Skip to main content
SpringerLink
Log in

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

  • Article
  • Published:
Science China Physics, Mechanics & Astronomy Aims and scope Submit manuscript

Abstract

The three-body problem can be traced back to Newton in 1687, but it is still an open question today. Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem. Although triple systems are common in astronomy, practically all observed periodic triple systems are hierarchical (similar to the Sun, Earth and Moon). It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star, and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce. However, we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses; 13315 among them are stable. Compared with the narrow mass range (only 10−5) in which stable “Figure-eight” periodic orbits of three-body systems exist, our newly found stable periodic orbits have fairly large mass region. We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations. This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice. Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses. Note that our approach has general meaning: in a similar way, every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Reipurth, and S. Mikkola, Nature 492, 221 (2012), arXiv: 1212.1246.

    ADS  Google Scholar 

  2. I. Newton, Mathematical Principles of Natural Philosophy (Royal Society Press, London, 1687).

    MATH  Google Scholar 

  3. V. R. Garsevanishvili, and D. G. Mirianashvili, Rep. Math. Phys. 11, 89 (1977).

    ADS  Google Scholar 

  4. A. M. Archibald, N. V. Gusinskaia, J. W. T. Hessels, A. T. Deller, D. L. Kaplan, D. R. Lorimer, R. S. Lynch, S. M. Ransom, and I. H. Stairs, Nature 559, 73 (2018), arXiv: 1807.02059.

    ADS  Google Scholar 

  5. G. Torres, R. P. Stefanik, and D. W. Latham, Astrophys. J. 885, 9 (2019), arXiv: 1909.04668.

    ADS  Google Scholar 

  6. J. H. Poincaré, Acta Math. 713, 1 (1890).

    Google Scholar 

  7. N. C. Stone, and N. W. C. Leigh, Nature 576, 406 (2019), arXiv: 1909.05272.

    ADS  Google Scholar 

  8. M. Šuvakov, and V. Dmitrašinović, Phys. Rev. Lett. 110, 114301 (2013), arXiv: 1303.0181.

    ADS  Google Scholar 

  9. X. M. Li, and S. J. Liao, Sci. China-Phys. Mech. Astron. 60, 129511 (2017), arXiv: 1705.00527.

    ADS  Google Scholar 

  10. X. Li, Y. Jing, and S. Liao, Publ. Astron. Soc. JPN 70, 64 (2018).

    ADS  Google Scholar 

  11. V. Dmitrašinović, A. Hudomal, M. Shibayama, and A. Sugita, J. Phys. A-Math. Theor. 51, 315101 (2018), arXiv: 1705.03728.

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  13. A. Chenciner, and R. Montgomery, Ann. Math. 152, 881 (2000).

    MathSciNet  Google Scholar 

  14. C. Simó, in Celestial Mechanics: Dedicated to Donald Saari for His 60th Birthday: Proceedings of an International Conference on Celestial Mechanics, December 15–19, 1999 (Northwestern University, Evanston, 2002). p. 209.

