Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A statistical solution to the chaotic, non-hierarchical three-body problem

Abstract

The three-body problem is arguably the oldest open question in astrophysics and has resisted a general analytic solution for centuries. Various implementations of perturbation theory provide solutions in portions of parameter space, but only where hierarchies of masses or separations exist. Numerical integrations1 show that bound, non-hierarchical triple systems of Newtonian point particles will almost2 always disintegrate into a single escaping star and a stable bound binary3,4, but the chaotic nature of the three-body problem5 prevents the derivation of tractable6 analytic formulae that deterministically map initial conditions to final outcomes. Chaos, however, also motivates the assumption of ergodicity7,8,9, implying that the distribution of outcomes is uniform across the accessible phase volume. Here we report a statistical solution to the non-hierarchical three-body problem that is derived using the ergodic hypothesis and that provides closed-form distributions of outcomes (for example, binary orbital elements) when given the conserved integrals of motion. We compare our outcome distributions to large ensembles of numerical three-body integrations and find good agreement, so long as we restrict ourselves to ‘resonant’ encounters10 (the roughly 50 per cent of scatterings that undergo chaotic evolution). In analysing our scattering experiments, we identify ‘scrambles’ (periods of time in which no pairwise binaries exist) as the key dynamical state that ergodicizes a non-hierarchical triple system. The generally super-thermal distributions of survivor binary eccentricity that we predict have notable applications to many astrophysical scenarios. For example, non-hierarchical triple systems produced dynamically in globular clusters are a primary formation channel for black-hole mergers11,12,13, but the rates and properties14,15 of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.

This is a preview of subscription content

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Non-hierarchical three-body scatterings.
Fig. 2: Topological maps of three-body scattering outcomes for run A.
Fig. 3: Marginal distribution of binary energy, dσ/dEB, as a function of dimensionless energy, EB/E0.
Fig. 4: Marginal distributions of binary eccentricity and orientation.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. 1.

    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 

  2. 2.

    Šuvakov, M. & Dmitrašinović, V. Three classes of Newtonian three-body planar periodic orbits. Phys. Rev. Lett. 110, 114301 (2013).

    ADS  Article  Google Scholar 

  3. 3.

    Standish, E. M. The dynamical evolution of triple star systems. Astron. Astrophys. 21, 185–191 (1972).

    ADS  Google Scholar 

  4. 4.

    Hut, P. & Bahcall, J. N. Binary–single star scattering. I – numerical experiments for equal masses. Astrophys. J. 268, 319–341 (1983).

    ADS  Article  Google Scholar 

  5. 5.

    Poincaré, H. Les Méthodes Nouvelles de la Mécanique Céleste (Gauthier-Villars et fils, 1892).

  6. 6.

    Sundman, K. F. Mémoire sur le problème des trois corps. Acta Math. 36, 105–179 (1913).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fermi, E. High energy nuclear events. Prog. Theor. Phys. 5, 570–583 (1950).

    CAS  ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Monaghan, J. J. A statistical theory of the disruption of three-body systems – I. Low angular momentum. Mon. Not. R. Astron. Soc. 176, 63–72 (1976).

    ADS  Article  Google Scholar 

  9. 9.

    Valtonen, M., Mylläri, A., Orlov, V. & Rubinov, A. Dynamics of rotating triple systems: statistical escape theory versus numerical simulations. Mon. Not. R. Astron. Soc. 364, 91–98 (2005).

    ADS  Article  Google Scholar 

  10. 10.

    Heggie, D. C. Binary evolution in stellar dynamics. Mon. Not. R. Astron. Soc. 173, 729–787 (1975).

    ADS  Article  Google Scholar 

  11. 11.

    Portegies Zwart, S. F. & McMillan, S. L. W. Black hole mergers in the Universe. Astrophys. J. Lett. 528, 17–20 (2000).

    ADS  Article  Google Scholar 

  12. 12.

    Rodriguez, C. L., Chatterjee, S. & Rasio, F. A. Binary black hole mergers from globular clusters: masses, merger rates, and the impact of stellar evolution. Phys. Rev. D 93, 084029 (2016).

    ADS  Article  Google Scholar 

  13. 13.

