Stability Regions of Equilibrium Points in Restricted Four-Body Problem with Oblateness Effects
aa r X i v : . [ a s t r o - ph . E P ] N ov Stability Regions of Equilibrium Points in Restricted Four-BodyProblem with Oblateness Effects
Reena KumariandBadam Singh Kushvah
Department of Applied Mathematics,Indian School of Mines, Dhanbad - 826004, Jharkhand,India [email protected]; [email protected]
ABSTRACT
In this paper, we extend the basic model of the restricted four-body problem introducing twobigger dominant primaries m and m as oblate spheroids when masses of the two primary bodies( m and m ) are equal. The aim of this study is to investigate the use of zero velocity surfacesand the Poincar´e surfaces of section to determine the possible allowed boundary regions and thestability orbit of the equilibrium points. According to different values of Jacobi constant C , wecan determine boundary region where the particle can move in possible permitted zones. Thestability regions of the equilibrium points expanded due to presence of oblateness coefficient andvarious values of C , whereas for certain range of t (100 ≤ t ≤ A (0 < A <
1) and A (0 < A < µ lies in the interval (0 . , . A and A respectively in the proposed model. Subject headings:
Restricted four-body problem; Poincar´e surface of section; Oblateness; Equilibriumpoints; Basins of attraction.
1. Introduction
To study the motion of celestial bodies, re-stricted four-body problem is one of the importantproblem in the dynamical system. An applicationof the restricted four-body problem is illustratedin the general behavior of the synchronous orbit inpresence of the Moon as well as the Sun whereascoupled restricted three-body problem is one ofthe example of restricted four-body problem. Theproblem is restricted in the sense that one of themasses is taken to be small, that the gravitationaleffect on the other masses by the fourth mass isnegligible. The smaller body is known as infinites-imal mass (body) and remaining three finite mas- sive bodies called primaries.The classical restricted four-body problem maybe generalized to include different types of ef-fect such as oblateness coefficient, radiation pres-sure force, Pyonting-Robertson drag etc. Vari-ous authors have studied the restricted four-bodyproblem and examined the existence of equi-librium points such as Hadjidemetriou (1980),Michalodimitrakis (1981), Kalvouridis et al (2007)and Papadakis (2007). Further, Baltagiannis and Papadakis(2011b) discussed the equilibrium points and theirstability in the restricted four-body problem.On the other hand, in recent years manyperturbing forces, such as oblateness, radiationforces of the primaries, Coriolis and centrifugal1orce, variation of the masses of the primariesetc. have been included in the study of re-stricted three-body problem (RTBP). The RTBPwith oblate effect has been studied by manyinvestigators such as Sharma and Rao (1975),Abouelmagd and El-Shaboury (2012), Khanna and Bhatnagar(1999), Douskos (2011) etc.Determination of the stability regions of theinfinitesimal body was introduced by Poincar´e(1892) during the study of periodic orbit of thesystem. This is very good technique to study thenature of trajectory of an infinitesimal body andalso known as surface of section method. Apartfrom that this method was used by Winter (2000)and Kumari and Kushvah (2013) to describe thelocation and stability of the equilibrium points inthe restricted three and four-body problem respec-tively.Here, we extend the basic model of restrictedfour-body problem by considering the dominantprimary m and m as oblateness body respec-tively. Our goal in this paper is to study theeffect of oblateness coefficient on the motion ofan infinitesimal body in the force field of mas-sive bodies. We also determine and presentbasins of attraction for the equilibrium points(attractors) of the problem created by NewtonRaphson method for their numerical computa-tion at sample values of the oblateness coeffi-cient parameter. The set of initial approxima-tion ( x, y ) which leads to a particular equilib-rium point, constitutes a convergence (or attract-ing) or divergence region. Douskos (2010) andCroustalloudi and Kalvouridis (2007) presented asimilar study of the basins of attraction in the xy -plane for the equilibrium points of Hill’s prob-lem with radiation and oblateness in restrictedthree body problem and of a ring problem of n + 1bodies.The Poincar´e surface of section of the proposedmodel is obtained with the help of the EventLocator Method. We have used Mathematica R (cid:13) Wolfram (2003) software package for numericaland algebraic computation of non-linear ordinarydifferential equations.This paper is organized as: we write the equa-tions of motion and find the Jacobi integral of thesystem in section (2). In section (3), we describethe zero velocity surfaces whereas in section (4) wedetermine equilibrium points. The stability of the Fig. 1.— Geometry of the problem.equilibrium points is examined in section (5) and(6) whereas in section (7) we present interestingbasins of attraction created by Newton Raphsonmethod applied for the solution of the equationswhose roots provide the locations of the equilib-rium points. Finally, section (8) includes the dis-cussion and conclusion of the paper.
