilibrium points for the varying values of any of the masses are parameters. The parameter or the mass relation only affects the position of the equilibrium points. This effect is best explained in Figure 18.
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In the case when and , the number of equilibrium points depends on and hence . For , there are only four equilibrium points as shown in Figure 19(a). Two of the four equilibrium points are on the axis. For , there are seven equilibrium points with three inside the kite and four outside the kite, as shown in Figure 19(b). For , there are nine equilibrium points with five inside the kite and four outside the kite, shown in Figure 19(c). For , there are eleven equilibrium points with five inside the kite and six outside the kite, shown in Figure 19(d). For , the number of equilibrium points increases to 13 with five inside and 8 outside the kite; however, for , the number of equilibrium points reduces to nine with three inside the kite and six outside the kite. There is a clear trend of dependence of equilibrium points on when ; however, the trend is broken when , and therefore, we cannot conclude that the number of equilibrium either depends on the parameter or the masses.
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5. 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 and consider a small displacement to the new position . Using Taylor’s series expansion, a new set of second-order linear differential equations is obtained.
The matrix form of the linearized equations iswhere
For and to be a stable solution, all the eigenvalues of must be pure imaginary. To find these eigenvalues, write the characteristic polynomial for aswhere and . For to be pure imaginary, we must either and or and . We will numerically identify regions where at least one of the above conditions is satisfied. In case I, when , the stability regions for two values of 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 have taken many values of and tested the equilibrium points for stability; however, none of the equilibrium points are stable. Some representative examples are given in Tables 2 to 4. Similarly, stability regions for case II and case III are given in Figure 21, and however, it has been numerically confirmed 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.
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6. Conclusion
In this paper, we studied the motion of a body of negligible mass under the Newtonian gravitational attraction of four bodies (primaries). The 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 configuration based on the masses of the four primaries placed at the vertices of a concave kite. Using analytical techniques, regions of central configurations are derived for all three cases. To complement the analytical results, these regions are also explored numerically. The fifth mass (negligible mass with respect to the primaries) is moving in the gravitational field of four primaries. We have obtained equations of motion of the infinitesimal 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 . The equations of motion of are nonlinear ordinary differential equations. We did a qualitative analysis of the equation of motion of and found equilibrium solutions for the infinitesimal mass for all three different cases of the central configuration of the four primaries. We investigated the uniqueness and existence of equilibrium points on the coordinate axes and numerically off the coordinate axes. We can confirm that the number of equilibrium points is between 7 and 13 for three cases of central configuration for different values of mass parameters. The linear stability analysis revealed that none of the equilibrium points are stable. Additionally, we have obtained the permissible region of motions for in the field of four primaries in kite configuration according to the Jacobian constant. The 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
The authors declare that they have no conflicts of interest.