Gravitational Recoil of Inspiralling Black-Hole Binaries to Second Post-Newtonian Order
Luc Blanchet, Moh'd S. S. Qusailah, Clifford M. Will
TL;DR
This work computes the gravitational recoil of non-spinning black-hole binaries to second post-Newtonian order, including the first appearance of the 1.5PN tail term and the subsequent 2PN correction in the linear momentum flux. By integrating the 2PN flux along an adiabatic inspiral up to the ISCO and modeling the plunge with Schwarzschild geodesics, the authors show that the plunge dominates the total kick, with inspiral contributing a modest 22 km/s maximum. They find a total recoil of about 100 ± 20 km/s for a mass ratio m2/m1 = 1/8 and around 250 ± 50 km/s near the maximum around a mass ratio of ~0.38, with a small-m ratio limit V/c ≈ 0.043 (m2/m1)^2; a simple phenomenological fit V/c ≈ 0.043 η^2 sqrt(1-4η) (1+η/4) captures the 2PN curve well. The results are compatible with prior Favata 2004 estimates and align with Lazarus-style perturbative/numerical results, providing tighter constraints on astrophysical recoil expectations and emphasizing the plunge phase in setting final kicks.
Abstract
The loss of linear momentum by gravitational radiation and the resulting gravitational recoil of black-hole binary systems may play an important role in the growth of massive black holes in early galaxies. We calculate the gravitational recoil of non-spinning black-hole binaries at the second post-Newtonian order (2PN) beyond the dominant effect, obtaining, for the first time, the 1.5PN correction term due to tails of waves and the next 2PN term. We find that the maximum value of the net recoil experienced by the binary due to the inspiral phase up to the innermost stable circular orbit (ISCO) is of the order of 22 km/s. We then estimate the kick velocity accumulated during the plunge from the ISCO up to the horizon by integrating the momentum flux using the 2PN formula along a plunge geodesic of the Schwarzschild metric. We find that the contribution of the plunge dominates over that of the inspiral. For a mass ratio m_2/m_1=1/8, we estimate a total recoil velocity (due to both adiabatic and plunge phases) of 100 +/- 20 km/s. For a ratio 0.38, the recoil is maximum and we estimate it to be 250 +/- 50 km/s. In the limit of small mass ratio, we estimate V/c to be approximately 0.043 (1 +/- 20%)(m_2/m_1)^2. Our estimates are consistent with, but span a substantially narrower range than, those of Favata et al. (2004).