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Stability Analysis of the Rhomboidal Restricted Six-Body Problem

Stability Analysis of the Rhomboidal Restricted Six-Body Problem Hindawi Advances in Astronomy Volume 2021, Article ID 5575826, 15 pages https://doi.org/10.1155/2021/5575826 Research Article 1 1 2 1 M. A. R. Siddique , A. R. Kahsif , M. Shoaib , and S. Hussain Department of Mathematics, Capital University of Science and Technology, Zone-V, Islamabad, Pakistan Smart and Scientific Solutions, 32 Allerdyce Drive, Glasgow G156RY, UK Correspondence should be addressed to M. A. R. Siddique; dmt163002@cust.pk Received 16 February 2021; Revised 2 June 2021; Accepted 21 June 2021; Published 1 July 2021 Academic Editor: Yue Wang Copyright © 2021 M. A. R. Siddique et al. *is 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. We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. *ese masses always move in rhomboidal CC with diagonals 2a and 2b. *e sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. *e masses of the primaries are m � m � m � m and m � m � m 􏽥 . *e masses m and m 􏽥 are written as functions of parameters a and b such that 1 2 0 3 4 they always form a rhomboidal central configuration. *e evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle m and identify the value of Jacobian constant C for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for √� b ∈ (1/ 3 , 1.1394282249562009), there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable. 8472 for the Newtonian four-body problem. *e homo- 1. Introduction graphic solutions to the rhombus four-body problem are the In the restricted n-body problem, a body of infinitesimal mass variational minimizers of the Lagrangian action confined on moves under the gravitational influence of n − 1 massive a holonomically constrained rhombus loop space, according bodies called primaries. *e force exerted by the infinites- to Mansur and Offin [4]. *e approach used by Mansur et al. imal mass on the primaries can be ignored as it has a [5] to demonstrate spectral instability for the entire pa- negligible mass, whereas primaries revolve perpetually in rameter ranges of mass ratio and eccentricity e≥ 0 for the concentric circles around their common center of mass by homographic family of rhombus solutions within the four retaining a particular central configurations (CCs). *e CC degree of freedom parallelogram four-body problem is based play a vital role in the understanding of the n-body problem on a topological invariant called the Maslov index. Shi and in celestial mechanics. It can be used to find simple or special Xie [6], using analytical approach, have shown that there is solutions of the n-body problem since the geometry formed exactly one family of concave and one family of convex by the arrangement of the primaries remains constant for all central configurations in addition to the family of equilateral time. triangle configurations. Llibre and Mello [7] classified the *e CC for n> 3 has been investigated extensively after central configurations of the four-body problem. Liu and st being identified as the problem of the 21 century by Arnol’d Zhou [8] investigated the four-body problem with three [1]. Xia [2] used the method of analytical continuation to masses forming a Lagrangian triangle, used the bifurcation find exact number of central configurations for n positive diagram of linearly stable and unstable regions, and found masses. Hampton and Moeckel [3] studied the finiteness of two linearly stable subregions with respect to α, β, and e. relative equilibria of the four-body problem and showed that Deng et al. [9] investigated the CC of the four-body problem the number of relative equilibria is always between 32 and with equal masses and showed that, for the planar 2 Advances in Astronomy Newtonian four-body problem having adjacent equal masses another Eulerian collinear central configuration at any in- i.e., m � m ≠ m � m and equal lengths for the two di- stant in the three-body problem. Using the port-Hamilto- 1 2 3 4 nian approach, Liu and Dong [22] reformulated the Circular agonals, any convex noncollinear CC must have a symmetry and must be an isosceles trapezoid. *ey also showed that Restricted *ree-Body Problem (CRTBP) and obtained the when the length between m and m equals the length closed loop Hamiltonian by designing a control strategy 1 4 between m and m , the CC is also an isosceles trapezoid. (based on energy shaping and dissipation injection) as a 2 3 Marchesin [10] investigated the rhomboidal configuration candidate of the Lyapunov function that assures asymptotic stability with a central mass and two pairs of equal masses. stability. *is control technique also demonstrated global *e mass m is at the center of the configuration, and the stability within the CRTBP model’s application region. equilibria obtained in this case were all shown to be unstable. Corbera et al. [23] established that every four-body convex Shoaib et al. [11] established the central configuration for the center configuration with perpendicular diagonals will have rhomboidal 5-body problem and highlighted the regions in a kite configuration. Lara and Bengochea [24] investigated the phase plane where it is possible to have central con- the symmetric periodic orbits of the four-body system both theoretically and numerically. Alvarez-Ram´ırez and Medina figuration. On the axis of symmetry, Shoaib et al. [12] considered a symmetric five-body problem with three un- [25] studied the planar restricted five-body problem in equal collinear masses. *e remaining two masses were which the four primaries form a axisymmetric four-body symmetrically arranged on both sides of the axis of sym- central configuration and described the equilibrium points metry, and areas of feasible central configurations were that depend on the mass parameters of the primaries. identified analytically and numerically for the rhomboidal In the six-body problem, Mello et al. [26] demonstrated and triangular four- and five-bodies. Using Levi-Civita type the existence of three new families of stacked spatial central transformations, the equations of motion were regularized, configurations. Alsaedi et al. [27] used variational methods and the phase space for chaotic and periodic orbits was and computational algorithm to investigate the six-body explored using the Poincare surface of sections. Zotos [13] problem and its new style periodic solutions. Idrisia and numerically explored the restricted four-body problem with Ullah [28] studied the CC of the restricted six-body problem with the central body and showed that all the libration three equal masses with a dynamically stable triangular configuration and found that the linearly stable Lagrange (equilibrium) points exist on the concentric circles C , C , 1 2 points only exist when one of the three masses has a con- and C having center at the origin. *e libration points that siderably larger mass. Dewangan et al. [14] investigated the lie on circles C and C are unstable, while there are some 1 3 elliptic restricted four-body problem by considering radia- stable libration points on circle C . tion and oblateness effects; they considered a bigger primary In the present paper, we consider a restricted six-body as a radiation source and the other primaries to be of equal problem, where four of the primaries are at the vertices of a masses as oblate spheroid. *ey found that the equilibrium rhombus and the fifth mass is at the intersection of the two points to be linearly stable. Liu et al. [15] studied the four- diagonals. *ree