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

Abstract

Many binary systems of interest for gravitational-wave astronomy are orbited by a third distant body, which can considerably alter their relativistic dynamics. Precision computations are needed to understand the interplay between relativistic corrections and three-body interactions. We use an effective field theory approach to derive the effective action describing the long time-scale dynamics of hierarchical three-body systems up to 1PN quadrupole order. At this level of approximation, computations are complicated by the backreaction of small oscillations on orbital time-scales as well as deviations from the adiabatic approximation. We address these difficulties by eliminating the fast modes through the method of near-identity transformations. This allows us to compute for the first time the complete expression of the 1PN quadrupole cross-terms in generic configurations of three-body systems. We numerically integrate the resulting equations of motion and show that 1PN quadrupole terms can affect the long term dynamics of relativistic three-body systems.

Precession resonances in hierarchical triple systems

“…a 3 , e 3 , ω 3 , ι 3 , Ω 3 , u 3 ) consisting in the semimajor axis, eccentricity, argument of perihelion, inclination, longitude of ascending node, and true anomaly of the orbit. The outer orbit is built by replacing the inner binary with an effective point-particle located at its center-of-mass, as explained in [32]. The hierarchical assumption only assumes a a 3 , so that these elements evolve solwly in time due to the interactions between the two orbits We will always be interested in dynamics on timescales greater than the period of the inner binary, so we will average all quantites over one orbit of the inner binary; however, it will be crucial not to average over one outer orbit in order to account for the effect of precession resonances.…”

“…The hierarchical assumption only assumes a a 3 , so that these elements evolve solwly in time due to the interactions between the two orbits We will always be interested in dynamics on timescales greater than the period of the inner binary, so we will average all quantites over one orbit of the inner binary; however, it will be crucial not to average over one outer orbit in order to account for the effect of precession resonances. The Hamiltonian of the three-body system can then be written as [25,32]…”

“…The first two terms in this Hamiltonian corresponds to the Newtonian energies of the two orbits. Since a is conserved by virtue of averaging the Hamiltonian over one inner orbit [25,32], we could as well drop the first term. However, a 3 may be free to vary.…”

“…Before moving on, let us emphasize that in the Hamiltonian 1, we have neglected relativistic effects on the outer orbit, as well as the relativistic coupling between the angular momentums of the two orbits. In the power-counting rules presented in [32], this corresponds to an accuracy v 2 and ε 2 while dropping terms scaling as v 2 ε and higher, where v 2 = G N m/a and ε = a/a 3 . As we prove in Appendix B, this accuracy is sufficient to quantitatively describe precession resonances.…”

Detecting gravitational lensing in hierarchical triples in galactic nuclei with space-borne gravitational-wave observatories

“…For more compact BBHs with a separation of ∼10 −3 AU, Ref. [41] showed that the (post-Newtonian) de Sitter-like precession of the BBH's orbital plane [42][43][44] is the more critical correction to the waveform and can be detectable by space-borne GW detectors out to a cosmological distance of ∼1 Gpc. Combining the de Sitterlike precession and the Doppler shift, Ref.…”