Nonlinear Dynamics

, Volume 93, Issue 4, pp 2019–2038 | Cite as

Nonlinear librations of distant retrograde orbits: a perturbative approach—the Hill problem case

  • Martin LaraEmail author
Original Paper


The non-integrability of the Hill problem makes its global dynamics be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem has been thoroughly investigated by perturbation methods in two cases: when the motion is close to the origin, and in the case of motion about the libration points. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes progress in the perturbation approach beyond the lower orders of the solution really difficult. An alternative perturbation approach is proposed here that allows to provide a very simple low-order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.


Hill problem Perturbation methods Lindstedt–Poincaré method Elliptic functions Distant retrograde orbits Quasi-satellite orbits 



Support by the Spanish State Research Agency and the European Regional Development Fund under Projects ESP2016-76585-R and ESP2017-87271-P (MINECO/ AEI/ERDF, EU) is recognized.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


  1. 1.
    Abell, P.A., Mazanek, D.D., Reeves, D.M., Chodas, P.W., Gates, M.M., Johnson, L.N., Ticker, R.L.: NASA’s Asteroid Redirect Mission (ARM). In: Lunar and Planetary Science Conference, Lunar and Planetary Institute Technical Report, vol. 48, p. 2652 (2017)Google Scholar
  2. 2.
    Armellin, R., San-Juan, J.F., Lara, M.: End-of-life disposal of high elliptical orbit missions: the case of INTEGRAL. Adv. Space Res. 56(3), 479–493 (2015). (Advances in Asteroid and Space Debris Science and Technology–Part 1 )CrossRefGoogle Scholar
  3. 3.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)CrossRefGoogle Scholar
  4. 4.
    Benest, D.: Libration effects for retrograde satellites in the restricted three-body problem. I—Circular plane Hill’s case. Celest. Mech. 13, 203–215 (1976). CrossRefzbMATHGoogle Scholar
  5. 5.
    Bezrouk, C., Parker, J.S.: Long term evolution of distant retrograde orbits in the Earth-Moon system. Astrophys. Space Sci. 362, 176 (2017). CrossRefGoogle Scholar
  6. 6.
    Boccaletti, D., Pucacco, G.: Theory of Orbits. Volume 2: Perturbative and Geometrical Methods. Astronomy and Astrophysics Library, 1st edn. Springer, Berlin (2002)zbMATHGoogle Scholar
  7. 7.
    Boccaletti, D., Pucacco, G.: Theory of Orbits. Volume 1: Integrable Systems and Non-Perturbative Methods. Astronomy and Astrophysics Library. Springer, Berlin (2004)zbMATHGoogle Scholar
  8. 8.
    Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (1960). CrossRefzbMATHGoogle Scholar
  9. 9.
    de la Fuente, Marcos C., de la Fuente, Marcos R.: Asteroid 2014 OL\(_{339}\): yet another Earth quasi-satellite. Mon. Not. R. Astron. Soc. 445, 2985–2994 (2014). arxiv:1409.5588 CrossRefGoogle Scholar
  10. 10.
    Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gil, P.J.S., Schwartz, J.: Simulations of quasi-satellite orbits around phobos. J. Guid. Control Dyn. 33, 901–914 (2010). CrossRefGoogle Scholar
  12. 12.
    Gómez, G., Lo, M., Masdemont, J.: Libration Point Orbits and Applications. World Scientfic, Singapore (2003). URL
  13. 13.
    Gómez, G., Marcote, M., Mondelo, J.M.: The invariant manifold structure of the spatial Hill’s problem. Dyn. Syst. 20(1), 115–147 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hénon, M.: Numerical exploration of the restricted problem, V. Hill’s case: periodic orbits and their stability. Astron. Astrophys. 1, 223–238 (1969)zbMATHGoogle Scholar
  15. 15.
    Hénon, M.: Numerical exploration of the restricted problem. VI. Hill’s case: non-periodic orbits. Astron. Astrophys. 9, 24–36 (1970)zbMATHGoogle Scholar
  16. 16.
    Hénon, M., Petit, J.M.: Series expansion for encounter-type solutions of Hill’s problem. Celest. Mech. 38, 67–100 (1986). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hill, G.W.: Researches in the Lunar Theory. Am. J. Math. 1, 5–26 (1878)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hori, G.: Theory of general perturbation with unspecified canonical variables. Publ. Astron. Soc. Jpn. 18(4), 287–296 (1966)Google Scholar
  19. 19.
    Lam, T., Whiffen, G.: Exploration of Distant Retrograde Orbits Around Europa (AAS 05-110). In: Vallado, D.A., Gabor, M.J., Desai, P.N. (eds.) AAS/AIAA Spaceflight Mechanics Meeting 2005, American Astronautical Society. Advances in the Astronautical Sciences, vol. 120, pp. 135–153. Univelt, Inc., Escondido (2005)Google Scholar
  20. 20.
    Lara, M.: Simplified equations for computing science orbits around planetary satellites. J. Guid. Control Dyn. 31(1), 172–181 (2008). CrossRefGoogle Scholar
  21. 21.
    Lara, M.: Three-body dynamics around the smaller primary: application to the design of science orbits. J. Aerosp. Eng. Sci. Appl. 2(1), 53–65 (2010). Google Scholar
  22. 22.
