# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 5 Dynamics of Exterior Polar Resonance, the \(2/3\) Case  
Now, we focus on the dynamics of exterior polar resonance except \(1/n\) resonance (\(n\) are positive integers), taking \(2/3\) as an example. Since the methods are similar to interior resonance, we mainly present relevant conclusions here.  
The polar \(2/3\) resonant configuration is depicted in Figure 7. The most obvious difference from the \(2/1\) resonance (Figure 4) is the emergence of the dominated resonant equilibriums at \(\sigma=180^{\circ}\) (corresponding to the blue part in Figure 7) for \(2/3\) resonance. For the high-eccentric region where \(e>0.7\), the dominated resonant equilibriums have libration centers at \(\sigma=0^{\circ}\), while in other cases the dominated libration centers are at \(\sigma=180^{\circ}\). Therefore, when we analyze the Kozai-Lidov dynamics inside polar \(2/3\) resonance, both of the \(e-\omega\) portraits at \(\sigma=0^{\circ}\) and \(\sigma=180^{\circ}\) are required. Figure 8 depicts the lifetime of \(5000\) particles in numerical experiments (panel a), and Kozai-Lidov dynamics inside exact polar \(2/3\) resonance with \(\sigma=0^{\circ}\) (panel b) and with \(\sigma=180^{\circ}\) (panel c). As shown in panels b and c of Figure 8, the evolutions of several particles in numerical integrations are depicted and correspond very well with the \(e-\omega\) portraits. Like the numerical results of \(2/1\) resonance, almost all \(5000\) particles are trapped in polar \(2/3\) resonance with the perturber. The unstable particles die due to their eccentricities rising to \(1\). Combining Figures 7 and 8, the secular dynamics inside polar \(2/3\) resonance should be understood as follows: For the region with eccentricity \(e<0.7\) where their dominated resonance is around \(\sigma=180^{\circ}\), their Kozai-Lidov dynamics are described by the corresponding region (\(e<0.7\)) in panel c of Figure 8. The \(\omega\) is circulation, and eccentricity is limited to a small range of oscillations, so particles in this region are stable in numerical experiments (see panel a of Figure 8). Conversely, for the high-eccentric region (\(e>0.7\)) where their dominated resonant equilibriums are at \(\sigma=0^{\circ}\), their secular dynamics should be understood by the corresponding region (\(e>0.7\)) in panel b of Figure 8. For the extreme high-eccentric region, the \(\omega\) oscillates while eccentricity rises to \(1\) and leads to instability. The Kozai-Lidov dynamics inside \(2/3\) resonance described by \(e-\omega\) portraits can explain the numerical phenomena. This further verifies that the phase-space portraits from the semi-analytical method represent the real secular dynamics inside resonance. As shown in Figure 8a, there appear to be several low-eccentricity particles (\(e\leq 0.4\)) with very short lifetimes. The dynamical evolution of these atypical particles is depicted in Figure 9. There are two  
Figure 6: Pure secular dynamics of polar orbits outside the \(2/1\) resonance. (**Left** panel): The lifetime ratio of the \(5000\) particles in numerical integrations. (**Right** panel): Phase-space portrait of pure secular dynamics. The levels of Hamiltonian in Equation (7) are plotted on \(e-\omega\) plane. The thick black dashed lines denote collision curves with the perturber.  
We have also investigated the dynamics of other exterior polar resonances, such as \(2/5\), \(7/9\). They have similar resonant secular dynamics with \(2/3\) resonance. As for the exterior polar \(1/n\) resonances, considering the effect of asymmetric libration (\(\sigma\) librates neither \(0^{\circ}\) nor \(180^{\circ}\)) [88; 89; 90], there will be some new challenges that need further discussion. We will try to investigate the dynamics of polar \(1/n\) resonance in our next work.