Abstract
For coplanar or nearly coplanar orbits, the widths in semimajor axes of mean motion resonances (MMRs) with planets have been analytically characterized as functions of eccentricity, but no satisfactory analytical models exist to consider the resonance widths for the inclined case. The present work intends to develop an analytical model with the aim of producing 3D phase structures of MMRs. For a p:q resonance at an arbitrary inclination, we define the characteristic resonance argument as σ = pλp − qλ + (q − p)Ω, whose amplitude in the disturbing function is a good indicator of representing the total resonance strength. We adopt the characteristic argument as the resonant angle and formulate a multi-harmonics pendulum model, governing the evolution of σ, under the assumption that the mean eccentricity, inclination, and argument of pericentre remain constant during the resonant time-scale. The pendulum model is then applied to the 3:1 and 2:1 resonances with Jupiter, and the phase space structures, dynamical separatrices, and resonance widths are identified as functions of the mean eccentricity, inclination, and argument of pericentre. For the 3:1 resonance, the analytical separatrices are compared to the corresponding numerical structures, and an excellent agreement between them is observed, indicating that the multi-harmonics pendulum model with σ as the resonant angle is applicable in producing 3D phase structures of MMRs.
Published Version
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