Celestial mechanics

, Volume 10, Issue 3, pp 375–388 | Cite as

Families of periodic orbits in the three-body problem

  • M. Hénon


We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.


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  1. Bray, T. A. and Goudas, C. L.: 1967,Adv. Astron. Astrophys. 5, 71.Google Scholar
  2. Broucke, R.: 1969, Technical Report 32-1360, Jet Propulsion Laboratory, Pasadena, California.Google Scholar
  3. Bulirsch, R. and Stoer, J.: 1966,Numerische Mathematik 8, 1.Google Scholar
  4. Hadjidemetriou, J. D. and Christides, Th.: 1974, submitted toCeles. Mech. Google Scholar
  5. Jefferys, W.: 1966,Astron. J. 71, 566.Google Scholar
  6. Poincaré, H.: 1892,Les méthodes nouvelles de la mécanique céleste, vol. I, Gauthier-Villars, Paris.Google Scholar
  7. Schubart, J.: 1956,Astron. Nachr. 283, 17.Google Scholar
  8. Siegel, C. L. and Moser, J. K.: 1971,Lectures on Celestial Mechanics, Springer-Verlag, Berlin, p. 138.Google Scholar
  9. Standish, E. M.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, p. 375.Google Scholar
  10. Szebehely, V.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, p. 382.Google Scholar
  11. Szebehely, V.: 1973, in B. D. Tapley and V. Szebehely (eds.),Recent Advances in Dynamical Astronomy, D. Reidel Publ. Co., Dordrecht, Holland, p. 75.Google Scholar
  12. Szebehely, V. and Feagin, T.: 1973,Celes. Mech. 8, 11.Google Scholar
  13. Szebehely, V. and Peters, C. F.: 1967,Astron. J. 72, 1187.Google Scholar
  14. Waldvogel, J.: 1972,Celes. Mech. 6, 221.Google Scholar
  15. Whittaker, E. T.: 1937,Analytical Dynamics of Particles and Rigid Bodies, fourth edition, Cambridge University Press, p. 386, § 167.Google Scholar

Copyright information

© D. Reidel Publishing Company 1974

Authors and Affiliations

  • M. Hénon
    • 1
  1. 1.Observatoire de NiceFrance

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