Analyses on a Relativistic Hierarchical Resonance with the Hamiltonian Approach

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.