Skip to main content
Log in

A family of periodic solutions of the planar three-body problem, and their stability

  • Published:
Celestial mechanics Aims and scope Submit manuscript

Abstract

We describe a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with Schubart's (1956) rectilinear orbit and ends in retrograde revolution, i.e. a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the ‘interplay’ type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Similar content being viewed by others

References

  • Agekyan, T. A. and Anosova, Zh. P.: 1967,Astron. Zh. 44, 1261 =Sov. Astron. 11, 1006.

    Google Scholar 

  • Agekyan, T. A. and Anosova, Zh. P.: 1968,Astrofizika 4, 31 =Astrophys. 4, 11.

    Google Scholar 

  • Arnold, V. I.: 1963,Uspeki Mat. Nauk 18, 13 =Russ. Math. Surveys 18, 9.

    Google Scholar 

  • Birkhoff, G. D.: 1927,Dynamical Systems, Am. Math. Soc. Publ., Providence, R. I., page 290.

    Google Scholar 

  • Bozis, G. and Christides, Th.: 1975,Celes. Mech. 12, 277.

    Google Scholar 

  • Bray, T. A. and Goudas, C. L.: 1967,Adv. Astron. Astrophys. 5, 111.

    Google Scholar 

  • Broucke, R.: 1969,American Institute of Aeronautics and Astronautics J. 7, 1003.

    Google Scholar 

  • Broucke, R.: 1975,Celes. Mech. 12, 439.

    Google Scholar 

  • Broucke, R. and Boggs, D.: 1975,Celes. Mech. 11, 13.

    Google Scholar 

  • Hadjidemetriou, J. D.: 1975,Celes. Mech. 12, 255.

    Google Scholar 

  • Hadjidemetriou, J. D. and Christides, Th.: 1975,Celes. Mech. 12, 175.

    Google Scholar 

  • Harrington, R. S.: 1968,Astron. J. 73, 190.

    Google Scholar 

  • Harrington, R. S.: 1969,Celes. Mech. 1, 200.

    Google Scholar 

  • Harrington, R. S.: 1972,Celes. Mech. 6, 322.

    Google Scholar 

  • Hénon, M.: 1965,Ann. Astrophys. 28, 992.

    Google Scholar 

  • Hénon, M.: 1973,Astron. Astrophys. 28, 415.

    Google Scholar 

  • Hénon, M.: 1974,Celes. Mech. 10, 375.

    Google Scholar 

  • Hénon, M.: 1975, in preparation.

  • Moser, J.: 1962,Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 1, 1.

    Google Scholar 

  • Schubart, J.: 1956,Astron. Nachr. 283, 17.

    Google Scholar 

  • Siegel, C. L. and Moser, J. K.: 1971,Lectures on Celestial Mechanics, Springer-Verlag, Berlin.

    Google Scholar 

  • 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 

  • Standish, E. M.: 1972,Astron. Astrophys. 21, 185.

    Google Scholar 

  • Strömgren, E.: 1933,Bull. Astron., Paris 9, 87.

    Google Scholar 

  • Szebehely, V.: 1967,Theory of Orbits—The Restricted Problem of Three Bodies, Academic Press, New York.

    Google Scholar 

  • Szebehely, V.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability, and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, p. 382.

    Google Scholar 

  • Szebehely, V.: 1971,Celes. Mech. 4, 116.

    Google Scholar 

  • Szebehely, V.: 1972,Celes. Mech. 6, 84.

    Google Scholar 

  • Szebehely, V.: 1973, in B. Tapley and V. Szebehely (eds.),Recent Advances in Dynamical Astronomy, D. Reidel Publ. Co., Dordrecht, Holland, p. 75.

    Google Scholar 

  • Szebehely, V.: 1974,Celes. Mech. 9, 359.

    Google Scholar 

  • Szebehely, V. and Feagin, T.: 1973,Celes. Mech. 8, 11.

    Google Scholar 

  • Szebehely, V. and Peters, C. F.: 1967,Astron. J. 72, 1187.

    Google Scholar 

  • Whittaker, E. T.: 1937,Analytical Dynamics of Particles and Rigid Bodies, fourth edition, Cambridge University Press.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hénon, M. A family of periodic solutions of the planar three-body problem, and their stability. Celestial Mechanics 13, 267–285 (1976). https://doi.org/10.1007/BF01228647

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01228647

Keywords

Navigation