Abstract
The one-dimensional Newtonian three-body problem is known to have stable (quasi-)periodic orbits when the masses are equal. The existence and size of the stable region is discussed here in the case where the three masses are arbitrary. We consider only the stability of the periodic (generalized) Schubart's (1956) orbit. If this orbit is linearly stable it is almost always surrounded by a region of stable quasi-periodic orbits and the size and shape of this stable region depends on the masses. The three-dimensional linear stability of the periodic orbits is also determined. Final results show that the region of stability has a complicated shape and some of the stable regions in the mass-plane are quite narrow. The non-linear three-dimensional stability is studied independently by extensive numerical integrations and the results are found to be in agreement with the linear stability analysis. The boundaries of stable region in the mass-plane are given in terms of polynomial approximations. The results are compared with a similar work by Héenon (1977).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aarseth, S.J. and Zare, K.: 1974, Cel. Mech. 10, 185–205
Arnold, V.I.: 1989, Mathematical Methods of Classical Mechanics, second edition, Springer-Verlag, New York
Bulirsch, R. and Stoer, J.: 1966, Nun. Math. 8, 1–13
Hénon, M.: 1976, Cel. Mech. 13, 267–285
Hénon, M.: 1977, Cel. Mech. 15, 243–261
Hénon, M.: 1983, in Iooss, G., Helleman, R.H.G. and Stora, R., ed(s)., Chaotic Behaviour of Deterministic Systems, Elsevier, Amsterdam, 53–170
Herrick, S.: 1972, Astrodynamics, VoIII, Van Nostrand Reinhold, London, 79–124
Marchal, C.: 1988, in M.J. Valtonen, ed(s)., The Few Body Problem, Kluwer, Dordrecht, 5–25
McGehee, R.: 1974, Inventiones math. 27, 191–227
Mikkola, S. and Hietarinta, J.: 1989, Cel. Mech. 46, 1–18
Mikkola, S. and Hietarinta, J.: 1990, Cel. Mech. 47, 321–331
Press, W.H., Flannery, B.P., Teukolsky, G.A. and Vetterling, W.T.: 1986, Numerical Recipes, Cambridge University Press, 563–568
Schubart, J.: 1956, Astron. Nachr. 283, 17–22
Whittaker, E.T.: 1937, Analytical Dynamics of Particles and Rigid Bodies, fourth edition, Cambridge University Press
Author information
Authors and Affiliations
Additional information
We thank the referee for pointing out this reference to us.
Rights and permissions
About this article
Cite this article
Mikkola, S., Hietarinta, J. A numerical investigation of the one-dimensional Newtonian three-body problem. Celestial Mech Dyn Astr 51, 379–394 (1991). https://doi.org/10.1007/BF00052929
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00052929