Analyses on a Relativistic Hierarchical Resonance with the Hamiltonian Approach

TL;DR

This paper investigates a unique relativistic resonance in a triple system involving an EMRI and a massive black hole, using an extended Hamiltonian approach to predict resonance characteristics.

Contribution

The authors extend the Hamiltonian approach to analyze a relativistic hierarchical resonance, deriving a simplified model that aligns well with numerical simulations.

Findings

Analytical predictions match numerical simulations for moderate eccentricities.

The resonance involves relativistic apsidal precession supporting a hierarchical configuration.

The approach provides insights into the resonance capture probability.

Abstract

We study dynamical evolution of a resonant triple system formed by an inner EMRI and an additional outer MBH. The relevant resonant state (\lambda_2-\varpi_1 ~ const) is supported by the relativistic apsidal precession of the inner EMRI, and, unlike standard mean motion resonances, the triple system can have a hierarchical orbital configuration (but different from the Kozai process). In order to analyze this unusual resonant system, we extend the so-called Hamiltonian approach, and derive a mapping from the EMRI-MBH triple system to a simple one-dimensional Hamiltonian. With the derived mapping, we make analytical predictions for characteristic quantities of the resonance, such as the capture probability, and find that they reasonably agree with numerical simulations up to moderate eccentricities.