Periodic orbits around the collinear libration points

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1 Available online at J. Nonlinear Sci. Appl. XX, XX XX Research Article Periodic orbits around the collinear libration points Elbaz I. Abouelmagd a,b, Faris Alzahrani b, Aatef Hobin b, J. L. G. Guirao c,, M. Alhothuali b a Celestial Mechanics Unit, Astronom Department, National Research Institute of Astronom and Geophsics (NRIAG). Helwan, Cairo, Egpt. b Nonlinear Analsis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz Universit, Jeddah, Saudi Arabia. c Departamento de Matemática Aplicada Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 3003 Cartagena, Región de Murcia, Spain. Abstract The locations for the collinear libration points in the framework of the restricted three-bod problem are determined when the bigger primar is a triaial rigid bod. The analsis of the periodic motion around these points is performed and given up to second order in the case that the initial state of the motion gives rise to periodic orbits. Moreover, some numerical results for the locations of collinear points are provided and the graphical investigations for the periodic motion are plotted, as well. It is worth mentioning that the collinear libration points and associated periodic orbits are considered for the optimal placement to transfer a spacecraft to the nominal periodic orbits or to an associated stable manifold. Kewords: Restricted three-bod problem, collinear points, periodic orbits. 00 MSC: 70E7, 70E0, 70E40, 37C7.. Introduction The restricted three-bod problem is one of the most important branches in celestial mechanics. This is due to its valuable applications for orbit design and space navigation flights. The literature is rich in works about finding the positions of the libration pints, studing their linear stabilit, and obtaining the orbits of motion around them. Since the land mark work of Szebehel (967), which studied the original problem consisting of the restricted three-bod problem with the primaries considered spherical, man researches have been made looking for a generalization of the problem b considering the elliptic problem rather than the circular problem, see [3]. While in the frame of the deviation of the shape of the primaries from the Corresponding author addresses: eabouelmagd@gmail.com, eabouelmagd@kau.edu.sa (Elbaz I. Abouelmagd), juan.garcia@upct.es (J. L. G. Guirao) Received September 4, 05

2 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX complete sphere, or their phsical nature being radiating or having a magnetic field, were introduced in [8, 9]. In [], the were studied the periodic orbits generated b Lagrangian solutions of the restricted threebod problem whenever the bigger primar is radiating and the smaller is an oblate spheroid. The eistence and stabilit of the libration points in the restricted problem, when the three bodies are oblated and the primaries are radiating, were studied in []. The also provided the periodic orbits around the triangular points. In addition, it was proved in [] that the locations for the triangular points as well as their linear stabilit depend on the parameters regarding the first two even zonal harmonic of the more massive primar in the planar circular restricted three-bod problem. Moreover, it was also showed therein that the triangular points become stable for 0 < µ < µ c, and unstable provided that µ c µ /, where µ c is the critical mass parameter, which depends on the coefficients of zonal harmonic. Further, some numerical values regarding the positions of the triangular points, mass ratio and critical mass for the planets sstems in our solar sstem, were provided. The motion around the collinear equilibria when the bigger primar is an oblate spheroid was studied in [5]. Further, [] deals with the motion around the collinear equilibrium points in the restricted three-bod problem when the larger primar is radiating and the second primar is an oblate spheroid. For additional details, see [6, 8, 9, 0]. Furthermore, in [0], it was presented a comprehensive analtical stud regarding the eistence of the libration points and their linear stabilit in the restricted three-bod problem under the effect of the first two even zonal harmonic parameters with respect to both primaries. The also found out the periodic orbits around the libration points, the epressions for semi-major and semi-minor aes, the eccentricities, and the periods of elliptical orbits, as well as the orientation of the principal aes. Moreover, there is an etensive and analtical stud which has a great scientific sound in the framework of the restricted three-bod problem. These studies on the eistence of libration points, their linear stabilit, the periodic and secular solutions, as well as the solutions through numerical integration methods under the effects of the lack of spherecit and radiation pressure, has been introduced b E.I. Abouelmagd and his research group. See for eample, but not limited to, [,, 3, 4, 5, 7]. However, the motion in the proimit of collinear libration points in the framework of the restricted three-bod problem has attracted the interest of man other works, including [4, 7]. The periodic orbits around the collinear libration points represent an interesting case of motion in astrodnamics, since the moving bod remains for infinite time in the proimit of an unstable equilibrium point. These orbits have a special importance, since the ma be considered as the limiting case of asmptotic orbits to the Lapunov periodic orbits (for further details, see [5]). In this work, we present an analtical stud regarding the periodic motion in the proimit of collinear libration points provided that the bigger primar is triaial. This motion has been proved to be generall unstable in the case that the bigger primar is triaial and radiating [0]. This also remains true whether the radiation effect is neglected. Nevertheless, for certain initial values for the velocit, it holds that periodic motions could eist around the three collinear points. The positions of the collinear points are obtained b considering the triaialit of the bigger primar as a perturbation for the motion, and periodic orbits around them are obtained up to second order. Numerical results are provided for some chosen values for the parameter µ and the triaialit terms of the bigger primar which are in turn functions of the moments of inertia of the bigger primar and the distance between the two primaries, respectivel.. Equations of motion In this paper, the are used the notations and terminolog of Szebehel (967). We will take the distance between the primaries equal to one the sum of the masses of the primaries is also taken as the unit of mass. The unit of time is also chosen as to make the gravitational constant is unit. Using dimensionless variables, the equations of motion of the infinitesimal mass in a snodic coordinate sstem (, ) can be written in the following form (see [6]):

