Chaos over Order: Mapping 3D Rotation of Triaxial Asteroids and Minor Planets

TL;DR

This paper investigates the chaotic and orderly rotation regimes of triaxial minor planets in a two-body system, analyzing how shape and eccentricity influence the transition to chaos using numerical methods.

Contribution

It presents a comprehensive analysis of 3D rotational dynamics of triaxial asteroids, including new numerical solutions and the impact of shape and eccentricity on chaos boundaries.

Findings

Chaos dominates at high eccentricities for most shapes.

The main 1:1 resonance disappears at eccentricities as low as 0.3 for certain shapes.

Stable regions near resonances shrink and vanish as eccentricity increases.

Abstract

Celestial bodies approximated with rigid triaxial ellipsoids in a two-body system can rotate chaotically due to the time-varying gravitational torque from the central mass. At small orbital eccentricity values, rotation is short-term orderly and predictable within the commensurate spin-orbit resonances, while at eccentricity approaching unity, chaos completely takes over. Here, we present the full 3D rotational equations of motion around all three principle axes for triaxial minor planets and two independent methods of numerical solution based on Euler rotations and quaternion algebra. The domains of chaotic rotation are numerically investigated over the entire range of eccentricity with a combination of trial integrations of Euler's equations of motion and the GALI($k$) method. We quantify the dependence of the order--chaos boundaries on shape by changing a prolateness parameter, and find that the main 1:1 spin-orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as 0.3. The island of short-term stability around the main 1:1 resonance shrinks with increasing eccentricity at a fixed low degree of prolateness and completely vanishes at approximately 0.8. This island is also encroached by chaos on longer time scales indicating longer Lyapunov exponents. Trajectories in the close vicinity of the 3:2 spin-orbit resonance become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as 0.8. Initial perturbations of rotational velocity in latitude away from the exact equilibrium result in a spectrum of free libration, nutation, and polar wander, which is not well matched by the linearized analysis omitting the inertial terms.