Effective two-body approach to the hierarchical three-body problem

TL;DR

This paper introduces an effective field theory approach to simplify the relativistic hierarchical three-body problem by modeling the inner binary as an effective spinning point-particle, aiding in the derivation of analytic waveforms.

Contribution

It adapts effective field theory techniques from two-body problems to the three-body context, providing a new method to analyze relativistic hierarchical three-body systems.

Findings

Mapping between binary osculating elements and effective spin established

Reduction of three-body problem to a two-body problem with spin

Foundation for deriving analytic gravitational waveforms

Abstract

The motion of three bodies can be solved perturbatively when a tightly bound inner binary is orbited by a distant perturber, giving rise for example to the well-known Kozai-Lidov oscillations. We propose to study the relativistic hierarchical three-body orbits by adapting the Effective Field Theory techniques used in the two-body problem. This allows us to conveniently treat the inner binary as an effective point-particle, thus reducing the complexity of the three-body problem to a simpler spinning two-body motion. We present in details the mapping between the inner binary osculating elements and the resulting spin of the effective point-particle. Our study builds towards a derivation of three-body analytic waveforms.