On the spatial collinear restricted four-body problem with non-spherical primaries
Md Sanam Suraj, Rajiv Aggarwal, Amit Mittal, Om Prakash Meena, Md Chand Asique
tto be inserted manuscript No. (will be inserted by the editor)
On the Spatial Collinear Restricted Four-Body Problem WithNon-Spherical Primaries
Md Sanam Suraj · Rajiv Aggarwal · Amit Mittal · Om Prakash Meena · Md ChandAsique
Received: date / Accepted: date
Abstract
In the present work a systematic study has beenpresented in the context of the existence of libration points,their linear stability, the regions of motion where the thirdparticle can orbit and the domain of basins of convergencelinked to libration points in the spatial configuration of thecollinear restricted four-body problem with non-sphericalprimaries (i.e., the primaries are oblate or prolate spheroid).The parametric evolution of the positions of the librationpoints as function of the oblateness and prolateness param-eters of the primaries and the stability of these points in lin-ear sense are illustrated numerically. Moreover, the numeri-cal investigation shows that the only libration points whichlie on either of the axes are linearly stable for several com-binations of the oblateness parameter and mass parameterwhereas the non-collinear libration points are found linearlyunstable, consequently unstable in nonlinear sense also, for
Md Sanam SurajDepartment of Mathematics, Sri Aurobindo College, University ofDelhi, New Delhi-110017, Delhi, IndiaE-mail: [email protected]: [email protected] AggarwalDepartment of Mathematics, Deshbandhu College, University ofDelhi, New Delhi-110019, Delhi, IndiaE-mail: rajiv [email protected] MittalDepartment of Mathematics, ARSD College, University of Delhi, NewDelhi-110021, Delhi, IndiaE-mail: [email protected] Prakash MeenaDepartment of Mathematics, Deshbandhu College, University ofDelhi, New Delhi-110019, Delhi, IndiaMd Chand AsiqueDeshbandhu College, University of Delhi, New Delhi-110019, Delhi,IndiaE-mail: [email protected] studied value of mass parameter and oblateness parameter.Moreover, the regions of possible motion are also depicted,where the infinitesimal mass is free to orbit, as function ofJacobian constant. In addition, the basins of convergence(BoC) linked to the libration points are illustrated by usingthe multivariate version of the Newton-Raphson (NR) itera-tive scheme.
Keywords
Collinear restricted four-body problem · Equilibrium points · Linear stability · Zero-velocity curves · Basins of Convergence
The restricted problem of the four bodies with various per-turbations have been fascinated by a large group of astronomers,researchers and scientists over the globe in few decades.The restricted problem of four bodies in planar case is re-ferred as restricted ( N + ) where N =
3, body problem. Inthis problem the fourth body does not influence the motionof the three primaries, and consequently the fourth parti-cle can be assumed as a dynamical system composed of aninfinitesimal mass (i.e., the test particle) together with thethree main primaries. The central configuration of the three-body problem contains two type of configurations: the Eulerconfiguration (i.e., straight-line configuration) and the La-grange configuration (i.e., the equilateral triangle configura-tion) (see Leandro (2006), Hamilton (2016)). The classicalrestricted three-body problem with various perturbations hasbeen studied by many researchers in the few decades. Forexample the existence as well as stability of the equilibriumpoints (e.g., Simmons et al. (1985), Abouelmagd (2012),Abouelmagd and Abdullah (2019a), Selim et al. (2019)),when the primaries are non spherical in shape (e.g., Sharmaand Subba Rao (1979), Arredondo et al. (2012)), the analy-sis of periodic orbits (e.g., Abouelmagd et al. (2019b) ), the a r X i v : . [ n li n . C D ] M a y Md Sanam Suraj et al. basins of convergence and classifications of orbits (e.g., Zo-tos (2015a), Zotos (2015b)) have been studied in restrictedthree-body problem.In the few decades, the Lagrange and Euler configura-tions of the planar restricted four-body problem (PR4BP)have been discussed by various researchers by modifyingthe effective potential by adding some extra terms whichoccur due to various perturbations to unveil the existenceof equilibrium points and their stability (e.g., Michalodimi-trakis (1981), Kalvouridis et. al. (2006), Medvedev and Perov(2008), Baltagiannis and Papadakis (2011a), Barrab´es et al.(2017), Suraj et al. (2018c)), the study of the motion ofthe test particle when its mass is variable (e.g., Suraj et al.(2019a), Suraj et al. (2018)), the computations of periodicorbits (e.g., Baltagiannis and Papadakis (2011b), Palacioset al. (2019) ), the study of fractal basins of convergence(e.g., Zotos (2017a), Suraj et al. (2018b)), or the study of or-bital dynamics of escape and collisions (e.g., Zotos (2016),Maranhˇao and Libre (1998)).In various research papers, the scientists have also stud-ied the existence of libration points and their stability in thesame dynamical model by taking the shape of primaries.Asique et al. (2015) have studied the existence of libra-tion points in the photogravitational version of the restrictedfour-body problem when one of the primary is an oblate/prolatespheroid, whereas Asique et al. (2016) have discussed thesame model by taking a triaxial rigid body as primary. Ku-mari and Kushvah (2013) have included the effect of solarwind drag to discuss the libration points and zero velocitysurfaces in the restricted problem of four bodies. In Ref. Zo-tos (2015c), the author has compared the orbital dynamicsin three models which describe the various properties of astar cluster which rotates in a circular orbit around its parentgalaxy. Moreover, Zotos and Jung (2018) have discussed theorbit and escape dynamics in barred galaxies and illustratedthe basins of escape linked to the escape through the escapechannels in the vicinity of the libration points L , whichare