The resilience of the sailboat stable region

TL;DR

This paper addresses the existence and extent of the sailboat stability region for S-type orbits in binary systems, showing that stability with high eccentricities up to $e \approx 0.9$ occurs only for mass ratios $μ \in [0.05,0.22]$ and is strongly suppressed by secondary eccentricity $e_s$ via $e_{s,\max} \approx 0.016 + 0.614 \exp(-25.6 μ)$. The authors combine ~1.2 million direct integrations of the elliptic three-body problem with XGBoost-based supervised classification to predict stability across ~10^9 initial conditions, and validate with Poincaré surfaces of section and Lyapunov (MEGNO) analysis. They show the sailboat region tolerates particle inclinations up to $90^ ext{\circ}$ and is constrained to narrow $ω$ windows around $0^ ext{\circ}$ and $180^ ext{\circ}$, shrinking as $μ$ grows. Among Solar System dwarfs, Pluto–Charon, Orcus–Vanth, and Varda–Ilmarë are identified as plausible hosts, illustrating the practical relevance for debris distributions and mission design in binary systems.

Abstract

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, allows 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. 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. 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 $10^9$ initial conditions. We validated our results using Poincaré surface of section and Lyapunov exponent analysis to confirm the dynamical mechanisms underlying the stability. The sailboat region exists only for binary mass ratios $μ= [0.05, 0.22]$. Secondary body eccentricity severely constrains the region, following an exponential decay: $e_{s,\mathrm{max}} \approx 0.016 + 0.614 \exp(-25.6μ)$. The region tolerates particle inclinations up to $90^\circ$ and persists in retrograde configurations for $μ\leq 0.16$. Stability requires specific argument of pericenter values within $\pm 10^\circ$ to $\pm 30^\circ$ of $ω= 0^\circ$ and $180^\circ$. Our machine learning models achieved over 97\% accuracy in predicting stability. The sailboat region shows strong sensitivity to system parameters, particularly secondary body eccentricity. Among Solar System dwarf planet binaries, Pluto-Charon, Orcus-Vanth and Varda-Ilmarë systems could harbor such regions.

Figures (12)

• Figure 1: Diagram of a binary star system ($d \times \mu$), where $d$ represents the separation distance between the stars and $\mu$ is the binary mass ratio. The diagram categorizes the binaries as follows: blue for Detached type, green for Semi-Detached type, and red for Contact type. The black point corresponds to Pluto-Charon system.
• Figure 2: Stability maps in the $(a,e)$ plane for binary systems with mass ratios $\mu = [0.01-0.23]$. Stable initial conditions are shown in orange, unstable in blue. All particles are in coplanar orbits around the primary body with $\omega = \Omega = f = 0^\circ$.
• Figure 3: The sailboat region for a binary system with $\mu = 0.12$ (the closest case with Pluto-Charon system, where Charon has eccentricity). The red dashed line represents the pericenter of the particle in the same distance of Pluto's radius. There is other stable regions in the range $a = [0.45 - 0.7]$, not showed in the plot.
• Figure 4: Maximum eccentricity of secondary body (blue dots) versus mass ratio of the system. The red curve represents a fitted model obtained using the Least Squares method. Green squares denote binary systems of dwarf planets and their satellites, while the black square represents the Pluto-Charon system.
• Figure 5: Stability maps $(a \times e)$ showing the evolution of the sailboat region with changing secondary body eccentricity for binary systems with mass ratios of $\mu = 0.05$, $0.12$, and $0.22$.