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 + 8b /a τ(a, b) � . (13) equations (5) and (6) in components’ form, we get 3/2 3 3 2 2 8b /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 + 1μ + 1 − 8μ + 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. Li, “Stability of the coplanar regions of motion are partially disconnected and fully planetary four-body system,” Research in Astronomy and disconnected. Astrophysics, vol. 20, no. 9, p. 144, 2020. [16] M. Ismail, S. H. Younis, and G. F. Mohamdien, “Stability analysis in the restricted four body problem with oblatness Data Availability and radiation pressure,” Al-Azhar Bulletin of Science, vol. 31, No data were used in the study. no. 1-B, pp. 12–22, 2020. [17] R. Wang and F. Gao, “Numerical study of the zero velocity surface and transfer trajectory of a circular restricted ﬁve- Conflicts of Interest body problem,” Mathematical Problems in Engineering, vol. 2018, Article ID 7489120, 10 pages, 2018. *e authors declare that they have no conﬂicts of interest. [18] M. S. Suraj, E. I. Abouelmagd, R. Aggarwal, and A. Mittal, “*e analysis of restricted ﬁve-body problem within frame of variable mass,” New Astronomy, vol. 70, pp. 12–21, 2019. References [19] X. Li and S. Liao, “More than six hundred new families of [1] V. I. Arnol’d, “Small denominators and problems of stability Newtonian periodic planar collisionless three-body orbits,” of motion in classical and celestial mechanics,” Uspekhi Science China Physics, Mechanics & Astronomy, vol. 60, no. 12, Matematicheskikh Nauk, vol. 18, no. 6, pp. 91–192, 1963. pp. 1–7, 2017. [2] Z. Xia, “Central conﬁgurations with many small masses,” [20] S. P. Sosnitskii, “On the Lagrange stability of motion in the Journal of Diﬀerential Equations, vol. 91, no. 1, pp. 168–179, planar restricted three-body problem,” Advances in Space 1991. Research, vol. 59, no. 10, pp. 2459–2465, 2017. [3] M. Hampton and R. Moeckel, “Finiteness of relative equilibria [21] L. Ding, J. M. Sanchez-Cerritos, ´ and J. Wei, “Eulerian col- of the four-body problem,” Inventiones Mathematicae, linear conﬁguration for 3-body problem,” 2019, https://arxiv. vol. 163, no. 2, pp. 289–312, 2006. org/abs/1910.00367. [4] A. M. Mansur and D. C. Oﬃn, “A minimizing property of [22] C. Liu and L. Dong, “Stabilization of Lagrange points in homographic solutions,” Acta Mathematica Sinica, English circular restricted three-body problem: a port-Hamiltonian Series, vol. 30, no. 2, pp. 353–360, 2014. approach,” Physics Letters A, vol. 383, no. 16, pp. 1907–1914, [5] A. Mansur, D. Oﬃn, and M. Lewis, “Instability for a family of homographic periodic solutions in the parallelogram four [23] M. Corbera, J. M. Cors, and G. E. Roberts, “A four-body body problem,” Qualitative Ceory of Dynamical Systems, convex central conﬁguration with perpendicular diagonals is vol. 16, no. 3, pp. 671–688, 2017. necessarily a kite,” Qualitative Ceory of Dynamical Systems, [6] J. Shi and Z. Xie, “Classiﬁcation of four-body central con- vol. 17, no. 2, pp. 367–374, 2018. ﬁgurations with three equal masses,” Journal of Mathematical [24] R. Lara and A. Bengochea, “A restricted four-body problem Analysis and Applications, vol. 363, no. 2, pp. 512–524, 2010. for the eight ﬁgure choreography,” 2019, https://arxiv.org/abs/ [7] J. Llibre and L. F. Mello, “New central conﬁgurations for the 1910.12822. planar 7-body problem,” Nonlinear Analysis: Real World [25] M. Alvarez-Ram´ırez and M. Medina, “Overview and com- Applications, vol. 10, no. 4, pp. 2246–2255, 2009. parison of approaches towards the planar restricted ﬁve-body [8] B. Liu and Q. Zhou, “Linear stability of elliptic relative problem with primaries forming an axisymmetric four-body equilibria of restricted four-body problem,” Journal of Dif- central conﬁguration,” Astrophysics and Space Science, ferential Equations, vol. 269, no. 6, 2020. vol. 365, no. 2, pp. 1–10, 2020. Advances in Astronomy 15 [26] L. F. Mello, F. E. Chaves, A. C. Fernandes, and B. A. Garcia, “Stacked central conﬁgurations for the spatial six-body problem,” Journal of Geometry and Physics, vol. 59, no. 9, pp. 1216–1226, 2009. [27] A. Alsaedi, F. Yousef, S. Bushnaq, and S. Momani, “New styles of periodic solutions of the classical six-body problem,” Mathematics and Computers in Simulation, vol. 159, pp. 183–196, 2019. [28] M. J. Idrisi and M. S. Ullah, “Central-body square conﬁgu- ration of restricted six-body problem,” New Astronomy, vol. 79, no. 4, Article ID 101381, 2020. [29] M. Kulesza, M. Marchesin, and C. Vidal, “Restricted rhom- boidal ﬁve-body problem,” Journal of Physics A: Mathematical and Ceoretical, vol. 44, no. 48, 2011. [30] H. D. Curtis, Orbital Mechanics for Engineering Students, Butterworth-Heinemann, Oxford, UK, 2013.
Advances in Astronomy – Hindawi Publishing Corporation
Published: Jul 1, 2021
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