Columbia Astrophysics Laboratory, Columbia University, New York, NY, USA.
Racah Institute of Physics, The Hebrew University, Jerusalem, Israel.
Nature. 2019 Dec;576(7787):406-410. doi: 10.1038/s41586-019-1833-8. Epub 2019 Dec 18.
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 integrations show that bound, non-hierarchical triple systems of Newtonian point particles will almost always disintegrate into a single escaping star and a stable bound binary, but the chaotic nature of the three-body problem prevents the derivation of tractable analytic formulae that deterministically map initial conditions to final outcomes. Chaos, however, also motivates the assumption of ergodicity, 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' encounters (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 mergers, but the rates and properties of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.
三体问题可以说是天体物理学中最古老的开放性问题，几个世纪以来一直没有得到普遍的解析解。各种摄动理论的实现方法在部分参数空间中提供了解，但仅在质量或分离的层次结构存在的情况下。数值积分表明，牛顿点粒子的非层次三体束缚系统几乎总是会分解为单个逃逸恒星和稳定的束缚双星，但三体问题的混沌性质阻止了可推导出的解析公式的推导，这些公式可以确定性地将初始条件映射到最终结果。然而，混沌也激发了遍历性的假设，即结果的分布在可及的相体积内是均匀的。在这里，我们报告了一种非层次三体问题的统计解决方案，该解决方案是使用遍历性假设推导出来的，并且在给定运动守恒积分时，可以提供结果（例如，二进制轨道元素）的闭式分布。我们将我们的结果分布与大量的数值三体积分进行比较，并发现只要我们将自己限制在“共振”（经历混沌演化的散射的大约 50%），就可以很好地吻合。在分析我们的散射实验时，我们将“混乱”（不存在任何二进制对的时间段）识别为遍历非层次三体系统的关键动力学状态。我们预测的幸存二进制偏心率的一般超热分布在许多天体物理场景中都有显著的应用。例如，在球状星团中动态产生的非层次三体系统是黑洞合并的主要形成渠道，但产生的引力波的速率和性质取决于后分解偏心率的分布。
