The analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries
tto be inserted manuscript No. (will be inserted by the editor)
The analysis of periodic orbits generated by Lagrangian solutions of therestricted three-body problem with non-spherical primaries
Amit Mittal · Md Sanam Suraj · Rajiv Aggarwal
Received: date / Accepted: date
Abstract
The present paper deals with the periodic orbitsgenerated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies.We have illustrated the periodic orbits for different valuesof µ , h , σ and σ ( h is energy constant, µ mass ratio of thetwo primaries, σ and σ are oblateness factors). These or-bits have been determined by giving displacements along thetangent and normal to the mobile coordinates as defined byKarimov and Sokolsky (19). We have applied the predictor-corrector algorithm to construct the periodic orbits in an at-tempt to unveil the effect of oblateness of the primaries bytaking the fixed values of parameters µ , h , σ and σ . Keywords
Restricted three-body problem · Periodic orbit · Oblateness · Libration points
In the field of Celestial mechanics, the restricted problem ofthree bodies is most searched and fascinated problem for themost of astrophysicists. The restricted three-body problemdescribes the motion of the infinitesimal test particle whichmoves in the gravitational field of two main primaries where
Amit MittalDepartment of Mathematics, ARSD College, University of Delhi, NewDelhi-110021, Delhi, IndiaE-mail: [email protected]
Md Sanam SurajDepartment of Mathematics, Sri Aurobindo College, University ofDelhi, New Delhi-110017, Delhi, IndiaE-mail: [email protected]
E-mail: [email protected]
Rajiv AggarwalDepartment of Mathematics, Deshbandhu College, University ofDelhi, New Delhi-110019, Delhi, IndiaE-mail: [email protected] they move in the circular or elliptic orbits around their com-mon center of mass. The infinitesimal test particle does notinfluence the motion of the primaries. The available litera-ture on the restricted three-body problem unveiled the factthat the scientists and researchers have included various per-turbation terms in the effective potential to obtain a realis-tic model. Indeed, this model can be used to study the so-lar system dynamics and kinematics as well as in the studyof stellar system. In particular, the model of the restrictedthree-body problem can be used in the space missions in anattempt to study the motion of the spacecraft in the Earth-Moon system.A plethora of research articles are available where the ef-fect of various perturbations due to radiations of primaries,the non-spheroid primaries, the effect of Coriolis and cen-trifugal forces and also the effect of variable mass is consid-ered to study the dynamics of the test particle, (e.g., (1; 2; 3;4; 5; 6; 7; 8; 9; 10), (30), (32; 33; 34; 35; 36; 37; 38)).The study of the families of periodic orbits in the re-stricted three-body problem has great importance in the fieldof Celestial mechanics. In the beginning of last century, thefamilies of the periodic orbits were studied by (13) and (25)firstly, but it was not complete. In the few decades, a num-ber of research paper were published where the families ofperiodic orbits were studied (e.g.,(21; 22), (16), (14), (11),(20)). Recently, a plethora of research papers are availablewhere the periodic orbits in the restricted four and five bodyproblems are also investigated (e.g., (40), (28), (18)).Ref. (12) and Ref. (29) proved the existence of two fami-lies of small periodic motions near the Lagrangian solutionsin the planar circular restricted three-body problem, with ar-bitrary values of the mass parameter. Ref. (31) completedthe analytical investigation of periodic motions. The resultson periodic motions of circular restricted three-body prob-lem are presented in the famous book entitled ”Theory oforbits” i.e., Ref. (39). Ref. (15) considered more results on a r X i v : . [ n li n . C D ] M a y Amit Mittal et al. periodic motions in their paper whereas Ref. (23; 24) inves-tigated the small periodic motions generated by Lagrangiansolutions for all values of the mass parameter and for smallvalues of energy constant for which the conditions of holo-morphic integral theorem are valid. Ref. (17) has presentedan exhaustive review of periodic solutions which are of in-terest to Dynamical Astronomy and their relation to actualsystems. Ref. (19) studied the periodic motions generatedby Lagrangian solutions of the circular restricted three-bodyproblem with the help of mobile co-ordinates by taking dis-placement along tangent and normal. Further, (26) have ex-tended their study by taking one of the primary as an oblatespheroid whereas the effects of radiation of the primary onthe periodic orbits are investigated in (27).Indeed, the celestial bodies are not spherical but in gen-eral axis-symmetric bodies therefore, we thought of takinginto account the shape of the bodies as well. The replace-ment of mass point by rigid-body is quite important becauseof its wide applications. Moreover, the re-entry of artificialsatellite has shown the importance of periodic orbits.That is why we have thought of studying, in this paper,to determine the periodic orbits generated by Lagrangian so-lutions of the restricted three-body problem when both theprimaries are oblate bodies. We have generated the periodicorbits by giving the tangential and normal displacements tothe mobile co-ordinates. In an attempt to disclose the effectof oblateness factor of both the primaries, and varying valueof energy constant, we have drawn the family of periodicorbits by fixing mass ratio of two primaries by applying thepredictor-corrector algorithm.The structure of the paper is as follows: the descriptionof the mathematical model and the equations of motion isexplored in Sec. 2. The description of the normal and tan-gent variable are presented in the following Section 3. Themethodology and the algorithm are described in the Section4 for numerical simulations in an attempt to analyse the ef-fect of oblateness of the primaries on the trajectories of theperiodic orbits. The paper finally ends with Section 5, wherethe discussion and the conclusions are presented.