3 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX 3 where { Ω = ẍ nẏ Ω = ÿ + nẋ, Ω = n [( µ) r + µ r ] + µ r + µ r + µ (σ r 3 σ ) + r = ( µ) + r = ( + µ) + µ = m m +m. 3 ( µ) r 5 (σ σ ) Note that µ (0, /) is the parameter of mass ratio, m, m are the masses of the primaries within m m, σ = A C 5R, σ = B C 5R, σ, σ <<, A, B, and C are the semi-aes of the triaial rigid bod, and R is the distance between the primaries, as well. Further, n is the mean motion of the primaries, which is governed b n = + 3 (σ σ ). It is worth mentioning that for an oblate bod, σ = σ. (.) 3. Locations of the collinear points If the collinear points are located in = 0, then the derivatives of Ω with respect to and, resp., are given b { Ω = n µ r Ω = 0. µ r 3( µ)(σ σ ) r 4 This leaves us a small parameter in the problem, namel, ε = σ σ, which takes the problem from the original case of a point mass to our case consisting of a tri-aial rigid bod. Thus, let us epand the locations of the collinear libration points as follows: J = J,0 + i>0 ε i J,i, J =,, 3, (3.) where J,0 are the coordinates of the collinear points whether σ = σ = 0. The above epansion is used to solve Eq. (3.) in order to get the net locations for the collinear libration points: J = J,0 + n ε i P J,i, where < µ < < µ < 3, and P J,i s depend on µ and j s. i=

4 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX 4 4. Motion around the collinear points The three collinear points are proved to be generall unstable whether the bigger primar becomes triaial (see [0]). However, for certain initial conditions, periodic and stable orbits ma eist. We stud the motion regarding the collinear points as follows. Let = J + ξ, = η, where ξ, η <<. Thus, substituting in Eq. (.) and epanding Ω, Ω in Talor series up to second order around the collinear points, we obtain { ξ n η = a ξ + a ξ + a 3 η η + n ξ (4.) = a 4 η + a 3 ξη, where a, a, a 3, and a 4 are the non-vanishing coefficients in that epansion. Their values are given below: a = Ω j = n + ( µ) r 3 a = 3 Ω j = 6( µ) 3 r 4 6µ r 4 a 3 = 3 Ω j = 3( µ) + 3µ r 4 r 4 a 4 = Ω j = n µ r 3 µ r 3 + µ r 3 + 6ε( µ) r 5 30ε( µ) r 6 + 5ε( µ)(+e) r 6 3ε( µ)(+e), r 5 and E = σ /σ. To solve Eq. (4.), the involved coefficients are written in the net form: { n = n 0 + ε n a i = a i0 + ε a i, where i =,, 3, 4, the values of a i0 and a i are the terms free of the small parameter ε, and the coefficients of it, respectivel. Net, let us epand the variables ξ, η as { ξ = ε ξ + ε ξ η = ε η + ε (4.) η. Substituting Eq. (4.) into Eq. (4.) and equating terms of the same order in ε with keeping the terms up to second order of ε, we obtain { ξ η = a 0 ξ { ξ 3 η = a 0 ξ + a ξ + a 0 ξ First order: η + ξ Second order: + a 30 η = a 40 η. η + 3 ξ = a 40 η + a 4 η + a 30 ξ η. (4.3) The solution of the first order is simpl the linearized motion about the equilibrium collinear points in the classical sense. Seeking a periodic motion about the collinear points, the initial conditions are taken such that onl the terms resulting in periodic motion remain. This is accomplished b choosing ξ 0 = s3 s 3 + a η 0, η 0 = (s + a ) η 0, and hence, { ξ = ξ 0 cos st + η 0 Γ sin st η = η 0 cos st Γ ξ 0 sin st, (4.4)

