Stability of Hilda asteroids at 3:2 resonance point in restricted three-body problem

TL;DR

This paper investigates the stability of Hilda asteroids at the 3:2 resonance with Jupiter, showing their orbits are approximately stable and integrable, unlike other resonances which exhibit chaotic behavior.

Contribution

The study demonstrates that the Hamiltonian near the 3:2 resonance can be approximated by a single-resonance Hamiltonian, indicating stable and integrable asteroid orbits.

Findings

Hilda asteroid orbits are stable and approximately integrable.

Hamiltonian approximation is valid around the 3:2 resonance.

Contrasts with chaotic behavior at 3:1 and 2:1 resonances.

Abstract

Stability of Hilda Asteroids in the solar system around the 3:2 resonance point is analyzed in terms of the Sun-Jupiter-asteroid elliptic restricted three-body problem. We show that the Hamiltonian of the system is well-approximated by a single-resonance Hamiltonian around the 3:2 resonance. This implies that orbits of the Hilda asteroids are approximately integrable, thus their motion is stable. This is in contrast to other resonances such as the 3:1 and the 2:1 resonances at which Kirkwood gaps occur. Indeed, around the 3:1 and the 2:1 resonances, the Hamiltonians are approximated by double-resonance Hamiltonians that are nonintegrable and thus indicate chaotic motions. By a suitable canonical transformation, we reduce the number of degrees of freedom for the system and derive a Hamiltonian which has two degrees of freedom. As a result, we can analyze the stability of the motion by constructing Poincare surface of section.