Analytical and Numerical Methods for Circumbinary Disk Dynamics -- II: Inclined Disks

TL;DR

The paper analyzes circumbinary disk dynamics around inclined black hole binaries using 2D Newtonian hydrodynamics, exploring how mass ratio $q$ and inclination $\iota$ shape gap formation. It develops a harmonic decomposition of the inclined binary potential and couples epicyclic perturbations with viscous torques, comparing an orbital-instability (Lyapunov) view with a resonant-torquing picture. Numerically, two regimes emerge: a stable, quasi-steady sector and an unstable sector near $\iota\approx45^{\circ}$, with density maxima that shift from rings to spirals as $\iota$ grows; the gap size correlates best with the orbital-instability prediction across parameter space. The results suggest that disk evolution and potential observational signatures (electromagnetic and gravitational-wave) are governed more by orbital instabilities than by resonant torques in the explored viscosity range, though vertical 3D effects may modify the picture in some regimes.

Abstract

(Abridged) To gain insight into the dynamical influence of a supermassive black hole binary on a circumbinary accretion disk, we investigate the binary and viscous torque densities throughout such a disk, with emphasis on the final density distribution, particularly the size and stability of the central gap between the binary and the inner edge of the disk. We limit ourselves to the simplified case of a massless viscous thin accretion disk under the influence of the gravitational potential from a binary system whose orbital plane is inclined relative to the disk. The orbital plane could be inclined if it is not coeval with the disk, or the black holes have spin angular momentum misaligned with respect to the disk's orbital angular momentum, so that the binary can precess to an inclined orientation. We employ 2D Newtonian hydrodynamics simulations to examine the influence of two model parameters: the mass ratio of the binary and the inclination angle between the binary and the disk. We investigate their impact on the density and torque distribution. In our analytical approach, we consider the stability of epicycles induced by the perturbative effect of the asymmetric inclined binary gravitational potential on Keplerian circular orbits. Through our simulations, we observe that certain configurations never attain a quasi-steady state, where the density profile averaged over many orbits stabilizes. This instability occurs when the inclination is close to 45 degrees. Furthermore, we identify configurations where there is never a persistent balance between the dynamical and viscous torque densities, as well as cases where the location of this balance oscillates or exhibits other time-dependent behavior over viscous timescales. These findings have implications for understanding both the expected gravitational-wave signal and electromagnetic counterparts from supermassive black hole binaries.

Figures (16)

• Figure 1: The configuration we are considering, where the orbital plane of the binary is inclined by an angle $\iota$ relative to the plane of the disk.
• Figure 2: The plots show the comparison of our standard approach where we intentionally, due to computational cost, excluded the inner region $r < a$ from our numerical domain (red curves), and a test run without any limits of this kind (green curves). The left plot shows the instantaneous (solid lines) and averaged mass density distributions (dotted lines) for both approaches. The right plot shows that the location where the dynamical (solid lines) and viscous torque densities (dotted lines) intersect does not depend on the choice for the inner boundary. Both plots present a stage of the evolution at $t=6000 \delta$.
• Figure 3: The averaged mass density as a function of radius is shown for two arbitrarily chosen cases: $q=1/1$, $\iota=30^\circ$ (upper row) and $q=2/3$, $\iota=75^\circ$ (lower row). In the left panels, the colored lines represent times prior to reaching the quasi-steady state. The final color curve (black) represents the density distribution at the quasi-steady state, specifically at $t=6000\delta$, while the gray lines with different patterns correspond to later stages. The left panels are zoomed in on the regions closer to the peak density, while right panels present the same data over a wider range of radii to show the evolution of the averaged density distribution throughout the disk.
• Figure 4: Averaged density distribution in the central part of the system for the cases with $q=1$ and $q=1/4$ together with the initial data and single-mass case fore reference.
• Figure 5: Left column: An averaged density distribution in the central part of the system for cases with $q = 1, 1/4, 1/10$. Right column: Instantaneous density (averaged over the azimuthal angle) for the same cases.