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

TL;DR

This paper models how turbulence in active galactic nucleus disks influences the orbital eccentricity and inclination of embedded stellar-mass black holes, revealing that turbulence prevents full orbit circularization and alignment, impacting merger rates.

Contribution

It introduces a semi-analytical stochastic framework for orbital evolution of black holes in turbulent AGN disks, validated by numerical simulations, and provides closed-form expressions for variances in different damping regimes.

Findings

Turbulence leads to steady-state Rayleigh distributions of eccentricity and inclination.

Analytical predictions match well with numerical simulations.

Turbulence prevents complete orbit circularization and alignment in AGN disks.

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.