A Hamiltonian model for the long-term evolution of high-inclination resonant small bodies
Abstract
The long-term dynamics of mean motion resonance significantly influence the dynamical structure of the main belt asteroids and the Jupiter Trojans. In this study, we establish a semi-analytical dynamical model for high-inclination resonant small bodies within the framework of the circular restricted three-body problem. Utilizing the adiabatic invariance approximation, we derive a nearly integrable Hamiltonian system. Based on the Hamiltonian, we construct phase diagrams of the adiabatic invariants and compare them with numerical integration trajectories. For non-1:n resonant and retrograde 1:1 resonant small bodies, the phase diagrams align well with numerical orbits across a broad range of eccentricity space. Conversely, for prograde 1:1 resonant small bodies, the phase diagram predictions are valid for high eccentricity scenarios. The trajectories in the phase diagrams provide a global view of the long-term dynamical evolution of eccentricity, orbital inclination, and argument of periapsis, highlighting potential regions of chaos. By applying the semi-analytical model to real small bodies in the 3:2, 2:1, prograde 1:1, and retrograde 1:1 resonances, we compare the phase diagrams with integral trajectories from the elliptical restricted three-body system. Our findings indicate that the long-term perturbations caused by Jupiter's eccentricity have a minimal impact on high-inclination small bodies in the 3:2 and 2:1 resonances.
