The existence of a new class of inclined periodic orbits of the collision restricted three–body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet

Barrabés Vera, Esther

Cors Iglesias, Josep Maria

Pinyol Pérez, Conxita

Soler VIllanueva, Jaume

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–4)J2 effect and the Collision Restricted Three–Body ProblemE. Barrabe´s1, J.M. Cors2, C. Pinyol3, J. Soler11 Deptartament d’Informa`tica i Matema`tica Aplicada, Universitat de Girona, 17071 Girona, Spain.E-mails: esther.barrabes@ima.udg.es, jaume.soler@ima.udg.es.2 Departament de Matema`tica Aplicada III, Universitat Polite`cnica de Catalunya, 08242 Manresa, Spain.E-mail: cors@epsem.upc.edu.3 Departament d’Economia i Histo`ria Econo`mica, Universitat Auto`noma de Barcelona, 08193 Bellaterra,Spain. E-mail: Conxita.Pinyol@uab.es.Palabras clave: Collision restricted three-body problem, periodic orbits, symmetric orbits, criticalinclination, continuation method.ResumenThe existence of a new class of inclined periodic orbits of the collision restrictedthree–body problem is shown. The symmetric periodic solutions found are perturba-tions of elliptic kepler orbits and they exist only for special values of the inclinationand are related to the motion of a satellite around an oblate planet.1. IntroductionFrom Keplerian orbit theory, without perturbations, elliptic orbits are always confinedon an planet centered plane known as orbital plane. Six parameters a, e, i, h, g and f ,called the classical orbital elements, completely characterize a conic orbit (see, for example,[3]).For Earth–orbit design, the main effect that must be included is the so called J2 term,named after the coefficient of a planet’s geopotential harmonic expansion. The J2 termcorresponds to the planet’s equatorial bulge and has two important effects on the orbit.Due this extra mass the satellite reach its ascending node sooner than it would withoutthe perturbation. Hence the node appears to move backward and the effect is called nodalregression. The second effect is the precession of the argument of perigee. J2 causes theperigee to rotate around the orbit’s normal vector. It is apparent (see, for example, pages503–504 of [3]), that there exist a critical inclination angle, i ' 63◦, such that perigeeis fixed. Critically inclined elliptic orbits are very useful. The most famous of these is1E. Barrabe´s, J.M. Cors, C. Pinyol, J. Solerthe Molniya orbit, which is a highly elliptic 12–hour–period orbit the Soviets originallydesigned to observe the northern hemisphere.The existence of a class of inclined periodic solutions in the circular restricted three–body problem was shown by Jefferys in [6]. He showed the existence of families of ellipticorbits for any value of the eccentricity and a critical inclination. In fact, when one of theprimaries is small enough, and the infinitesimal body is far away from the primaries, thesmall primary moves very fast and the potential looks almost like that of an oblate planet.This is why Jefferys finds a critical inclination for these orbits. The method used in [6]was the continuation method developed by Poincare´ (see [9]), which is one of the mostfrequently used methods for proving the existence of periodic orbits.The case dealt with in this communication (see the complete text in [2]) is differentfrom Jefferys because the primaries move on an elliptic collision orbit along the z-axis.Heuristically speaking, however, it can be expected that far away from the primaries thepotential will be similar to that of a very eccentric prolate ellipsoid, so that a J2 effect, withits critical inclination, will exist. We show the existence of periodic solutions of Jefferystype: large semi-axis compared to the that of the primaries, arbitrary eccentricity andinclination close to critical.The perturbed problem is degenerate due to the fast motion of the primaries, andthe equations are no longer analytic when the parameter equals zero, which precludesthe use of standard implicit function theorem. We overcome the difficulty by using Arens-torf’s theorem, where weaker assumptions of differentiability are needed (see [1]). A planarconfiguration of this problem is studied in [7].In our case, the problem has a rotational symmetry around the z-axis (which containsthe colliding primaries). This symmetry would be lost if we considered elliptic non collisionorbits for the primaries. See [4] and [5], where the elliptic restricted three-body problem isconsidered. In those papers, the periodic orbits are perturbations of the circular solutionsof the Kepler problem having large radii on a plane perpendicular to that of the primaries.If the unperturbed orbit is a polar one, the precession induced by a small variation of theinclination can be used to