    Hong, J. et al. Binary black hole mergers from globular clusters: the impact of globular cluster properties. Mon. Not. R. Astron. Soc. 480, 5645–5656 (2018).

    CAS  ADS  Article  Google Scholar 

  14. 14.

    Samsing, J., MacLeod, M. & Ramirez-Ruiz, E. The formation of eccentric compact binary inspirals and the role of gravitational wave emission in binary–single stellar encounters. Astrophys. J. 784, 71 (2014).

    ADS  Article  Google Scholar 

  15. 15.

    Rodriguez, C. L. et al. Post-Newtonian dynamics in dense star clusters: formation, masses, and merger rates of highly-eccentric black hole binaries. Phys. Rev. D 98, 123005 (2018).

    CAS  ADS  Article  Google Scholar 

  16. 16.

    Portegies Zwart, S. F. & Boekholt, T. C. N. Numerical verification of the microscopic time reversibility of Newton’s equations of motion: fighting exponential divergence. Commun. Nonlinear Sci. Numer. Simul. 61, 160–166 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Hut, P. The topology of three-body scattering. Astron. J. 88, 1549–1559 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Samsing, J. & Ilan, T. Topology of black hole binary–single interactions. Mon. Not. R. Astron. Soc. 476, 1548–1560 (2018).

    ADS  Article  Google Scholar 

  19. 19.

    Bohr, N. Neutron capture and nuclear constitution. Nature 137, 344–348 (1936).

    CAS  ADS  Article  Google Scholar 

  20. 20.

    Monaghan, J. J. A statistical theory of the disruption of three-body systems – II. High angular momentum. Mon. Not. R. Astron. Soc. 177, 583–594 (1976).

    ADS  Article  Google Scholar 

  21. 21.

    Nash, P. E. & Monaghan, J. J. A statistical theory of the disruption of three-body systems – III. Three-dimensional motion. Mon. Not. R. Astron. Soc. 184, 119–125 (1978).

    ADS  Article  Google Scholar 

  22. 22.

    Geller, A. M., Leigh, N. W. C., Giersz, M., Kremer, K. & Rasio, F. A. In search of the thermal eccentricity distribution. Astrophys. J. 872, 165 (2019).

    ADS  Article  Google Scholar 

  23. 23.

    Pomeau, Y. & Manneville, P. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980).

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Mardling, R. A. & Aarseth, S. J. Tidal interactions in star cluster simulations. Mon. Not. R. Astron. Soc. 321, 398–420 (2001).

    ADS  Article  Google Scholar 

  25. 25.

    Leigh, N. W. C. & Geller, A. M. The dynamical significance of triple star systems in star clusters. Mon. Not. R. Astron. Soc. 432, 2474–2479 (2013).

    ADS  Article  Google Scholar 

  26. 26.

    Leonard, P. J. T. & Fahlman, G. G. On the origin of the blue stragglers in the globular cluster NGC 5053. Astron. J. 102, 994 (1991).

    ADS  Article  Google Scholar 

  27. 27.

    Leigh, N., Sills, A. & Knigge, C. An analytic model for blue straggler formation in globular clusters. Mon. Not. R. Astron. Soc. 416, 1410–1418 (2011).

    ADS  Article  Google Scholar 

  28. 28.

    Ivanova, N. et al. Formation and evolution of compact binaries in globular clusters – I. Binaries with white dwarfs. Mon. Not. R. Astron. Soc. 372, 1043–1059 (2006).

    ADS  Article  Google Scholar 

  29. 29.

    Pooley, D. & Hut, P. Dynamical formation of close binaries in globular clusters: cataclysmic variables. Astrophys. J. Lett. 646, 143–146 (2006).

    ADS  Article  Google Scholar 

  30. 30.

    Ivanova, N., Heinke, C. O., Rasio, F. A., Belczynski, K. & Fregeau, J. M. Formation and evolution of compact binaries in globular clusters – II. Binaries with neutron stars. Mon. Not. R. Astron. Soc. 386, 553–576 (2008).