symmetrical in nature and further established the rela-tion with the corresponding distribution of the escape timesof the orbits. In Zotos et al. (2018), the motion of the testparticle in a non-spinning binary black hole system is inves-tigated numerically when the masses are equal. Further theyhave classified the initial conditions into three different cate-gories, i.e., bounded, escaping and displaying close encoun-ters by using the smaller alignment index chaos indicator,the orbits are classified into regular, sticky and chaotic.In the literature, there is plethora of papers available wherethe spatial collinear restricted four-body problem (CR4BP)has been discussed. Ref. Arribas et al. (2016a), have dis-cussed the planar motion of the infinitesimal body movingunder the system of three main primaries situated in a collinearconfiguration (Euler 1767) with a symmetry and the periph-eral primaries are also source of radiation. In addition, they have considered the case where the gravitational force is lessthan the radiation force. In the mentioned setup, they havediscussed the analytic study of the position and stability ofthe libration points for the involve parameters. In continu-ation of their study, Ref. Arribas et al. (2016b) have dis-cussed the out-of-plane libration points, i.e., the equilibriumpoints which lie out side the configuration plane of primariesin the same dynamical system. Barrab´es et al. (2017) havediscussed the relative equilibria and their stability of the in-finitesimal mass, moving under the gravitational attractionof three primaries which are in a syzygy, under the repulsiveManev potential. Zotos (2017b) has investigated the BoClinked with the equilibrium points by applying the NR iter-ative scheme in the collinear restricted four-body problemwith angular velocity. He has illustrated the parametric evo-lution of the positions of equilibrium points and their sta-bility when the values of the mass parameter and the an-gular velocity are considered in the fixed intervals. Palacioset al. (2019) have illustrated the evolution of families ofsymmetric periodic orbits as the function of mass param-eter and evolution of spiral points, which show the con-nection between heteroclinic orbits and equilibrium points.Alvarez-Ramirez et. al. (2019) have illustrated the high or-der parametrizations of the stable and unstable manifoldsconnected with the libration points to allocate the ejectionorbits that eject from quadruple collision, in addition, theyhave also shown the existence of ejection-direct escape or-bits analytically.In the present manuscript we proposed to discuss themotion of the infinitesimal mass, moving in the gravitationalinfluence of the main primaries which are in a syzygy, thissetup always referred as collinear restricted four-body prob-lem. In this study we extend the work of authors Palacios etal. (2019) by taking the peripheral primaries as oblate/prolatespheroid.The present manuscript has following structure: the de-scription of the mathematical model and the equations ofmotion of the test particle are discussed in Sec. 2. The Sec.3 deal with the existence of the libration points as functionof parameters whereas stability of these libration points isanalyzed in the following section. The regions of possiblemotions are presented in Sec. 5. The BoC linked with thelibration points are illustrated in Sec 6. The paper ends withSec. 7 where the concluding remarks are presented. The dynamics of the infinitesimal mass (also referred astest particle) moving under the gravitational influence of thethree primaries P i , i = , , P and P , n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 3 having the same mass m = m = m , are oblate/prolate bod-ies and the oblateness/prolateness parameters A = A = A are also same while the central primary P is spherical inshape. Moreover, we have assumed that the primaries are incollinear central configuration where the primaries P and P are situated symmetrically with respect to the central pri-mary P . Also β m , taken as the mass of the central primary P where β denotes the so called mass parameter, is placedat the center of masses of the system which is taken as originof the reference frame. The peripheral bodies P , describethe circular orbit around the central primary P with sameangular velocity ω .In the synodic frame of reference with the same origin,the line passes through origin and joining the center of massof P and P is assumed as x − axis and Oy − axis is a line per-pendicular to Ox − axis and passing through origin while theline through origin and perpendicular to instantaneous planeof the primaries is assumed as Oz − axis. Consequently, inthe synodic frame of reference, the coordinates of the pri-maries P i , i = , , ( x i , , ) , x = , x = − x = , and Gm = β . The configuration of the primaries remains invari-ant, if the sum of the total gravitational force exerted by P and P on P are equal to the centrifugal force, i.e., m ω (cid:107) P P (cid:107) = Gm m (cid:107) P P (cid:107) + Gm (cid:107) P P (cid:107) I ∗ + Gm m (cid:107) P P (cid:107) + Gm (cid:107) P P (cid:107) I ∗ + Gm (cid:107) P P (cid:107) I (cid:48) ∗ , where I ∗ = ( I + I + I − I ) , I (cid:48) ∗ = ( I (cid:48) + I (cid:48) + I (cid:48) − I (cid:48) ) , whereas I i and I (cid:48) i , i = , , P and P at its center of mass O , respec-tively while I is the moment of inertia about the line joiningthe center of mass of the primary P and the primary P or P and I (cid:48) is the moment of inertia about the line joining thecenter of mass of the primary P and the primary P or P .Therefore, the angular velocity of the synodic frame is ω = Λ = ( + β ) + A ( + β ) , (1)where A = a − c R , A = a − c R and in the Copenhagen casewe have taken A = A = A . Also, when the oblateness of theprimaries are neglected i.e., A =
0, the ω reduces to sameas in Eq.(1) in Ref. Arribas et al. (2016b).The units of distances, mass and time are chosen in sucha way that (cid:107) P P (cid:107) =