of the primaries have equal masses of m � body problem and found that the boundaries of possible m � m � m and are located on the horizontal axis, and two 2 0 motions obey the change in parameter c E, that is, if the other equal mass primaries are located on the vertical axis 2 2 value of c E is less than or equal to a critical value (c E) , with masses m � m � m. Before dividing the equations of cr 3 4 then the system is stable. Ismail et al. [16] studied the four- motion for the restricted 6-body in Section 3, we find body problem by considering the effects of radiation pres- continuous families of central configurations for the sure and oblateness and used the Lyapunov function to show rhomboidal 5-body problem. *e rest of the paper is or- the stability of equilibrium points. Wang and Gao [17] did a ganized as follows: we investigate the Hill region and pos- numerical study of the restricted five-body problem re- sible region of motion of m according to the Jacobi constant garding the zero velocity surface and transfer trajectory by in Section 4. In Sections 5 and 6, we show the existence, considering four equal masses (primaries) forming a regular uniqueness, and stability of equilibrium points, respectively. tetrahedron configuration and the fifth (infinitesimal) mass Conclusions are given in Section 7. moving under the gravitational influence of the four pri- maries. *ey numerically simulated the zero velocity surface of the infinitesimal mass in the three-dimensional space and 2. Rhomboidal Central Configurations designed the transfer trajectory of the infinitesimal mass. In this section, we prove the existence and uniqueness of Suraj et al. [18] studied the five-body problem to investigate central configuration of a rhomboidal 5-body problem for the effects of perturbation parameter on the positions, positive masses. *e mass ratio is written as a function of “a” motion, and stability of the libration points due to the and “b” which can be used to find regions of central con- variable mass of the fifth mass. Li and Liao [19] obtained 695 figuration for the rhomboidal 5-body problem. *e classical families of Newtonian periodic planar collisionless orbits of equation of motion for the n-body problem has the following three-body systems with equal mass and zero angular form: momentum numerically. In the planar restricted three-body problem, Sosnitskii [20] investigated Lagrange stability and r − r .. j i m r � 􏽘 proved a theorem on the Lagrange stability of the infini- m m , 􏼌 􏼌 i i (1) i j 􏼌 􏼌3 􏼌 􏼌 j�0,j≠ i 􏼌 􏼌 r − r tesimal particle particularly for the circular restricted three- 􏼌 j i􏼌 body problem. Ding et al. [21] proved that there exists Advances in Astronomy 3 where the units are chosen so that the gravitational constant is equal to one. A central configuration is a particular m (x, y) configuration of the n bodies where the acceleration vector of each body is proportional to its position vector, and the m (0, b) constant of proportionality is the same for the n bodies. *erefore, a CC is a configuration that satisfies the following m (–a, 0) m (a, 0) equation: 2 m (0,0) 1 n x r − r 2 j i − ω r − c 􏼁 � 􏽘 m m , i 􏼌 􏼌 i j (2) 􏼌 􏼌 􏼌 􏼌 j�0,j≠ i 􏼌 􏼌 r − r 􏼌 􏼌 j i m (0, –b) where ω is angular speed and 􏽐 m r i i i�1 Figure 1: *e restricted six-body problem. c � , (3) 􏽐 m i�1 i which is the center of mass of the n bodies. We take the generality, we take ω � 1 and solve equations (7) and (8) position vector (see Figure 1) of the five primaries m , where simultaneously for m(a, b) and m 􏽥 (a, b): j � 0, 1, . . . , 4 as r � 0, N (a, b) m(a, b) � , iωt D (a, b) r � ae , (9) iωt N (a, b) r � − ae , (4) m 􏽥 m 􏽥 (a, b) � , D (a, b) iωt r � bie , iωt where r � − bie . 2 3/2 3 2 2 2 2 3 For i � 1 and 3 in equation (2), the CC equation for N (a, b) � 4a 􏼐a + b 􏼑 􏼒􏼐a + b 􏼑 − 8b 􏼓, (10) masses m and m is 1 3 􏽥 3/2 m(− a, 0) m(− 2a, 0) m(− a, b) 2 2 7 5 2 4 3 3 4 N (a, b) � 4 a + b − 4a − 8a b + 5a b − 4a b − ω (a, 0) � + + 􏼐 􏼑 􏼒 m 􏽥 3 3 3/2 2 2 a 8a a + b 􏼐 􏼑 􏽱������ 2 5 3 3 2 2 7 (5) + 10a b − 8a b a + b + 5b , m 􏽥 (− a, − b) + , 3/2 (11) 2 2 􏼐a + b 􏼑 7 5 2 3 4 6 D (a, b) � − 32a − 64a b − 32a b + 5a m(0, − b) m(a, − b) m(− a, − b) 􏽱������ − ω (0, b) � + + 3/2 3/2 2 2 2 2 4 2 3 3 2 4 6 2 2 􏼐a + b 􏼑 􏼐a + b 􏼑 +15a b − 64a b + 15a b + 5b 􏼑􏼠 a + b 􏼡. (6) m(0, − 2b) + , (12) 8b Defining τ � (m 􏽥 /m), then equations (9)–(11) give where m � m � m � m and m � m � m 􏽥 . *e positions 1 2 0 3 4 3/2 3 3 2 2 3 3 of the primaries are taken from equation (4). Writing 􏼐4 − 􏼐5b /a 􏼑􏼑􏼐1 + 􏼐b /a 􏼑􏼑 + 8􏼐b /a 􏼑 τ(a, b) � . (13) equations (5) and (6) in components’ form, we get 3/2 3 3 2 2 8􏼐b /a 􏼑 − 􏼐1 + 􏼐b /a 􏼑􏼑 5m 2m 􏽥 ω � + , 3 3/2 (7) 2 2 Again, let μ � (b/a) in equation (13); we get an alternate 4a 􏼐a + b 􏼑 form of equation (13): 3/2 1 2 m 􏽥 3 3 2 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 8μ + 􏼐4 − 5μ 􏼑􏼐1 + μ 􏼑 ω � m + + . (8) 3 3 3/2 2 2 τ(μ) � . (14) b 4b 3/2 􏼐a + b 􏼑 3 2 8μ − 􏼐1 + μ 􏼑 Because of the symmetry of the problem, the equations for m and m are identical to equations (7) and (8). *e 2 4 equation for m is identically zero. We have extended the Lemma 1. τ(μ) given by (13) is a continuous and strictly astrophysical model given in [29] by keeping the mass m at decreasing positive function in the interval √� √� the center. *e results of the model given in [29] can be ((1/ 3 ), 1.1394282249562009), lim τ(μ) � ∞, and μ⟶1/ 3 reproduced by taking m � 0 in our model. Without loss of τ(1.1394282249562009) � 0. 0 4 Advances in Astronomy Proof. One can easily check the continuity and 1.2 √� lim τ(μ) � ∞ and τ(1.1394282249562009) � 0 di- μ⟶1/ 3 1.0 rectly from equation (14). It means the value of τ(μ) which is the mass ratio will remain positive i.e., (τ ∈ [0,∞)) as the 0.8 ratio of the distance parameters varies between √� ((1/ 3 ), 1.1394282249562009]. Now we need to prove that 0.6 the function is decreasing and positive for the given interval. Taking the derivative of τ(μ), 0.4 ′ ′ τ (μ) � 5κ (μ), 0.2 􏽱����� 5/2 2 2 2 5 15μ μ + 1􏼒􏼐μ + 1􏼑 − 8􏼐μ + 1􏼑􏼓 (15) –1.0 –0.8 –0.5 –0.3 0.3 0.5 0.8 1.0 κ (μ) � , 3/2 2 3 –0.2 􏼒􏼐μ + 1􏼑 − 8μ 􏼓 –0.4 ′ ′ where τ (μ) is a constant multiple of κ (λ) given in [29] by Kulesza et al. *e proof will therefore be similar and can be –0.6 seen in [29]. Here, we present some particular cases for different –0.8 values of τ and μ. We show these different shapes of rhombus for different values of τ and μ in Figure 2. –1.0 (i) If τ � 1, then μ � 1; from equation (13), we get the –1.2 shape of true square with m � m 􏽥 � 1 and a � b � 1 Figure 2: Some possible shapes for the rhombus for different values (ii) If τ � 0.67, then μ � 1.04232; from equation (13), we of τ and μ. get m � 0.67m and b � 1.04232a (iii) If τ � 0.97, then μ � 1.00367; from equation (13), we where get m 􏽥 � 0.97m and b � 1.00367a 2 2 􏼐x + y 􏼑 *ere are two cases in which the RR6BP central con- 1 1 1 1 1 U(x, y) � + m + + + m 􏽥 + , 􏼠 􏼡 􏼠 􏼡 figuration degenerates. 2 r r r r r 50 51 52 53 54 (i) If μ � 1.1394282249562009, then τ � 0; this implies (19) m 􏽥 � 0 and b � 1.1394282249562009a √� which is the effective potential. *e mutual distances of m (ii) If μ � 1/ 3, then τ � ∞; this implies m � 0 and √� from the primaries in the corotating frame are b � (1/ 3 )a □ 􏽱������ 2 2 r � x + y , 3. Equations of Motion 􏽱���������� � 2 2 r � (x − 1) + y , In this section, we describe the motion of the infinitesimal 􏽱���������� � 2 2 body, m , under the gravitational attraction of the five (20) r � (x + 1) + y , primaries. We assume that the sixth body has a significantly 􏽱���������� � 2 2 smaller mass compared to the masses of the primary r � x + (y − b) , (m ≪ m , m , m , m , m ). On this basis, the sixth body acts 􏽱���������� � 5 0 1 2 3 4 2 2 as an infinitesimal test particle, and therefore, it does not r � x + (y + b) . influence the motion of the five primaries. In the RR6BP, the equations of motion of m are *e Jacobian constant is given by Curtis [30]: . . 