    CAS  ADS  Article  Google Scholar 

Download references

Acknowledgements

We acknowledge discussions with D. Heggie, P. Hut, R. Sari and S. Portegies-Zwart, as well as feedback from E. Michaely and O. C. Winter. N.C.S. received financial support from NASA, through Einstein Postdoctoral Fellowship Award number PF5-160145 and the NASA Astrophysics Theory Research Program (grant NNX17AK43G; Principal Investigator, B. Metzger). N.C.S. also thanks the Aspen Center for Physics for its hospitality during early stages of this work. N.W.C.L. acknowledges support by Fondecyt Iniciacion grant number 11180005. We thank the Chinese Academy of Sciences for hosting us as we completed our efforts. We thank M. Valtonen and H. Karttunen, whose book on the three-body problem motivated much of this work.

Author information

Affiliations

Authors

Contributions

N.C.S. led the analytic work, to which N.W.C.L. contributed significantly. The FEWBODY simulations were performed by N.W.C.L. The comparison between the simulations and the analytic theory was performed jointly by the two authors.

Corresponding author

Correspondence to Nicholas C. Stone.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Erez Michaely and Othon Cabo Winter for their contribution to the peer review of this work.

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

Extended data figures and tables

Extended Data Fig. 1 Marginal distribution of binary energies, dσ/dEB.

Colours show dimensionless angular momenta \({\tilde{L}}_{0}\); upper and lower black dashed lines are asymptotic power laws for \({\tilde{L}}_{0}\) = 1 and \({\tilde{L}}_{0}\) ≈ 1, respectively. a, Ergodic outcome distributions using the ‘apocentric escape’ (AE) criterion; that is, assuming that disintegration of metastable triples occurs within a strong interaction region of size R = αaB(1 + eB). Here we take α = 2. Solid lines represent equal-mass scattering ensembles (ma = mb = ms) and dotted lines extreme-mass-ratio ensembles (ma = mb = 10ms). b, As in a, but for a ‘simple escape’ (SE) criterion, R = αaB. c, Intermediate-mass-ratio scattering ensembles (ma = mb = 3ms). Solid lines correspond to α = 2 and dotted lines to α = 5. d, As in c, but for ma = mb = 10ms. Note that \({\tilde{L}}_{0}\) is a dimensionless angular momentum normalized by the circular orbit angular momentum of a binary with energy E0 and masses ma and mb.

Extended Data Fig. 2 Marginal distribution of binary eccentricity, dσ/deB.

Line styles and assumptions are as in Extended Data Fig. 1, except for the upper and lower black dashed lines, which here show the \({\tilde{L}}_{0}\approx 1\) and \({\tilde{L}}_{0}\ll 1\) limits of the dσ/deB distribution, respectively (unlike for dσ/dEB, these limits differ significantly in the AE and SE regimes). In comparable-mass AE calculations, mildly super-thermal outcomes arise from geometric effects when \({\tilde{L}}_{0}\approx 1\); by contrast, angular-momentum starvation produces extremely super-thermal outcomes when \({\tilde{L}}_{0}\ll 1\). Small ms values foreclose parts of eB space, as LB ≈ L0.

Extended Data Fig. 3 Marginal distribution of binary orientation, dσ/dCB.

Assumptions and line styles are as in Extended Data Fig. 1, except that the black dashed lines show (i) an isotropic outcome configuration and (ii) an analytic approximation for dσ/dCB, as labelled in a (for an equal-mass triple with \({\tilde{L}}_{0}=0.5\)). For \({\tilde{L}}_{0}\ll 1\), surviving binaries are distributed isotropically (as symmetry dictates). Otherwise, binary orientations \({C}_{{\rm{B}}}={\hat{{\bf{L}}}}_{{\rm{B}}}\cdot {\hat{{\bf{L}}}}_{0}\) are biased towards prograde outcomes. For extreme mass ratios and large \({\tilde{L}}_{0}\), retrograde outcomes may be entirely prohibited.

Extended Data Table 1 Numerical (binary–single) scattering ensembles used for comparison to analytic theory

Supplementary information

Supplementary Information

This Supplementary Information file contains: (1) Chaotic Escape in the Three-Body Problem; (2) Outcomes of Non-Hierarchical Three Body Encounters; (3) Comparison to Numerical Scattering Experiments; (4) Discussion; and associated references.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stone, N.C., Leigh, N.W.C. A statistical solution to the chaotic, non-hierarchical three-body problem. Nature 576, 406–410 (2019). https://doi.org/10.1038/s41586-019-1833-8

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing