# Flux-based statistical prediction of three-body outcomes

• Original Article
• Published:

## 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, log in via an institution to check access.

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

## 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)

• 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)

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

• 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

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

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

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

• 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)

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

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

• 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)

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

• 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

• 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]]

• 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)

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

• 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]]

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

• 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)

• 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

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

• 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)

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

• 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

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

• 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

• 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)

• 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

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

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

• 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)

• Wikipedia, Characteristic function (probability theory)

## 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

### Corresponding author

Correspondence to Barak Kol.

### Publisher's Note

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

## Rights and permissions

Reprints and permissions

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