    Google Scholar 

  15. J. Galán, F. J. Muñoz-Almaraz, E. Freire, E. Doedel, and A. Vanderbauwhede, Phys. Rev. Lett. 88, 241101 (2002).

    ADS  MathSciNet  Google Scholar 

  16. R. Montgomery, Ergod. Th. Dynam. Sys. 27, 1933 (2007).

    Google Scholar 

  17. J. D. Hadjidemetriou, Celest. Mech. 12, 255 (1975).

    ADS  Google Scholar 

  18. M. Henon, Celest. Mech. 13, 267 (1976).

    ADS  MathSciNet  Google Scholar 

  19. M. R. Janković, V. Dmitrašinović, and M. Šuvakov, Comput. Phys. Commun. 250, 107052 (2020).

    MathSciNet  Google Scholar 

  20. S. C. Farantos, J. Mol. Struct.-Theochem. 341, 91 (1995).

    Google Scholar 

  21. M. Lara, and J. Peláez, Astron. Astrophys. 389, 692 (2002).

    ADS  Google Scholar 

  22. S. Liao, Tellus A 61, 550 (2009).

    ADS  Google Scholar 

  23. S. Liao, Commun. Nonlinear Sci. Numer. Simul. 19, 601 (2014), arXiv: 1305.6094.

    ADS  MathSciNet  Google Scholar 

  24. S. J. Liao, and P. F. Wang, Sci. China-Phys. Mech. Astron. 57, 330 (2014), arXiv: 1305.4222.

    ADS  Google Scholar 

  25. X. M. Li, and S. J. Liao, Sci. China-Phys. Mech. Astron. 57, 2121 (2014), arXiv: 1312.6796.

    ADS  Google Scholar 

  26. Z. L. Lin, L. P. Wang, and S. J. Liao, Sci. China-Phys. Mech. Astron. 60, 014712 (2017), arXiv: 1612.00120.

    ADS  Google Scholar 

  27. T. Hu, and S. Liao, J. Comput. Phys. 418, 109629 (2020), arXiv: 1910.11976.

    MathSciNet  Google Scholar 

  28. G. Corliss, and Y. F. Chang, ACM Trans. Math. Softw. 8, 114 (1982).

    Google Scholar 

  29. Y. F. Chang, and G. F. Corhss, Comput. Math. Appl. 28, 209 (1994).

    MathSciNet  Google Scholar 

  30. R. Barrio, F. Blesa, and M. Lara, Comput. Math. Appl. 50, 93 (2005).

    MathSciNet  Google Scholar 

  31. O. Portilho, Comput. Phys. Commun. 59, 345 (1990).

    ADS  Google Scholar 

  32. X. Li, and S. Liao, New Astron. 70, 22 (2019), arXiv: 1805.07980.

    ADS  Google Scholar 

  33. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer-Verlag, Berlin, 1993).

    MATH  Google Scholar 

  34. R. Montgomery, Nonlinearity 11, 363 (1998).

    ADS  MathSciNet  Google Scholar 

  35. E. L. Allgower, and K. Georg, Introduction to Numerical Continuation Methods, Vol. 45 (SIAM, New York, 2003).

    MATH  Google Scholar 

  36. V. Dmitrašinović, and M. Šuvakov, Phys. Lett. A 379, 1939 (2015), arXiv: 1507.08096.

    ADS  MathSciNet  Google Scholar 

  37. M. R. Janković, and V. Dmitrašinović, Phys. Rev. Lett. 116, 064301 (2016), arXiv: 1604.08358.

    ADS  Google Scholar 

  38. T. Kapela, and C. Simó, Nonlinearity 20, 1241 (2007).

    ADS  MathSciNet  Google Scholar 

  39. W. Dimitrov, H. Lehmann, K. Kamiński, M. K. Kamińska, M. Zgórz, and M. Gibowski, Mon. Not. R. Astron. Soc. 466, 2 (2017).

    ADS  Google Scholar 

  40. V. Szebehely, Proc. Natl. Acad. Sci. USA 58, 60 (1967).

    ADS  Google Scholar 

  41. F. Marcadon, T. Appourchaux, and J. P. Marques, Astron. Astrophys. 617, A2 (2018), arXiv: 1804.09296.

    Google Scholar 

  42. T. Prusti, et al. (Gaia Collaboration), Astron. Astrophys. 595, A1 (2016), arXiv: 1609.04153.

    Google Scholar 

  43. V. Dmitrašinović, M. Šuvakov, and A. Hudomal, Phys. Rev. Lett. 113, 101102 (2014), arXiv: 1501.03405.

    ADS  Google Scholar 

  44. Y. Meiron, B. Kocsis, and A. Loeb, Astrophys. J. 834, 200 (2017), arXiv: 1604.02148.

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou, 510632, China

    XiaoMing Li

  2. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, 02139, USA

    XiaoMing Li

  3. School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, 510641, China

    XiaoChen Li

  4. Department of Engineering Science, University of Oxford, Oxford, OX1 3PJ, UK

    XiaoChen Li

  5. Center of Advanced Computing, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    ShiJun Liao

  6. School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China

    ShiJun Liao

Authors
  1. XiaoMing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XiaoChen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. ShiJun Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to XiaoChen Li or ShiJun Liao.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12002132, 11702099, and 91752104), China Postdoctoral Science Foundation (Grant No. 2020M673058), and the International Program of Guangdong Provincial Outstanding Young Researcher. This work was carried out on TH-1A (in Tianjin) and TH-2 (in Guangzhou) at National Supercomputer Center, China.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Li, X. & Liao, S. One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems. Sci. China Phys. Mech. Astron. 64, 219511 (2021). https://doi.org/10.1007/s11433-020-1624-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11433-020-1624-7

Keywords

Search

Navigation