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.
Similar content being viewed by others
References
Agekyan, T. A. and Anosova, Zh. P.: 1967,Astron. Zh. 44, 1261 =Sov. Astron. 11, 1006.
Agekyan, T. A. and Anosova, Zh. P.: 1968,Astrofizika 4, 31 =Astrophys. 4, 11.
Arnold, V. I.: 1963,Uspeki Mat. Nauk 18, 13 =Russ. Math. Surveys 18, 9.
Birkhoff, G. D.: 1927,Dynamical Systems, Am. Math. Soc. Publ., Providence, R. I., page 290.
Bozis, G. and Christides, Th.: 1975,Celes. Mech. 12, 277.
Bray, T. A. and Goudas, C. L.: 1967,Adv. Astron. Astrophys. 5, 111.
Broucke, R.: 1969,American Institute of Aeronautics and Astronautics J. 7, 1003.
Broucke, R.: 1975,Celes. Mech. 12, 439.
Broucke, R. and Boggs, D.: 1975,Celes. Mech. 11, 13.
Hadjidemetriou, J. D.: 1975,Celes. Mech. 12, 255.
Hadjidemetriou, J. D. and Christides, Th.: 1975,Celes. Mech. 12, 175.
Harrington, R. S.: 1968,Astron. J. 73, 190.
Harrington, R. S.: 1969,Celes. Mech. 1, 200.
Harrington, R. S.: 1972,Celes. Mech. 6, 322.
Hénon, M.: 1965,Ann. Astrophys. 28, 992.
Hénon, M.: 1973,Astron. Astrophys. 28, 415.
Hénon, M.: 1974,Celes. Mech. 10, 375.
Hénon, M.: 1975, in preparation.
Moser, J.: 1962,Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 1, 1.
Schubart, J.: 1956,Astron. Nachr. 283, 17.
Siegel, C. L. and Moser, J. K.: 1971,Lectures on Celestial Mechanics, Springer-Verlag, Berlin.
Standish, E. M.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability, and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, p. 375.
Standish, E. M.: 1972,Astron. Astrophys. 21, 185.
Strömgren, E.: 1933,Bull. Astron., Paris 9, 87.
Szebehely, V.: 1967,Theory of Orbits—The Restricted Problem of Three Bodies, Academic Press, New York.
Szebehely, V.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability, and Resonances, D. Reidel Publ. Co., Dordrecht, Holland, p. 382.
Szebehely, V.: 1971,Celes. Mech. 4, 116.
Szebehely, V.: 1972,Celes. Mech. 6, 84.
Szebehely, V.: 1973, in B. Tapley and V. Szebehely (eds.),Recent Advances in Dynamical Astronomy, D. Reidel Publ. Co., Dordrecht, Holland, p. 75.
Szebehely, V.: 1974,Celes. Mech. 9, 359.
Szebehely, V. and Feagin, T.: 1973,Celes. Mech. 8, 11.
Szebehely, V. and Peters, C. F.: 1967,Astron. J. 72, 1187.
Whittaker, E. T.: 1937,Analytical Dynamics of Particles and Rigid Bodies, fourth edition, Cambridge University Press.
Author information
Authors and Affiliations
Rights 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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01228647