The resilience of the sailboat stable region

¹ LIRA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CY Cergy Paris Université, CNRS, 92190 Meudon, France
² São Paulo State University (UNESP), School of Engineering and Sciences, Guaratinguetá, SP 12516-410, Brazil
³ Eberhard Karls Universität Tübingen, Auf der Morgenstelle, 10, 72076 Tübingen, Germany
⁴ Observatório Nacional (ON)/MCTI, Rio de Janeiro, RJ 20921-400, Brazil

^★ Corresponding author: rafael.sfair@unesp.br

Received: 10 June 2025
Accepted: 2 October 2025

Abstract

Context. Binary systems host complex orbital dynamics where test particles can occupy stable regions, despite strong gravitational perturbations. The sailboat region, discovered in the Pluto–Charon system, hosts highly eccentric S-type orbits at intermediate distances between the two massive bodies. This region challenges traditional stability concepts by supporting eccentricities up to 0.9 in a zone typically dominated by chaotic motion.

Aims. We investigate the sailboat region’s existence and extent across different binary system configurations. We examine how variations in mass ratio, secondary body eccentricity, particle inclination, and argument of pericenter affect this stable region.

Methods. We performed 1.2 million numerical simulations of the elliptic three-body problem to generate four datasets exploring different parameter spaces. We trained XGBoost machine learning models to classify stability across approximately 10⁹ initial conditions. We validated our results using Poincaré surface of section and Lyapunov exponent analysis to confirm the dynamical mechanisms underlying the stability.

Results. The sailboat region exists only for binary mass ratios of μ = [0.05, 0.22]. Secondary body eccentricity severely constrains the region, following an exponential decay of e_s,max ≈ 0.016 + 0.614exp(−25.6μ). The region tolerates particle inclinations up to 90° and persists in retrograde configurations for μ ≤ 0.16. Stability requires a specific argument of pericenter within ±10° to ±30° of ω = 0° and 180°. Our machine learning models have achieved over 97% accuracy in predicting stability.

Conclusions. The sailboat region shows strong sensitivity to system parameters, particularly secondary body eccentricity. Among Solar System dwarf planet binaries, we find that Pluto–Charon, Orcus–Vanth and Varda–Ilmarë systems could harbor such regions. The combination of numerical simulations and machine learning provides an efficient approach for mapping stability in complex gravitational systems.