2 2 2 .. r − r 5 j (21) C + U � 􏼒x + y 􏼓 � v . r � − 􏽘 m 􏼌 􏼌 , 5 (16) 􏼌 􏼌3 2 􏼌 􏼌 j�0 􏼌 􏼌 r − r 􏼌 􏼌 5 j For a given value of the Jacobi constant, v is only a where dot represents the derivative with respect to time. function of position in the rotating frame. Since v cannot be From here onward, without loss of generality, we take the negative, it must be true that value of a � 1. *e equations of motion of m in the syn- C + U≥ 0. (22) odical coordinates x and y are *e boundaries between forbidden and allowed regions x € − 2y _ � U , (17) of motion are found by setting v � 0, i.e., _ (18) y + 2x � U , C + U � 0. (23) y Advances in Astronomy 5 It is now trivial to show that C(x, y) is the first integral of C≥ − 2.20, − 2.85, and − 2.56, as shown in Figures 5–7, for motion of system (11) by proving that C(x, y) � 0. the above values of b. For the increasing values of C, the allowed region of motion (white region) becomes partially 4. The Hill Regions connecting at C � − 2.574, − 3.328, and − 3.342 and com- pletely connected at C � − 3.2, − 3.9, and − 3.5. It can be seen *e region of permitted motion is also known as the Hill from Figures 5–7 that the transition of motion from totally region, and the curves found by equation (16) for various disconnected to completely connected occurs in six stages values of C are known as the zero velocity curves. for b � 0.67, 0.97, and 1.13. For these values of b, m can *e zero velocity curves when freely move in the gravitational field of the CC region for b � 0.67, m � 0.07717, and m 􏽥 � 0.7879 and when C≥ − 2.2, − 2.85, and − 2.56 and cannot reach any of the b � 0.97, m � 0.4588, and m 􏽥 � 0.5766 are given in Figure 4, primaries for C≤ − 3.2, − 3.9, and − 3.5. and the corresponding Hill regions are given in Figure 3. We also give the region of possible motion of m for six different 5 5. Equilibrium Solutions values of Jacobi constants C in Figures 5–7 for mass pa- rameters b � 0.67, 0.97, and 1.13. *e shaded regions rep- Equilibrium solutions of the RR6BP are the solutions of resent the forbidden regions of motion for the infinitesimal U (x, y) � 0 and U (x, y) � 0. *e derivative U and U of x y x y mass m . It is numerically confirmed that the permitted the effective potential, given in equation (19), are found as regions are completely disconnected for follows: mx x − a x + 1 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ U � x − − m + 3/2 3/2 3/2 2 2 2 2 2 2 􏼐x + y 􏼑 􏼐(x − a) + y 􏼑 􏼐(x + 1) + y 􏼑 (24) 1 1 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ − mx 􏽥 + , 3/2 3/2 2 2 2 2 􏼐(y − b) + x 􏼑 􏼐(y + b) + x 􏼑 my 1 1 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ U � y − − my + 3/2 3/2 3/2 2 2 2 2 2 2 􏼐x + y 􏼑 􏼐(x − 1) + y 􏼑 􏼐(x + 1) + y 􏼑 (25) y − b b + y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ − m 􏽥 + . 3/2 3/2 2 2 2 2 􏼐(y − b) + x 􏼑 􏼐(y + b) + x 􏼑 √� 5.1. Equilibrium Solutions on the Coordinates’ Axes. Since the 5.2. 0< y< b; (1/ 3 )< b< 1.1394282249562009. Rewrite potential given in equation (19) is invariant under the the right hand side of equation (25) by taking into account symmetry (x, − y), (− x, y), and (− x, − y), we will restrict our that y ∈ (0, b): computation to the first quadrant: x≥ 0 and y≥ 0. Initially, we study the existence and number of equilibrium solutions m 2my 1 1 f (y) � y − − + m 􏽥 􏼠 − 􏼡. 2 3/2 2 2 on the axes and then off the coordinate axes. To study the y (y − b) (y + b) 􏼐1 + y 􏼑 equilibrium solutions on the y-axis, let x � 0; then, equa- (27) tions (24) and (25) are given as U � 0, At y ≈ 0, f (y)< 0; and, at y ≈ b, f (y)> 0; therefore, 1 1 by the mean value theorem, there is at least one zero of f (y) my 2my U � y − − when y ∈ (0, b). *e derivative of f (y) is given by 3/2 3/2 2 2 􏼐y 􏼑 􏼐1 + y 􏼑 (26) df (y) 1 1 3y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ � 1 + 2m + − 3 3/2 5/2 y − b b + y 2 2 dy ⎜ ⎟ y ⎛ ⎝ ⎞ ⎠ 􏼐y + 1􏼑 􏼐y + 1􏼑 − m 􏽥 + . 3/2 3/2 2 2 (28) 􏼐(y − b) 􏼑 􏼐(y + b) 􏼑 1 1 + 2m􏼠 − 􏼡. To solve U � 0, divide y into subintervals 0< y< b and 3 3 (y + b) (y − b) y> b. 6 Advances in Astronomy 4.5 4.0 3.5 1 1 3.0 0 0 –2 y –2 –1 –1 –1 –1 0 0 x x 1 1 –2 –2 (a) (b) Figure 3: *e effective potential. (a) b � 0.67; m � 0.0778; m 􏽥 � 0.7879. (b) b � 0.97; m � 0.4588; m 􏽥 � 0.5766. 2 2 5/2 *e only term (− (3y /(y + 1) )) that can make 5.4. Equilibrium Solutions off the Coordinates Axes. It is df (y)/dy negative for y ∈ (0, b) is dominated by the rest of numerically confirmed that, for b � 0.67, 0.97, and 1.13, the term in equation (26); therefore, (df (y)/dy)> 0. *is there are always a total of 12 equilibrium points. As shown in proves the existence of the unique equilibrium solution Figure 9, four of the equilibrium points are on the x-axis, inside rhombus on the y-axis. four on the y-axis, and remaining four of the equilibrium points are off the axes. Since the gravitational field is a function of mass parameters m(b), therefore the equilibrium 5.3. y> b. Now, consider the case y> b. Use equation (26) points change their positions around the primaries for and rewrite as changing values of b. It is numerically confirmed that ma- jority of equilibrium points are around the primaries along √� m 2my 1 1 the horizontal axis if b is around (1/ 3 ) as the masses on the f (y) � y − − − m 􏽥 􏼠 + 􏼡. 2 2 2 3/2 horizontal axis are dominant (see Figures 9(a)). For b≥ 1, the y (y − b) (y + b) 􏼐1 + y 􏼑 equilibrium points concentrated around the primaries on (29) the vertical axis (see Figure 9(b)). At y ≈ b, f (y)< 0; and, at y ≈ ∞, f (y)> 0; therefore, 2 2 by mean value theorem, there is at least one zero of f (y) 6. Stability Analysis when y ∈ (b,∞). *e derivative of f (y) is given by To study the stability of the equilibrium points obtained in the previous section, we will follow the standard linearization df (y) 1 1 3y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ procedure by linearizing the equation of motion of the in- � 1 + 2m + − 3/2 5/2 2 2 dy finitesimal mass. Let the location of an equilibrium point in 􏼐y + 1􏼑 􏼐y + 1􏼑 (30) the RR6BP be denoted by (x, y), and consider a small dis- placement (X, Y) from the point such that x + X and y + Y 1 1 + 2m 􏽥 􏼠 + 􏼡. 3 3 are the new position of the infinitesimal. Using Taylor’s series (y + b) (y − b) expansion in equations (17) and (18), we obtain a new set of second-order linear differential equations: Following the same procedure as given for 0< y< b, one can easily prove the uniqueness of the equilibrium solution € _ X − 2Y � XU + YU , xx xy for y> b. (31) € _ We discuss here two special cases of CC for Y + 2X � XU + YU . xy yy √� √� b ∈ ((1/ 3 ), 1.1394282249562009). When b � (1/ 3 ), *e matrix form of the linearized equations is then the masses on the horizontal axis are zero, i.e., m � 0. In this case, we only get the two equilibrium points along y-axis. _ 0 0 1 0 When b � 1.1394282249562009, then m � 0. In this case, we ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎟ ⎜ ⎟⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ get four equilibrium points along x-axis. *e positions of the ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ _ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 1 ⎟⎜ Y ⎟ ⎜ Y ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ � ⎜ ⎟⎜ ⎟. (32) masses and the corresponding equilibrium points for these ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ € ⎟ ⎜ ⎟⎜ _ ⎟ ⎜ ⎟ ⎜ U U 0 2 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ X ⎟ ⎜ X ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ xx xy ⎝ ⎠ two cases are shown in Figure 8. *e stability of these cases will be discussed in Section 6. € U U 0 − 2 Y xy yy Advances in Astronomy 7 2 2 4.5 4.5 1 1 4.0 4.0 3.5 y 0 y 0 3.0 3.5 –1 –1 2.5 3.0 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 x x (a) (b) Figure 4: *e evolution of zero velocity curves. (a) b � 0.67; m � 0.07717; m 􏽥 � 0.7879. (b) b � 0.97; m � 0.4588; m 􏽥 � 0.5766. 