# Dynamics of Polar Resonances and Their Effects on Kozai-Lidov Mechanism
## 1 Introduction
[58] developed a new resonant secular model to describe the dynamics of TNOs in MMR with Neptune. [PERSON] et al. [59] applied this model to study the long-term dynamics of TNOs in exterior high-order MMRs and whose orbits are entirely beyond Neptune. They believed that the resonant order has little impact on the structure of resonant secular phase portraits. Further, [PERSON] and [PERSON] [60] studied the long-term dynamics of the known TNOs in MMR with Neptune, and they confirmed that MMRs completely alter the secular dynamics of a subset of objects. [PERSON] and [PERSON] [61] studied the secular dynamics facilitated by resonant interactions using semi-analytical theory. Via double numerical averaging [PERSON] [62] explored the long-term dynamics of \"jumping\" Trojans [63] in the restricted planar elliptical model. Further, we investigated how the retrograde resonant dynamics affects Kozai-Lidov cycle [64]. [PERSON] and [PERSON] [65] studied the Kozai dynamics inside inclined MMRs through the 3D \((e,\omega,\phi)\) phase space. Recently, [PERSON] and [PERSON] [66] introduced an analytical model to study the long-term dynamics in MMRs in the 3D motion. However, how the Kozai-Lidov mechanism interacts with the minor body near-polar orbit (\(i\approx 90\)) has received less attention. Our curiosity about Kozai-Lidov dynamics inside polar resonance has been sparked. Highly inclined objects are not rare in the Solar System and exosolar systems. To date, 187 minor bodies with \(60^{\circ}<i<120^{\circ}\) have been discovered in the Solar System. [PERSON] and [PERSON] [67] proposed that the hypothetical Planet Nine may produce the high inclined TNOs. Limiting the inclination range down to \(80^{\circ}<i<90^{\circ}\), there are still 29 minor bodies with polar orbits (from JPL Small-Body Database ([[https://ssd.jpl.nasa.gov](https://ssd.jpl.nasa.gov)]([https://ssd.jpl.nasa.gov](https://ssd.jpl.nasa.gov))), retrieved on 29 September 2020). [PERSON] and [PERSON] [68] developed a new disturbing function for objects with arbitrary inclination. [PERSON] and [PERSON] [69] reported the first near-polar TNO, nicknamed Niku, which is trapped in 7/9 polar resonance with Neptune. [PERSON] et al. [25] identified several retrograde asteroids trapped in or that will be captured in polar resonances (\(90^{\circ}<i<110^{\circ}\)). [PERSON] and [PERSON] [70] provided a series expansion algorithm of the disturbing function in the vicinity of polar orbit, which demonstrated that the force amplitude of a \(k^{\prime}/k\) resonance does not depend on its resonant order. Here, we focus on the resonant secular dynamics of a polar orbit with \(i=90^{\circ}\), and apply them to explain numerical experiments' results. This article is structured as follows: In Section 2, we adapt the semi-analytical method described in [PERSON] [13] and [PERSON] et al. [21] to the case of polar mean motion resonances. We also deduce proper Hamiltonian canonical transformations to tackle the secular dynamics inside polar resonances. Section 3 gives a full description of the numerical model and experiments carried out in this study. In Sections 4 and 5, we study the interior and exterior polar resonant dynamics and their effects on the Kozai-Lidov mechanism, respectively. The last section contains our conclusions.