Accretion rates of stellar-mass compact objects embedded in AGN discs

TL;DR

This work replaces the conventional Bondi/BHL treatment of CO accretion in AGN discs with a physically motivated viscous-disc framework that accounts for angular momentum from disc differential rotation. By separating the CO–gas relative motion into a gradient-driven component and a bulk component, the authors derive a unified viscous accretion rate Mdot_vis and show that the actual CO accretion rate is Mdot_CO = min{Mdot_vis, Mdot_BHL}, with Mdot_vis scaling as α ξ (r_H/r_BHL)^3 Mdot_BHL. The framework yields subsonic and supersonic regimes, a lower limit on the CO-to-SMBH mass ratio q for viscosity dominance, and Toomre Q criteria for stability, and it is illustrated via slim-disc and eccentric-orbit cases. The approach improves upon Bondi/BHL prescriptions and provides a robust basis for studying CO growth, migration, and EM/GW signatures in the AGN disc channel, including potential feedback and gap-opening considerations.

Abstract

Stellar-mass compact objects (COs) embedded in active galactic nucleus (AGN) discs are commonly assumed to accrete via Bondi or Bondi-Hoyle-Lyttleton (BHL) prescriptions, neglecting gas angular momentum. We show that differential rotation in AGN discs can impart non-negligible angular momentum, in which case accretion proceeds through a viscous disc rather than Bondi/BHL flow. Our model provides a new framework estimating the CO accretion rate as $\dot{M}_\mathrm{CO} = \min\{\dot{M}_\mathrm{vis}, \dot{M}_\mathrm{BHL}\}$, where the viscous rate $\dot{M}_\mathrm{vis}$ accounts for gas--CO relative motion decomposed into a local gradient term (due to differential rotation) and bulk motion (from differing orbital parameters). This rate can be expressed as $\dot{M}_\mathrm{vis} = αξ(r_\mathrm{H}/r_\mathrm{BHL})^3\dot{M}_\mathrm{BHL}$, where $ξ$ is a coefficient of order unity. It can also be approximately scaled to the global AGN accretion rate as $\dot{M}_\mathrm{vis} \propto \dot{M}_1$, with the scaling coefficients in both forms determined by the specific dynamical configuration. The accretion is viscosity-limited when $q > [αξ(1+\mathcal{M}^2)^{3}/3]^{1/2} h^3$, where $q$ is the mass ratio between the CO and the supermassive black hole, $α$ the viscosity parameter, $\mathcal{M}$ the Mach number of the bulk relative motion, and $h$ the aspect ratio of the AGN disc. In thin AGN discs this condition is satisfied for most stellar-mass or more massive COs. Our framework also naturally allows for the inclusion of established outflow corrections, thereby enabling a more realistic treatment of super-Eddington flows. Our formulation thus improves upon Bondi/BHL prescriptions and offers a more physically motivated basis for studying CO evolution in AGN environments.

Figures (5)

• Figure 1: Velocity field of the surrounding gas relative to the CO in the rest frame. The central red dot represents the CO. The unit of length is much smaller than $R_2$. The $x$- and $y$-directions correspond to $\bm{e}_\Phi$ and $\bm{e}_R$ at the CO.
• Figure 2: Velocity field of the surrounding gas relative to the CO in the corotating frame. The central red dot represents the CO. The unit of length is much smaller than $R_2$. The $x$- and $y$-directions correspond to $\bm{e}_\Phi$ and $\bm{e}_R$ at the CO.
• Figure 3: Profiles of accretion rates ($\dot{M}_\mathrm{vis}$, $\dot{M}_\mathrm{prev}$, $\dot{M}_\mathrm{B}$), characteristic radii ($r_\mathrm{H}$, $r_\mathrm{rel}$, $r_\mathrm{B}$, $r_\mathrm{K}$), and Toomre $Q$ as functions of the SMBH--CO separation $R_2$. The accretion rates are normalised to $L_\mathrm{Edd2}/c^2$. The dashed line in the left panel represents the accretion rate of the AGN disc, $\dot{M}_1$. The input parameters are $M_1=10^8M_\odot$, $M_2=10M_\odot$, $\alpha=0.1$, $\dot{M}_1=L_\mathrm{Edd1}/c^2$ (corresponding to $10^7L_\mathrm{Edd2}/c^2$), $\mu=0.6$, and $\kappa_\mathrm{es}=0.35$ cm$^2$ g$^{-1}$.
• Figure 4: Profiles of dimensionless parameters ($\xi$, $\eta$, $\mathcal{M}$), accretion rates ($\dot{M}_\mathrm{vis}$, $\dot{M}_\mathrm{BHL}$, $\dot{M}_\mathrm{B}$), characteristic radii ($r_\mathrm{H}$, $r_\mathrm{BHL}$, $r_\mathrm{B}$, $r_\mathrm{K}$), and Toomre $Q$ as functions of the SMBH--CO separation $R_2$, for the slim-disc case with $M_1=10^8M_\odot$, $M_2=10M_\odot$, $\alpha=0.1$, $\dot{M}_1=100L_\mathrm{Edd1}/c^2$ (corresponding to $10^9L_\mathrm{Edd2}/c^2$), $\mu=0.6$, and $\kappa_\mathrm{es}=0.35$ cm$^2$ g$^{-1}$. The CO is assumed to be on a circular Keplerian orbit ($f_\mathrm{CO}=1$, $v_{R,\mathrm{CO}}\approx0$).
• Figure 5: Variations of dimensionless parameters ($\xi$, $\eta$, $\mathcal{M}$), accretion rates ($\dot{M}_\mathrm{vis}$, $\dot{M}_\mathrm{BHL}$, $\dot{M}_\mathrm{B}$), characteristic radii ($r_\mathrm{H}$, $r_\mathrm{BHL}$, $r_\mathrm{B}$, $r_\mathrm{K}$), and Toomre $Q$ along the CO orbital phase. The horizontal axis shows the true anomaly $\nu$, with $\nu=0$ (and $2\pi$) corresponding to pericentre, and $\nu=\pi$ corresponding to apocentre. The upper and lower panels correspond to eccentricity $e=0.001$ and $e=0.1$, respectively, both with semi-major axis $a=100R_\mathrm{g}$. The AGN disc is the same as that in Fig. \ref{['fig3']} in section \ref{['accretion']}.