# The Eccentric Kozai-Lidov mechanism for Outer Test Particle

## TL;DR

This paper investigates the complex dynamics of a hierarchical three-body system with a test particle, revealing chaotic behavior and orbital flips at the octupole level, with implications for debris disk evolution and retrograde object formation.

## Contribution

It extends the secular approximation to the octupole level for a test particle, analyzing chaotic dynamics and long-term evolution including general relativity effects.

## Key findings

- Chaotic orbital flips occur at the octupole level.
- Long-term evolution can produce retrograde objects.
- Debris disks can become puffed up over Gyr timescales.

## Abstract

The secular approximation of the hierarchical three body systems has been proven to be very useful in addressing many astrophysical systems, from planets, stars to black holes. In such a system two objects are on a tight orbit, and the tertiary is on a much wider orbit. Here we study the dynamics of a system by taking the tertiary mass to zero and solve the hierarchical three body system up to the octupole level of approximation. We find a rich dynamics that the outer orbit undergoes due to gravitational perturbations from the inner binary. The nominal result of the precession of the nodes is mostly limited for the lowest order of approximation, however, when the octupole-level of approximation is introduced the system becomes chaotic, as expected, and the tertiary oscillates below and above 90deg, similarly to the non-test particle flip behavior (e.g., Naoz 2016). We provide the Hamiltonian of the system and investigate the dynamics of the system from the quadrupole to the octupole level of approximations. We also analyze the chaotic and quasi-periodic orbital evolution by studying the surfaces of sections. Furthermore, including general relativity, we show case the long term evolution of individual debris disk particles under the influence of a far away interior eccentric planet. We show that this dynamics can naturally result in retrograde objects and a puffy disk after a long timescale evolution (few Gyr) for initially aligned configuration.