The parametric oscillator model for the case of resonant argument circulations

Abstract

The goal of this paper is to obtain an approximate solution of the restricted three-body problem in the case of small perturbations in the vicinity of, but not in exact resonance. In this paper, we study the restricted threebody problem known as planetary type (i.e., when the eccentricity of the test particle is small). A method of linearizing the equation of motion close to (but not in) resonance is proposed under the assumption of small perturbations. In other words, we study orbits when the resonant argument circles the resonance. In the practically interesting case of resonant perturbations we can restrict our study to a perturbation with a single frequency with the largest amplitude, and reduce the problem to the Mathieu equation. The model qualitatively describes the behavior of the perturbation in the vicinity of the resonance. It can be used to estimate the exact position of the resonance and the boundaries between neighboring resonances.