# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 4 Dynamics of Interior Polar Resonance, the \(2/1\) Case
### Phase-Space Structure of Polar \(2/1\) Resonance  
The resonant dynamics are the basis for our research on the secular mechanism inside MMRs. However, the polar resonant configurations have remained unclear until now. Let us first investigate the essential dynamical maps of polar MMRs, including the libration centers and resonant width. In Section 2.1, we introduce a set of canonical action-angles and explain that \(a-\sigma\) phase-space portraits can understand the resonant dynamics well. Given the particular set of \((e,\omega)\), the averaged Hamiltonian can be calculated in the \(a-\sigma\) plane by Equation (4). Figure 3 shows three examples of phase-space portraits of polar \(2/1\) resonance with a different set of \((e,\omega)\), including the low- (panel a), medium- (panel b), and high-eccentric case (panel c).  
Figure 1: The lifetime of 5000 particles in \(2/1\) polar resonance on \(e-\omega\) plane. The color bar represents the ratio of the lifetime of the particle and whole integration time span.  
Figure 3: Phase-space portraits of \(2/1\) polar resonance in \(a-\sigma\) plane under three different sets of parameters \((e,\omega)\) (plane **(a)**: \(e=0.3\), \(\omega=90^{\circ}\); plane **(b)**: \(e=0.5\), \(\omega=45^{\circ}\); plane **(c)**: \(e=0.95\), \(\omega=0^{\circ}\)). The variances of the color and contours represent the level of the averaged Hamiltonian in Equation (4). \(\Delta a\) is the resonant width of dominated libration center.  
Figure 2: Dynamical evolution of the representatives of the surviving (plane **(a)**) and dead (plane **(b)**) particles. From top to bottom panel: semi-major axis \(a\), eccentricity \(e\), inclination \(i\), argument of pericenter \(\omega\), \(2/1\) critical angle \(\sigma\), respectively.  
As shown in Figure 3, the positions corresponding to the minimum Hamiltonian is the resonant equilibrium. The resonant width \(\Delta a\) is defined by the librational amplitude around the resonant equilibrium. Then, we can obtain the resonant equilibriums and width of the resonance \(\Delta a\). Here, we define the dominated resonant equilibrium as the center with the larger resonant width. For the cases which have more than one equilibrium, such as panels a and c in Figure 3, the resonant width \(\Delta a\) refers to the larger one. Therefore, all of the dominated libration centers of four cases depicted in Figure 3 are \(\sigma=0^{\circ}\). For each set of the grid on the \(e-\omega\) plane, we plot the phase-space portrait and disturbing function as in Figure 3. Then, we can obtain the dominated libration centers and resonant width \(\Delta a\) for each set of \((e,\omega)\). Figure 4 systematically presents the polar \(2/1\) resonant dynamical map, including resonant libration center and resonant width. The alpha of markers represents the resonant width, and the color indicates its dominated libration center. The dominated libration centers of polar \(2/1\) resonance are always around \(\sigma=0^{\circ}\) for all the sets of \((e,\omega)\), which provides an essential guide for our further study of the Kozai mechanism in polar \(2/1\) resonance. This is the first time that the dynamics of polar \(2/1\) resonance have been systematically studied.