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

¹ Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
e-mail: aatrani@gmail.com
² Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Japan
³ Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
⁴ Departamento de Astronomía, Facultad Ciencias Físicas y Matemáticas, Universidad de Concepción, Avenida Esteban Iturra, Casilla 160-C, Concepción 4030000, Chile
⁵ Department of Astrophysics, American Museum of Natural History, Central Park West and 79th Street, New York, NY 10024, USA
⁶ NASA Ames Research Center, Moffett Field, CA 94035, USA
⁷ Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, The Netherlands

Received: 5 March 2024
Accepted: 25 June 2024

Abstract

Context. 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.

Aims. This study probes the presence of regular (i.e. non-chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories.

Methods. Using three-body simulations performed with the accurate, regularized code TSUNAMI, we established criteria for identifying regular trajectories and analysed their impact on statistical outcomes.

Results. Our analysis reveals that regular trajectories occupy a significant fraction of the phase space, ranging from 28% to 84% depending on the initial setup, 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.

Conclusions. 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.