The present system consists of two primaries P i , i = , x − axis. The dimensionless massesof the primaries are m = µ and m = − µ , where themass parameter µ = m / ( m + m ) is same as Ref. (19).In addition, the centre of the primaries are on the horizontalaxis with coordinates ( − µ , ) and ( − µ , ) whereas ( x , y ) is the synodic rectangular dimensionless co-ordinates of theinfinitesimal mass.The equations of motion of the test particle with La-grangian function L are given by ddt ∂ L ∂ ˙ x − ∂ L ∂ x = , (1a) ddt ∂ L ∂ ˙ y − ∂ L ∂ y = , (1b)where L = (cid:16) ˙ x + ˙ y (cid:17) + n ( x ˙ y − ˙ xy ) + n (cid:16) x + y (cid:17) + − µ r + µ r + σ µ r + σ − µ r + U , r = (cid:113) ( x + µ ) + y , r = (cid:113) ( x + µ − ) + y , are the distances of the third body from the respective pri-maries m i , i = , P i , i = , σ i , i = ,
2, having masses m i , while the mean motion n is given by n = + σ + σ , σ i << , (2)where, σ i = a i − c i R , i = , , R = is dimensional distance between primaries , a i , c i = semi axes of the rigid body of mass m i . In addition, U is a constant to be chosen in such a mannerthat the energy constant h vanishes at the triangular librationpoint L .The coordinates of L are x L = ( − µ ) + ( σ − σ ) , y L = √ (cid:26) − ( σ + σ ) (cid:27) , and U = − ( − µ + µ ) − σ ( − µ + µ ) − σ ( − µ + µ ) . he analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries 3 Fig. 1
The zero velocity surfaces in which the shaded regions show theforbidden regions of motion. The blue dots indicate the location of theprimaries, the black dots show the non-collinear libration points, whilegreen dots denote the collinear libration points.