1, and Gm =
1. The equations of motionof the test particle after a change of time ds = ω dt , whereboth of the peripheral bodies are oblate, can be written as:¨ x − y = ∂ U ∂ x , (2a)¨ y + x = ∂ U ∂ y , (2b)¨ z = ∂ U ∂ z , (2c) Fig. 1
The bifurcation curve in the ( β , A ) plane. (colour figure online). where U ( x , y , z ) = ( x + y ) + Λ (cid:40) β r + ∑ i = (cid:18) r i + A i r i (cid:19)(cid:41) , r i = (cid:113) ( x − x i ) + y + z , i = , , . Furthermore, analogous to the restricted problem of threebodies, the system of equations in Eqs. 2a-2c possesses thefirst integral, always considered as Jacobi integral, and ex-pressed as C = U ( x , y , z ) − ( ˙ x + ˙ y + ˙ z ) , (3)where the Jacobian constant is represented by C . The libration points can be obtained by solving the equations U x = , U y = , U z = , (4)where U x , U y , U z are as follows: U x = x − Λ (cid:26) β xr + ∑ i = ˜ x i (cid:16) r i + A r i (cid:17)(cid:27) , (5a) U y = y (cid:20) − Λ (cid:26) β r + ∑ i = (cid:16) r i + A r i (cid:17)(cid:27)(cid:21) , (5b) U z = − z Λ (cid:26) β r + ∑ i = (cid:16) r i + A r i (cid:17)(cid:27) , (5c)˜ x i = x − x i . (5d) Md Sanam Suraj et al.
Fig. 2
Evolution of the positions of collinear libration points L + x , , , and L − x , , , as function of parameter A ; (a) left : for the fixed value of β = .
1. The horizontal dashed magenta lines show the critical value of A = A = − . A , there exist eight collinearlibration points. (b) right : for β = magenta lines show the critical value of A = − . A ∈ ( − . , − . ) and ( , . ) , there exist four collinear libration points whereas in A ∈ ( − . , ) there exist no collinearlibration points. (colour figure online). From the Eq.(1), we observe that Λ >
0, therefore theparameter A must satisfy the condition A > − ( + β ) ( + β ) . InFig. 1, the evolution of the bifurcation curve are presentedin the ( β , A ) plane. In the area below the curve there arenon-permissible value of ( β , A ) , while in the area above thecurve there are permissible values of the combination of ( β , A ) . The red dashed line shows the permissible value of A when β =
0. It can be observed that for large value of β ,the value of A becomes almost constant, i.e., for β > A = − . xy − plane. When we consider the libration pointson the xy − plane, i.e., z = U x = U y = z =
0. Thereexist two type of libration points in the xy − plane, i.e., thecollinear libration points and the non-collinear libration points. The collinear libration points are solutions of the equation5a, when y = z =
0, i.e., x − Λ (cid:20) β x | x | + R − | R − | + R + | R + | + A (cid:26) R − | R − | + R + | R + | (cid:27)(cid:21) = x − / = R − and x + / = R + .In this subsection, we wish to discuss the effect of oblate-ness or prolateness, and the mass parameter on the positionsof collinear libration points. According to Douskos et al.(2012), A < A > A =
0, the problem converts to the classical case.The parametric evolution of the positions of collinear li-bration points are illustrated in Fig. 2 for two different valuesof β . It is observed that the positions as well as the numberof libration points on x − axis strongly depend on the com-bination of the values of β and A . For β = .
1, there exist8 collinear libration points for A ∈ ( − . , ) whereasthere exist only 4 libration points for A ∈ [ , ) and A ∈ [ − . , − . ) . The libration points L + x , and L − x , originate from the vicinity of the primaries m and m re-spectively for slightly negative value of A , and L + x , L − x co-incide with L + x , L − x respectively and disappear completelyat A = − . L + x , L − x exist for A < − . n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 5 which we again labeled as L + x , L − x for A ∈ [ − . , − . ) .In addition, when β = A ∈ ( − . , − . ) and A ∈ ( , ) whereas there exist no collinear libration point when A ∈ ( − . , ) . In addition, we can notice that there orig-inate a pair of collinear libration points in the vicinity of theeach peripheral primaries, i.e., there exist four collinear li-bration points in total. The non-collinear libration are solutions of Eqs (5a, 5b) when y (cid:44)
0. In this subsection, we discuss the locations of the li-bration points evaluated numerically, by well known Newton-Raphson iterative scheme. This method operates good con-cerning the convergence and the reliability of the results forthe particular class of equations and this is exactly the reasonwhy we have used this method. It is necessary to note thatthe Newton-Raphson method behaves well on the properchoice of the initial condition and consequently, for the goodchoice of initial conditions, we have used the efficient andsmart technique introduced by Arribas et al. (2016a).In Fig. 3, the parametric evolution of the positions of li-bration points on the ( x , y ) -plane are illustrated for the differ-ent values of β and A . In Fig. 3(a-c), the value of β = . A decreases. It is observed that there ex-ist 14 libration points in which four libration points originatein the vicinity of the each of peripheral primaries. As thevalue of A decreases, the libration points L + x , L + x and L − x , L − x move towards each other and collide and finally disap-pear completely (see Fig. 3b). Further decrease in the valueof A leads to the result that the libration points L + xy , and L − xy , disappear completely (see Fig. 3c) and consequentlythere exist the libration points which lie only on the axes.In Fig. 3(d-e), we have illustrated the in-plane librationpoints for the value of β = β and A . It is observed that the librationpoints L + x , L + x and L − x , L − x move towards each other anddisappear completely as the value of A increases and conse-quently there exists no libration point on the x -axis (see Fig.3f).In Fig.4, we have illustrated the parametric evolutionof the positions of the libration points for varying value ofthe oblatenes/proletness parameter and fixed value of themass parameter β = .
