Skip to main content
Log in

Search for periodic orbits in the general three-body problem

  • Published:
Astronomy Reports Aims and scope Submit manuscript

Abstract

An original method for searching for regions of initial conditions giving rise to close to periodic orbits is proposed in the framework of the general three-body problem with equal masses and zero angular momentum. Until recently, three stable periodic orbits were known: the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. Recent studies have also found new periodic orbits for this problem. The proposed method minimizes a functional that calculates the sum of squared differences between the initial and current coordinates and the velocities of the bodies. The search is applied to short-period orbits with periodsT < 10τ, where τ is the mean crossing time for the components of the triple system. Twenty one regions of initial conditions, each corresponding to a particular periodic orbit, have been found in the current study. A criterion for the reliability of the results is that the initial conditions for the previously known stable periodic orbits are located inside the regions found. The trajectories of the bodies with the corresponding initial conditions are presented. The dynamics and geometry of the orbits constructed are described.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Poincaré, Selected Works, Vol. 1: New Methods of Celestial Mechanics (Am. Inst. of Physics, 1997; Nauka, Moscow, 1971).

    Google Scholar 

  2. C. Marchal, The Three-Body Problem (Elsevier, Amsterdam, 1990).

    MATH  Google Scholar 

  3. M. Valtonen and H. Karttunen, The Three-Body Problem (Cambridge Univ. Press, Cambridge, 2006).

    Book  MATH  Google Scholar 

  4. A. I. Martynova, V. V. Orlov, A. V. Rubinov, L. L. Sokolov, and I. I. Nikiforov, Dynamics of Triple Systems (SPb. Gos. Univ., St. Petersburg, 2010) [in Russian].

    Google Scholar 

  5. J. von Schubart, Astron. Nachr. 283, 17 (1956).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. R. Broucke, Astron. Astrophys. 73, 303 (1979).

    ADS  MATH  Google Scholar 

  7. C. Moore, Phys. Rev. Lett. 70, 3675 (1993).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. A. I. Martynova, V. V. Orlov, and A. V. Rubinov, Astron. Rep. 53, 710 (2009).

    Article  ADS  Google Scholar 

  9. M. Šuvakov and V. Dmitrašinović, Phys. Rev. Lett. 110, 114301 (2013).

    Article  Google Scholar 

  10. X. Li and S. Liao, arXiv:1312.6796 [astro-ph.EP] (2013).

  11. M. Šuvakov, Celest. Mech. Dyn. Astron. (2014, in press); arXiv:1312.7002 (2013).

    Google Scholar 

  12. P. P. Iasko and V. V. Orlov, Astron. Zh. 91 (2014, in press).

  13. R. Montgomery, arXiv:math/0601269 [math.DS] (2006).

  14. R. Bulirsch and J. Stoer, Num. Math. 8, 1 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  15. S. J. Aarseth and K. Zare, Celest. Mech. 10, 185 (1974).

    Article  ADS  MATH  Google Scholar 

  16. S. J. Aarseth, Gravitational N-Body Simulations. Tools and Algorithms (Cambridge Univ. Press, Cambridge, 2003).

    Book  MATH  Google Scholar 

  17. Three-Body Gallery (Institute of Physics, Belgrade). http://suki.ipb.ac.rs/3body/

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. P. Iasko.

Additional information

Original Russian Text © P.P. Iasko, V.V. Orlov, 2014, published in Astronomicheskii Zhurnal, 2014, Vol. 91, No. 11, pp. 978–988.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iasko, P.P., Orlov, V.V. Search for periodic orbits in the general three-body problem. Astron. Rep. 58, 869–879 (2014). https://doi.org/10.1134/S1063772914110080

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063772914110080

Keywords

Navigation