Chaotic Dynamics and Zero-Velocity Structures in the Pluto-Charon CR3BP

TL;DR

This study investigates the Pluto–Charon system within the planar Circular Restricted Three-Body Problem to test Trojan stability and reveal chaotic transport at a high mass ratio $\mu \approx 0.109$. Using RK4 integration, zero-velocity curves, and Jacobi-constant dynamics, it analyzes canonical tadpole and horseshoe orbits and examines the sensitivity of trajectories to small changes in energy near the $L_1$ neck. The results show that Pluto–Charon cannot support long-lived Trojan companions; instead, the system exhibits chaotic evolution with initial-condition sensitivity and potential low-energy transport pathways between lobes. The findings have implications for mission design around high-$\mu$ binaries and motivate extending the analysis to spatial and elliptic formulations to better understand instability and transport in such systems.

Abstract

Pluto and Charon are a dwarf binary system with a high mass ratio $μ$, preventing Trojan companions. This instability creates ideal intersections for low-energy pathways that spacecraft can traverse and serves as an important test case for stability at Lagrange points for high $μ$ binaries. This study models the Pluto-Charon system in the planar Circular Restricted Three-Body Problem (CR3BP) and compares the tadpole and horseshoe orbits of a massless particle with known low-$μ$ orbits. Moreover, it compares the trajectories for instances where the L1 neck is opened and closed, and exhibits the corresponding zero-velocity curves; RK4 integration was used to update the position and velocity of the particle. The simulations consistently showed chaotic and unpredictable trajectories, where small changes in initial parameters could completely alter the results; trajectories displayed the influence of higher values of $μ$ on the stability of binary systems. These findings confirm that Pluto-Charon cannot host long-lived Trojan companions and instead behaves as a system of chaotic transport. Future work could extend the study to include the spatial or elliptic three-body problem, further refining the understanding of instability in high-$μ$ binaries.

Figures (7)

• Figure 1: Purple contours mark the level set $C=2.9031+\delta$, with $\delta = 1\times10^{-3}$. Pluto (dark gray) is at $(-\mu,0)$ and Charon (light gray) is at $(1-\mu,0)$
• Figure 2: Figures show $C=3.6210+ \delta$, with $\delta=\pm1\times10^{-2}$. All $L$ points are in the forbidden region in $(a)$; A difference of only $2\delta$ in $C$ can alter connectivity between regions.
• Figure 3: Example of a tadpole orbit librating around $L_4$. The primaries are shown as black circles. Integration parameters: $h=5\times 10^{-4}$, $C_{\text{target}}=C_{L_4}+\delta$, with $\delta=-5\times 10^{-4}$ tangential initial velocity, starting coordinates $(L_{4_x}-0.04,L_{4_y}-0.03)$. Drift in $C$: $\sim 6.7\times 10^{-15}$. The trajectory is plotted in blue.
• Figure 4: Example of a horseshoe orbit. The primaries are shown as black circles. Integration parameters: $h=5\times 10^{-4}$, $C_{\text{target}}=C_{L_3}+\delta$, with $\delta=-5\times10^{-5}$, vertical initial velocity, starting coordinates $(L_{3_x}-0.027,L_{3_y})$. Drift in $C$: $\sim 4.9\times 10^{-15}$. The trajectory is plotted in blue.
• Figure 5: Attempt at tadpole orbit for Pluto--Charon. Pluto (dark gray) is at $(-\mu,0)$ and Charon (light gray) is at $(1-\mu,0)$. Integration parameters: $h=5\times 10^{-4}$, $C_{\text{target}}=C_{L_4}+\delta$, with $\delta=-5\times10^{-4}$, tangential initial velocity, starting coordinates $(L_{4_x}-0.04,L_{4_y}-0.03)$. Drift in $C$: $\sim 5.1\times 10^{-9}$. The trajectory is plotted in blue.