Therefore, in dimensionless synodic co-ordinate system, theequations of motion of the test particle can also be writtenas:¨ x − n ˙ y = W x , (3a)¨ y + n ˙ x = W y , (3b)where W = n (cid:16) x + y (cid:17) + − µ r + µ r + σ µ r + σ − µ r + U + h . The corresponding Jacobi integral is read as: C = (cid:16) ˙ x + ˙ y (cid:17) − n (cid:16) x + y (cid:17) − − µ r − µ r − σ µ r − σ − µ r − U ≡ h , (4)which is the only Jacobian integral of motion, exists for thisdynamical system. The regions of possible motion are il-lustrated in Fig. 1, and it is observed that as the energy con-stant decreases, the regions of possible motion increase. Theshaded region shows the region where the motion of the testparticle is forbidden whereas the boundary of these forbid-den regions shown in cyan color describes the zero velocitycurves. We continue our study with the system of generalized coor-dinates Q = ( x , y ) T which depend upon four parameters P =( µ , h , σ , σ ) T . The corresponding differential equations aregiven by system of Eqs. 3a, 3b, with Jacobi integral givenby Eq. 4. We determine the solutions of the Eqs. 3a, 3b,for which C vanishes. If we consider the solutions of theEqs. 3a, 3b, given by Eqs. (5) for some fixed parameters val-ues P = ( µ , h , σ , σ ) T then there may exist another solutiongiven by Eqs. (6) with P ∗ = ( µ ∗ , h ∗ , σ ∗ , σ ∗ ) T close to P . Wehave x = x ( t , µ , h , σ , σ ) , y = y ( t , µ , h , σ , σ ) , ˙ x = ˙ x ( t , µ , h , σ , σ ) , ˙ y = ˙ y ( t , µ , h , σ , σ ) , (5)and x ∗ = x ( t , µ ∗ , h ∗ , σ ∗ , σ ∗ ) , y ∗ = y ( t , µ ∗ , h ∗ , σ ∗ , σ ∗ ) , ˙ x ∗ = ˙ x ( t , µ ∗ , h ∗ , σ ∗ , σ ∗ ) , ˙ y ∗ = ˙ y ( t , µ ∗ , h ∗ , σ ∗ , σ ∗ ) . (6)The solutions in Eq. (6) will reduce to the solutions in Eq. (5)as P ∗ → P . We give the displacements ∆ P = P ∗ − P and ξ = Q ∗ − Q to the parameters and the coordinates respectively asfollows: ∆ µ = µ ∗ − µ , (7a) ∆ h = h ∗ − h , (7b) ∆ σ = σ ∗ − σ , (7c) ∆ σ = σ ∗ − σ , (7d) ξ = x ∗ − x , (7e) ξ = y ∗ − y , (7f)where Q ∗ = ( x ∗ , y ∗ ) T and ξ = ( ξ , ξ ) T .We consider that ∆ P and ξ are small quantities of thesame order. Then, we have the following variational equa-tions¨ ξ = W xx ξ + W xy ξ + n ˙ ξ + W x µ ∆ µ + W xh ∆ h + W x σ ∆ σ + W x σ ∆ σ , (8a)¨ ξ = W yx ξ + W yy ξ − n ˙ ξ + W y µ ∆ µ + W yh ∆ h + W y σ ∆ σ + W y σ ∆ σ , (8b)with the integral, constructed from the Eq. (4) by retainingthe members of the first order only, given by C = ˙ x ˙ ξ + ˙ y ˙ ξ − W x ξ − W y ξ − W µ ∆ µ − W h ∆ h − W σ ∆ σ − W σ ∆ σ . (9) Amit Mittal et al.
We consider V ( t ) = | ˙ Q ( t ) | = (cid:112) ˙ x + ˙ y , as momentary ve-locity on the orbit.We assume that (6) is not corresponding to the equilib-rium state, i.e., V ( t ) (cid:44) V ( t ) (cid:44) x and y become the mobile coordinates.We will, now, use the mobile coordinate system to draw theperiodic orbits by resolving one of the axes along the veloc-ity vector X = ( ˙ x , ˙ y ) T and the other axis along the normalvector Y = ( − ˙ y , ˙ x ) T . In the new coordinate system, we de-fine the transition matrix S as follows: s ( t ) = X ( t ) / V ( t ) = the unit vector along the tangent to the orbit is taken as thelast column of the matrix S , whereas, the first column of S is r ( t ) = Y ( t ) / V ( t ) = the unit vector, normal to the orbit whichis orthogonal to the vector s ( t ) .Therefore,, we have S = { r , s } , with dimension 2 × ( r ) = ×