2 2 2 1 1 1 m m m 3 3 m m m m m 0 m y 0 y 0 1 y 1 1 2 2 2 0 m m m 4 4 –1 –1 –1 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) 2 2 2 1 1 1 m m 3 3 m m m m m y 0 1 y 0 y 0 2 1 2 1 2 m m m 0 0 0 m m m 4 4 4 –1 –1 –1 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 5: *e regions of motion of m (white region) for energy interval C ∈ (− 3.2, − 2.20) when b � 0.67 from the top-left corner to the bottom-right corner. (a) C � − 3.2. (b) C � − 3.157. (c) C � − 2.574. (d) C � − 2.2832. (e) C � − 2.269. (f) C � − 2.20. 8 Advances in Astronomy 2 2 2 m m 3 3 1 1 1 m m y 0 y 0 y 0 1 2 m m m m m m m 1 2 1 2 0 0 0 –1 –1 –1 m 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) 2 2 2 m m 3 3 1 1 1 m m m m m y 0 1 m y 0 y 0 1 1 2 2 m m 0 0 –1 –1 –1 m m 4 4 m –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 6: *e regions of motion of m (white region) for energy interval C ∈ (− 3.5, − 2.85) when b � 0.97 from the top-left corner to the bottom-right corner. (a) C � − 3.5. (b) C � − 3.47. (c) C � − 3.328. (d) C � − 3.1868. (e) C � − 3.121. (f) C � − 2.85. 2 2 2 m 3 1 1 1 m m m m m m 1 2 1 2 1 2 y 0 y 0 y 0 m m 0 0 –1 –1 –1 m m 4 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) Figure 7: Continued. Advances in Astronomy 9 2 2 2 m m m 3 3 3 1 1 1 m m m m y 0 y 0 1 2 y 0 1 2 m m m m 0 2 0 0 –1 –1 –1 m m m 4 4 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 7: *e regions of motion of m (white region) for energy interval C ∈ (− 3.9, − 2.56) when b � 1.13 from the top-left corner to the bottom-right corner. (a) C � − 3.9. (b) C � − 3.3842. (c) C � − 3.342. (d) C � − 2.655. (e) C � − 2.6360. (f) C � − 2.56. 1.5 1.0 0.5 L m L m L m L m m m 2 2 4 0 3 1 1 2 0 1 0.0 y 0 –0.5 –1 –1.0 –1.5 –2 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 –2 –1 0 1 2 x x (a) (b) Figure 8: (a) Equilibrium points (red color) along y-axis when m � 0. (b) Equilibrium points (red color) along x-axis when m 􏽥 � 0. *ese equations can also be written in the following *e characteristic polynomial for A is matrix form: 4 2 Λ + αΛ + β � 0, (35) (33) Ψ � AΨ, where α � 4 − U − U and β � U U − U . Let xx yy xx yy xy where Λ � λ; then, equation (35) reduces to ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ _ ⎟ λ + αλ + β � 0. (36) ⎜ ⎟ ⎜ Y ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟, Ψ � ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ € ⎟ ⎝ ⎠ Now, in order for a Lagrange point to be linearly stable to a small perturbation, all four roots, Λ, of equation (35) must (34) 0 0 1 0 be purely imaginary. *us, in turn, it implies that the two ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 1 ⎟ roots of equation (36). ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A � ⎜ ⎟. ⎜ ⎟ 􏽱������ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ U U 0 2 ⎠ xx xy − α ± α − 4β (37) U U 0 − 2 xy yy λ � , 2 10 Advances in Astronomy 1.5 1.0 1 L L L 10 9 0.5 L L L L m m m L m L m L L 8 6 5 7 m 2 0 1 8 2 6 0 5 7 y 0.0 y 0 –0.5 L L 12 11 –1 12 m –1.0 L 4 –1.5 –2 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 –2 –1 0 1 2 (a) (b) 1 L m L L L m L m 2 6 7 8 0 5 1 L L –1 12 11 –2 –2 –1 0 1 2 (c) Figure 9: (a–c) 12 equilibrium points (red colors) for different values of b � 0.67, 0.97, and 1.13, respectively. must be real and negative. For λ < 0, we must have (left to right) give the stability region in case (i) and case (ii), and their projections for b � 0.67, b � 1, and b � 1.13 are (i) α> 0 and 0< β≤ , shown in Figures 11 and 12, respectively. (38) We have tested a large number of equilibrium points for many values of b and found all of them unstable. In other or, (ii) α> 0 and α − 4β � 0. words, the intersection of U � 0 and U � 0 within x y 2 2 􏼈α> 0 and 0< β≤ (α /4)􏼉 and 􏼈α> 0 and α − 4β � 0􏼉 is an We will numerically identify regions when either con- empty set. Representative examples are given in Tables 1–3. dition (i) or condition (ii) is satisfied. Figures 10(a) and 10(b) Advances in Astronomy 11 2 2 –2 1.0 y 0 0.8 –1 0.6 0.6 0.8 1.0 –2 –2 0 –2 –1 0 1 2 x x (a) (b) Figure 10: (a-b) Shaded regions represent stability regions for cases (i) and (ii), respectively. 1.7 2.0 1.6 1.9 1.5 1.8 y y 1.4 1.7 1.3 1.6 1.2 –1.0 –0.5 0.0 0.5 1.0 –0.5 0.0 0.5 x x (a) (b) Figure 11: Continued. 12 Advances in Astronomy 2.1 2.0 1.9 1.8 1.7 1.6 –0.5 0.0 0.5 (c) Figure 11: Projection of stability regions for fixed values of the parameter b � 0.67, 1, and 1.13. 2 2 1 1 y 0 y 0 –1 –1 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 x x (a) (b) Figure 12: Continued. Advances in Astronomy 13 y 0 –1 –2 –2 –1 0 1 2 (c) Figure 12: Case (ii): projection of stability regions for fixed values of the parameter b � 0.67, 1, and 1.13. Table 1: Equilibrium points and stability analysis for b � 0.67. Equilibrium point Eigenvalues Stability L � ( ±1.342241, 0) ±1.660648, ±1.525769i Unstable 1,2 L � ( ±0.678932, 0) ±1.943106, ±1.168505i Unstable 3,4 L � (0, ±1.472145) ±1.218669, ±1.351617i Unstable 5,6 L � (0, ±0.180864) ±6.347782, ±4.5925990i Unstable 7,8 L � (±1.023953, ±0.479505) ∓0.913372 ±0.998959i Unstable 9,10,11,12 Table 2: Equilibrium points and stability analysis for b � 0.97. Equilibrium point Eigenvalues Stability L � ( ±1.637980, 0) ±1.480313, ±1.458732i Unstable 1,2 L � ( ±0.509436, 0) ±3.583445, ±2.544469i Unstable 3,4 L � (0, ±1.660744) ±1.454216, ±1.451670i Unstable 5,6 L � (0, ±0.464669) ±4.047077, ±2.908229i Unstable 7,8 L � (±0.950136, ± 0.874620) ∓0.909938 ±0.998607i Unstable 9,10,11,12 Table 3: Equilibrium points and stability analysis for b � 1.13. Equilibrium point Eigenvalues Stability L � ( ±1.751564, 0) ±1.5080962, ±1.499923i Unstable 1,2 L � ( ±0.495101, 0) ±4.863781, ±3.594754i Unstable 3,4 L � (0, ±1.413901) ±1.797668, ±1.600502i Unstable 5,6 L � (0, ±0.874617) ±2.535781, ±1.929722i Unstable 7,8 L � (±0.423616, ±1.103657) ∓0.903618 ±0.9990127i Unstable 9,10,11,12 14 Advances in Astronomy [9] Y. Deng, B. Li, and S. Zhang, “Four-body central configu- 7. Conclusions rations with adjacent equal masses,” Journal of Geometry and We have modeled and studied the Rhomboidal Restricted Physics, vol. 114, pp. 329–335, 2017. [10] M. Marchesin, “Stability of a rhomboidal configuration with a Six-Body Problem which has four masses at the vertices of central body,” Astrophysics and Space Science, vol. 362, no. 1, the rhombus, and the fifth mass is at the intersection of the two diagonals placed at the origin. It is assumed that m � [11] M. Shoaib, I. Faye, and A. Sivasankaran, “Some special so- m � m � m and m � m � m 􏽥 . *e primaries always move 2 0 3 4 lutions of the rhomboidal five-body problem,”vol. 1482, in the rhomboidal configuration. *e sixth mass m ≪ m , 5 i pp. 496–501, in Proceedings of the AIP Conference Proceed- where i � 0, . . . , 4, is moving in the gravitational field of the ings, vol. 1482, , American Institute of Physics, College Park, five primaries. To get rid of the time dependency, the MD, USA, June 2012. equation of motion of m is written in the rotating