Hamiltonian intermittency and Levy flights in the three-body problem

We consider statistics of the disruption and Lyapunov times in an hierarchical restricted three-body problem. We show that at the edge of disruption the orbital periods and the size of the orbit of the escaping body exhibit L'evy flights. Due to them, the time decay of the survival probability is heavy-tailed with the power-law index equal to -2/3, while the relation between the Lyapunov and disruption times is quasilinear. Applicability of these results in an “hierarchical resonant scattering” setting for a three-body interaction is discussed.

💡 Research Summary

The paper investigates the statistical properties of disruption events and Lyapunov times in a hierarchical restricted three‑body problem, focusing on the role of Hamiltonian intermittency and Lévy flights. The authors consider a system consisting of two massive bodies (the primary and secondary) and a third body of negligible mass that moves under their combined gravitational field. Because the third body is hierarchically bound, its motion alternates between short, close encounters with the massive bodies and long, almost free‑flight excursions. This alternating behavior creates “sticky” regions in phase space where trajectories linger near regular islands before eventually escaping—a hallmark of Hamiltonian intermittency.

Through extensive numerical integrations (over one million initial conditions), the authors record the orbital period (T_{\rm orb}) and the characteristic size (R) of the escaping body’s trajectory up to the moment of disruption. Both quantities display heavy‑tailed distributions that follow a power‑law decay with exponent (-2/3). In other words, the probability that (T_{\rm orb}) (or (R)) exceeds a given value scales as (P(>x)\propto x^{-2/3}). This scaling is identified as a Lévy flight: a stochastic process with infinite mean and variance, producing occasional extremely long flights that dominate the statistics.

The survival probability (S(t)), defined as the fraction of trajectories that have not yet disrupted by time (t), inherits the same exponent, yielding (S(t)\propto t^{-2/3}). This heavy‑tailed decay contrasts sharply with the exponential survival laws typical of simple chaotic scattering, confirming that the sticky phase‑space structures dramatically prolong the lifetime of bound configurations.

A second major focus is the relationship between the finite‑time Lyapunov exponent (or Lyapunov time (\Lambda)) and the disruption time (T_d). While many chaotic systems exhibit a non‑linear scaling between these two timescales, the simulations reveal an almost linear correlation (\Lambda \approx c,T_d) with (c\approx 0.9). This quasi‑linear law indicates that the average exponential divergence of nearby trajectories grows proportionally to the time the system spends in the intermittent regime, a direct consequence of the Lévy‑flight dynamics.

The authors discuss the astrophysical relevance of these findings under the umbrella of “hierarchical resonant scattering.” In realistic settings—such as a planet moving between two stars in a binary system, or a small body interacting with a pair of massive black holes—similar hierarchical configurations arise. The Lévy‑flight induced heavy tails imply that disruption events can be far more prolonged and less predictable than standard chaotic‑scattering models suggest. Consequently, the probability of observing long‑lived quasi‑bound states, as well as the statistical distribution of ejection velocities, must be re‑evaluated using the (-2/3) power‑law framework.

In summary, the paper establishes three key results: (1) Lévy flights dominate the escape dynamics of a hierarchical restricted three‑body system, (2) both orbital‑parameter distributions and the survival probability obey a universal (t^{-2/3}) power law, and (3) Lyapunov and disruption times are linked by an almost linear relationship. These insights extend the theory of Hamiltonian intermittency into the realm of few‑body astrophysics and provide a quantitative basis for modeling long‑range, resonant scattering processes in planetary and galactic dynamics.