# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 4 Dynamics of Interior Polar Resonance, the \(2/1\) Case
### Kozai-Lidov Dynamics inside Polar 2/1 Resonance  
In our numerical experiments, as shown in Figure 1, resonant particles have a different destiny, The secular dynamics inside MMR need to be considered to explain the results of our numerical experiments. As we have introduced in Section 2.2, the secular dynamics on the \(e-\omega\) space can be obtained simply by the set \(a=a_{0}\). It has been confirmed that all the dominated resonant equilibriums are around \(\sigma=0^{\circ}\) for the polar \(2/1\) resonance. If we consider the exact resonance case, the resonant angle is always equal to \(\sigma=0^{\circ}\). The averaged Hamiltonian can be calculated by Equation (5). If we consider the effect of libration amplitude \(\sigma_{\rm amp}\) of the resonant angle, the averaged Hamiltonian is given by Equation (6). The phase-space portraits in three different resonant amplitudes (\(\sigma_{\rm amp}=0^{\circ},60^{\circ},150^{\circ}\)) are  
Figure 4: The scatter map of the resonant width of polar \(2/1\) resonance on the \(e-\omega\) plane. The filled dots and pentagons represent their corresponding phase-space portraits that have one or two equilibriums, respectively. The color reflects the position of the libration center. The red means that the dominated libration center is around \(\sigma=0^{\circ}\). In other cases, such as interior \(2/3\) resonance we will introduce later, there is a blue part, which means its dominated resonant equilibrium is around \(\sigma=180^{\circ}\). The alpha of the marker represents the resonant width under the corresponding parameters. The higher the transparency, the narrower the resonant width. depicted in Figure 5. Comparing these three portraits, we know that the effects of libration amplitude can be ignored when the libration amplitude is small. Dynamical evolutions of several representative particles in numerical integrations are also plotted in panel a. The solid black dots represent the initial positions of test particles, and the arrows depict the evolution of unstable particles in numerical integrations. The numerical results correspond very well with the level curves of the Hamiltonian. Therefore, we use the \(e-\omega\) portrait at exact resonance to analyze the Kozai mechanism directly. The most typical dynamics reflected by the \(e-\omega\) portrait under exact \(2/1\) polar resonance are as follows: Firstly, for the high-eccentric region (around \(e>0.5\)), the \(\omega\) is limited to oscillations while the eccentricity will rise to around 1. Therefore, the particles in this high-eccentric region should be unstable. Instead, the variations in eccentricity are maintained within a small range, and the \(\omega\) is in circulation in the lower eccentricity region. Therefore, the particles in the low-eccentric region should have long-term stability. As some assumptions are made to obtain the one d.o.f approximation of this problem, the validity of the \(e-\omega\) portraits above should be checked by the numerical experiments. Comparing Figures 1, 2, and 5, the secular dynamics given by \(e-\omega\) can explain the dynamical evolution of the test particles in the numerical experiments (the existence of some dead particles in the low-eccentric region can be understood by considering the effects of resonant amplitude \(\sigma_{\text{amp}}\)), and thus the \(e-\omega\) portraits can represent the real Kozai-Lidov dynamics inside polar \(2/1\) resonance. We have investigated the secular dynamics inside polar \(2/1\) resonance using numerical integrations and the semi-analytical method by mutual authenticating. As a supplement, we study the purely polar secular dynamics further to discuss the effects of polar \(2/1\) resonance. In our numerical experiments, the maximum libration width of polar \(2/1\) resonance is no more than \(0.05\) au. We repeat the numerical integrations in Section 4.1 with initial semi-major axis \(a=a_{0}+0.05\) while leaving the other parameters unchanged.