coor- [12] M. Shoaib, A. R. Kashif, and I. Szucs-Csillik, ¨ “On the planar dinate system. It is shown that, for the mass parameter √� central configurations of rhomboidal and triangular four- and b ∈ ((1/ 3 ), 1.1394282249562009), the total number of five-body problems,” Astrophysics and Space Science, vol. 362, equilibrium points is always 12. Out of the 12 equilibrium no. 10, p. 182, 2017. points, 4 are on the x-axis and 4 are on the y-axis. As the [13] E. E. Zotos, “Exploring the location and linear stability of the mass parameters vary, the equilibrium points also change equilibrium points in the equilateral restricted four-body their positions around the five primaries, and the number of problem,” International Journal of Bifurcation and Chaos, equilibrium points remains same. *e linear stability vol. 30, no. 10, Article ID 2050155, 2020. [14] R. R. Dewangan, A. Chakraborty, and A. Narayan, “Stability analysis revealed that none of the equilibrium points are of generalized elliptic restricted four body problem with ra- stable. *e permissible region of motion is also discussed diation and oblateness effects,” New Astronomy, vol. 78, according to the variation of Jacobian constant of C. We Article ID 101358, 2020. have identified the values of C at which the permissible [15] C. Liu, S.-P. Gong, and J.-F. 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Stability Analysis of the Rhomboidal Restricted Six-Body Problem

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Hindawi Advances in Astronomy Volume 2021, Article ID 5575826, 15 pages https://doi.org/10.1155/2021/5575826 Research Article 1 1 2 1 M. A. R. Siddique , A. R. Kahsif , M. Shoaib , and S. Hussain Department of Mathematics, Capital University of Science and Technology, Zone-V, Islamabad, Pakistan Smart and Scientific Solutions, 32 Allerdyce Drive, Glasgow G156RY, UK Correspondence should be addressed to M. A. R. Siddique; dmt163002@cust.pk Received 16 February 2021; Revised 2 June 2021; Accepted 21 June 2021; Published 1 July 2021 Academic Editor: Yue Wang Copyright © 2021 M. A. R. Siddique et al. *is 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. We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. *ese masses always move in rhomboidal CC with diagonals 2a and 2b. *e sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. *e masses of the primaries are m � m � m � m and m � m � m 􏽥 . *e masses m and m 􏽥 are written as functions of parameters a and b such that 1 2 0 3 4 they always form a rhomboidal central configuration. *e evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle m and identify the value of Jacobian constant C for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for √� b ∈ (1/ 3 , 1.1394282249562009), there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable. 8472 for the Newtonian four-body problem. *e homo- 1. Introduction graphic solutions to the rhombus four-body problem are the In the restricted n-body problem, a body of infinitesimal mass variational minimizers of the Lagrangian action confined on moves under the gravitational influence of n − 1 massive a holonomically constrained rhombus loop space, according bodies called primaries. *e force exerted by the infinites- to Mansur and Offin [4]. *e approach used by Mansur et al. imal mass on the primaries can be ignored as it has a [5] to demonstrate spectral instability for the entire pa- negligible mass, whereas primaries revolve perpetually in rameter ranges of mass ratio and eccentricity e≥ 0 for the concentric circles around their common center of mass by homographic family of rhombus solutions within the four retaining a particular central configurations (CCs). *e CC degree of freedom parallelogram four-body problem is based play a vital role in the understanding of the n-body problem on a topological invariant called the Maslov index. Shi and in celestial mechanics. It can be used to find simple or special Xie [6], using analytical approach, have shown that there is solutions of the n-body problem since the geometry formed exactly one family of concave and one family of convex by the arrangement of the primaries remains constant for all central configurations in addition to the family of equilateral time. triangle configurations. Llibre and Mello [7] classified the *e CC for n> 3 has been investigated extensively after central configurations of the four-body problem. Liu and st being identified as the problem of the 21 century by Arnol’d Zhou [8] investigated the four-body problem with three [1]. Xia [2] used the method of analytical continuation to masses forming a Lagrangian triangle, used the bifurcation find exact number of central configurations for n positive diagram of linearly stable and unstable regions, and found masses. Hampton and Moeckel [3] studied the finiteness of two linearly stable subregions with respect to α, β, and e. relative equilibria of the four-body problem and showed that Deng et al. [9] investigated the CC of the four-body problem the number of relative equilibria is always between 32 and with equal masses and showed that, for the planar 2 Advances in Astronomy Newtonian four-body problem having adjacent equal masses another Eulerian collinear central configuration at any in- i.e., m � m ≠ m � m and equal lengths for the two di- stant in the three-body problem. Using the port-Hamilto- 1 2 3 4 nian approach, Liu and Dong [22] reformulated the Circular agonals, any convex noncollinear CC must have a symmetry and must be an isosceles trapezoid. *ey also showed that Restricted *ree-Body Problem (CRTBP) and obtained the when the length between m and m equals the length closed loop Hamiltonian by designing a control strategy 1 4 between m and m , the CC is also an isosceles trapezoid. (based on energy shaping and dissipation injection) as a 2 3 Marchesin [10] investigated the rhomboidal configuration candidate of the Lyapunov function that assures asymptotic stability with a central mass and two pairs of equal masses. stability. *is control technique also demonstrated global *e mass m is at the center of the configuration, and the stability within the CRTBP model’s application region. equilibria obtained in this case were all shown to be unstable. Corbera et al. [23] established that every four-body convex Shoaib et al. [11] established the central configuration for the center configuration with perpendicular diagonals will have rhomboidal 5-body problem and highlighted the regions in a kite configuration. Lara and Bengochea [24] investigated the phase plane where it is possible to have central con- the symmetric periodic orbits of the four-body system both theoretically and numerically. Alvarez-Ram´ırez and Medina figuration. On the axis of symmetry, Shoaib et al. [12] considered a symmetric five-body problem with three un- [25] studied the planar restricted five-body problem in equal collinear masses. *e remaining two masses were which the four primaries form a axisymmetric four-body symmetrically arranged on both sides of the axis of sym- central configuration and described the equilibrium points metry, and areas of feasible central configurations were that depend on the mass parameters of the primaries. identified analytically and numerically for the rhomboidal In the six-body problem, Mello et al. [26] demonstrated and triangular four- and five-bodies. Using Levi-Civita type the existence of three new families of stacked spatial central transformations, the equations of motion were regularized, configurations. Alsaedi et al. [27] used variational methods and the phase space for chaotic and periodic orbits