Accretion Disks Around Binary Black Holes: A Quasistationary Model

Tidal torques acting on a gaseous accretion disk around a binary black hole can create a gap in the disk near the orbital radius. At late times, when the binary inspiral timescale due to gravitational wave emission becomes shorter than the viscous timescale in the disk, the binary decouples from the disk and eventually merges. Prior to decoupling the balance between tidal and viscous torques drives the disk to a quasistationary equilibrium state, perturbed slightly by small amplitude, spiral density waves emanating from the edges of the gap. We consider a black hole binary with a companion of smaller mass and construct a simple Newtonian model for a geometrically thin, Keplerian disk in the orbital plane of the binary. We solve the disk evolution equations in steady state to determine the quasistationary, (orbit-averaged) surface density profile prior to decoupling. We use our solution, which is analytic up to simple quadratures, to compute the electromagnetic flux and approximate radiation spectrum during this epoch. A single nondimensional parameter Td/Tvis, equal to the ratio of the tidal to viscous torque at the orbital radius, determines the disk structure, including the surface density profile, the extent of the gap, the existence of an inner disk, and the accretion rate. The solution reduces to the Shakura-Sunyaev profile for a stationary accretion disk around a single black hole in the limit of small Td/Tvis. Our solution may be useful for choosing physical parameters and setting up quasistationary disk initial data for detailed numerical simulations that begin prior to decoupling and track the subsequent evolution of a black hole binary-disk system.

💡 Research Summary

The paper presents a simple Newtonian model for a geometrically thin, Keplerian accretion disk that lies in the orbital plane of a binary black hole system with a small mass‑ratio companion. The authors focus on the epoch preceding the so‑called decoupling, when the inspiral timescale driven by gravitational‑wave emission becomes shorter than the viscous diffusion timescale of the disk. In this regime the tidal torque exerted by the secondary black hole and the internal viscous torque balance each other, driving the disk toward a quasistationary, orbit‑averaged equilibrium that is only weakly perturbed by low‑amplitude spiral density waves launched at the gap edges.

Starting from the standard mass‑conservation and angular‑momentum equations, the authors add two torque terms: the viscous torque (T_{\rm vis}) (modeled after the classic (\alpha)‑disk prescription) and the tidal torque (T_{\rm d}) produced by the binary’s gravitational potential. Assuming (\partial\Sigma/\partial t\simeq0) (steady state) they obtain a first‑order differential equation for the surface density (\Sigma(r)) that can be integrated analytically up to a simple quadrature. The solution is divided into two radial domains – inside and outside the gap – and is fully determined by a single dimensionless parameter, \