2. Equations of motion
In this problem, we suppose that the motionof an infinitesimal mass ( m ) is governed by thegravitational force of the oblate spheroid m , m and third body m with m > m ≥ m (1).The oblateness factor of the primaries ( m , m )are also taking into account. It is assumed thatthe influence of infinitesimal mass on the motionof primaries moving under their mutual gravita-tional attraction is negligible. We normalize theunits with the supposition such that the sum ofthe masses and separation between the primariesboth be unity and unit of time is taken as the timeperiod of rotating frame moving with the meanmotion ( n ). Hence, we have G ( m + m + m ) =1. Let the co-ordinates of infinitesimal mass be( x, y ) and masses m , m and m are ( √ µ, − √ (1 − µ ) , − ) and ( − √ (1 − µ ) , ) respec-tively, relative to rotating frame Oxyz , where µ = m m + m + m = m m + m + m is the mass param-eter and we assume that µ = 0 .
2. The perturbedmean motion n = q ( A + A ), where A i =2 ei − R pi R , i = 1 , m and m respectively with R e i and R p i as equatorial and polar radii and R is separationbetween the primaries.The equations of motion of the infinitesimalmass in the rotating co-ordinate system is givenas ¨ x − n ˙ y = Ω x , (1)¨ y + 2 n ˙ x = Ω y , (2)where Ω = n ( x + y )2 + (1 − µ ) r + µr + µr + (1 − µ ) A r + µA r , (3)with r = q ( x − √ µ ) + y ,r = vuut x + √
32 (1 − µ ) ! + (cid:18) y − (cid:19) ,r = vuut x + √
32 (1 − µ ) ! + (cid:18) y + 12 (cid:19) ,r = p x + y . The suffixes x and y indicate the partial deriva-tives of Ω with respect to x and y respectively.The well known energy integral of the problemgiven as: C = − ˙ x − ˙ y + 2Ω , (4)where C is known as Jacobi constant. We observe(from 4) that 2Ω − C ≥
0. The curves of zerovelocity are defined through the expression 2Ω = C ; such a relation defines a boundary, called Hill’ssurface, which separates regions where motion isallowed or forbidden.In Fig.2, four frames represent the orbit of theinfinitesimal body. First two frames show the orbitin absence of oblateness effect whereas last twoframes show orbit in presence of oblateness effect.The orbit of the infinitesimal body represents infirst frame when 0 ≤ t ≤
200 whereas second framewhen 100 ≤ t ≤ - -
505 x y H a L - -
505 x y H b L - -
50 0 50 100 - - y H c L - - - - - -
202 x y H d L Fig. 2.— Orbits of the restricted four-body prob-lem with and without oblateness effect.looks like cote’s spiral. However, with effect ofoblateness, orbit becomes regular when 0 ≤ t ≤
200 which is shown in third frame while fourthframe shows the orbit when 0 ≤ t ≤
3. Zero velocity surfaces
Eq.(4) represents a relation between square ofvelocity and the coordinates of the infinitesimalbody in the rotating coordinate system. The Ja-cobi constant C is determined numerically usinginitial conditions. Therefore equation (4) deter-mines the boundaries of the regions where thebody can move from one allowed region to an-other one. In particular, if we take velocity of theinfinitesimal body equal to zero then surfaces ob-tained in xy -plane known as zero relative velocitysurfaces which are given as follows: C = 2Ω (5)or n ( x + y ) + 2(1 − µ ) r + 2 µr + 2 µr + (1 − µ ) A r + µA r = C. (6)The above solution gives much information aboutthe possible dynamics at a given Jacobi constant3 . In particular, if A = A = 0 in equation (6)we obtain the classical zero velocity surfaces of thesystem, to study the behavior of the zero velocitysurfaces in the vicinity of the singular point and inthe vicinity of the main bodies for increasing anddecreasing values