Skip to main content

Flux-based statistical prediction of three-body outcomes

Celestial Mechanics and Dynamical Astronomy volume 133, Article number: 17 (2021) Cite this article

Abstract

Since Poincaré, the three-body problem is known to be chaotic and is believed to lack a general deterministic solution. Instead, decades ago a statistical solution was marked as a goal. Yet, despite considerable progress, all extant approaches display two flaws. First, probability was equated with phase space volume, thereby ignoring the fact that significant regions of phase space describe regular motion, including post-decay motion. Secondly and relatedly, an adjustable parameter, the strong interaction region, which is a sort of cutoff, was a central ingredient of the theory. This paper introduces remedies and presents for the first time a statistical prediction of decay rates, in addition to outcomes. Based on an analogy with a particle moving within a leaky container, the statistical distribution is presented in an exactly factorized form. One factor is the flux of phase-space volume, rather than the volume itself, and it is given in a cutoff-independent closed form. The other factors are the chaotic absorptivity and the regularized phase space volume. The situation is analogous to Kirchhoff’s law of thermal radiation, also known as greybody radiation. In addition, an equation system for the time evolution of the statistical distribution is introduced; it describes the decay rate statistics while accounting for sub-escape excursions. Early numerical tests indicate a leap in accuracy.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1
Fig. 2
Fig. 3

Notes

  1. We note that Valtonen and Karttunen (2006) includes also the empirical expression \(P_s =\frac{1}{m_s^q}\) with \(q=3/(1+2 L^2/L_{\max }^2),\, L_{\max }=5/2 [(m_1 m_2 + m_2 m_3 + m_3 m_1)/3]^{5/4}/\sqrt{E}\), see equations (7.30, 7.33, 2.67, 7.28) there. We find it appropriate to compare our derived expression with a derived expression, rather than an empirical one. In addition, a comparison to the empirical expression can be found in table 2 of [38], which leads to similar conclusions.

References

  • Agekyan, T.A., Anosova, Z.P.: A study of the dynamics of triple systems by means of statistical sampling. Astron. Zh. 44, 1261 (1967)

    ADS  Google Scholar 

  • Agekyan, T.A., Anosova, ZhP, Orlov, V.V.: Decay time of triple systems. Astrophysics 19, 66 (1983). https://doi.org/10.1007/BF01005813. Translation of Astrofizika 19, 111 (1983)

    ADS  Article  Google Scholar 

  • Barrow-Green, J.: Poincaré and the Three Body Problem. American Mathematical Society, London (1996)

    Book  Google Scholar 

  • Boyd, P.T., McMillan, S.L.W.: Chaotic scattering in the gravitational three-body problem. Chaos 3, 507 (1993). https://doi.org/10.1063/1.165956

    ADS  Article  Google Scholar 

  • Bunimovich, L.A., Yurchenko, A.: Where to place a hole to achieve a maximal escape rate. Isr. J. Math. 182, 229 (2011)

    MathSciNet  Article  Google Scholar 

  • Chernov, N., Markarian, R., Troubetzkoy, S.: Invariant measures for Anosov maps with small holes. Ergod. Theory Dyn. Syst. 20, 1007 (2000)

    MathSciNet  Article  Google Scholar 

  • Cordani, B.: Geography of Order and Chaos in Mechanics. Springer, Birkhäuser (2013)

    Book  Google Scholar 

  • Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi commentarii academiæ scientarum Petropolitanæ 11, 144151 (1767). in Oeuvres, Seria Secunda tome XXV Commentationes Astronomicæ (p. 286)

  • Fowler, R.H.: Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge (1936)

    MATH  Google Scholar 

  • Gaspard, P.: Chaos, Scattering and Statistical Mechanics. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  • Gibbs, J.W.: Statistical Mechanics. Charles Scribner’s Sons, New York (1902)

    MATH  Google Scholar 

  • Jacobi, C.G.J.: Sur le movement d’un point et sur un cas particulier du problème des trois corps. C. R. Acad. Sci. Paris 3, 59 (1836)

    Google Scholar 

  • Jeans, J.H.: Astronomy and Cosmogony. Dover Publications Inc., New York (1929)

    MATH  Google Scholar 

  • Lagrange, J.L.: Essai sur le Problème des Trois Corps, Prix de l’Académie Royale des Sciences de Paris, tome IX (1772), in vol. 6 of Oeuvres (p. 292)

  • Landau, L., Lifshitz, E.M.: Mechanics. Pergamon Press, London (1960). eq. (49.6) in 2nd edition

    MATH  Google Scholar 

  • Manwadkar, V., Kol, B., Trani, A.A., Leigh, N.W.C.: Testing the Flux-based statistical prediction of the Three-Body Problem. [arXiv:2101.03661 [astro-ph.EP]], under review in MNRAS

  • Manwadkar, V., Trani, A.A., Leigh, N.W.C.: Chaos and Lévy flights in the three-body problem. Mon. Not. Roy. Ast. Soc. 497, 3694 (2020). https://doi.org/10.1093/mnras/staa1722. [arXiv:2004.05475 [astro-ph.EP]]

