Accretion disks around binary black holes: A quasistationary model

Yuk Tung Liu and Stuart L. Shapiro

Phys. Rev. D 82, 123011 – Published 28 December 2010

Abstract

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 time scale due to gravitational wave emission becomes shorter than the viscous time scale 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 the 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 $\tilde{g}$ , 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 $\tilde{g}$ . 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.

Received 26 October 2010

DOI:https://doi.org/10.1103/PhysRevD.82.123011

Authors & Affiliations

Yuk Tung Liu and Stuart L. Shapiro^*

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

^*Also at the Department of Astronomy and NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

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Issue

Vol. 82, Iss. 12 — 15 December 2010

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Images

Figure 1
Accretion rate $\dot{m}$ as a function of the tidal torque parameter $\tilde{g}$ for various values of the viscosity power-law index $- 1 \leq n \leq 2$ . Results are shown for binary orbital radius $a = 100 M$ , disk thickness $h / r = 0.1$ in the vicinity of the disk edge at $r \approx a$ [see Eq. (12)], disk outer radius $r_{out} = 10^{5} M$ and disk ISCO at $r_{isco} = 6 M$ . Solid (black) line represents the case for $n = - 1$ , dotted (red) line for $n = 0$ , dashed (blue) line for $n = 1 / 2$ , long-dashed (magenta) line for $n = 1$ , dot-dashed (green) line for $n = 3 / 2$ and dot-long-dashed (cyan) line for $n = 2$ .Reuse & Permissions
Figure 2
Accretion rate $\dot{m}$ as a function of the tidal torque parameter $\tilde{g}$ for various values of disk thickness $h / r$ . Results are shown for the viscosity power-law index $n = 1 / 2$ (top) and $n = 3 / 2$ (bottom), with binary orbital radius $a = 100 M$ , disk outer radius $r_{out} = 10^{5} M$ and disk ISCO at $r_{isco} = 6 M$ . Solid (black) line represents the case for $h / r = 0.01$ , dotted (red) line for $h / r = 0.05$ , dashed (blue) line for $h / r = 0.1$ , and long-dashed (magenta) line for $h / r = 0.15$ .Reuse & Permissions
Figure 3
The surface density profile $Σ$ plotted for different values of the tidal torque parameter $\tilde{g}$ . All results assume $h / r = 0.1$ , $n = 1 / 2$ , $a = 100 M$ , $r_{out} = 10^{5} M$ and $r_{isco} = 6 M$ . Top: $\tilde{g} = 5000$ (black solid line), $\tilde{g} = 500$ (red dotted line), $\tilde{g} = 50$ (blue short-dash line) and $\tilde{g} = 15$ (magenta long-dash line). Middle: $\tilde{g} = 5000$ (black solid), $\tilde{g} = 15$ (magenta long-dash line), $\tilde{g} = 5$ (green dot-short-dash line), $\tilde{g} = 2.5$ (cyan dot-long-dash line), and $\tilde{g} = 0$ (orange long-dash-short-dash line). Bottom: Same as the middle figure but plotted in linear scale in $r / M$ .Reuse & Permissions
Figure 4
Surface density of disks around a binary black hole just before decoupling. Shown here are cases for the binary mass ratios (a) $q = 0$ (black solid line), (b) $q = 10^{- 3}$ (red dotted line), (c) $q = 2 \times 10^{- 3}$ (blue short-dash line), (d) $q = 3 \times 10^{- 3}$ (magenta long-dashed line), (e) $q = 5 \times 10^{- 3}$ (green dot-short-dash line), (f) $q = 10^{- 2}$ (cyan dot-long-dash line), and (g) $q = 10^{- 1}$ (orange long-dash-short-dash line).Reuse & Permissions
Figure 5
Ratio of tidal heating to viscous heating $D_{tid} / D_{vis}$ as a function of radius for disks around a binary black hole just before decoupling for binary ratios (b) $q = 10^{- 3}$ (red dotted line), (c) $q = 2 \times 10^{- 3}$ (blue short-dash line), (d) $q = 3 \times 10^{- 3}$ (magenta long-dashed line), (e) $q = 5 \times 10^{- 3}$ (green dot-short-dash line), (f) $q = 10^{- 2}$ (cyan dot-long-dash line), and (g) $q = 10^{- 1}$ (orange long-dash-short-dash line).Reuse & Permissions
Figure 6
Electromagnetic spectrum of disks around a binary black hole at redshift $z$ just before decoupling. Shown here are cases for the binary mass ratios (a) $q = 0$ (black solid line), (b) $q = 10^{- 3}$ (red dotted line), (c) $q = 2 \times 10^{- 3}$ (blue short-dash line), (d) $q = 3 \times 10^{- 3}$ (magenta long-dashed line), (e) $q = 5 \times 10^{- 3}$ (green dot-short-dash line), (f) $q = 10^{- 2}$ (cyan dot-long-dash line), and (g) $q = 10^{- 1}$ (orange long-dash-short-dash line). Note that lines (c) and (d) are barely distinguishable.Reuse & Permissions

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