was and computational algorithm to investigate the six-body explored using the Poincare surface of sections. Zotos [13] problem and its new style periodic solutions. Idrisia and numerically explored the restricted four-body problem with Ullah [28] studied the CC of the restricted six-body problem with the central body and showed that all the libration three equal masses with a dynamically stable triangular configuration and found that the linearly stable Lagrange (equilibrium) points exist on the concentric circles C , C , 1 2 points only exist when one of the three masses has a con- and C having center at the origin. *e libration points that siderably larger mass. Dewangan et al. [14] investigated the lie on circles C and C are unstable, while there are some 1 3 elliptic restricted four-body problem by considering radia- stable libration points on circle C . tion and oblateness effects; they considered a bigger primary In the present paper, we consider a restricted six-body as a radiation source and the other primaries to be of equal problem, where four of the primaries are at the vertices of a masses as oblate spheroid. *ey found that the equilibrium rhombus and the fifth mass is at the intersection of the two points to be linearly stable. Liu et al. [15] studied the four- diagonals. *ree of the primaries have equal masses of m � body problem and found that the boundaries of possible m � m � m and are located on the horizontal axis, and two 2 0 motions obey the change in parameter c E, that is, if the other equal mass primaries are located on the vertical axis 2 2 value of c E is less than or equal to a critical value (c E) , with masses m � m � m. Before dividing the equations of cr 3 4 then the system is stable. Ismail et al. [16] studied the four- motion for the restricted 6-body in Section 3, we find body problem by considering the effects of radiation pres- continuous families of central configurations for the sure and oblateness and used the Lyapunov function to show rhomboidal 5-body problem. *e rest of the paper is or- the stability of equilibrium points. Wang and Gao [17] did a ganized as follows: we investigate the Hill region and pos- numerical study of the restricted five-body problem re- sible region of motion of m according to the Jacobi constant garding the zero velocity surface and transfer trajectory by in Section 4. In Sections 5 and 6, we show the existence, considering four equal masses (primaries) forming a regular uniqueness, and stability of equilibrium points, respectively. tetrahedron configuration and the fifth (infinitesimal) mass Conclusions are given in Section 7. moving under the gravitational influence of the four pri- maries. *ey numerically simulated the zero velocity surface of the infinitesimal mass in the three-dimensional space and 2. Rhomboidal Central Configurations designed the transfer trajectory of the infinitesimal mass. In this section, we prove the existence and uniqueness of Suraj et al. [18] studied the five-body problem to investigate central configuration of a rhomboidal 5-body problem for the effects of perturbation parameter on the positions, positive masses. *e mass ratio is written as a function of “a” motion, and stability of the libration points due to the and “b” which can be used to find regions of central con- variable mass of the fifth mass. Li and Liao [19] obtained 695 figuration for the rhomboidal 5-body problem. *e classical families of Newtonian periodic planar collisionless orbits of equation of motion for the n-body problem has the following three-body systems with equal mass and zero angular form: momentum numerically. In the planar restricted three-body problem, Sosnitskii [20] investigated Lagrange stability and r − r .. j i m r � 􏽘 proved a theorem on the Lagrange stability of the infini- m m , 􏼌 􏼌 i i (1) i j 􏼌 􏼌3 􏼌 􏼌 j�0,j≠ i 􏼌 􏼌 r − r tesimal particle particularly for the circular restricted three- 􏼌 j i􏼌 body problem. Ding et al. [21] proved that there exists Advances in Astronomy 3 where the units are chosen so that the gravitational constant is equal to one. A central configuration is a particular m (x, y) configuration of the n bodies where the acceleration vector of each body is proportional to its position vector, and the m (0, b) constant of proportionality is the same for the n bodies. *erefore, a CC is a configuration that satisfies the following m (–a, 0) m (a, 0) equation: 2 m (0,0) 1 n x r − r 2 j i − ω r − c 􏼁 � 􏽘 m m , i 􏼌 􏼌 i j (2) 􏼌 􏼌 􏼌 􏼌 j�0,j≠ i 􏼌 􏼌 r − r 􏼌 􏼌 j i m (0, –b) where ω is angular speed and 􏽐 m r i i i�1 Figure 1: *e restricted six-body problem. c � , (3) 􏽐 m i�1 i which is the center of mass of the n bodies. We take the generality, we take ω � 1 and solve equations (7) and (8) position vector (see Figure 1) of the five primaries m , where simultaneously for m(a, b) and m 􏽥 (a, b): j � 0, 1, . . . , 4 as r � 0, N (a, b) m(a, b) � , iωt D (a, b) r � ae , (9) iωt N (a, b) r � − ae , (4) m 􏽥 m 􏽥 (a, b) � , D (a, b) iωt r � bie , iωt where r � − bie . 2 3/2 3 2 2 2 2 3 For i � 1 and 3 in equation (2), the CC equation for N (a, b) � 4a 􏼐a + b 􏼑 􏼒􏼐a + b 􏼑 − 8b 􏼓, (10) masses m and m is 1 3 􏽥 3/2 m(− a, 0) m(− 2a, 0) m(− a, b) 2 2 7 5 2 4 3 3 4 N (a, b) � 4 a + b − 4a − 8a b + 5a b − 4a b − ω (a, 0) � + + 􏼐 􏼑 􏼒 m 􏽥 3 3 3/2 2 2 a 8a a + b 􏼐 􏼑 􏽱������ 2 5 3 3 2 2 7 (5) + 10a b − 8a b a + b + 5b , m 􏽥 (− a, − b) + , 3/2 (11) 2 2 􏼐a + b 􏼑 7 5 2 3 4 6 D (a, b) � − 32a − 64a b − 32a b + 5a m(0, − b) m(a, − b) m(− a, − b) 􏽱������ − ω (0, b) � + + 3/2 3/2 2 2 2 2 4 2 3 3 2 4 6 2 2 􏼐a + b 􏼑 􏼐a + b 􏼑 +15a b − 64a b + 15a b + 5b 􏼑􏼠 a + b 􏼡. (6) m(0, − 2b) + , (12) 8b Defining τ � (m 􏽥 /m), then equations (9)–(11) give where m � m � m � m and m � m � m 􏽥 . *e positions 1 2 0 3 4 3/2 3 3 2 2 3 3 of the primaries are taken from equation (4). Writing 􏼐4 − 􏼐5b /a 􏼑􏼑􏼐1 + 􏼐b /a 􏼑􏼑 + 8􏼐b /a 􏼑 τ(a, b) � . (13) equations (5) and (6) in components’ form, we get 3/2 3 3 2 2 8􏼐b /a 􏼑 − 􏼐1 + 􏼐b /a 􏼑􏼑 5m 2m 􏽥 ω � + , 3 3/2 (7) 2 2 Again, let μ � (b/a) in equation (13); we get an alternate 4a 􏼐a + b 􏼑 form of equation (13): 3/2 1 2 m 􏽥 3 3 2 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ 8μ + 􏼐4 − 5μ 􏼑􏼐1 + μ 􏼑 ω � m + + . (8) 3 3 3/2 2 2 τ(μ) � . (14) b 4b 3/2 􏼐a + b 􏼑 3 2 8μ − 􏼐1 + μ 􏼑 Because of the symmetry of the problem, the equations for m and m are identical to equations (7) and (8). *e 2 4 equation for m is identically zero. We have extended the Lemma 1. τ(μ) given by (13) is a continuous and strictly astrophysical model given in [29] by keeping the mass m at decreasing positive function in the interval √� √� the center. *e results of the model given in [29] can be ((1/ 3 ), 1.1394282249562009), lim τ(μ) � ∞, and μ⟶1/ 3 reproduced by taking m � 0 in our model. Without loss of τ(1.1394282249562009) � 0. 0 4 Advances in Astronomy Proof. One can easily check the continuity and 1.2 √� lim τ(μ) � ∞ and τ(1.1394282249562009) � 0 di- μ⟶1/ 3 1.0 rectly from equation (14). It means the value of τ(μ) which is the mass ratio will remain positive i.e., (τ ∈ [0,∞)) as the 0.8 ratio of the distance parameters varies between √� ((1/ 3 ), 1.1394282249562009]. Now we need to prove that 0.6 the function is decreasing and positive for the given interval. Taking the derivative of τ(μ), 0.4 ′ ′ τ (μ) � 5κ (μ), 0.2 􏽱����� 5/2 2 2 2 5 15μ μ + 1􏼒􏼐μ + 1􏼑 − 8􏼐μ + 1􏼑􏼓 (15) –1.0 –0.8 –0.5 –0.3 0.3 0.5 0.8 1.0 κ (μ) � , 3/2 2 3 –0.2 􏼒􏼐μ + 1􏼑 − 8μ 􏼓 –0.4 ′ ′ where τ (μ) is a constant multiple of κ (λ) given in [29] by Kulesza et al. *e proof will therefore be similar and can be –0.6 seen in [29]. Here, we present some particular cases for different –0.8 values of τ and μ. We show these different shapes of rhombus for different values of τ and μ in Figure 2. –1.0 (i) If τ � 1, then μ � 1; from equation (13), we get the –1.2 shape of true square with m � m 􏽥 � 1 and a � b � 1 Figure 2: Some possible shapes for the rhombus for different values (ii) If τ � 0.67, then μ � 1.04232; from equation (13), we of τ and μ. get m � 0.67m and b � 1.04232a (iii) If τ � 0.97, then μ � 1.00367; from equation (13), we where get m 􏽥 � 0.97m and b � 1.00367a 2 2 􏼐x + y 􏼑 *ere are two cases in which the RR6BP central con- 1 1 1 1 1 U(x, y) � + m + + + m 􏽥 + , 􏼠 􏼡 􏼠 􏼡 figuration degenerates. 