Astronomy7 papers
Mean-motion resonances in 2025
by Evgeny Smirnov
All Reviews
AstronomyN/A
On the limits of application of mean motion resonant normal forms of the three-body problem for crossing orbits and close encounters: On the limits of application of mean motion resonant...
Liu, Xiang & Guzzo, Massimiliano (2025)
- Published
- Feb 1, 2025
- Journal
- Celestial Mechanics and Dynamical Astronomy
- DOI
- 10.1007/s10569-024-10232-0
At a Glance
MMR normal forms remain valid for orbit-crossing dynamics in the restricted three-body problem up to surprisingly high eccentricities.
Summary
This paper analyzes where canonical perturbation theory for mean-motion resonance normal forms breaks down for orbits that cross or nearly cross the secondary body's orbit in the planar circular restricted three-body problem. Using numerically computed canonical transformations, regularized fast Lyapunov indicators, and quasi-integral preservation analysis, the authors show that for the external 1:2 MMR, the averaging method remains reliable even for chaotic orbits with eccentricities up to ~0.55. This establishes concrete validity boundaries for a widely used analytical tool in celestial mechanics, particularly relevant for planet-crossing asteroids and TNOs.
Method Snapshot
theoretical (normal-forms, canonical transformations, Hamiltonians) and numerical methods (e.g., fast Lyapunov indicator)
Background
Hamiltonian perturbation theory, canonical transformations and normal forms, mean-motion resonances, and stability/chaos diagnostics such as Lyapunov indicators
AI Abstract
This paper studies the limits of canonical perturbation theory, specifically the validity of normal form approximations, for mean motion resonances (MMRs) in the planar circular restricted three-body problem in domains containing orbits that cross or closely approach the secondary body's orbit. The authors first discuss analytic issues in defining an MMR normal form on such phase-space domains and then compute the canonical transformations defining the MMR normal forms using numerical methods. The validity limits of the averaging method are investigated by combining phase-portrait representations of the normal form Hamiltonian with analyses of the preservation of quasi-integrals obtained after one perturbative step and with computations of fast Lyapunov indicators regularized for close encounters. Numerical examples use the external 1:2 and 5:6 MMRs with mass ratios representative of the Sun–Jupiter and Sun–Neptune systems, and for the 1:2 MMR the averaging method is effective even for chaotic orbits with eccentricity up to approximately 0.55.
Semi-analytical study (theoretical + numerical); useful for anyone who wants to get semi-analytical results on resonances
— ES