Hydrodynamic simulations of black hole evolution in AGN discs II: inclination damping for partially embedded satellites

TL;DR

This work demonstrates that BHs with small inclinations in AGN discs experience rapid, gas-gravity–driven inclination damping, with a characteristic timescale $ au_d \,\approx\ 2.43\,P_{\rm SMBH}$ in the fiducial regime. Across 27 simulations spanning nine disc environments, damping is strongest in high ambient Hill-mass environments and exhibits an exponential decay for $i<3H_0R_0^{-1}$, while damping weakens for larger inclinations. A simple power-law fit links fractional inclination changes per disc crossing to the ambient Hill mass, and a knee near $ ilde{i}_c$ separates regimes of efficient damping from weaker damping. Comparisons with Hill-limited BHL accretion reproduce the damping magnitude well, whereas gas dynamical friction overestimates damping by an order of magnitude, especially at low inclinations. The results imply that most gas-captured BH binaries form from components with negligible inclination to the disc, and provide practical analytic fits to predict damping across a broad range of disc environments, advancing our understanding of the AGN channel for BH mergers.

Abstract

We investigate the evolution of black holes on orbits with small inclinations ($i < 2^\circ$) to the gaseous discs of active galactic nuclei. We perform 3D adiabatic hydrodynamic simulations within a shearing frame, studying the damping of inclination by black hole-gas gravitation. We find that for objects with $i<3H_0R_0^{-1}$, where $H_0R_0^{-1}$ is the disc aspect ratio, the inclination lost per midplane crossing is proportional to the inclination preceding the crossing, resulting in a net exponential decay in inclination. For objects with $i>3H_0R_0^{-1}$, damping efficiency decreases for higher inclinations. We consider a variety of different AGN environments, finding that damping is stronger for systems with a higher ambient Hill mass: the initial gas mass within the BH sphere-of-influence. We provide a fitting formula for the inclination changes as a function of Hill mass. We find reasonable agreement between the damping driven by gas gravity in the simulations and the damping driven by accretion under a Hill-limited Bondi-Hoyle-Lyttleton prescription. We find that gas dynamical friction consistently overestimates the strength of damping, especially for lower inclination systems, by at least an order of magnitude. For regions in the AGN disc where coplanar binary black hole formation by gas dissipation is efficient, we find that the simulated damping timescales are especially short with $τ_d < 10P_\mathrm{SMBH}$. We conclude that as the timescales for inclination damping are shorter than the expected interaction time between isolated black holes, the vast majority of binaries formed from gas capture should form from components with negligible inclination to the AGN disc.

Figures (10)

• Figure 1: Logarithmic density plots for the fiducial system, slicing the domain in the $y-z$ plane at various points in time. The BH starts well above the midplane at $z_{\mathrm{BH},0} = 5H_0$ in a very low density region. As it falls towards the midplane it shocks the disc gas, forming a bow shock. The BH deforms the disc at it punches through, driving hot winds that disrupt the bow shock. Each disc crossing reduces the BH's inclination as, for the majority of the transit, the BH velocity is anti-parallel to the acceleration by gas gravity, see Figures \ref{['fig:spec']} & \ref{['fig:accel_map']} for a detailed energetic chronology. By $t=5P_\mathrm{SMBH}$, the BH is well embedded within the disc and the remaining inclination ($i \ll H_0 R_0^{-1}$) has little effect on the gas morphology. Grey windows in the left of the lower panel indicate the transition from no gas gravity to limited feedback to full feedback (see text).
• Figure 2: Evolution of fiducial BH trajectory, with panels for the BH position $z_\mathrm{BH}$, velocity $\dot{z}_\mathrm{BH}$, vertical acceleration due to gas gravity $a_{z,\mathrm{gas}}$, dissipation due to gas $\epsilon_{z,\mathrm{gas}}$ and inclination $i$ as functions of time. The gas acceleration lags behind the velocity, resulting in brief periods of energy/inclination injection, but a net dissipative effect. The decrease in inclination is well modelled by exponential decay with a characteristic timescale $\tau_d = 2.43\pm0.04 P_\mathrm{SMBH}$. The grey zones on the left side of the plot indicates the epoch where gas feedback is turned off.
• Figure 3: Inclination lost per disc crossing compared to the inclination preceding the crossing for the fiducial model. We observe a strong linear correlation, indicating that the secular inclination evolution is well modelled by a simple exponential decay. This relationship is substantiated by the other simulations in the suite, see Figure \ref{['fig:all_deltas']}.
• Figure 4: Spatial maps for regions of gas damping (blue) and exciting (red) inclination during three epochs of the fiducial BHs evolution, with the circle showing the size of the Hill sphere. In the lower panels, the evolution of: BH position $z_\mathrm{BH}$ and velocity $\dot{z}_\mathrm{BH}$, acceleration by gas gravity $a_{z,\mathrm{gas}}$ and dissipation by gas gravity $\epsilon_{z,\mathrm{gas}}$. At $t=t_A$, the BH experiences maximal damping shortly after punching through the disc; the BH is travelling rapidly and experiences a strong attraction to the disc gas that it has entrained alongside it: hence the dissipation is strong. At $t=t_B$, the BH still experiences strong gas attraction, but as it approaches its apex it slows, resulting in decreased dissipation. At $t=t_C$, the BH experiences a brief period of inclination excitation as it falls back towards the disc. However, as the disc has relaxed back to the midplane (ahead of the BH), the gravitational attraction on the BH has weakened and the energy injection is minor.
• Figure 5: Evolution of BH position $z_\mathrm{BH}$ over time for all 27 simulations in the suite, separated by AGN environment: by row, Eddington fraction $l_E$ and by column, distance from SMBH $R_0$. Each environment is labelled by the initial inclinations of the 3 simulations run within it $i_0\in \left[1,2,5\right]H_0R_0^{-1}$, by the ambient Hill mass $m_\mathrm{H,0}$ and the damping timescale $\tau_d$ averaged across the 3 simulations. The damping timescales shows anti-correlation with the ambient Hill mass, see Sections \ref{['sec:inc_changes']} for further analysis.