The Eccentric Kozai–Lidov Mechanism for Outer Test Particle

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 90 • , 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.

Inverse Lidov-Kozai resonance for an outer test particle due to an eccentric perturber

“…According to such an study, a natural result observed in these systems is the generation of particles with prograde and retrograde orbits whose ascending node longitude Ω circulates, and particles whose orbit plane flips 1 from prograde to retrograde and back again throughout the evolution with coupled librations associated with Ω. It is worth noting that the comparative analysis carried out by Zanardi et al (2017) shows that the numerical results obtained in their research are supported by the analytical expressions derived by Naoz et al (2017).…”

“…Recently, Naoz et al (2017) analyzed in detail the secular dynamics of an outer test particle evolving under the gravitational influence of an inner and eccentric perturber of mass m 1 orbiting a central star of mass m 0 . If the orbital parameters e 2 , i 2 , ω 2 , and Ω 2 represent the eccentricity, the inclination with respect to the inner orbit, the argument of pericenter, and the ascending node longitude of the outer test particle relative to the inner perturber's periapse, respectively, the equations of motion can be written as partial derivatives of an energy function f in the following way…”

A Hidden Friend for the Galactic Center Black Hole, Sgr A*

“…These large eccentricity values can lead to instability by plunging the star so close to the SMBH binary that it experiences a three-body scattering event. To avoid this we require that general relativistic (GR) precessions be strong enough to suppress quadrupole, and thus octupole, excitations (e.g., Naoz et al 2013bNaoz et al , 2017. We include the octupole level of approximation in the numerical results presented in Appendix A.…”

Orbital dynamics of circumbinary planets

“…That is, the longitude of the ascending node covers a lim-ited range of angles less than 360 • and the nodal precession rate changes sign. In the test particle case, the angular momentum vector of the test particle librates about the binary eccentricity vector (or binary semi-major axis) and undergoes tilt oscillations (Verrier & Evans 2009;Farago & Laskar 2010;Doolin & Blundell 2011;Naoz et al 2017;de Elía et al 2019). The minimum inclination required for libration decreases with increasing binary eccentricity.…”