5 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX 5 where s is the angular frequenc of motion, and ξ 0, η 0 are selected to match the initial conditions. In this contet, the values of Γ and s will be controlled through the following epressions: Γ = s +a 0 s = s (4 a 0 + a 40 ) + 4 (4 a 0 + a 40 ) a 0 a 40. A particular solution for Eq. (4.3) (nd order), will be ruled b { ξ = α 0 + k= [αc k cos(ks)t + αs k sin(ks)t] η = β 0 + k= [βc k cos(ks)t + βs k sin(ks)t]. (4.5) Note that Eq. (4.5) describes the motion about the collinear points arising from the second order of the triaialit. Substituting Eq. (4.4) into the sstem in Eq. (4.3), then we get the coefficients in Eq. (4.5), which are listed below: α 0 = β 0 = 0 α C = ξ 0 Γ a 0 [(Γ a 0 + Γ 4 a 30 ) ξ 0 + (a 0 + Γ a 30 ) η 0] a s + 3Γ a 4 s + a a 4 s 4 + s (a 0 + a 40 9) + a 0 a 40 α C = a 30 (η0 Γ ξ0 ) Γs α S = η 0 Γ α S = η 0ξ 0 Γ β C = η 0 Γ β C = η 0ξ 0 Γ a s +3Γa 4 s+a a 4 s 4 +s (a 0 +a 40 9)+a 0 a 40 c 4 s 4 +c 3 s 3 +c s +c s+c 0 Γ (96s 5 +4(a 0 +a 40 9)s 3 +6a 0 a 40 s) 4(a 0 Γ a 30 )s +6Γa 30 s+a 0 a 40 Γ a 30 a 40 6s 4 +4s (a 0 +a 40 9)+a 0 a 40 Γa 4 s +3a s+γa 0 a 4 s 4 +(a 0 +a 40 9)s +a 0 a 40 4Γa 30 s +6(a 0 Γ a 30 )s+γa 0 a 30 6s 4 +4(a 0 +a 40 9)s +a 0 a 40 β S = ξ Γa 4 s + 3a s + a 0 Γa 4 0 s 4 + (a 0 + a 40 9)s + a 0 a 40 β S = 4Γa 30(η0 Γ ξ0 )s + 6[(a 0 a 30 Γ )η0 + (a 30Γ 4 a 0 Γ )ξ0 ]s + Γa 0a 30 (η0 Γ ξ0 ) Γ [6s 4 + 4(a 0 + a 40 9)s, + a 0 a 40 ] where the superscripts C, S of α and β refer to the coefficients of cosines and sines: c 0 = a 0a 40 c 4 6 c = a 40 4 c 3 c = 4Γa 30 (η0 Γ ξ0 )(a 0 + a 40 ) c 3 = 4((a 0 Γ a 30 ) η0 (a 0 Γ a 30 )Γ ξ0 ) c 4 = 6Γa 30 (η0 Γ ξ0 ). After determining the quantities of variations ξ, ξ, η, η, we can state that Eq. (4.) describes the possible periodic motion about the collinear equilibrium points up to second order. 5. Numerical results In this section, some numerical results are provided for the problem in each point. In the first subsection, the locations of the collinear points within the eistence of triaial terms of order 0.0 and 0. are given in comparison with the case of a spherical bod (corresponding to ε = 0). These locations are calculated for

6 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX 6 µ ε = 0 ε = 0.0 ε = Table : Locations of. µ ε = 0 ε = 0.0 ε = Table : Locations of. different values of the parameter, whose mass ratio is µ = 0., 0., 0.3, 0.4,. In the second subsection, the orbits of the motion are plotted for ε = 0.0 at µ = 0., 0., 0.3, 0.4,. The calculations are carried out at the initial position (0., 0.), when σ = 0., σ = 0.9. It is worth mentioning that when the calculations are made at other values of triaial parameters also satisfing ε = 0.0, then the graphs showed no difference within an accurac for the calculations of the order of Comparison results for the locations of the collinear points In this subsection, the locations of the collinear points,, and 3, will be determined for two different values which represent the effect of triaialit. To investigate the influence of the non-sphericit, such locations have been also calculated whether the bigger primar has spherical shape see forthcoming Tables Curves of motion around the collinear points One of the ke facts related to periodic trajectories around collinear libration points to be checked is if the moving bod remains an infinite time in the vicinit of collinear points. Moreover, it holds that these orbits can be used to transfer the trajector of the infinitesimal bod (for instance, a spacecraft) to periodic orbits close to the nominal or to associate stable manifolds. The basic idea behind this approach depends on the fact that making a spacecraft to follow a reference periodic orbit is often not critical, while it is much more important to save enough fuel for station-keeping during the planned mission term. Thus, a special attention has been paid in this subsection on the graphs of periodic orbits in the proimit of collinear points having several values for mass ratio µ and period T, where T is the dimensionless periodic time given b T = π/s, see Figs. ()-(5). It should be noted that, since the motion in the vicinit of the collinear points are unstable in general, there is a special interest to find their locations and associated periodic orbits, as well. These points provide µ ε = 0 ε = 0.0 ε = Table 3: Locations of 3.

7 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX Figure : Periodic orbits around, when µ = 0. and T = Figure : Periodic orbits around, when µ = 0. and T = Figure 3: Periodic orbits around, when µ = 0.3 and T = Figure 4: Periodic orbits around, when µ = 0.4 and T =.047.

8 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX Figure 5: Periodic orbits around, when µ = and T = Figure 6: Periodic orbits around, when µ = 0. and T = Figure 7: Periodic orbits around, when µ = 0. and T = Figure 8: Periodic orbits around, when µ = 0.3 and T =

9 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX Figure 9: Periodic orbits around, when µ = 0.4 and T = Figure 0: Periodic orbits around, when µ = and T = Figure : Periodic orbits around 3, when µ = 0. and T = Figure : Periodic orbits around 3, when µ = 0. and T =.9955.

10 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX Figure 3: Periodic orbits around 3, when µ = 0.3 and T = Figure 4: Periodic orbits around 3, when µ = 0.4 and T = Figure 5: Periodic orbits around 3, when µ = and T =