of Jacobi constant.In Fig. 3, frame (a) shows zero velocity curves(ZVC) for different values of Jacobi constant C whereas frame (b) indicates ZVC for various val-ues of oblateness coefficients A and A . For ex-ample, in frame (a) curves are labeled as C i , i =1 , , , C = 3 . , C = 2 . , C = 1 . C = 1 . A ( A =0 . , A = 0 . , A ( A = 0 . , A = 0 . A ( A = 0 . , A = 0 . C is very large thenthe three primary bodies are separated with eachother where the particle cannot move from oneregion to another. Again, when the values of C are small, connections open at two points wheremotion is possible and the body can never escapefrom the system. Further, we take C even smallerthen all the possible connections are opened i.e.inner and outer regions are opened and the parti-cle can freely move from one allowed regions to an-other allowed region. On the other hand in frame(b), for increasing values of oblateness coefficients A and A respectively, their corresponding pos-sible boundary regions increase where the parti-cle can freely move from one side to another side.Therefore, we say that possible boundary regiondepends on the Jacobi constant as well as oblate-ness coefficients and observed that how does theconnection open for decreasing values of Jacobiconstant and increasing values of oblateness coef-ficients A and A respectively with other fixedvalues of the parameters.
4. Equilibrium points
The coordinates of equilibrium points of theproblem are obtained by equating R.H.S. of (1) and (2) to zero i.e. Ω x = Ω y = 0. In other words n x − (1 − µ )( x − √ µ ) r − A (1 − µ )( x − √ µ )2 r − ( x + √ (1 − µ )) µr − A ( x + √ (1 − µ )) µ r − ( x + √ (1 − µ )) µr = 0 , (7)and n y − (1 − µ ) yr − A (1 − µ ) y r − ( y − ) µr − A ( y − ) µ r − ( y + ) µr = 0 . (8)Solving above equations for µ = 0 . A and A , we ob-tain two collinear L , points on the x -axis and sixnon-collinear equilibrium points L i , i = 3 , , ..., y = 0The equilibrium points at x -axis are the solu-tions of Eqs. (7) and (8) when y = 0, which give f ( x,
0) = n x − (1 − µ )( x − √ µ ) | x − √ µ | − A (1 − µ )( x − √ µ )2 | x − √ µ | − x + √ (1 − µ )) µ (cid:16) ( x + √ (1 − µ )) + (cid:17) − A ( x + √ (1 − µ )) µ (cid:16) ( x + √ (1 − µ )) + (cid:17) = 0 . (9)Now, solving the above expression using initialconditions, we get equilibrium points for variousvalues of the oblateness coefficients. We observedthat it has only two real roots and other are com-plex conjugates. Also, we noticed that for fixedvalues of at A = 0 . A (0 < A < x -axis shifted from left to right, whereas for fixedvalues of A = 0 . A (0 < A < L shifted formleft to right while L point is shifted form right toleft which are shown in Table 1.We plot graph of equation (7) when y = 0 andfixed values of parameters µ = 0 . , A = 0 . x -axis A = 0 . A L L A = 0 . A L L A = 0 . L = − . L = 1 . A and A , number ofequilibrium points remain same. The non-collinear points are the solutions ofEqs. (7) and (8) when y = 0, which gives f ( x, y ) = n x − (1 − µ )( x − √ µ ) r − ( x + √ (1 − µ )) µr − A (1 − µ )( x − √ µ )2 r − A ( x + √ (1 − µ )) µ r − ( x + √ (1 − µ )) µr = 0 , (10)and g ( x, y ) = n y − (1 − µ ) yr − A (1 − µ ) y r − ( y − ) µr − A ( y − ) µ r − ( y + ) µr = 0 . (11)Solving equation (10) and (11), we get non-collinear equilibrium points for different values ofthe oblateness coefficients A and A respectively.For fixed value of A and increasing values of A as well as for fixed A and increasing values of A , co-ordinates of non-collinear points L i , i = 3 , , ..., A increases from 0 . . A = 0 . A increases from 0 . .
9, the problem has then seven equilibrium pointsbecause L approaches to L point. Also, when A = 0 . A = 1 . L and L coincide on the collinearpoint L and in consequence problem has six equi-librium points. However, when A = 1 . A =0 . L reaches L point, whereas oblateness coefficient A increases form 0 . . A = 0 . µ = 0 . A = A = 0 then our results agree with theresults of (Papadouris and Papadakis 2013), theirconfiguration was the mirror image of our config-uration as depicted in Fig. 7.For fixed A = 0 . , A = 0 . A = 0 . , A = 0 .
0, we observe that secondand third primary bodies form dumbell shape ofthe curve( Figs. 5 and 6). However, the lowerloop of the third primary body is disconnected,whereas one of the loop of second primary bodyreduces due to an increase in value of A for fixed A (Fig. 8). On the other hand, the dumbell shapeof the second and third primary bodies are less af-fected due to increasing value of A for fixed valueof A (Fig. 9). In Figs. 8 and 9, we have usedsize of point to show the shifting of equilibriumpoints i.e. the equilibrium points shifted towardsthe large point size or along with increasing point-size due to presence of oblateness coefficients. For A = 0 . A (0 < A < L , L and L are attracted to second primary body, whereas L , L and L are attracted towards the first pri-mary body and it happens due to the attractionof the oblate bulge. Also, we see that L and L have very less effect of the parameters (Fig. 8).Further, for A = 0 . A (0 < A < L , L and L are attracted towards the secondprimary body while L , L and L are attractedtowards the third primary. Moreover, L has veryless effect of the parameters but L is attracted bythe first primary body due to same mass param-5ter values of second and third primary as shownin Fig. 9.
5. Linear stability of non-collinear points
To analyze the possible motions of the infinites-imal body in a small displacement of the equilib-rium points ( x , y ), we first make infinitesimalchange ξ and η in its coordinates i.e. x = x + ξ and y = y + η such that the displacement becomes ξ = P e λt , η = Qe λt , (12)where P , Q are constants and λ is parameter. Sub-stituting these values into equations (1) and (2),we get differential equations of second order in ξ and η respectively (Murray and Dermott 1999)¨ ξ − n ˙ η = ξ Ω xx + η Ω xy , ¨ η + 2 n ˙ ξ = ξ Ω yx + η Ω yy , (13)where superfix 0 indicates that the values are com-puted at the equilibrium point ( x , y ). Again,substituting ξ = P e λt , η = Qe λt in equation(13) and simplifying, we obtain( λ − Ω xx ) P + ( − nλ − Ω xy ) Q = 0 , (14)(2 nλ − Ω yx ) P + ( λ − Ω yy ) Q = 0 . (15)Now, the condition of nontrivial solution is thatthe determinant of the coefficients matrix of theabove system should be zero i.e. (cid:12)(cid:12)(cid:12)(cid:12) λ − Ω xx − nλ − Ω xy nλ − Ω yx λ − Ω yy (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Therefore, from above matrix we obtain a quadraticequation in λ known as characteristic equation: λ + (4 n − Ω xx − Ω yy ) λ +(Ω xx Ω yy − Ω xy ) = 0 . (16)The four roots of characteristic equation (16) playa crucial role to determine the orbits of equilib-rium points. An equilibrium point will be stable ifthe above equation evaluated at the equilibrium,has four pure imaginary roots or complex rootswith negative real parts. This happens if the fol-lowing conditions(4 n − Ω xx − Ω yy ) − xx Ω yy − (Ω xy ) ) > , (4 n − Ω xx − Ω yy ) > , Ω xx Ω yy − (Ω xy ) > , (17) are satisfied simultaneously.Now, using the determinant of the characteris-tic equation(16) we obtain(4 . . A − . A ) − (30 . . A + 31 . A ) µ − (191 . . A + 244 . A ) µ > , (18)which is a quadratic equation in µ . Therefore, itsroot are given as µ = s .
351 + 951 . A + 244 . A ) ,µ = s .
351 + 951 . A + 244 . A ) , where s , = − (30 . . A + 31 . A ) ∓ p (4082 .
61 + 33286 . A − . A ) . These roots satisfy condition (18) if either (i) µ − µ > µ − µ > µ − µ < µ − µ <
0, which implies that µ > max ( µ , µ )and µ < min ( µ , µ ) and therefore roots lie inbetween µ < µ < µ . For numerical results weuse x = 0 . , y = 0 . , < A < < A < µ = − . µ = 0 . m m + m m + m m ( m + m + m ) < , (19)where m , m and m are masses of the three pri-maries body. As we assumed m ≤ m and theleft term of equation (19) inequality is monotoni-cally increasing in m , ∀ m ∈ (0 , m ), with max-imum at m = m . Therefore the stability condi-tion becomes − m + 54 m < , ∀ m , conse-quently we get (3 − √ ≤ m ≤ (3 + 2 √ . ≤ µ ≤ . µ which shows thatnon-collinear points are stable.In Fig.10, we have depicted the graph A verses µ for different fixed values of A and it is observedthat for increasing values of A and A , value of µ decreases consequently stability region decreasesmonotonically.6 . Poincar´e surfaces of section In the restricted four-body problem, Poincar´esurface of section is very useful for finding stableperiodic and quasi-periodic orbits around the pri-maries. In order to determine Poincar´e surface ofsection (PSS) of the infinitesimal body at any in-stant, it is necessary to know its position ( x, y )and velocity ( ˙ x, ˙ y ), which correspond to a pointin a four dimensional phase space. We have con-structed surface of section on the x ˙ x -plane by tak-ing y = ˙ x = 0 and ˙ y > R (cid:13) Wolfram (2003).This is a good technique to determine the regularor chaotic nature of the trajectory. On the otherhand, if there are smooth, well defined islands,then the behavior of the trajectory is likely to beregular. Whereas, if the curves shrink down to apoint, it represents a periodic orbit. Apart formthat, we have obtained PSS at the values of Jacobiconstant C for a certain values of x and ˙ x whileeach orbit is determined with initial conditions: x = x , y = 0 , ˙ x = 0 , ˙ y = q b + n x − ˙ x − C, (20)where b = 2(1 − µ )( x − √ µ ) + 3 µ (cid:16) ( x + √ (1 − µ )) + (cid:17) + (1 − µ ) A ( x − √ µ ) + µA (cid:16) ( x + √ (1 − µ )) + (cid:17) . Since in the above proposed system key quantitiesare the values of
C, A and A respectively. There-fore, we plot the graph of Poincar´e surfaces of sec-tion for specific initial values x = 0 . , ˙ x = 0 . y = − . A and A re-spectively, then the region expands (as shown in frame 11(b)). Similarly if we increase the valuesof C i.e. C = 2 . , .
99 and C = 3 .
5, then thebounded region spans (as shown in frame 11(a)).For a particular values of initial conditions x = 0 . , ˙ x = 0 . y = − . C , we observe that near the points A (0 . , − . , B (0 . , . , P (0 . , . Q (0 . , − .
