Isles of regularity in a sea of chaos amid the gravitational three-body problem

TL;DR

This paper investigates the coexistence of regular and chaotic trajectories in the three-body problem, revealing that regular trajectories occupy a significant phase space portion and challenge the assumptions of statistical escape theories, with implications for astrophysics.

Contribution

It provides new criteria for identifying regular trajectories in the three-body problem and analyzes their impact on statistical predictions, highlighting the limitations of existing theories.

Findings

Regular trajectories occupy up to 32% of phase space.

Regular trajectories' outcomes often contradict statistical escape predictions.

Integration errors can artificially induce chaos, affecting individual simulations.

Abstract

The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. This study probes the presence of regular (i.e. non chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Using numerical simulations, we establish criteria for identifying regular trajectories and analyse their impact on statistical outcomes. Our analysis reveals that regular trajectories occupy up to 32% of the phase space, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.