11 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX a useful platform to investigate both the solar sstem and the universe. This importance comes from a growing interest in missions to collinear libration points and associated periodic orbits. 6. Conclusion In this paper, the equations of motion in the framework of the restricted three-bod problem are found whether the bigger primar is a triaial rigid bod. The locations of the collinear libration points have been also evaluated. The analtical stud on the periodic solution around these points is constructed and given up to second order in the case that the initial state of the motion gives rise to periodic orbits. Furthermore, some numerical results are established for the locations of each collinear libration point with the eistence of triaial terms. These results are compared with the case consisting of the bigger primar has a spherical shape for different values regarding the mass ratio. Moreover, the curves of motion in the proimit of these points have been also plotted. Finall, we would like to point out that the model of the restricted three-bod problem results interesting in space dnamics, and in particular, if the model includes the effect of the triaialit. In that case, in fact, significant improvements for theories regarding the motion for certain satellites in our solar sstem will follow. Acknowledgements: All the authors of this work ecept the fourth one would like to thank the Deanship of Scientific Research (DSR) at King Abdulaziz Universit, Jeddah, which funded this stud under grant no. ( RG). The authors, therefore, acknowledge with thanks DSR technical and financial support. The fourth author was partiall supported b MICINN/FEDER: grant number MTM0-587 MINE- CO: grant number MTM P, and Fundación Séneca de la Región de Murcia: grant number 99/PI/4. References [] Abouelmagd E. I., Alhothuali M. S., Guirao J. L. G., Malaikah H. M., On the periodic structure in the planar photogravitational Hill problem, Applied Mathematics & Information Science. (05a) 9 (5): [] Abouelmagd E. I., Alhothuali M. S., Guirao J. L. G., Malaikah H. M., Periodic and secular solutions in the restricted threebod problem under the effect of zonal harmonic parameters, Applied Mathematics & Information Science. (05b) 9 (4):. [3] Abouelmagd E. I., Alhothuali M. S., Guirao J. L. G., Malaikah H. M., The effect of zonal harmonic coefficients in the framework of the restricted three-bod problem, Advances in Space Research (05c) 55: [4] Abouelmagd E. I., Guirao Juan L. G., Mostafa A., Numerical integration of the restricted thee-bod problem with Lie series, Astrophsics Space Science (04a) 354: [5] Abouelmagd E. I., Awad M. E., Elzaat E. M. A., Abbas I. A., Reduction the secular solution to periodic solution in the generalized restricted three-bod problem, Astrophsics Space Science (04b) 350: [6] Abouelmagd E. I., Asiri H. M., Sharaf M. A., The effect of oblateness in the perturbed restricted three-bod problem, Meccanica (03) 48: [7] Abouelmagd E. I., Mostafa A., Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-bod problem with variable mass, Astrophsics Space Science (05) 357 (): 58. [8] Abouelmagd E. I., Stabilit of the triangular points under combined effects of radiation and oblateness in the restricted three-bod problem, Earth Moon and Planets (03a) 0: [9] Abouelmagd E. I., The effect of photogravitational force and oblateness in the perturbed restricted three-bod problem, Astrophsics Space Science (03b) 346: [0] Abouelmagd E. I., Sharaf M. A., The motion around the libration points in the restricted three-bod problem with the effect of radiation and oblateness, Astrophsics Space Science (03) 344: [] Abouelmagd E. I., Eistence and stabilit of triangular points in the restricted three-bod problem with numerical applications, Astrophsics Space Science (0) 34: [] Abouelmagd E. I., El-Shabour S. M., Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Astrophsics Space Science (0) 34: [3] El-Shabour S.M., Mostafa A., The singl averaged elliptical restricted three-bod problem, Astrophsics Space Science (03) 348 ():

12 E.I. Abouelmagd et al., J. Nonlinear Sci. Appl. XX, XX XX [4] Gmez G., Mondelo J. M., The dnamics around the collinear equilibrium points of the RTBP, Phsica D (00) 57(4) [5] Kalantonis V.S., Douskos C.N., Perdios E.A., Numerical determination of homoclinic and heteroclinic orbits at collinear equilibria in the restricted Three-Bod Problem with Oblateness, Celest. Mech. And Dn. Astr. (006) 94 (): [6] Khanna M., Bhatnagar K. B., Eistence and stabilit of libration points in the restricted three-bod problem when the smaller primar is a triaial rigid bod, Indian J. Pure. Appl. Math. (998) 9 (0): 0. [7] Llibre J., Martnez R., Sim C., Transversalit of the invariant manifolds associated to the Lapunov famil of the periodic orbits near L in the restricted three-bod problem, J. Diff. Eqs. (985) 58, [8] Sharma R. K., Subba Rao P.V., Collinear equilibria and their characteristic eponents in the restricted three-bod problem when the primaries are oblate spheroids, Celest. Mech. And Dn. Astr. (975) (): [9] Sharma R. K., Subba Rao P.V., Stationar solutions and their characteristic eponents in the restricted three-bod problem when the more massive primar is an oblate spheroid, Celest. Mech. And Dn. Astr. (976) 3(): [0] Sharma R. K., Taqvi Z. A., Bhatnagar K. B., Eistence and stabilit of libration points in the restricted three bod problem when the bigger primar is a triaial rigid bod and a source of radiation, Indian J. Pure Appl. Math. (00) 3(), [] Mittal A., Ahmad I., Bhatnagar K. B., Periodic orbits in the photogravitational restricted problem with the smaller primar an oblate bod, Astrophsics Space Science (009), 33: [] Tsirogiannis G. A., Douskos C. N., Perdios E.A., Computation of the Liapunov orbits in the photogravitational RTBP with oblateness, Astrophsics Space Science (006) 305: