Three-body periodic collisionless equal-mass free-fall orbits revisited (2308.16159v1)

Abstract: Li and Liao announced (2019) discovery of 313 periodic collisionless orbits' initial conditions (i.c.s), 30 of which have equal masses, and 18 of these 30 orbits have physical periods (scale-invariant periods) $T^{*}=T|E|^{3/2}<80$. That work left a lot to be desired, however, both in terms of logical consistency and of numerical efficiency. We have conducted a new search for periodic free-fall orbits, limited to the equal-mass case. Our search produced 24,582 i.c.s of equal-mass periodic orbits with scale-invariant period $T^{*}<80$, corresponding to 12,409 distinct solutions, 236 of which are self-dual.

• The paper introduces refined numerical methods that identify 12,409 distinct free-fall orbits in equal-mass three-body systems.
• It expands the orbit catalog from 30 to 24,582 initial conditions, showcasing the power of high-performance computing in celestial mechanics.
• The research reveals 236 self-dual orbits and employs symbolic dynamics to highlight underlying topological symmetries.

Three-Body Periodic Collisionless Equal-Mass Free-Fall Orbits Revisited

The paper "Three-body periodic collisionless equal-mass free-fall orbits revisited," explores the exploration and numerical analysis of periodic orbits within the framework of the three-body problem, specifically focusing on the equal-mass case under free-fall, collisionless conditions. This domain, despite its historical significance, has not been extensively cataloged compared to other facets of celestial mechanics, prompting the authors to explore its intricacies using modern computational methods.

The authors identify a striking contrast in the number of periodic orbits previously discovered by Li and Liao—30 orbits— and the significantly larger number discovered in their paper—24,582 initial conditions leading to 12,409 distinct orbits, of which 236 are self-dual. This discrepancy can be attributed to the limitations in logical consistency and numerical efficiency found in prior research.

The paper reaffirms the relevance of free-fall orbits as potentially applicable to astronomical bodies such as the Saturn-Janus-Epimetheus and Sun-Earth-Cruithne systems, which belong to the same topological family as the orbits studied. Moreover, the paper expands upon the theoretical groundwork laid by Agekyan and Anosova who initially characterized the domain for such orbits, enabling a comprehensive numerical exploration through the use of advanced computational tools and high-performance computing resources.

Numerical Methodology and Findings

1. Methodological Advances: The research employs a refined numerical approach, enhancing previous search algorithms and adopting high-accuracy numerical methods. By utilizing a high-performance computer cluster, the authors efficiently generate and validate a vast set of initial conditions for free-fall orbits.
2. Increase in Known Orbits: The effort yields a comprehensive database, enhancing the known catalog of periodic free-fall orbits from 30 to 24,582 initial conditions. The 12,409 distinct solutions underpin a substantial numerical evidence of the intricate structure embedded within the equal-mass three-body problem.
3. Self-Dual Orbits: The identification of 236 self-dual orbits is particularly noteworthy, suggesting a deeper underlying symmetry in such dynamical systems. These self-dual solutions correspond to orbits where the spatial configuration is symmetric or reflects spatiotemporal invariances upon completion of half the orbital period.
4. Symbolic Dynamics: To categorize these orbits, the paper applies a symbolic sequencing technique developed by Tanikawa and Mikkola, effectively delineating different dynamical behaviors through syzygy counts, aligned with the symbolic dynamics approach. This enhances the understanding of the geometrical and topological characteristics of the orbit space.
5. Scale-Invariant Period Correlations: A dependence of the scale-invariant period on the symbolic sequence length is identified, adhering to previously confirmed linear patterns in similar dynamical systems. This relationship provides a predictive framework to estimate orbit periods based on sequence characteristics.

Implications and Future Directions

The paper significantly bolsters the foundational understanding of the three-body problem in the context of collisionless free-fall dynamics, demonstrating both a methodological enhancement and theoretical enrichment of the field. By counterposing the results against the backdrop of earlier studies, the authors clarify the numerical landscape and artisanal boundaries of the domain.

The findings set the stage for further theoretical and computational inquiries into the stability, bifurcation phenomena, and intricate structures of self-dual orbits. This could potentially yield novel insights into not only celestial mechanics but also broader dynamical systems. The exploration of higher-dimensional analogs, exhaustive stability analysis, and the detailed characterization of the symbolic sequence-coupled dynamics are natural extensions of this research. As the field advances, continued scrutiny of both the algorithmic techniques and the physical understandings underpinning these complex dynamical dances will undoubtedly uncover further hidden patterns and structures within the mathematical elegance of celestial mechanics.