7. Basins of attraction
We determine basins of attraction of the equi-librium points with the help of Newton-Raphsonmethod, provided an initial point ( x, y ) and themass parameter µ as well as oblateness coefficient A and A respectively are given.It is a good technique to find the convergenceof trajectory originated from neighborhood of anequilibrium point. We present basins of attrac-tion of a fixed points, means that the set of pointsconverge towards a fixed point under successive it-erations of some transformation. The set of points( x, y ) that are created as follows:Ω x ( x, y, µ, A , A ) = 0 , Ω y ( x, y, µ, A , A ) = 0 , (21)from which we obtain the equilibrium points of theproblem. The algorithm of our problem takes theform x ( n ) = x ( n − − Ω x Ω yy − Ω y Ω xy Ω yy Ω xx − Ω xy | x ( n − ,y ( n − ,y ( n ) = y ( n − + Ω x Ω yx − Ω y Ω xx Ω yy Ω xx − Ω xy | x ( n − ,y ( n − , (22)where x n and y n are the values of x and y at the n th step of the Newton-Raphson method.Now, if the starting point ( x, y ) convergesrapidly to a specific root of the algebraic equa-tion (21), then this point ( x, y ) is a member ofthe basin of attraction of the specific root. TheNewton-Raphson method stops when the result-ing successive approximation converges to an at-tractor, the convergence being terminated whenthe repetition is happened. If the iteration di-verges, then the process is terminated after 1007terations. The regions of the basins of attractionare constructed by applying a dense grid of nodepoints in the xy -plane as starting points for theiteration.In Fig.12, we present the basins of attraction ofthe equilibrium points in the restricted four bodyproblem which are shown in frame (a) whereasother frames are zoom portions of frame (a). Foreach basins of attraction we use different color andthe equilibrium points are indicated by small stars.The existence of one very large body and other twosmall ones effects the structure of the basins sub-stantially. The points of the attracting domain ofthe central zone are organized in diamond shapedparts, whose wavy sides have vague boundaries.Inside, these areas lie the equilibrium positions ofthat zone. The boundaries of the central part arenot clearly defined. They look like a ”chaotic sea”.Again, outside the central zone the points of at-tractor is organized in mushroom shaped regionswhere the equilibrium points contain in this zone.The boundaries of the mushroom shaped regionsare dispersed points. The dispersed points of thisclass are densely allocated on the boundaries of thedense areas of the attracting regions. In presenceof oblateness coefficients A and A , there is veryless difference in absence of oblateness coefficients A and A respectively which are shown in frame(b) and frame (d). On the other hand, we can saythat different combination of oblateness coefficientgives same nature of the problem. However, frame(c) indicates the zoom part of frame (a) when theoblateness coefficients are absent.
8. Discussion and conclusion
We have studied restricted four-body problem(RFBP) introducing first two bigger primaries asoblate spheroids. The boundary regions for themotion of an infinitesimal body are obtained withthe help of zero velocity surfaces at different valuesof Jacobi constant and fixed values of oblatenesscoefficients. We have found that the allowed pos-sible regions of the motion of infinitesimal bodydecrease with increases values of the Jacobi Inte-gral C . We have investigated orbit of the RFBPand found that in absence of oblateness coeffi-cients, orbit looks like cote’s spiral in the timeinterval 100 ≤ t ≤ ≤ t ≤ y = 0 and non-collinear pointsat y = 0, which depend on oblateness coefficient A and A . We have noticed that for fixed valueof A = 0 . A (0
1) as well as for fixed value of A = 0 . A (0 < A < y = 0 has only two real roots called collinearpoints, whereas at y = 0 it has six real roots callednon-collinear points. The oblateness coefficientsaffect the existence of the equilibrium points ofthe problem in hand, since for A = 0 . A from 0 . . L disap-pears by coalescing at the L and consequently theproblem has seven equilibrium points. However,when the oblateness coefficient A increases from0 . . A = 0 . A = 0 . A (0 < A < L , L and L are attractedby second primary, whereas L , L and L are at-tracted towards the first primary and this happensdue to the attraction of the oblate bulge. Also, wehave seen that L and L have very less effect ofthe parameters. Furthermore, for A = 0 . A (0 < A < L , L and L are attractedtowards the second primary while L , L and L are attracted towards the third primary. The L point have very less effect of the parameters but L is attracted by the first primary body due tosame mass parameter values of second and thirdprimary bodies respectively.The non-collinear points are stable if the massparameter µ belongs to the interval (0 . , . x = 0 . , ˙ x = 0 . y = − . A (0 . , − . , B (0 . , . , P (0 . , . Q (0 . , − . xy -plane, showing the attractor ofthe Newton iteration. Due to the presence ofoblateness coefficients, we have found that bound-aries of the basins of attraction for the equilibraare not clearly defined which shows the chaoticnature. Also, we observed that there is very lessdifference in basins of attraction compare to ab-sence of oblateness coefficients. Since it is difficultto obtain an exact boundaries of the equilibra ofthe restricted four-body problem (Douskos 2010;Baltagiannis and Papadakis 2011a), further workis needed in this regard. This work may be ap-plicable to study the motion of a test particle inthe Sun-Earth-Moon-spacecraft as well as Sun-Jupiter-Trojan-spacecraft system.We are thankful to IUCAA, Pune for partiallyfinancial support to visit library and to use com-puting facility. We are also thankful to Prof.Bhola Ishwar, B.R.A. Bihar University, Muzaf-farpur (India) and Mr. Ashok Kumar Pal, ISM,Dhanbad (India) for their valuable suggestionsduring the preparation of the manuscript. REFERENCES Abouelmagd EI, El-Shaboury SM (2012) Peri-odic orbits under combined effects of oblate-ness and radiation in the restricted problemof three bodies. Ap&SS341:331–341, DOI 10.1007/s10509-012-1093-7Baltagiannis AN, Papadakis KE (2011a) Equi-librium Points and Their Stability in theRestricted Four-Body Problem. InternationalJournal of Bifurcation and Chaos 21:2179, DOI10.1142/S0218127411029707Baltagiannis AN, Papadakis KE (2011b) Fami-lies of periodic orbits in the restricted four-body problem. Ap&SS336:357–367, DOI 10.1007/s10509-011-0778-7Croustalloudi M, Kalvouridis T (2007) Attract-ing domains in ring-type N-body formations.Planet. Space Sci.55:53–69, DOI 10.1016/j.pss.2006.04.008Douskos CN (2010) Collinear equilibriumpoints of Hill’s problem with radiationand oblateness and their fractal basinsof attraction. Ap&SS326:263–271, DOI10.1007/s10509-009-0213-5 Douskos CN (2011) Equilibrium points of therestricted three-body problem with equalprolate and radiating primaries, and theirstability. Ap&SS333:79–87, DOI 10.1007/s10509-010-0584-7Gascheau M (1843) Examen d’une classed’equations differentielles et application a uncas paticulier du probleme des trois corps. C RAcad Sci 16:393–394Hadjidemetriou JD (1980) The restricted plan-etary 4-body problem. Celestial Mechanics21:63–71, DOI 10.1007/BF01230248Kalvouridis TJ, Arribas M, Elipe A (2007) Para-metric evolution of periodic orbits in the re-stricted four-body problem with radiation pres-sure. Planet. Space Sci.55:475–493, DOI 10.1016/j.pss.2006.07.005Khanna M, Bhatnagar KB (1999) Existence andstability of libration points in the restrictedthree body problem when the smaller primaryis a triaxial rigid body