The parametric oscillator model for the case of resonant argument circulations

TL;DR

The paper addresses the planar restricted three-body problem near resonance with a circulating resonant argument and small radial perturbations $x$, proposing a linearization that reduces the dynamics to a $\mathrm{Mathieu}$ equation when a single dominant frequency is present. This yields analytic insight into resonance locations and stability boundaries, with the key contribution being the identification of two characteristic perturbation frequencies, a parametric instability zone, and an offset of the resonance center under strong forcing within a circulation regime. The authors derive the averaged resonant problem $\ddot x + \omega_0^2(1 - h_j\cos\nu t)x = f(t)$ and its asymptotic solution, showing how amplitude scales with the perturbing mass and distance to resonance, and validate the framework via numerical integrations of the Sun–Jupiter–Asteroid system near the 3:2 resonance. The results provide a qualitative, analytically tractable model for circulation-type resonant dynamics, offering estimates of resonance positions and boundaries that complement nonlinear libration analyses and aid interpretation of asteroid belt and exoplanetary system dynamics.

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.

Figures (5)

• Figure 1: Exterior resonance geometry(left) and interior resonance geometry (right)
• Figure 2: Numerical solution of the Mathieu equation (red) and its approximation (blue) (in arbitrary units)
• Figure 3: The behavior of the frequency of the approximated solution of the Mathieu equation in dependence of distance to resonance. Here the values of frequency $\nu$ (see equation(\ref{['E31']})) are on the vertical axis and the distance to resonance (in frequency) is on the horizontal axis.
• Figure 4: The dependence of the amplitude of the numerical solution of the Mathieu equation on the mass of the perturbing planet (at a distance to resonance $\delta_n=0.2n_{res}$).
• Figure 5: The frequency of the asymptotic solution of the equation (\ref{['E26']}) in arbitrary units at different values of $h_j=0.01$ (black), $h_j=0.02$ (red), $h_j=0.05$ (blue). The vertical lines indicate the positions of the center of the unstable zones for each case. (see text for a details)