Celestial Mechanics and Dynamical Astronomy

, Volume 82, Issue 4, pp 323–361 | Cite as

A Perturbative Treatment of The Co-Orbital Motion

  • D. Nesvorný
  • F. Thomas
  • S. Ferraz-Mello
  • A. Morbidelli


We develop a formalism of the non-singular evaluation of the disturbing function and its derivatives with respect to the canonical variables. We apply this formalism to the case of the perturbed motion of a massless body orbiting the central body (Sun) with a period equal to that of the perturbing (planetary) body. This situation is known as the ‘co-orbital’ motion, or equivalently, as the 1/1 mean motion commensurability. Jupiter's Trojan asteroids, Earth's co-orbital asteroids (e.g., (3753) Cruithne, (3362) Khufu), Mars' co-orbital asteroids (e.g., (5261) Eureka), and some Jupiter-family comets are examples of the co-orbital bodies in our solar system. Other examples are known in the satellite systems of the giant planets. Unlike the classical expansions of the disturbing function, our formalism is valid for any values of eccentricities and inclinations of the perturbed and perturbing body. The perturbation theory is used to compute the main features of the co-orbital dynamics in three approximations of the general three-body model: the planar-circular, planar-elliptic, and spatial-circular models. We develop a new perturbation scheme, which allows us to treat cases where the classical perturbation treatment fails. We show how the families of the tadpole, horseshoe, retrograde satellite and compound orbits vary with the eccentricity and inclination of the small body, and compute them also for the eccentricity of the perturbing body corresponding to a largely eccentric exoplanet's orbit.

restricted three-body problem disturbing function mean motion resonances Lagrange equilibrium points co-orbital motion Trojan asteroids 


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  1. Beaugé, C.: 1994, 'Asymmetric librations in exterior resonances', Celest. Mech. & Dyn. Astr. 60, 225-248.Google Scholar
  2. Christou, A. A.: 2000, 'A numerical survey of transient co-orbitals of the terrestrial planets', Icarus 144, 1-20.CrossRefGoogle Scholar
  3. Ferraz-Mello, S.: 1987, 'Expansion of the disturbing force-function for the study of high-eccentric librations', Astron. Astrophys. 183, 397-402.Google Scholar
  4. Ferraz-Mello, S.: 1994, 'The convergence domain of the Laplacian expansion of the disturbing function', Celest. Mech. 58, 37-52.CrossRefGoogle Scholar
  5. Ferraz-Mello, S. and Sato: 1989, 'The very-high-eccentricity asymmetric expansion of the disturbing function near resonances of any order', Astron. Astrophys. 225, 541-547.Google Scholar
  6. Giacaglia, G. E. O. and Nacozy, P. E.: 1970, 'Resonances in the elliptic restricted problem', In: G. E. O. Giacaglia (ed.), Periodic Orbits, Stability and Resonances, D. Reidel Publishing Company, Dordrecht Holland, pp. 96-127.Google Scholar
  7. Hénon, M. and Petit, J.-M.: 1986, 'Series expansion for encounter-type solutions of Hill's problem', Celest. Mech. & Dyn. Astr. 38, 67-100.Google Scholar
  8. Henrard, J.: 1990, 'A semi-numerical perturbation method for separable Hamiltonian systems', Celest. Mech. & Dyn. Astr. 49, 43.Google Scholar
  9. Hori, G. I.: 1966, 'General perturbations theory with unspecified canonical variables', Publ. Astron. Soc. Jap. 18, 287.Google Scholar
  10. Kozai, Y.: 1985, 'Secular perturbations of resonant asteroids', Celest. Mech. & Dyn. Astr. 36, 47-69.Google Scholar
  11. Message, J.: 1958, 'The search for asymmetric periodic orbits in the restricted problem of three bodies', Astron. J. 63, 443-448.CrossRefGoogle Scholar
  12. Message, J.: 1959, 'Some periodic orbits in the restricted problem of three bodies and their stabilities', Astron. J. 64, 226-236.CrossRefGoogle Scholar
  13. Moons, M.: 1993, On the resonant Hamiltonian in the restricted three-body problem, Report 19, Facultés Universitaires de Namur, Belgique.Google Scholar
  14. Moons, M.: 1994, 'Extended Schubart averaging', Celest. Mech. & Dyn. Astr. 60, 173-186.Google Scholar
  15. Morbidelli, A.: 2001, Modern Celestial Mechanics (in press).Google Scholar
  16. Morbidelli, A. and Moons, M.: 1993, 'Secular resonances in mean-motion commensurabilities. The 2/1 and 3/2 cases', Icarus 102, 316-332.CrossRefGoogle Scholar
  17. Morbidelli, A., Thomas, F. and Moons, M: 1995, 'The resonant structure of the Kuiper belt and the dynamics of the first five trans-Neptunian objects', Icarus 118, 322-340.CrossRefGoogle Scholar
  18. Namouni, F.: 1999, 'Secular interactions of coorbiting objects', Icarus 137, 293-314.CrossRefGoogle Scholar
  19. Namouni, F., Christou, A. A. and Murray, C. D.: 1999, 'Coorbital dynamics at large eccentricity and inclination', Phys. Rev. Lett. 83, 2506-2509.CrossRefGoogle Scholar
  20. Poincaré, H.: 1902, 'Les solutions périodiques et les plan`etes du type d'H´ecube', Bull. Astron. 19, 177.Google Scholar
  21. Press, W. H., Teukolsky, S. A., Vettering, W. T. and Flannery, B. P.: 1986, Numerical Recipes, Cambridge University Press, Cambridge, UK.Google Scholar
  22. Roig, F. V., Simula, A., Ferraz-Mello, S. and Tsuchida, M.: 1998, 'The high-eccentricity asymmetric expansion of the disturbing function for non-planar resonant problems', Astron. Astrophys. 329, 339-349.Google Scholar
  23. Schubart, J.: 1964, 'Long-period effects in nearly commensurable cases of the restricted three-body problem', In: Smithsonian Institution Astrophysical Observatory Research in Space Science, Special Report 149, Cambridge, Massachusetts.Google Scholar
  24. Sundman, K.: 1916, 'Öfversigt at Finska Vetenskaps-Societetens Förhandlingar', 58A(24), 1Google Scholar
  25. Wisdom, J.: 1985, 'A perturbative treatment of motion near the 3/1 commensurability', Icarus 63, 272.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • D. Nesvorný
    • 1
    • 2
  • F. Thomas
    • 1
  • S. Ferraz-Mello
    • 1
    • 3
  • A. Morbidelli
    • 2
  1. 1.Instituto Astronômico e GeofísicoUniversidade de São PauloBrasil
  2. 2.Observatoire de la Côte d'AzurNiceFrance
  3. 3.Observatório NacionalRio de JaneiroBrasil

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