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.

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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.

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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

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.

Correspondence to Nicholas C. Stone.

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The authors declare no competing interests.

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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.

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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

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