1, and dim ( s ) = ×
1. We may verify that s T s = , (10a) r T s = , (10b) S − s = e = ( , ) T . (10c)Finally, S can be written as: S = V ( t ) (cid:32) − ˙ y ˙ x ˙ x ˙ y (cid:33) , with its inverse as: S − = S T = V ( t ) (cid:32) − ˙ y ˙ x ˙ x ˙ y (cid:33) . We also define r ∗ = r T = V ( t ) ( − ˙ y , ˙ x ) , as the first line of S − , s ∗ = s T = V ( t ) ( ˙ x , ˙ y ) , as the last line of S − . In the new coordinate system, we introduce α = (cid:32) NM (cid:33) , dim ( N ) = × , anddim ( M ) = × , where N is displacement along the normal to the orbit and M is displacement along the tangent to the orbit.Then, the new coordinates are given by the formulas ξ = S α = ( r , s ) (cid:32) NM (cid:33) = rN + sM , (11a) α = S − ξ , (11b) N = r ∗ ξ , (11c) M = s ∗ ξ . (11d) Now, differentiating Eq. 11a, we get˙ ξ = ˙ S α + S ˙ α = ˙ rN + r ˙ N + ˙ sM + s ˙ M , which can also be written as: ξ = V ( − ˙ yN + ˙ xM ) , (12a) ξ = V ( ˙ xN + ˙ yM ) , (12b)˙ ξ = V (cid:0) − ¨ yN + ¨ xM − ˙ y ˙ N + ˙ x ˙ M (cid:1) − ˙ VV (cid:18) − ˙ yN + ˙ xM (cid:19) , (12c)˙ ξ = V (cid:0) ¨ xN + ¨ yM + ˙ x ˙ N + ˙ y ˙ M (cid:1) − ˙ VV ( ˙ xN + ˙ yM ) , (12d)where˙ V = ˙ x ¨ x + ˙ y ¨ yV = W x ˙ x + W y ˙ yV . Substituting these values into the integral (9), we have C = V (cid:16) W x ˙ y − W y ˙ x + nV (cid:17) N − (cid:0) ˙ V M − ˙ MV (cid:1) − W µ ∆ µ − W h ∆ h − W σ ∆ σ − W σ ∆ σ ≡ , or C = WV (cid:0) ˙ MV − M ˙ V (cid:1) + V (cid:18) W x ˙ y − W y ˙ x + ¨ x ˙ y − ˙ x ¨ y (cid:19) N + N − W µ ∆ µ − W h ∆ h − W σ ∆ σ − W σ ∆ σ ≡ . (13)We may note that C = V − W , ˙ C = V ˙ V − W x ˙ x − W y ˙ y ≡ , we have WV ≡ C ≡ M as˙ M = M ˙ VV − W (cid:0) W x ˙ y − W y ˙ x + ¨ x ˙ y − ˙ x ¨ y (cid:1) N + V (cid:0) W µ ∆ µ + W h ∆ h + W σ ∆ σ + W σ ∆ σ (cid:1) . (14)Now, using the equations¨ x = n ˙ y + W x , ¨ y = − n ˙ x + W y , ... x = n ¨ y + W xx ˙ x + W xy ˙ y , ... y = − n ¨ x + W xy ˙ x + W yy ˙ y , he analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries 5 and substituted the values of ˙ MV − M ˙ V from Eq.14, theequations of motion Eqs. 3a-3b, in the normal and tangentcoordinates, can be written as: S ¨ α = ¨ ξ − S ˙ α − ¨ S α , or¨ α = S − (cid:32) F N N + F ˙ N ˙ N + F ∆ P ∆ P + ¨ VV Ms (cid:33) , where F N = (cid:32) F N F N (cid:33) , F ˙ N = (cid:32) F N F N (cid:33) , and the remaining symbols are available in the appendix.Since α = (cid:32) NM (cid:33) therefore, the equations of motion innormal and tangent co-ordinates can be written as¨ N = r ∗ (cid:0) F N N + F ˙ N ˙ N + F ∆ P ∆ P (cid:1) , (15)and¨ M = ¨ VV M + s ∗ (cid:0) F N N + F ˙ N ˙ N + F ∆ P ∆ P (cid:1) . (16)The differential equation in the normal coordinate N canalso be written as:¨ N = V W xx ˙ y − W xy ˙ x ˙ y + W yy ˙ x + W (cid:0) − W x ˙ y + W y ˙ x (cid:1) (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) − n ( ¨ x ˙ y − ˙ x ¨ y ) − W ( ¨ x ˙ y − ˙ x ¨ y ) (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) + ¨ x + ¨ y − ˙ V N + V − W x µ ˙ y + W y µ ˙ x + W µ W (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) ∆ µ + W h W (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) ∆ h + − W x σ ˙ y + W y σ ˙ x + W σ W (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) ∆ σ + − W x σ ˙ y + W y σ ˙ x + W σ W (cid:16) − nV + ¨ x ˙ y − ˙ x ¨ y (cid:17) ∆ σ . For the sake of simplicity, we use the first order differ-ential equation 14 in ˙ M instead of using the second orderdifferential equation 16 in ¨ M . The matrix S ( t ) can be taken as periodical and the Eqs. (15) and (16) are linear differen-tial equations with periodical coefficients at ∆ P → N in Eq. 15 is in-dependent of the tangential coordinate M and homogeneous. We develop an algorithm to find the periodic solution (6)in two stages. First predictor and then corrector. In the pre-dictor part, we find linear displacements with respect to theparameters’ increments for the initial conditions x ( , µ , h , σ , σ ) = x ( T , µ , h , σ , σ ) , y ( , µ , h , σ , σ ) = y ( T , µ , h , σ , σ ) , ˙ x ( , µ , h , σ , σ ) = ˙ x ( T , µ , h , σ , σ ) , ˙ y ( , µ , h , σ , σ ) = ˙ y ( T , µ , h , σ , σ ) , where T is the time period of periodic solution.In the corrector part for the non-linear nature of param-eters’ increments, we use convergent iteration procedure tofind the linear corrections (with respect to the parameters) inthe initial conditions and the period.4.1 Predictor PartWe introduce displacements ξ by using the formula givenin Eqs. 7e-7f and then normal and tangential displacementsby using formula given in Eq. 11a. We use the Eq. (11a) toderive the Eqs. (14), (15) and (16). All the coefficients of N and M in these equations are periodic of period T .The period T ∗ of the desired solution can be written as T ∗ = T + τ , where T = T ( P ) , T ∗ = T ( P ∗ ) . Assuming ξ and ∆ P as first order small terms. By using the conditionsof periodicity of the solutions for normal and tangential dis-placements, we have the following boundary conditions: N ( ) = N ( T ) , ˙ N ( ) = ˙ N ( T ) , (17a) M ( ) = M ( T ) + V ( ) τ , (17b)˙ M ( ) = ˙ M ( T ) + ˙ V ( ) τ . (17c)The displacements in N , M and τ can be written in linearcombinations of varied parameters as follows: N = N ∆ µ + N ∆ h + N ∆ σ + N ∆ σ , (18a) M = M ∆ µ + M ∆ h + M ∆ σ + M ∆ σ , (18b) τ = τ ∆ µ + τ ∆ h + τ ∆ σ + τ ∆ σ . (18c)Using the independence of the parameter increments ∆ P k into the Eqs. 13 and 14 along with the boundary conditionsgiven by Eqs. 17a-17c, we have determined the value of N k , M k and τ k , ( k = , , , ) . Amit Mittal et al.
Fig. 2
The periodic orbits: the color code are as follows: 1: red, 2: blue, 3: green, 4: rubin red, 5: black for µ = . σ = . , σ = . σ = . , σ = . σ = . , σ = . σ = . , σ = . Now, we determine the boundary problem for the normaldisplacement: dv k dt = (cid:32) I J r ∗ F N r ∗ F ˙ N (cid:33) v k + (cid:32) r ∗ F ∆ P k (cid:33) , v k = (cid:32) N k ˙ N k (cid:33) , v k ( ) = v k ( T ) , ( k = , , . . . , K ) , K = J = , (19)where F ∆ P k is the k th column of the matrix F ∆ P and I J is theunit matrix of dimension J .The general solution of Eq. 19 can be written as: v k ( t ) = Z ( t ) v k ( ) + v ∆ P k ( t ) , (20)where Z ( t ) is the matrix of fundamental solutions of homo-geneous system with initial condition Z ( ) = I J , v k ( ) = initial conditions for v k ( t ) and v ∆ P k ( t ) = a particular solu-tion of inhomogeneous equations with zero initial condi-tions, i.e., v ∆ P k ( t ) = v k ( ) = − (cid:0) Z ( T ) − I J (cid:1) − v ∆ P k ( T ) . (21)The Eqs. (20) together with Eqs. (21) give the solution ofthe boundary value problem given by Eqs. 19.We determine displacement τ in the period by using theEqn. 13 in the form˙ M k = ˙ VV M k − V W (cid:0) g v v k + g P k (cid:1) , (22) he analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries 7 where g v = (cid:0) g N g ˙ N (cid:1) , g N = V (cid:0) W x ˙ y − W y ˙ x + ¨ x ˙ y − ˙ x ¨ y (cid:1) , g ˙ N = ( ˙ x ˙ y ) V (cid:32) − ˙ y ˙ x (cid:33) = , g P k = k th element of row matrix , g P = (cid:0) − W µ − W h − W σ − W σ (cid:1) . Then the general solution of Eqn. (22) is of the form: M k ( t ) = V ( t ) V ( ) M k ( ) + µ ( t ) v k ( ) + µ ∆ P k ( t ) , (23)where row-vector µ ( t ) and µ ∆ P k ( t ) are the solutions of Cauchyproblem 26e with 26i of system ( ) and M k ( ) = τ k = − V ( ) M k ( T ) , = − V ( ) (cid:0) µ ( T ) v k ( ) + µ ∆ P k ( T ) (cid:1) . (24)Using τ k by Eqn. (23) and differentiating Eqn. 22, we obtain˙ M k ( ) = − V ( ) W ( ) (cid:0) g v ( ) v k ( ) + g P ( ) (cid:1) . (25)The new periodic motion can be determined by calculatingthe quantities in the right-hand sides of (20), (23), (24) and(25). Therefore, we integrate the following differential equa-tions from t = t = T :¨ x = n ˙ y + W x , (26a)¨ y = − n ˙ x + W y , (26b)˙ Z j = (cid:32) I J r ∗ F N r ∗ F ˙ N (cid:33) Z j , (26c) Z j ( ) = e j , ( j = , , . . . J ) , (26d)˙ µ j = ˙ VV µ j − V W g v Z j , (26e) µ j ( ) = , (26f)˙ v ∆ P k = (cid:32) I J r ∗ F N r ∗ F ˙ N (cid:33) v ∆ P k + (cid:32) r ∗ F ∆ P k (cid:33) , (26g) v ∆ P k ( ) = , ( k = , , . . . , K ) , (26h)˙ µ ∆ P k = ˙ VV µ ∆ P k − V W (cid:0) g v v ∆ P k + g P k (cid:1) , (26i) µ ∆ P k ( ) = . (26j)Here, I J = ( e , . . . , e J ) , Z = ( Z , . . . , Z J ) , µ = ( µ , . . . , µ J ) ,and the initial conditions x ( ) , y ( ) , ˙ x ( ) and ˙ y ( ) are known.The order of the above system is 2 ( I + J ) + J ( J + I ) +( J + ) K , i.e., twenty two. Fig. 3
The periodic orbit for µ = . , σ = . , σ = . On solving the system of Eqs. 26a-26j, we can find Z ( T ) , v ∆ P k ( T ) , µ ( T ) and µ ∆ P k ( T ) . Then, we determine N k ( ) , ˙ N k ( ) , ˙ M k ( ) and τ k by using (21), (25) and (24) respectively. We, further, cal-culate the values of N ( ) , ˙ N ( ) , M ( ) , ˙ M ( ) and τ fromEqs. 18a-18c. The Eqs. 12a-12d and (9a) give the values of ξ ( ) and ˙ ξ ( ) . Finally, we determine the initial conditionsand period for new periodic solution with new parameter P ∗ = P + ∆ P , using formulae: x ∗ ( ) = x ( ) + ξ ( ) , y ∗ ( ) = y ( ) + ξ ( ) , ˙ x ∗ ( ) = a (cid:16) ˙ x ( ) + ˙ ξ ( ) (cid:17) , ˙ y ∗ ( ) = a (cid:16) ˙ y ( ) + ˙ ξ ( ) (cid:17) , T ∗ = T ( P ∗ ) = T + τ , (27)where a = (cid:115) W ( Q ∗ ( ) , P ∗ ) ˙ Q T ( ) ˙ Q ∗ ( ) , is correction coefficient satisfying the energy integral givenby Eq. 4 if a ≡
1. But, C (cid:44) x ∗ ( T ∗ ) − x ∗ ( ) , y ∗ ( T ∗ ) − y ∗ ( ) , ˙ x ∗ ( T ∗ ) − ˙ x ∗ ( ) and˙ y ∗ ( T ∗ ) − ˙ y ∗ ( ) are non-zero but small quantities of the sec-ond order with respect to ∆ P in the previous step. To refinethe initial conditions and the period, we use the correctorpart only.Let the solution 5 obtained by the predictor part be non-periodic solution of the Eqs. 3a-3b but in its close vicinity Amit Mittal et al. in the phase space there exists the periodic solution 6 withsame parameter values. Our goal is to find the periodic solu-tion given by 6 by taking the solution 5 as the initial approx-imation.We give the displacements by using the Eqs. 7e-7f todetermine displacements in initial conditions ξ ( ) , ξ ( ) ,˙ ξ ( ) and ˙ ξ ( ) and the period τ at ∆ P ≡ ξ ( t ) and ξ ( t ) are satisfying the Eqs.8a-8b along with the Jacobi integral given by Eq. 9. Fromthe normal and tangential displacements given by Eqs. 11c-11d, we obtain the differential equations 13 and 14.We consider the quantities ∆ x = x ( T ) − x ( ) (cid:44) , ∆ y = y ( T ) − y ( ) (cid:44) , ∆ ˙ x = ˙ x ( T ) − ˙ x ( ) (cid:44) , ∆ ˙ y = ˙ y ( T ) − ˙ y ( ) (cid:44) , as the small quantities of same order as that of ξ , ξ , ˙ ξ , ˙ ξ and τ . Thus, the new boundary conditions for the Equations13 and 14 are: N ( ) = N ( T ) + r ∗ ( ) ∆ Q , (28)˙ N ( ) = ˙ N ( T ) + ˙ r ∗ ( ) ∆ Q + r ∗ ( ) ∆ ˙ Q , M ( ) = M ( T ) + V ( ) τ + s ∗ ( ) ∆ Q , (29)˙ M ( ) = ˙ M ( T ) + ˙ V ( ) τ + ˙ s ∗ ( ) ∆ Q + s ∗ ( ) ∆ ˙ Q , where ∆ Q = (cid:32) ∆ x ∆ y (cid:33) and ∆ ˙ Q = (cid:32) ∆ ˙ x ∆ ˙ y (cid:33) .The general solution of the Equation 14 is determined inthe same fashion as in the predictor-part by using formula20 and rejecting high index k , i.e., v ∆ P k ≡
0. Then by usingboundary conditions given in Eqs. 28 and 29, we get v ( ) = (cid:32) N ( ) ˙ N ( ) (cid:33) = − (cid:0) Z ( T ) − I J (cid:1) − (cid:32) a a (cid:33) , (30)where a = V ( ) (cid:0) − ˙ y ( ) ∆ x + ˙ x ( ) ∆ y (cid:1) , a = V ( ) (cid:0) − ¨ y ( ) ∆ x + ¨ x ( ) ∆ y (cid:1) − ˙ V ( ) V ( ) − ˙ y ( ) ∆ x + ˙ x ( ) ∆ y + V ( ) (cid:0) − ˙ y ( ) ∆ x + ˙ x ( ) ∆ y (cid:1) . The boundary problem for tangential displacement can befound by substituting the value of normal displacement ob-tained by Eqs. 30 in Eqn. 13. Now, we can find the generalsolution in the form of (23) by rejecting the index k and as-suming that µ ∆ P k ≡
0. if we set initial displacement along the orbit as zero, weget M ( ) = , ˙ M ( ) = − V ( ) W ( ) g v ( ) v ( ) . (31)Using the periodic conditions given by Eqs. 28 in 29, thedisplacement for the period is given by τ = − V ( ) (cid:0) µ ( T ) v ( ) + s ∗ ( ) ∆ Q (cid:1) , = − V ( ) (cid:0) µ ( T ) N ( ) + µ ( T ) ˙ N ( ) (cid:1) + V ( ) (cid:0) ˙ x ( ) ∆ x + ˙ y ( ) ∆ y (cid:1) . (32)In corrector part, we integrate 26a-26f of system 26 from t = t = T . Hence, we calculate the values of N ( ) , ˙ N ( ) , M ( ) , ˙ M ( ) and τ by using the Eqns. (30), (31) and (32).Then, finally, we determine displacements ξ ( ) and ˙ ξ ( ) byusing 8a-8b, and new initial conditions by using (27). Theabove process is repeated again and again until a periodicorbit is drawn. We have determined periodic orbits in Figures 2 and 3 forfixed value of the mass parameter µ , different values of oblate-ness parameters σ , and varying values of the energy con-stant h . These orbits have been numbered 1 , , , h mentioned in each panel of the Figs. 2 and3. It is observed that the family in each case continues evenif the orbit (number 5) touches the point L .In this paper, we have studied the periodic orbits asso-ciated with the mobile coordinates where V ( t ) (cid:44) U (simi-lar to Karimov and Sokolsky (19)) in the lagrangian function L such that energy constant h vanishes at the non-collinearlibration