compensate for any variation of the orbital plane due to theperturbation.Periodic orbits in the spatial elliptic restricted three–body are also studied using doubleaveraging in [8].2. The Collision Restricted Three Body ProblemThe Collision Restricted Three Body Problem describes the motion of a massless par-ticle under the attraction of two primaries with equal masses, m1 = m2 = 1/2, movingon a collision elliptic orbit. In order to avoid a triple collision, we consider that the thirdbody is far from the primaries compared to the distance between them. This fact can beintroduced in the equations of motion by making the primaries very close to each other andlooking for solutions of the massless particle at distance of order unity to the primaries.Let µ be a small parameter. The distance between both primaries is given byρ = µ(1− cosEp(t)),where Ep = Ep(t) is the eccentric anomaly of m1 and it is related to its mean anomaly `p2J2 effect and the Collision RTBPthrough Kepler’s equationEp − sinEp = `p, (1)where `p = µ−3/2t. The period of the motion of the primaries is Tp = 2piµ3/2, so thatEp = kpi when t = pikµ3/2.The equations of motion, in spherical coordinates, for the infinitesimal body can bewritten as a non autonomous Hamiltonian system depending on the parameter µ asH =12(p2r +p2φr2+p2θr2 cos2 φ)− 12(1R1+1R2), (2)with R1 and R2 given byR21 = r2 +(µ2)2 (1− cosEp)2 − rµ(1− cosEp) sinφ,R22 = r2 +(µ2)2 (1− cosEp)2 + rµ(1− cosEp) sinφ.Notice that Ep as given by Equation (1), is a function of time t and µ, which is not definedfor µ = 0. So, neither the Hamiltonian (2) is defined.Since R1 and R2 do not depend on θ, p˙θ = 0 and the angular momentum pθ = Θis constant. Thus, it can be calculated from the initial conditions and the equation for θcan be decoupled from the other equations. We call, the problem in variables (r, φ, pr, pφ),reduced problem.3. Main resultTHEOREM. Consider the Three–Dimensional Collision Restricted Three Body Problemwith masses m1 = m2 = 1/2, and primaries’s semimajor axis µ/2. If µ = k−2/3, where k isa positive integer large enough, there exist initial conditions such that the infinitesimal bodymoves in a symmetric periodic orbit of the reduced problem, of period 2pi, near a Keplerianelliptic orbit. The inclination of the orbit is close to the “critical value” cos i = 1/√5.Remark: all the orbits found are on an integral resonance with the motion of theprimaries, i.e. the primaries undergo k complete orbits in one orbit of the infinitesimalbody. If k = p/q is an irreducible rational then similar arguments show that in q completeorbits of the infinitesimal the primaries undergo p complete orbits.AgradecimientosE. Barrabe´s has been supported by grant MTM2006-05849/Consolider (including aFEDER contribution), J.M. Cors by grant number MTM 2005-06098-C02-01 and by aCIRIT grant number 2001SGR00173, C. Pinyol by grant SEJ2006/00712ECO and J. Solerby grant MTM2005-07660-C02-02.3E. Barrabe´s, J.M. Cors, C. Pinyol, J. SolerReferencias[1] R.F. Arenstorf, A new method of perturbation theory and its application to the satellite problemof celestial mechanics, Journal fu¨r die Reine und Angewandte Math, 221 (1966), pp. 113–145.[2] E. Barrabe´s J.M. Cors, C. Pinyol, and J. Soler, J2 effect and the elliptic inclined periodicorbits in the collision restricted three–body problem, SIAM J. Appl. Dyn. Syst. (2007). To appear.[3] Richard H. Battin, An introduction to the mathematics and methods of astrodynamics, AIAAEducation Series, American Institute of Aeronautics and Astronautics (AIAA), Washington, DC,1987. With a foreword by J. S. Przemieniecki.[4] J.M. Cors, C. Pinyol, and J. Soler, Periodic solutions in the spatial elliptic restricted three-bodyproblem, Phys. D, 154 (2001), pp. 195–206.[5] , Analytic continuation in the case of non-regular dependency on a small parameter with anapplication to celestial mechanics, J. Differential Equations, 219 (2005), pp. 1–19.[6] W.H. Jefferys, A new class of periodic solutions of the three-dimensional restricted problem, TheAstronomical Journal, 71 (1966), pp. 99–102.[7] J. Llibre & D. Pasca, Periodic orbits of the planar collision restricted 3-body problem, Celestialmechanics and dynamical astronomy, 96 (2006), pp. 19–29.[8] J.F. Palacian, P. Yanguas, S. Fernandez, and M.A. Nicotra, Searching for periodic orbits ofthe spatial elliptic restricted three-body problem by double averaging, Physica D: Nonlinear Phenome-na, 213 (2006), pp. 15–24.[9] H. Poincare´, Les Me´thodes Nouvelles de la Me´canique Ce´leste. Tome III, Gauthier-Vilars, Paris,1899.4

J2 effect and the collision restricted three–body problem

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J2 effect and the collision restricted three–body problem

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