The hierarchical three-body problem at $\mathcal{O} (G^2)$

TL;DR

This paper develops a new analytical approach using scattering amplitudes and effective field theory to derive the three-body potential at second order in gravitational constant for hierarchical triples, applicable to astrophysical systems.

Contribution

It introduces a systematic expansion method for the three-body problem at $ ext{O}(G^2)$, providing new analytic results in position space for arbitrary masses and velocity configurations.

Findings

Derived new analytic three-body potentials at $ ext{O}(G^2)$

Expanded results to arbitrary mass configurations in the rest frame

Provided exact velocity results applicable to astrophysical systems

Abstract

Employing techniques from scattering amplitudes and effective field theory, we model the dynamics of hierarchical triples, which are three-body systems composed of two bodies separated by a distance $r$ and a third body a distance $\rho$ away, with $r \ll \rho$. We apply the method of regions to systematically expand in the small ratio $r/\rho$ and illustrate this approach for evaluating Fourier transform integrals, which have been the bottleneck for deriving complete results in position space. In the limit where the distant third body is much heavier than the other two, we derive new analytic results in position space for the three-body conservative potential at $\mathcal{O}(G^2)$ and at leading and next-to-leading order in $r/\rho$. We also derive new results for arbitrary masses in the rest frame of the distant particle. Our results are exact in velocity, and can be used in analyses involving both bound and unbound hierarchical triples in astrophysical systems.