and the bigger one anoblate spheroid. Indian Journal of Pure and Ap-plied Mathematics 30:721–733Kumari R, Kushvah BS (2013) Equilibriumpoints and zero velocity surfaces in therestricted four-body problem with so-lar wind drag. Ap&SS344:347–359, DOI10.1007/s10509-012-1340-y, Michalodimitrakis M (1981) The circular re-stricted four-body problem. Ap&SS75:289–305,DOI 10.1007/BF00648643Murray C, Dermott S (1999) Solar SystemDynamics. Cambridge University Press, URL http://books.google.co.in/books?id=aU6vcy5L8GAC Papadakis KE (2007) Asymptotic orbitsin the restricted four-body problem.Planet. Space Sci.55:1368–1379, DOI10.1016/j.pss.2007.02.005Papadouris JP, Papadakis KE (2013) Equilibriumpoints in the photogravitational restricted four-body problem. Ap&SS344:21–38, DOI 10.1007/s10509-012-1319-8Poincar´e H (1892) Les methodes nouvelles de lamecanique celeste9outh EJ (1875) On Laplace’s Three Particles,with a Supplement on the Stability of SteadyMotion. Proceedings London Mathematical So-ciety, 1875, Volume 6, p 86-97 6:86–97, DOI10.1112/plms/s1-6.1.86Sharma RK, Rao PVS (1975) Collinear equilib-ria and their characteristic exponents in therestricted three-body problem when the pri-maries are oblate spheroids. Celestial Mechan-ics 12:189–201, DOI 10.1007/BF01230211Winter OC (2000) The stability evolution ofa family of simply periodic lunar orbits.Planet. Space Sci.48:23–28, DOI 10.1016/S0032-0633(99)00082-3Wolfram S (2003) The mathemat-ica book. Wolfram Media, URL http://books.google.co.in/books?id=dyK0hmFkNpAC This 2-column preprint was prepared with the AAS L A TEXmacros v5.2. C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C - - - - - - y H a L A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A - - - - - - y H b L Fig. 3.— Zero velocity curves for (a) C =3 . , C = 2 . , C = 1 . C = 1 . A ( A = 0 . , A = 0 . , A ( A = 0 . , A =0 . A ( A = 0 . , A = 0 . - - - H x L Fig. 4.— Equilibrium point at y = 0 for µ =0 . , A = 0 . A = 0 . L L L L L L L L m m m H a L - - - - - - y Fig. 5.— The eight equilibrium points for µ =0 . , A = 0 . 0, and A = 0 . L L L L L L L L m m m H b L - - - - - - y Fig. 6.— The eight equilibrium points for µ =0 . , A = 0 . A = 0 . L L L L L L L L m m m H c L - - - - - - y Fig. 7.— The eight equilibrium points for µ =0 . , A = 0 . A = 0 . L L L L L L L m m m L L L L L L L L m m m L L L L L L L L m m m L L L L L L L L m m m H d L - - - - - - y Fig. 8.— The eight equilibrium points for µ =0 . , A = 0 . A (0
08 (II) A = 0 . A = 0 . 04 (IV) A = 0 . 02, varying 0 < A < . = = = H a L - - x x A1 = A2 = = A2 = H b L - - - x x Fig. 11.— Poincar´e surface of section for the effectof Jacobi constant as well as oblateness. Fig. 12.— (a) The regions of different colors de-note the basins of attraction for the equilibriumpoints except collinear points which are shown inthe single color of the restricted four-body prob-lem when oblateness coefficient A = 0 . A = 0 . A = 0 . A = 0 . able 2Non-collinear equilibrium points A = 0 . A L L L L L L A = 0 . A0.0000 (-0.193927, -0.289496) (-0.876758, -0.829082) (-0.193927, 0.289496) (-0.876758, 0.829082) (0.168296, 0.913002) (0.168296, -0.913002)0.0015 (-0.193948, -0.289374) (-0.876402, -0.828690) (-0.192469, 0.288840) (-0.877506, 0.829856) (0.169129, 0.912330) (0.168010, -0.912197)0.0030 (-0.193970, -0.289252) (-0.876048, -0.828300) (-0.191051, 0.288205) (-0.878237, 0.830614) (0.169956, 0.911660) (0.167724, -0.911395)0.0045 (-0.193991, -0.289129) (-0.875695, -0.827912) (-0.189669, 0.287588) (-0.878951, 0.831356) (0.170776, 0.910990) (0.167438, -0.910594)0.0060 (0.194013, -0.289006) (-0.875343, -0.827524) (-0.188321, 0.286988) (-0.879650, 0.832082) (0.171590, 0.910321) (0.167153, -0.909796)0.0075 (-0.194035, -0.288883) (-0.874992, 0.827138) (-0.187006, 0.286405) (-0.880334, 0.832794) (0.172397, 0.909654) (0.166869, -0.908999)