    ADS  Article  Google Scholar 

  • Mathematica computing system, Wolfram Research

  • Monaghan, J.J.: Statistical-theory of the disruption of three-body systems—I. Low angular momentum. Mon. Not. Roy. Astron. Soc. 176, 63 (1976)

    ADS  Article  Google Scholar 

  • Monaghan, J.J.: Statistical-theory of the disruption of three-body systems - 2. High angular-momentum. Mon. Not. Roy. Astron. Soc. 177, 583 (1976)

    ADS  Article  Google Scholar 

  • Musielak, Z.E., Quarles, B.: The three-body problem. Rep. Prog. Phys. 77, 065901 (2014). https://doi.org/10.1088/0034-4885/77/6/065901. [arXiv:1508.02312 [astro-ph.EP]]

    ADS  MathSciNet  Article  Google Scholar 

  • Narnhofer, H., Thirring, W.: Canonical scattering transformation in classical mechanics. Phys. Rev. A 23, 1688 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  • Nash, P.E., Monaghan, J.J.: Statistical-theory of the disruption of three-body systems—3. 3-dimensional motion. Mon. Not. Roy. Astron. Soc. 184, 119 (1978)

    ADS  Article  Google Scholar 

  • Newton, I.: Philosophiæ Naturalis Principia Mathematica (1687)

  • Orlov, V.V., Rubinov, A.V., Shevchenko, I.I.: The disruption of three-body gravitational systems: lifetime statistics. Mon. Not. Roy. Ast. Soc. 408, 1623 (2010). https://doi.org/10.1111/j.1365-2966.2010.17239.x

    ADS  Article  Google Scholar 

  • Ott, E., Tél, T.: Chaotic scattering: an introduction. Chaos 3, 4 (1993)

    Article  Google Scholar 

  • Poincaré, H.: Les méthodes nouvelles de la méchanique céleste. Gauthier-Villars et fils (1892)

  • Saslaw, W., Valtonen, M.J., Aarseth, S.J.: Gravitational slingshot and structure of extra-galactic radio-sources. Astrophys. J. 190, 253 (1974)

    ADS  Article  Google Scholar 

  • Seoane, J.M., Sanjuan, M.A.F.: New developments in classical chaotic scattering. Rep. Prog. Phys. 76, 016001 (2013)

    ADS  Article  Google Scholar 

  • Shevchenko, I.I.: Hamiltonian intermittency and Lévy flights in the three-body problem. Phys. Rev. E 81, 066216 (2010). https://doi.org/10.1103/PhysRevE.81.066216

    ADS  MathSciNet  Article  Google Scholar 

  • Standish, E.M.: Dynamical evolution of triple star systems—numerical study. Astr. Astrophys. 21, 185 (1972)

    ADS  Google Scholar 

  • Stone, N.C., Leigh, N.W.C.: A statistical solution to the chaotic, non-hierarchical three-body problem. Nature 576(7787), 406 (2019). https://doi.org/10.1038/s41586-019-1833-8

    ADS  Article  Google Scholar 

  • Valtonen, M.J.: Statistics of three body experiments. In: Kozai, Y. (ed.) “The Stability of the Solar System and of Small Stellar Systems,” symposium proceedings, p. 211. Reidel, Dordrecht (1974)

    Chapter  Google Scholar 

  • Valtonen, M.J.: The general three-body problem in astrophysics. Vistas in Astron. 32, 23 (1988). https://doi.org/10.1016/0083-6656(88)90395-9

    ADS  MathSciNet  Article  Google Scholar 

  • Valtonen, M.J., Aarseth, S.J.: Numerical experiments on the decay of three-body systems. Rev. Mex. Astron. Astrofiz. 3, 163 (1977)

    ADS  Google Scholar 

  • Valtonen, M.J., Karttunen, H.: The Three-Body Problem. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  • Valtonen, M.J., Anosova, J., Kholshevnikov, K., Mylläri, A., Orlov, V., Tanikawa, K.: The Three-body Problem from Pythagoras to Hawking. Springer, New York (2016)

    Book  Google Scholar 

  • Wikipedia, Characteristic function (probability theory)

Download references

Acknowledgements

It is a pleasure to thank L. Lederer, N. Leigh, V. Manwadkar, S. Mazumdar, A. Ori, A. Schiller, U. Smilansky, N. Stone, R. Shir and A. Trani for discussions. I thank Sara Kol for linguistic editing help. This work is based in part on ideas on statistical predictions for the double pendulum, a chaotic mechanical system, developed by the author in January 2016, initiated by teaching a course on Analytical Mechanics, and tested against simulations in an unpublished internal report with A. Marmor “Predicting chaos statistically: the double pendulum” (August 2016).

Author information

Authors and Affiliations

  1. Racah Institute of Physics, Hebrew University, 91904, Jerusalem, Israel

    Barak Kol

Authors
  1. Barak Kol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barak Kol.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kol, B. Flux-based statistical prediction of three-body outcomes. Celest Mech Dyn Astr 133, 17 (2021). https://doi.org/10.1007/s10569-021-10015-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10569-021-10015-x

Keywords

  • Three-body problem
  • Statistical mechanics
  • Chaos