    Lara, M.: A Hopf variables view on the libration points dynamics. Celest. Mech. Dyn. Astron. 129(3), 285–306 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lara, M., Peláez, J.: On the numerical continuation of periodic orbits. An intrinsic, 3-dimensional, differential, predictor-corrector algorithm. Astron. Astrophys. 389, 692–701 (2002). CrossRefzbMATHGoogle Scholar
  24. 24.
    Lara, M., San-Juan, J.F.: Secular motion around synchronously orbiting planetary satellites. Chaos Interdiscip. J. Nonlinear Sci. 15(4), 043–101 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lara, M., Russell, R., Villac, B.F.: Classification of the distant stability regions at Europa. J. Guid. Control, Dyn. 30, 409–418 (2007a). CrossRefGoogle Scholar
  26. 26.
    Lara, M., Russell, R., Villac, B.F.: Fast estimation of stable regions in real models. Meccanica 42(5), 511–515 (2007b). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lara, M., Palacián, J., Russell, R.: Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter. Celest. Mech. Dyn. Astron. 108(1), 1–22 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lara, M., Palacián, J.F., Yanguas, P., Corral, C.: Analytical theory for spacecraft motion about Mercury. Acta Astronaut. 66(7–8), 1022–1038 (2010). CrossRefGoogle Scholar
  29. 29.
    Lara, M., Pérez, I., López, R.: Higher order approximation to the Hill problem dynamics about the libration points. Commun. Nonlinear Sci. Numer. Simul. (2017). Google Scholar
  30. 30.
    Lidov, M.L., Vashkov’yak, M.A.: Perturbation theory and analysis of the evolution of quasi-satellite orbits in the restricted three-body problem. Cosmic Res. 31, 187–207 (1993)Google Scholar
  31. 31.
    Lidov, M.L., Vashkov’yak, M.A.: On quasi-satellite orbits in a restricted elliptic three-body problem. Astron. Lett. 20, 676–690 (1994)Google Scholar
  32. 32.
    Lidov, M.L., Yarskaya, M.V.: Integrable cases in the problem of the evolution of a satellite orbit under the joint effect of an outside body and of the noncentrality of the planetary field. Cosmic Res. 12, 139–152 (1974)Google Scholar
  33. 33.
    Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  34. 34.
    Ming, X., Shijie, X.: Exploration of distant retrograde orbits around Moon. Acta Astronaut. 65, 853–860 (2009). CrossRefGoogle Scholar
  35. 35.
    Murdock, J.A.: Perturbations: Theory and Methods, Classics in Applied Mathematics, vol. 27. SIAM-Society for Industrial and Applied Mathematics, Philadelphia (1999). CrossRefGoogle Scholar
  36. 36.
    Namouni, F.: Secular interactions of coorbiting objects. Icarus 137, 293–314 (1999). CrossRefGoogle Scholar
  37. 37.
    Nayfeh, A.H.: Perturbation Methods. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2004)Google Scholar
  38. 38.
    Oberst, J., Willner, K., Wickhusen, K.: DePhine—the Deimos and Phobos interior explorer—a proposal to ESA’s cosmic vision program. European Planetary Science Congress 11:EPSC2017-539 (2017)Google Scholar
  39. 39.
    Perozzi, E., Ceccaroni, M., Valsecchi, G.B., Rossi, A.: Distant retrograde orbits and the asteroid hazard. Eur. Phys. J. Plus 132(8), 367 (2017). CrossRefGoogle Scholar
  40. 40.
    Petit, J.M., Hénon, M.: Satellite encounters. Icarus 66, 536–555 (1986). CrossRefGoogle Scholar
  41. 41.
    Pousse, A., Robutel, P., Vienne, A.: On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited. Celest. Mech. Dyn. Astron. 128, 383–407 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    San-Juan, J.F., Lara, M., Ferrer, S.: Phase space structure around oblate planetary satellites. J. Guid. Control Dyn. 29, 113–120 (2006). CrossRefGoogle Scholar
  43. 43.
    Sidorenko, V.V., Neishtadt, A.I., Artemyev, A.V., Zelenyi, L.M.: Quasi-satellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment. Celest. Mech. Dyn. Astron. 120, 131–162 (2014). arxiv:1409.0417 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Simó, C., Stuchi, T.J.: Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Phys. D Nonlinear Phenom. 140, 1–32 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stramacchia, M., Colombo, C., Bernelli-Zazzera, F.: Distant retrograde orbits for space-based near earth objects detection. Adv. Space Res. 58, 967–988 (2016). CrossRefGoogle Scholar
  46. 46.
    Szebehely, V.: Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press Inc., New York (1967). zbMATHGoogle Scholar
  47. 47.
    Zagouras, C., Markellos, V.V.: Three-dimensional periodic solutions around equilibrium points in Hill’s problem. Celest. Mech. 35, 257–267 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zotos, E.E.: Orbit classification in the Hill problem: I. The classical case. Nonlinear Dyn. 89(2), 901–923 (2017). MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Scientific Computing Group–GRUCACIUniversity of La RiojaLogroñoSpain
  2. 2.Space Dynamics GroupPolytechnic University of Madrid–UPMMadridSpain

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