1. It can be observed that, for slightlyhigher value of the prolateness parameter A , there exist fourcollinear libration points whereas in the vicinity of librationpoints L ± x there originate a pair of non-collinear librationpoints L ± xy , xy which first move away from x − axis as thevalue of A increases i.e., when A ∈ ( − . , B ) , four collinear libration points exist in which two libration pointslie on y − axis and four libration points lie on the xy -plane.The collinear libration points L ± x and L ± x move towards theperipheral primaries, the libration points L ± y move towardsthe central primary whereas the non-collinear libration points L ± xy and L ± xy move far from the x − axis first then again turntowards the x − axis as A ∈ ( − . , B ) and finally an-nihilate in the vicinity of the peripheral primaries as A ∈ ( B, C ) . Further, when A ∈ ( B, C ) we can observe that thereexist eight collinear libration points in total and two newcollinear libration points L ± x and L ± x originate in which thelibration points L ± x move towards peripheral primaries andat A = L ± x move towardsthe central primary as the value of parameter A increases andthe movement of the remaining libration points are same.Further, when A >
0, there exists no libration point on the xy − plane, on the contrary the libration points always existon either of the axes (shown in blue colour line). It can beobserved that the libration points L ± x and L ± y move towardsthe central primary m along the x − axis and y − axis respec-tively.In Fig. 5, we have illustrated the parametric evolutionof the positions of libration points for the fixed value of β = A ∈ ( − . , ) . The magenta, cyan, olive and orange dots represented by A, B,C, and D respectively show the critical values of the oblate-ness/prolateness parameters where the number of librationpoints changes. It is observed that as the value of A is slightlygreater than the permissible value i.e., − . L + x , x and L − x , x exist on x − axis and two libration points L + y , L − y exist on y − axis. As the value of A approaches to A (shown by magenta dots in Fig. 5), four non-collinear li-bration points namely L + xy , L − xy and L + xy , L − xy originate inthe vicinity of the collinear libration points L + x and L − x re-spectively, and it can be further observed that these librationpoints move away from the x − axis whereas the collinear li-bration points L ± x move towards and L ± x move away fromthe peripheral primaries as value of A increases. In addition,at B (shown by cyan dot) the collinear libration points L ± x collide with L ± x respectively, and disappear. Consequently,there exists no collinear libration point for prolateness pa-rameter A ∈ ( B , C ) (shown in darker blue line) and onlythe non-collinear libration points L ± xy , and L ± y exist in thisrange and at C , the non-collinear libration points L ± xy , col-lide with the peripheral primaries and disappear completely.Further, when A ∈ ( C , . ) , a pair of collinear librationpoints originate in the vicinity of each of the peripheral pri-maries and move far from them as the value of A increases(shown in darker gray line). In this interval, for the valuesof the oblateness parameter, there exist six libration pointsin total which lie on either of the axes (shown in darker gray Md Sanam Suraj et al.
Fig. 3
The evolution of the positions of the libration points, first row : for β = . A = − . A = − .
1; (c) right: A = − . second row : for β = A = − . A = − . A = − . line). It is noticed that as the value of oblateness/prolatenessparameter A increases the libration points L ± y always movetowards the origin along y − axis. In this section, we deal with the stability of the librationpoints in the configuration ( x , y ) -plane, where the effects ofthe parameters A and β on the stability of these librationpoints are illustrated. The stability of the libration pointscan be determined by linearizing the equations of motion ofthe test particle given in Eqs. 2a-2b about the libration point ( x ∗ , y ∗ ) . Consequently, the linearized equations for the testparticle near the libration points in the collinear restrictedfour-body problem are ˙ x = A x , where x = ( x , y , ˙ x , ˙ y ) T , and x is the state vector of the infinitesimal body with respect tothe libration points and A , the coefficient matrix is read as: A = (cid:32) O IB C (cid:33) , (7) where O = (cid:32) (cid:33) , I = (cid:32) (cid:33) , B = (cid:32) A A A A (cid:33) , C = (cid:32) − (cid:33) . Accordingly, the characteristic equation corresponding to thematrix given in Eq. 7 is λ + a λ + a = , (8)where a = − A − A , a = A A − A , A = − Λ (cid:18) β r + ∑ i = (cid:18) r i + A r i (cid:19)(cid:19) + Λ (cid:18) β x ∗ r + ∑ i = ˜ x i (cid:18) r i + A r i (cid:19)(cid:19) , n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 7 Fig. 4
The movement of the positions of the libration points on ( x , y ) -plane for β = . A ∈ ( − . , . ) , the yellow dots showthe value of the parameter A = A ≈ − . A = B = − . A = C = A = D = .
25. The arrow representsthe movement of the positions of libration points while the big bluedots show the positions of primaries. (colour figure online).
Fig. 5
The movement of the positions of the libration points for β = A ∈ ( − . , ) . The magenta dots showthe value of parameter A = A , cyan dots show the value of A = B = − . A = C =
0, and orange dotsshow the value of A = D ≈
1. The arrow shows the movement of thepositions of libration points. The big blue dots show the positions ofthe primaries. (colour figure online). A = − Λ (cid:18) β r + ∑ i = (cid:18) r i + A r i (cid:19)(cid:19) + y ∗ Λ (cid:18) β r + ∑ i = (cid:18) r i + A r i (cid:19)(cid:19) , A = y ∗ Λ (cid:18) β x ∗ r + ∑ i = ˜ x i (cid:18) r i + A r i (cid:19)(cid:19) , = A . ˜ x i = x ∗ − x i , r i = (cid:113) ( x ∗ − x i ) + y , i = , , . A libration point is said to be stable if the solution of theEq.8, evaluated at the libration point, has four pure imagi-nary roots. This is true only when the conditions, a − a > , a > , a > , (9)are satisfied simultaneously.In Fig. 6, we have illustrated the linear stability of thecollinear equilibrium points. In Fig. 6a and Fig. 6b, the sta-bility of the equilibrium points are depicted for constantvalue of β = . β = L ± x and L ± x arestable. However, as the value of β increases, the intervalsof the values of parameter A decreases in which the libra-tion points are stable. In Fig. 6c, the stability of collinearlibration points are presented for fixed value of A = − . β and observed that L ± x and L ± x arestable for β ∈ ( , . ) and β ∈ ( , . ) re-spectively.In Fig. 7, the stability of the libration points for β = . A .In Fig. 7a, the stability of the libration points is presentedfor mass parameter β = .
1. It is unveiled that the librationpoints L ± x are linearly stable for A ∈ ( A , B ) and A ∈ ( B, C ) (the stable equilibrium points are depicted in thick orange and green lines respectively). Further, the equilibrium points L ± x are linearly stable for A ∈ ( B, C ) .In Fig. 7b, the stability of the libration points is pre-sented for β = L ± x are stable for A ∈ ( A, B ) whereas L ± y are alsostable for A ∈ ( A, B ) . In addition, we have observed that the L ± y are stable for the value of A ∈ ( B, C ) and A ∈ ( C , ) also.It is also observed that none of the libration points which lieon ( x , y ) plane are stable, however, for some mass ratio theequilibrium points which lie on y − axis are linearly stable. By using the relation 3, we shall draw the evolution of thezero-velocity curves for fixed value of C , by assuming themotions of the particle on the xy -plane. The zero-velocitycurves bifurcate the regions of possible motion of those planes Md Sanam Suraj et al.
Fig. 6
The parametric evolution of the stability of collinear libration points: (a) left : for β = . ( x , A ∗ ) = ( . , − . ) ,and ( x , A ∗ ) = ( . , − . ) ; (b) middle : for β = ( x , A ∗ ) = ( . , − . ) , and ( x , A ∗ ) =( . , − . ) ; (c) right : for A = − .
02 and ( x , β ∗ ) = ( . , . ) , and ( x , β ∗ ) = ( . , . ) .The stable regions are depicted by green colour. (colour figure online). Fig. 7
The parametric evolution of the stability regions of the libration points for (a) left : β = . A ∈ ( − . , . ) , here the thickorange and green lines show the positions of the stable libration points. The value of A, B, C , and D are same as in Fig. 4; (b) right : β = A ∈ ( − . , ) , thick black lines show the positions of stable libation points. The values of A, B, C , and D are same as in Fig. 5. (colourfigure online). from the forbidden region where the test particle can not or-bit. In Fig. 8(a, b, c), the regions of possible motion aredepicted for the fixed value of β , the prolateness parame-ter A and increasing value of the Jacobian constant C . Thecoloured area represents the forbidden regions where the testparticle can not communicate. At C = C L + y = C L − y , there ex-ist two circular islands containing each of the peripheral pri-maries where the motion of the test particle is not possible.Therefore, the test particle can orbit from central primaryto any libration points whereas it can not orbit to peripheral primaries. In 8b, when the Jacobian constant C is increasedto C = C L + x = C L − x the forbidden region increased and twocrescent shapes appear which originate from L + x or L − x andcontain the libration points L + y or L − y , respectively. Conse-quently, the infinitesimal mass cannot approach to these li-bration points. Further increase in value of Jacobian constantto C = C L + x = C L − x leads to further increase in the forbiddenregions and consequently all the libration points fall insidethe forbidden region. The test particle can move either out-side the circular annulus shaped region or in the vicinity ofthe central primary. Therefore, we can conjuncture that as n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 9 Fig. 8
The evolution of the regions of possible motion: first row : for β = . , A = − .
22: (a)left: C = . C = . C = . second row : for C = . A = − . A = − .
05, (c) right: A = − .
1. (colour figureonline). the value of Jacobian constant increases, the forbidden re-gion also increases.In Fig. 8(d, e, f), the ZVCs are depicted for the fixed val-ues of β , C and the varying values of prolateness parameter A . It can be noticed that when A = − . y − axis. However, the motion of the infinitesimalmass is possible inside the pear shaped region which con-tains all the libration points except those which lie on the y − axis. As we decrease the value of parameter A = − . L + y and L − y shrink and further shrink to libration points L + y and L − y as the value of A approaches to ≈ − . A . Consequently, the test particle can not communicate fromperipheral primaries to any other and vice-versa.In Fig. 9(a-f), the regions of possible motion are depictedfor fixed value of the parameters A and β and increasing val-ues of the Jacobian constant. It is observed that the regions of possible motion decrease significantly when the value ofthe Jacobian constant increases. In the present manuscript a systematic study related to thebasins of convergence (BoC) associated with the librationpoints are presented when the peripheral primaries are non-spherical in shape. The well known method in multivariateversion, i.e., Newton-Raphson method is used to solve thenonlinear equations. In addition, we will use the procedureand methodology used by Zotos (2016) to illustrate the do-main of the basins of convergence in the in-plane case only.The associated multivariate iterative scheme is x n + = x n − J − f ( x n ) , (10)where f ( x n ) shows the system of equations, whereas the as-sociated inverse Jacobian matrix is given by J − . In our sys-tem 2a-2c, we have three equations. Fig. 9
The evolution of the regions of possible motion when A = − . β = .
1: (a) top left: C = . C = . C = . C = . C = . C = . Thus, we can use the multivariate NR iterative schemeon the system: U x ( x , y ) = , (11a) U y ( x , y ) = , (11b)and for the configuration ( x , y ) plane, for each coordinate,the bivariate version of the iterative scheme are read as: x n + = x n − U x n U y n y n − U y n U x n y n U x n x n U y n y n − U x n y n U y n x n , (12a) y n + = y n + U x n U y n x n − U y n U x n x n U x n x n U y n y n − U x n y n U y n x n . (12b)In the above equations, the values of x and y coordinatesat the n -th step of the iterative scheme are given by x n and y n in the Newton-Raphson method. Here, the correspondingsecond order partial derivatives of the potential function arerepresented by the subscripts of U ( x , y ) .The philosophy that works behind the Newton-Raphsoniterative scheme is as follows: the numerical code activates when the initial condition ( x , y ) is provided on the plane,whereas the iterative scheme continues until an attractor (i.e.,equilibrium point) is achieved, with the coveted predefinedaccuracy. Whenever the specific initial condition reached toone of the attractor (i.e., libration point) of the dynamicalsystem, we claim that for that specific initial condition, theiterative scheme converges. It can be noted that the iterativescheme does not converge equally well for each of the ini-tial conditions, in general. The collections of all those initialconditions which converge to the same attractor compile theso-called BoC or NR basins of attraction.It is well known that in the dynamical system such as N -body problem, it is not always possible to find the ana-lytic formulae to evaluate the coordinates of positions of theequilibrium points. In fact, for N >
3, we do not have anyanalytical formulae to evaluate the position of the librationpoint, thus, the one of the best means is to use the numericalmethods to evaluate their positions numerically. At this point n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 11
Fig. 10
The BoC linked with the libration points on ( x , y ) -plane for β = .
1, and for (a) A = − . A = − .
01, (c) A = − . A = − . L + x = red , L + x = cyan , L + x = magenta , L + x = blue , L − x = yellow , L − x = light pink , L − x = brown , L − x = pink , L + y = green , L − y = olive , L + xy = teal , L + xy = purple , L − xy = gray and L − xy = orange . The dots show the positions of libration points. (colourfigure online). of time it is necessary to note that every numerical methodstrongly depends on the choice of initial conditions. Indeed,the numerical methods may converge to one of the attrac-tors (i.e., the roots) or may need a vast number of iterationfor some of the initial conditions to converge at one of the at-tractor while for some of the initial conditions the numericalmethod may trapped into an endless cycle in a periodic oraperiodic manner or may diverge to infinity. This fact leads to the conclusion that the choice of initial condition must begood so that the method may converge to one of the attractor.The various literature concerning the iterative scheme sug-gest that those initial conditions need less number of the it-erations which lie in the regular domain of the basins of con-vergence, on the other hand the initial conditions which fallin the fractal region may need a huge number of iterations toconverge at one of the attractor. The above mentioned facts Fig. 11
The BoC linked with the libration points on ( x , y ) -plane for β = A = − . A = − . L + x = blue , L + x = cyan , L − x = yellow , L − x = pink , L + y = green and L − y = olive , L + xy = teal , L + xy = purple , L − xy = gray , L − xy = orange . The dots show the positions of libration points. (colour figure online). give us a considerable amount of reason to examine the BoCcorresponding to the equilibrium points. Consequently, wecan pick those initial conditions easily for which the itera-tive method need lowest number of iterations to converge atone of the specific attractor with predefined accuracy. More-over, the most intrinsic properties of the dynamical systemare unveiled by the analysis of the BoC corresponding to thelibration points. Since , the equations (12a, 12b) contain thefirst and second order derivatives which combine the dynam-ics of the test particle’s orbit together with the correspondingstability properties and therefore, it gives a strong reason forrevealing the basins of attraction in the present dynamicalmodel.Recently, various scientists and researchers have depletetheir time to analyze the NRBoC in different types of dy-namical systems, such as the restricted problem of three bod-ies (e.g., Douskos (2010), Zotos (2016), Zotos (2017a)), therestricted problem of four bodies (e.g., Zotos (2016), Surajet al. (2017a), Suraj et al. (2017b)), and the restricted prob-lem of five bodies (e.g., Zotos and Suraj (2017)).To illustrate the NRBoC, we deploy the following al-gorithm: we classify the configuration plane into dense uni-form grids of 1024 × ( x , y ) and per-form a double scan of the configuration ( x , y ) plane. Forthe present numerical computations, the maximum number N max of iterations allowed is set to 500 while the prede-fined accuracy is set to 10 − regarding the coordinates ofthe equilibrium poiints. At this point of time it is necessaryto make clear that the Newton-Raphson BoC must not be mistaken with the basins of attraction of that of dissipativesystems. It should be noted that the present manuscript dealswith numerical attractors and the BoC associated with them.6.1 In-plane basins of convergenceWe begin our numerical analysis with the in-plane case i.e.,for the case when libration points lie on ( x , y ) plane only.To classify each nodes on the configuration plane we haveused the color coded diagrams (CCDs), where each differentcolor is associated with each pixel, as per the final stage ofthe associated initial conditions. In Section 3, we have illus-trated that there exist different type of sets of libration pointsfor different combinations of the parameter β and A . On thisbasis we have divided our analysis in following subsections. In this subsection, we have discussed the BoC in those casesfor which the libration points exist on the axes as well ason the xy − plane. In Fig.10, we have illustrated the BoCfor fixed value of β = . A . The configuration plane is covered by well formedBoC linked to the equilibrium points. Moreover, the extentof BoC are infinite linked to all libration points. We observedthat the libration points linked to L − x and L + x are resemblewith exotic bugs with many legs and antennas. It is obvi-ous that the majority of area of the configuration plane are n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 13 Fig. 12
The BoC linked with the libration points on ( x , y ) -plane for β = A = . A = .
75. For β = .
1, and (c) for A = − . L + x = purple , L + x = cyan , L − x = green , L − x = olive , L + y = gray and L − y = orange . The dotsshow the positions of libration points. (colour figure online). covered by BoC linked to the libration points L ± x and L ± y .However, the basin boundaries are composed of the mix-ture of the initial conditions and looks like chaotic sea. Asthe value of parameter A decreases from − .
005 to − . L − x and L + x .In Fig.10c, the BoC is depicted for large scale of the config-uration plane ( x , y ) to have the idea of the BoC on broader scale. It can be noticed that multiple wings shaped regionlinked to the equilibrium points L ± x (magenta and brown) in-creases its wingspan and the boundaries of the basins lookshighly chaotic, infact the area between the wings shaped re-gion linked to L ± x and L ± y is highly chaotic which is com-posed of the mixture of various initial conditions. There-fore, it is very difficult to predict the final state of the ini-tial conditions which falling in these chaotic reasons. Aswe compare the BoC with the previous panels we can no-tice that as the value of A decreases, the libration points L ± x Fig. 13
The BoC linked with the libration points on ( x , y ) -plane for β = A = − . A = − . L + xy = teal , L + xy = purple , L − xy = gray , L − xy = orange , L + y = green and L − y = olive . The dots show the positions of librationpoints. (colour figure online). and L ± x come closer to each other respectively and conse-quently a major area of the domain of BoC of the config-uration plane is covered by those initial conditions whichconverge to these libration points. The domain of the BoCcorresponding to libration points L ± xyi , i = ,
2, looks likefour lobes originating from x − axis. The domain of the BoClinked to these libration points increase as the value of A de-crease. In Fig.10d, the BoC is illustrated for that value of A for which the libration points L ± x and L ± x collide and annihi-late completely, and consequently only ten libration pointsexist. We noticed that a major area of the configuration planelooks like chaotic sea, composed of the initial conditions,however the areas in the neighbourhood of the equilibriumpoints are regular. If we compare panels Fig.10a,b, to panelFig.10d, we notice that the area adjoining the exotic bugsshaped region which looks highly chaotic and appears el-liptic in shape, increases significantly. We believe that theinitial conditions which converge to libration points L ± x , when these points exist, now converge randomly to any ofthe existing libration points which is the reason why this re-gion turn into the chaotic sea. Infact, the regular island ofBoC which converges to the equilibrium points L ± x , turnedinto the chaotic sea. The zoomed view near the origin showsthat a elongated ”eight” shaped region exist from L − x to L + x is also chaotic except the four regular lobes shaped basinslinked to L ± xyi , i = , β = A for which there exist ten equib-rium points. We noticed that in both the panels, the extent of the BoC is infinite linked to all the equilibrium points. More-over, as the value of A increases from -0.142 to -0.09804989,the BoC linked to libration points L ± xyi , i = ,
2, becomesmore regular also the domain of the BoC linked to the equi-librium points L ± xi , i = ,
2, decrease. However, in both thecases a major part of the configuration plane is covered bychaotic sea composed of the various initial conditions.
In this subsection, we will discuss the BoC in those casefor which the libration points lie only on the axes. In Fig.12, we have discussed the BoC for two different values of β = . , A . Fig. 12a, when β = A = . x − axis while two equilibrium points on y − axis. We no-ticed that the configuration plane is covered by well-formedBoC which extended to infinity. However, the majority ofthe area is occupied by domain of the BoC linked to theequilibrium points L ± y , which looks like the multiple butter-fly wings. In addition, the domain of BoC which exists inthe vicinity of the equilibrium points L ± x looks like circu-lar island and the domain of BoC linked with the collinearequilibrium points L ± x looks like lobes originate from ori-gin along the x − axis. Further, as the value of the oblatenessparameter A = .
75 (see Fig. 12b) increases, it is seen thatthe butterfly wings shaped regions shrink in the vicinity of x − axis and consequently domain of the BoC associated withthe equilibrium points L ± x increases whereas there is negli- n the Spatial Collinear Restricted Four-Body Problem With Non-Spherical Primaries 15 gible change in the circular island shaped BoC linked with L ± x .Fig. 12c, the BoC is illustrated for β = . A = − . L ± x (see purple & green colors)looks like exotic bugs with many legs and antenna however,the BoC linked to libration points L ± x the two exotic bugsshaped region corresponding to each libration points (shownin cyan and olive color) exist in which one exists in the vicin-ity of the libration points and other exists adjacent to the do-main of BoC linked to the equilibrium points L ± x and wingsand antennas of these exotic bugs are very noisy. It is ob-served that these wings and antenna are chaotic mixture ofthe initial conditions which converge to different attractorsand consequently it is almost impossible to predict that towhich attractors these initial conditions are converging. In this subsection, we have illustrated the BoC for fixedvalue of β and various values of A for which there exist onlynon-collinear libration points (see Fig. 13). The extent of thedomain of BoC linked to each of the libration point is alsoinfinite in these cases. When the value of A = . L ± y . The wellformed domain of the BoC linked to the libration points L ± xyi , i = , , exist in the vicinity of the x − axis. However, theentire configuration plane look like a chaotic sea of initialconditions except for those BoC which are associate to L ± xyi (see Fig. 13a). The BoC changes drastically as the value ofthe A increase to A = − . L ± xyi increases significantlyand consequently the domain of the BoC linked to L ± y de-creases. However, the basins boundaries are very noisy and,indeed, it is impossible to predict the final state of the initialcondition falling inside these basins boundaries even after asufficient number of iterations. In the present problem, the collinear restricted four bodyproblem has been investigated where the peripheral primariesare non-spherical in shape. In particularly, the shape of theperipheral primaries are either oblate or prolate spheroid.It should be emphasized that a numerical investigationis performed in such a thorough and systematic manner tounveil the effect of the parameters β and A on the positionand stability of libration points, regions of possible motionand on the topology of BoC by applying the NR iterativescheme that all the illustrated results are novel, whereas theyenhance substantially to our knowledge related to the evolu-tion of the libration points. The most important outcomes of our numerical analysisare listed below: • The existence as well as the number of collinear librationpoints strongly depend on the particular values of param-eters β and A . For β = .
1, there exist eight collinearlibration points for A ∈ ( − . , ) while it reducesto four when A ∈ ( − . , − . ) and (0, 1), inaddition, for β = A ∈ ( − . , − . ) and [ , ) whereas there exist no collinear libration point for A ∈ ( − . , ) . It is further noticed that there ex-ist two type of non-collinear libration points, i.e., the li-bration points which lie on ( x , y ) plane and the librationpoints which lie on y − axis for the considered values of β = . , A . • The movement of the libration points which lie on y − axis remains toward the central primary as the value ofthe parameter A increases. The libration points L ± xy and L ± xy originate in the vicinity of the libration point L ± x and move far from the x − axis along the arc shaped tra-jectories and again turned to annihilate in the neighbour-hood of the peripheral primaries for A = • The stability analysis for the in-plane libration pointssuggests that only some of the libration points which lieon the x − axis are linearly stable for a particular valueof β and particular range of the parameter A . Whereasnone of the non-collinear libration points are found lin-early stable for any permissible value of the parameter A for the studied value of β . • The parametric evolution of the regions of possible mo-tion in the in-plane case is illustrated and found that asthe value of the Jacobian constant increases, the regionsof the possible motion, where the test particle is free tomove, decrease. Moreover, the region of forbidden mo-tion decreases in the in-plane case when the value of A decreases. • The evolution of the BoC as the function of the parame-ter A is presented to determine the effect of the parameter A on their topology. It is observed that in all the cases theextent of the domain of BoC linked to the libration pointsis infinite. The basins boundaries are always chaotic innature which are mainly composed of those type of ini-tial conditions which converge to any of the equilibriumpoints randomly.We have used the latest version 12 . (cid:114) toperform all numerical as well as graphical illustrations. Wehope that the presented numerical analysis and the obtainedresults to be useful in the real world where the primaries arecelestial bodies which are in collinear configuration. In fu-ture work it is interesting to unveil that how the topologyof the BoC linked to the out-of-plane equilibrium points, ifexists, is changed with the change when the value of param-eters β and A changes. Compliance with Ethical Standards - Funding: The author, Rajiv Aggarwal, has received theresearch grant by Department of Science and Technol-ogy, New Delhi, India.- Conflict of interest: The authors declare that they haveno conflict of interest.
Acknowledgements
This work is funded by
Department of Science and Tech-nology, India , under project scheme
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