Dynamics of Polar Resonances and Their Effects on Kozai–Lidov Mechanism

Abstract

The research on highly inclined mean motion resonances (MMRs), even retrograde resonances, has drawn more attention in recent years. However, the dynamics of polar resonance with inclination i≈90∘ have received much less attention. This paper systematically studies the dynamics of polar resonance and their effects on the Kozai–Lidov mechanism in the circular restricted three-body problem (CRTBP). The maps of dynamics are obtained through the numerical method and semi-analytical method, by mutual authenticating. We investigate the secular dynamics inside polar resonance. The phase-space portraits on the e−ω plane are plotted under exact polar resonance and considering libration amplitude of critical angle σ. Simultaneously, we investigate the evolution of 5000 particles in polar resonance by numerical integrations. We confirm that the e−ω portraits can entirely explain the results of numerical experiments, which demonstrate that the phase-space portraits on the e−ω plane obtained through the semi-analytical method can represent the real Kozai–Lidov dynamics inside polar resonance. The resonant secular dynamical maps can provide meaningful guidance for predicting the long-term evolution of polar resonant particles. As a supplement, in the polar 2/1 case, we analyze the pure secular dynamics outside resonance, and confirm that the effect of polar resonance on secular dynamics is pronounced and cannot be ignored. Our work is a meaningful supplement to the general inclined cases and can help us understand the evolution of asteroids in polar resonance with the planet.