2 r r r r r 50 51 52 53 54 (i) If μ � 1.1394282249562009, then τ � 0; this implies (19) m 􏽥 � 0 and b � 1.1394282249562009a √� which is the effective potential. *e mutual distances of m (ii) If μ � 1/ 3, then τ � ∞; this implies m � 0 and √� from the primaries in the corotating frame are b � (1/ 3 )a □ 􏽱������ 2 2 r � x + y , 3. Equations of Motion 􏽱���������� � 2 2 r � (x − 1) + y , In this section, we describe the motion of the infinitesimal 􏽱���������� � 2 2 body, m , under the gravitational attraction of the five (20) r � (x + 1) + y , primaries. We assume that the sixth body has a significantly 􏽱���������� � 2 2 smaller mass compared to the masses of the primary r � x + (y − b) , (m ≪ m , m , m , m , m ). On this basis, the sixth body acts 􏽱���������� � 5 0 1 2 3 4 2 2 as an infinitesimal test particle, and therefore, it does not r � x + (y + b) . influence the motion of the five primaries. In the RR6BP, the equations of motion of m are *e Jacobian constant is given by Curtis [30]: . . 2 2 2 .. r − r 5 j (21) C + U � 􏼒x + y 􏼓 � v . r � − 􏽘 m 􏼌 􏼌 , 5 (16) 􏼌 􏼌3 2 􏼌 􏼌 j�0 􏼌 􏼌 r − r 􏼌 􏼌 5 j For a given value of the Jacobi constant, v is only a where dot represents the derivative with respect to time. function of position in the rotating frame. Since v cannot be From here onward, without loss of generality, we take the negative, it must be true that value of a � 1. *e equations of motion of m in the syn- C + U≥ 0. (22) odical coordinates x and y are *e boundaries between forbidden and allowed regions x € − 2y _ � U , (17) of motion are found by setting v � 0, i.e., _ (18) y + 2x � U , C + U � 0. (23) y Advances in Astronomy 5 It is now trivial to show that C(x, y) is the first integral of C≥ − 2.20, − 2.85, and − 2.56, as shown in Figures 5–7, for motion of system (11) by proving that C(x, y) � 0. the above values of b. For the increasing values of C, the allowed region of motion (white region) becomes partially 4. The Hill Regions connecting at C � − 2.574, − 3.328, and − 3.342 and com- pletely connected at C � − 3.2, − 3.9, and − 3.5. It can be seen *e region of permitted motion is also known as the Hill from Figures 5–7 that the transition of motion from totally region, and the curves found by equation (16) for various disconnected to completely connected occurs in six stages values of C are known as the zero velocity curves. for b � 0.67, 0.97, and 1.13. For these values of b, m can *e zero velocity curves when freely move in the gravitational field of the CC region for b � 0.67, m � 0.07717, and m 􏽥 � 0.7879 and when C≥ − 2.2, − 2.85, and − 2.56 and cannot reach any of the b � 0.97, m � 0.4588, and m 􏽥 � 0.5766 are given in Figure 4, primaries for C≤ − 3.2, − 3.9, and − 3.5. and the corresponding Hill regions are given in Figure 3. We also give the region of possible motion of m for six different 5 5. Equilibrium Solutions values of Jacobi constants C in Figures 5–7 for mass pa- rameters b � 0.67, 0.97, and 1.13. *e shaded regions rep- Equilibrium solutions of the RR6BP are the solutions of resent the forbidden regions of motion for the infinitesimal U (x, y) � 0 and U (x, y) � 0. *e derivative U and U of x y x y mass m . It is numerically confirmed that the permitted the effective potential, given in equation (19), are found as regions are completely disconnected for follows: mx x − a x + 1 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ U � x − − m + 3/2 3/2 3/2 2 2 2 2 2 2 􏼐x + y 􏼑 􏼐(x − a) + y 􏼑 􏼐(x + 1) + y 􏼑 (24) 1 1 ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ − mx 􏽥 + , 3/2 3/2 2 2 2 2 􏼐(y − b) + x 􏼑 􏼐(y + b) + x 􏼑 my 1 1 ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ U � y − − my + 3/2 3/2 3/2 2 2 2 2 2 2 􏼐x + y 􏼑 􏼐(x − 1) + y 􏼑 􏼐(x + 1) + y 􏼑 (25) y − b b + y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ − m 􏽥 + . 3/2 3/2 2 2 2 2 􏼐(y − b) + x 􏼑 􏼐(y + b) + x 􏼑 √� 5.1. Equilibrium Solutions on the Coordinates’ Axes. Since the 5.2. 0< y< b; (1/ 3 )< b< 1.1394282249562009. Rewrite potential given in equation (19) is invariant under the the right hand side of equation (25) by taking into account symmetry (x, − y), (− x, y), and (− x, − y), we will restrict our that y ∈ (0, b): computation to the first quadrant: x≥ 0 and y≥ 0. Initially, we study the existence and number of equilibrium solutions m 2my 1 1 f (y) � y − − + m 􏽥 􏼠 − 􏼡. 2 3/2 2 2 on the axes and then off the coordinate axes. To study the y (y − b) (y + b) 􏼐1 + y 􏼑 equilibrium solutions on the y-axis, let x � 0; then, equa- (27) tions (24) and (25) are given as U � 0, At y ≈ 0, f (y)< 0; and, at y ≈ b, f (y)> 0; therefore, 1 1 by the mean value theorem, there is at least one zero of f (y) my 2my U � y − − when y ∈ (0, b). *e derivative of f (y) is given by 3/2 3/2 2 2 􏼐y 􏼑 􏼐1 + y 􏼑 (26) df (y) 1 1 3y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ � 1 + 2m + − 3 3/2 5/2 y − b b + y 2 2 dy ⎜ ⎟ y ⎛ ⎝ ⎞ ⎠ 􏼐y + 1􏼑 􏼐y + 1􏼑 − m 􏽥 + . 3/2 3/2 2 2 (28) 􏼐(y − b) 􏼑 􏼐(y + b) 􏼑 1 1 + 2m􏼠 − 􏼡. To solve U � 0, divide y into subintervals 0< y< b and 3 3 (y + b) (y − b) y> b. 6 Advances in Astronomy 4.5 4.0 3.5 1 1 3.0 0 0 –2 y –2 –1 –1 –1 –1 0 0 x x 1 1 –2 –2 (a) (b) Figure 3: *e effective potential. (a) b � 0.67; m � 0.0778; m 􏽥 � 0.7879. (b) b � 0.97; m � 0.4588; m 􏽥 � 0.5766. 2 2 5/2 *e only term (− (3y /(y + 1) )) that can make 5.4. Equilibrium Solutions off the Coordinates Axes. It is df (y)/dy negative for y ∈ (0, b) is dominated by the rest of numerically confirmed that, for b � 0.67, 0.97, and 1.13, the term in equation (26); therefore, (df (y)/dy)> 0. *is there are always a total of 12 equilibrium points. As shown in proves the existence of the unique equilibrium solution Figure 9, four of the equilibrium points are on the x-axis, inside rhombus on the y-axis. four on the y-axis, and remaining four of the equilibrium points are off the axes. Since the gravitational field is a function of mass parameters m(b), therefore the equilibrium 5.3. y> b. Now, consider the case y> b. Use equation (26) points change their positions around the primaries for and rewrite as changing values of b. It is numerically confirmed that ma- jority of equilibrium points are around the primaries along √� m 2my 1 1 the horizontal axis if b is around (1/ 3 ) as the masses on the f (y) � y − − − m 􏽥 􏼠 + 􏼡. 2 2 2 3/2 horizontal axis are dominant (see Figures 9(a)). For b≥ 1, the y (y − b) (y + b) 􏼐1 + y 􏼑 equilibrium points concentrated around the primaries on (29) the vertical axis (see Figure 9(b)). At y ≈ b, f (y)< 0; and, at y ≈ ∞, f (y)> 0; therefore, 2 2 by mean value theorem, there is at least one zero of f (y) 6. Stability Analysis when y ∈ (b,∞). *e derivative of f (y) is given by To study the stability of the equilibrium points obtained in the previous section, we will follow the standard linearization df (y) 1 1 3y ⎛ ⎜ ⎞ ⎟ ⎝ ⎠ procedure by linearizing the equation of motion of the in- � 1 + 2m + − 3/2 5/2 2 2 dy finitesimal mass. Let the location of an equilibrium point in 􏼐y + 1􏼑 􏼐y + 1􏼑 (30) the RR6BP be denoted by (x, y), and consider a small dis- placement (X, Y) from the point such that x + X and y + Y 1 1 + 2m 􏽥 􏼠 + 􏼡. 