point L , . We have drawn these periodic orbits byusing the well known method: predictor-corrector. We havegiven the displacements to the mobile coordinates along thenormal and the tangent directions to the orbit. We have plot-ted the five periodic orbits in a family by taking the fixedvalues of the mass parameter µ , oblateness parameters σ , and increasing values of the energy constant h . These peri-odic orbits are named as 1, 2, 3, 4, and 5 in each family. Inorder to draw these orbits, we have used the methodologyadopted by Karimov and Sokolsky (19).The most prominent observations on the periodic orbitsas well as on the energy constant h in the presence of theoblate primaries are as follows: he analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries 9
1. The energy constant h decreases if we consider both theprimaries as oblate bodies but it increases if we consider σ =
0, i.e., in the absence of oblateness of the first pri-mary.2. With the increase in the oblateness parameters σ , , theenergy constant h decreases.3. Each periodic orbit is non-symmetrical in the presenceof oblateness of the primaries.4. Family of periodic orbits do not terminate at the librationpoint L rather they continued which is contrary to thecase of Karimov and Sokolsky (1989).We emphasize that this study is significantly different fromothers in the sense that most of the natural and artificial bod-ies moving in space instead of point masses, they are oblatebodies. Thus, our model is more realistic. Besides takingboth the primaries as oblate bodies, we have used mobile-coordinates by giving the displacements along the normaland the tangent to the orbit which has wider applications inspace dynamics. Appendix F N = ˙ xW xy − W xx ˙ yV + WV (cid:18) − W x ˙ y + W y ˙ x (cid:19)(cid:18) n ˙ y − ¨ x + ˙ VV ˙ x (cid:19) + n ¨ xV − n ˙ VV ˙ x − WV ( ¨ x ˙ y − ˙ x ¨ y ) (cid:32) n ˙ y − ¨ x + ˙ VV ˙ x (cid:33) + W xy ˙ x + W yy ˙ y − n ¨ xV − VV ¨ y − ¨ VV ¨ y + V V ¨ y , F N = W yy ˙ x − W xy ˙ yV + WV (cid:0) − W x ˙ y + W y ˙ x (cid:1) (cid:32) − n ˙ x − ¨ y + ˙ VV ˙ y (cid:33) + n ¨ yV − n ˙ VV ˙ y − WV ( ¨ x ˙ y − ˙ x ¨ y ) (cid:32) − n ˙ x − ¨ y + ˙ VV ˙ y (cid:33) − W xx ˙ x + W xy ˙ y + n ¨ yV + VV ¨ x + ¨ VV ¨ x − V V ¨ x , F N = (cid:32) n ˙ xV + ¨ yV − ˙ V ˙ yV (cid:33) , F N = (cid:32) n ˙ yV − ¨ xV + ˙ V ˙ xV (cid:33) , F ∆ P = (cid:16) F ∆ µ F ∆ h F ∆σ F ∆σ (cid:17) , F ∆ µ = W x µ + W µ W (cid:16) n ˙ y + ˙ VV ˙ x − ¨ x (cid:17) W y µ + W µ W (cid:16) − n ˙ x + ˙ VV ˙ y − ¨ y (cid:17) , F ∆ h = W h W (cid:16) n ˙ y + ˙ VV ˙ x − ¨ x (cid:17) W h W (cid:16) − n ˙ x + ˙ VV ˙ y − ¨ y (cid:17) , F ∆σ = W x σ + W σ W (cid:16) n ˙ y + ˙ VV ˙ x − ¨ x (cid:17) W y σ + W σ W (cid:16) − n ˙ x + ˙ VV ˙ y − ¨ y (cid:17) , F ∆σ = W x σ + W σ W (cid:16) n ˙ y + ˙ VV ˙ x − ¨ x (cid:17) W y σ + W σ W (cid:16) − n ˙ x + ˙ VV ˙ y − ¨ y (cid:17) . Acknowledgments
The authors are thankful to Center for Fundamental Research in Spacedynamics and Celestial mechanics (CFRSC), New Delhi, India for pro-viding research facilities.
Compliance with Ethical Standards - Funding: The authors state that they have not received any researchgrants.- Conflict of interest: The authors declare that they have no conflict ofinterest.
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