Turbulent drag on stellar mass black holes embedded in disks of active galactic nuclei

Abstract

We investigate how AGN disk turbulence affects the orbital dynamics of a stellar-mass black hole (BH) initially located at a migration trap, focusing on the long-term behavior of eccentricity and inclination in the quasi-embedded regime. We develop a semi-analytical framework in which turbulence is modeled as a stochastic velocity field acting through a modified drag force. We integrate the resulting stochastic differential equations both in Cartesian coordinates and in orbital elements using a linearized perturbative approach, and compare these results with full numerical simulations. Eccentricity and inclination evolve toward steady-state Rayleigh distributions, with variances determined by the local disk properties and the ratio of the gas damping rate to the orbital frequency. The analytical predictions agree well with the numerical simulations. We provide closed-form expressions for the variances in both the fast and slow damping regimes. These results are directly applicable to Monte Carlo population models and can serve as physically motivated initial conditions for hydrodynamical simulations. Turbulent forcing prevents full circularization and alignment of BH orbits in AGN disks, even in the presence of strong gas drag. This has important implications for BH merger and binary formation rates, which are sensitive to the residual eccentricity and inclination. Our results highlight the need to account for turbulence-induced stochastic heating when modeling the dynamical evolution of compact objects in AGN environments.

Figures (5)

• Figure 1: Dependency of ostriker1999 the dynamical friction prescription as a function of the mach number $\mathcal{M}$ (Eq. \ref{['eq:prescription']}). The dotted line marks the linear interpolation used to circumvent the mathematical discontinuity at $\mathcal{M} = 1$. The dashed line indicates the linear subsonic limit.
• Figure 2: Radial profiles of the AGN disk employed in this work. From top to bottom: midplane gas density $\rho$, sound speed ${c}_\mathrm{s}$, disk aspect ratio $h/R$, and magnitude of the migration torque $\Gamma$. The migration torque assumes a secondary BH mass of ${m}_\mathrm{BH} = 20 \,{\mathrm{M}_\odot}$. The outer migration trap lies within the star formation region, which begins at $R \simeq 767 \,{\mathrm{au}}$.
• Figure 3: Evolution of the semimajor axis (top), the eccentricity (middle), and the inclination (bottom) for an inclined ($\iota_0 = 0.5^\circ$) and eccentric ($e_0 = 0.1$) BH (${m}_\mathrm{BH} = 20 \,{\mathrm{M}_\odot}$) undergoing dynamical friction in the AGN disk. The thick blue line indicates the evolution without the turbulent velocity field, while each thin red line includes a different realization of the turbulent velocity field. The BH is placed at the migration trap within an AGN disk around a $10^7 \, {\mathrm{M}_\odot}$ SMBH. The dotted black line is the mean evolution of the realizations including turbulence. In the top panel, the dot-dashed lines indicate the mean square change in semimajor axis $\langle \Delta a^2 \rangle$ (Eq. \ref{['eq:deltaa']}). In the middle and bottom panels, the dot-dashed green line is the ratio between the turbulent velocity dispersion $\sigma_\mathrm{turb}$ and the circular velocity $v_\mathrm{circ}$. The vertical dashed lines indicate the first three midplane crossings. The turbulence prevents the full circularization and alignment of the embedded BH.
• Figure 4: Distributions (normalized to one) of the semimajor axis (top), the eccentricity (middle) and the inclination (bottom) for $2.5\times 10^4$ realizations of an embedded BH, evaluated after 20 initial orbital periods. The initial conditions are identical to those in Fig. \ref{['fig:evolturb']}. In the top panel, the blue line marks the final semimajor axis from the nonturbulent simulation, while the green curve shows a Gaussian distribution with standard deviation given by Eq. (\ref{['eq:deltaa']}). In the middle and bottom panels, the dot-dashed green line is the ratio between the turbulent velocity dispersion $\sigma_\mathrm{turb}$ and the circular velocity $v_\mathrm{circ}$, while the green curves indicate Rayleigh distributions with mean value $\sigma_\mathrm{turb} / v_\mathrm{circ}$. The insets show the same distributions in logarithm scale, highlighting the presence of fatter tails with respect to the Rayleigh prediction.
• Figure 5: Ratio between the damping timescale $\tau_*$ (Eq. \ref{['eq:taustar']}) and the orbital period $P$, as a function of the distance from the central SMBH. The estimate assumes a BH with mass $m_{\rm BH}=20 \, \rm M_{\rm \odot}$ based on the AGN disk profile for SMBHs with masses $M_{\rm SMBH}=10^6, 3\times10^7, 7\times10^6,$ and $2\times 10^7 \, \rm M_{\rm \odot}$, with $\alpha = 0.1$. The vertical lines indicate the location of the migration traps.