3 3 are the new position of the infinitesimal. Using Taylor’s series (y + b) (y − b) expansion in equations (17) and (18), we obtain a new set of second-order linear differential equations: Following the same procedure as given for 0< y< b, one can easily prove the uniqueness of the equilibrium solution € _ X − 2Y � XU + YU , xx xy for y> b. (31) € _ We discuss here two special cases of CC for Y + 2X � XU + YU . xy yy √� √� b ∈ ((1/ 3 ), 1.1394282249562009). When b � (1/ 3 ), *e matrix form of the linearized equations is then the masses on the horizontal axis are zero, i.e., m � 0. In this case, we only get the two equilibrium points along y-axis. _ 0 0 1 0 When b � 1.1394282249562009, then m � 0. In this case, we ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎟ ⎜ ⎟⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ get four equilibrium points along x-axis. *e positions of the ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ _ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 1 ⎟⎜ Y ⎟ ⎜ Y ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ � ⎜ ⎟⎜ ⎟. (32) masses and the corresponding equilibrium points for these ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ € ⎟ ⎜ ⎟⎜ _ ⎟ ⎜ ⎟ ⎜ U U 0 2 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ X ⎟ ⎜ X ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ xx xy ⎝ ⎠ two cases are shown in Figure 8. *e stability of these cases will be discussed in Section 6. € U U 0 − 2 Y xy yy Advances in Astronomy 7 2 2 4.5 4.5 1 1 4.0 4.0 3.5 y 0 y 0 3.0 3.5 –1 –1 2.5 3.0 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 x x (a) (b) Figure 4: *e evolution of zero velocity curves. (a) b � 0.67; m � 0.07717; m 􏽥 � 0.7879. (b) b � 0.97; m � 0.4588; m 􏽥 � 0.5766. 2 2 2 1 1 1 m m m 3 3 m m m m m 0 m y 0 y 0 1 y 1 1 2 2 2 0 m m m 4 4 –1 –1 –1 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) 2 2 2 1 1 1 m m 3 3 m m m m m y 0 1 y 0 y 0 2 1 2 1 2 m m m 0 0 0 m m m 4 4 4 –1 –1 –1 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 5: *e regions of motion of m (white region) for energy interval C ∈ (− 3.2, − 2.20) when b � 0.67 from the top-left corner to the bottom-right corner. (a) C � − 3.2. (b) C � − 3.157. (c) C � − 2.574. (d) C � − 2.2832. (e) C � − 2.269. (f) C � − 2.20. 8 Advances in Astronomy 2 2 2 m m 3 3 1 1 1 m m y 0 y 0 y 0 1 2 m m m m m m m 1 2 1 2 0 0 0 –1 –1 –1 m 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) 2 2 2 m m 3 3 1 1 1 m m m m m y 0 1 m y 0 y 0 1 1 2 2 m m 0 0 –1 –1 –1 m m 4 4 m –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 6: *e regions of motion of m (white region) for energy interval C ∈ (− 3.5, − 2.85) when b � 0.97 from the top-left corner to the bottom-right corner. (a) C � − 3.5. (b) C � − 3.47. (c) C � − 3.328. (d) C � − 3.1868. (e) C � − 3.121. (f) C � − 2.85. 2 2 2 m 3 1 1 1 m m m m m m 1 2 1 2 1 2 y 0 y 0 y 0 m m 0 0 –1 –1 –1 m m 4 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (a) (b) (c) Figure 7: Continued. Advances in Astronomy 9 2 2 2 m m m 3 3 3 1 1 1 m m m m y 0 y 0 1 2 y 0 1 2 m m m m 0 2 0 0 –1 –1 –1 m m m 4 4 4 –2 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x x x (d) (e) (f) Figure 7: *e regions of motion of m (white region) for energy interval C ∈ (− 3.9, − 2.56) when b � 1.13 from the top-left corner to the bottom-right corner. (a) C � − 3.9. (b) C � − 3.3842. (c) C � − 3.342. (d) C � − 2.655. (e) C � − 2.6360. (f) C � − 2.56. 1.5 1.0 0.5 L m L m L m L m m m 2 2 4 0 3 1 1 2 0 1 0.0 y 0 –0.5 –1 –1.0 –1.5 –2 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 –2 –1 0 1 2 x x (a) (b) Figure 8: (a) Equilibrium points (red color) along y-axis when m � 0. (b) Equilibrium points (red color) along x-axis when m 􏽥 � 0. *ese equations can also be written in the following *e characteristic polynomial for A is matrix form: 4 2 Λ + αΛ + β � 0, (35) (33) Ψ � AΨ, where α � 4 − U − U and β � U U − U . Let xx yy xx yy xy where Λ � λ; then, equation (35) reduces to ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ _ ⎟ λ + αλ + β � 0. (36) ⎜ ⎟ ⎜ Y ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟, Ψ � ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ € ⎟ ⎝ ⎠ Now, in order for a Lagrange point to be linearly stable to a small perturbation, all four roots, Λ, of equation (35) must (34) 0 0 1 0 be purely imaginary. *us, in turn, it implies that the two ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 1 ⎟ roots of equation (36). ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A � ⎜ ⎟. ⎜ ⎟ 􏽱������ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ U U 0 2 ⎠ xx xy − α ± α − 4β (37) U U 0 − 2 xy yy λ � , 2 10 Advances in Astronomy 1.5 1.0 1 L L L 10 9 0.5 L L L L m m m L m L m L L 8 6 5 7 m 2 0 1 8 2 6 0 5 7 y 0.0 y 0 –0.5 L L 12 11 –1 12 m –1.0 L 4 –1.5 –2 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 –2 –1 0 1 2 (a) (b) 1 L m L L L m L m 2 6 7 8 0 5 1 L L –1 12 11 –2 –2 –1 0 1 2 (c) Figure 9: (a–c) 12 equilibrium points (red colors) for different values of b � 0.67, 0.97, and 1.13, respectively. must be real and negative. For λ < 0, we must have (left to right) give the stability region in case (i) and case (ii), and their projections for b � 0.67, b � 1, and b � 1.13 are (i) α> 0 and 0< β≤ , shown in Figures 11 and 12, respectively. (38) We have tested a large number of equilibrium points for many values of b and found all of them unstable. In other or, (ii) α> 0 and α − 4β � 0. words, the intersection of U � 0 and U � 0 within x y 2 2 􏼈α> 0 and 0< β≤ (α /4)􏼉 and 􏼈α> 0 and α − 4β � 0􏼉 is an We will numerically identify regions when either con- empty set. Representative examples are given in Tables 1–3. dition (i) or condition (ii) is satisfied. Figures 10(a) and 10(b) Advances in Astronomy 11 2 2 –2 1.0 y 0 0.8 –1 0.6 0.6 0.8 1.0 –2 –2 0 –2 –1 0 1 2 x x (a) (b) Figure 10: (a-b) Shaded regions represent stability regions for cases (i) and (ii), respectively. 1.7 2.0 1.6 1.9 1.5 1.8 y y 1.4 1.7 1.3 1.6 1.2 –1.0 –0.5 0.0 0.5 1.0 –0.5 0.0 0.5 x x (a) (b) Figure 11: Continued. 12 Advances in Astronomy 2.1 2.0 1.9 1.8 1.7 1.6 –0.5 0.0 0.5 (c) Figure 11: Projection of stability regions for fixed values of the parameter b � 0.67, 1, and 1.13. 2 2 1 1 y 0 y 0 –1 –1 –2 –2 –2 –1 0 1 2 –2 –1 0 1 2 x x (a) (b) Figure 12: Continued. Advances in Astronomy 13 y 0 –1 –2 –2 –1 0 1 2 (c) Figure 12: Case (ii): projection of stability regions for fixed values of the parameter b � 0.67, 1, and 1.13. Table 1: Equilibrium points and stability analysis for b � 0.67. Equilibrium point Eigenvalues Stability L � ( ±1.342241, 0) ±1.660648, ±1.525769i Unstable 1,2 L � ( ±0.678932, 0) ±1.943106, ±1.168505i Unstable 3,4 L � (0, ±1.472145) ±1.218669, ±1.351617i Unstable 5,6 L � (0, ±0.180864) ±6.347782, ±4.5925990i Unstable 7,8 L � (±1.023953, ±0.479505) ∓0.913372 ±0.998959i Unstable 9,10,11,12 Table 2: Equilibrium points and stability analysis for b � 0.97. Equilibrium point Eigenvalues Stability L � ( ±1.637980, 0) ±1.480313, ±1.458732i Unstable 1,2 L � ( ±0.509436, 0) ±3.583445, ±2.544469i Unstable 3,4 L � (0, ±1.660744) ±1.454216, ±1.451670i Unstable 5,6 L � (0, ±0.464669) ±4.047077, ±2.908229i Unstable 7,8 L � (±0.950136, ± 0.874620) ∓0.909938 ±0.998607i Unstable 9,10,11,12 Table 3: Equilibrium points and stability analysis for b � 1.13. 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Advances in AstronomyHindawi Publishing Corporation

Published: Jul 1, 2021

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