Mean Motion Resonances in AGN Disks

TL;DR

The paper develops an analytic framework to assess mean-motion resonances for stellar-mass black holes embedded in AGN disks, integrating resonant dynamics, general-relativistic precession, migration torques (Type I/II and thermal), GW emission, and stochastic forcing from MRI turbulence and NSC flybys. By reducing the resonant problem to a single degree of freedom and coupling diffusion to migration, it derives criteria for resonance disruption and maps stability across a wide range of MBH masses and disk conditions. The results reveal three MBH-mass regimes with distinct resonance behavior: high-mass AGN host unstable resonances, low-mass AGN sustain resonances, and intermediate-mass AGN show parameter-dependent stability; in general, resonances tend to occur in outward-migration regions between inner anti-traps and outer thermal-trap boundaries. These findings imply that high-mass AGN can form LVK-band mergers largely without resonant chains, while lower-mass AGN may harbor MMRs that influence merger pathways, with observable implications for gravitational waves and EMRIs; nonetheless, substantial disk physics uncertainties and potential metastable resonances call for numerical validation and more detailed disk models.

Abstract

Mean motion resonances (MMRs) are a generic outcome of convergent migration for bodies embedded in accretion disks around a central mass. Long studied in planetary systems, the same phenomenon should occur for stellar-mass black holes (BHs) in AGN disks. In this work, we derive simple analytic criteria describing when BH pairs are driven out of resonance, and use them to chart MMR stability across AGN parameter space, accounting for disruption from general-relativistic apsidal precession, hydrodynamic turbulence, and stellar stirring. Across plausible AGN disk models, we find three MBH mass regimes: (i) for $M/ M_\odot\gtrsim 10^{7.5}$, first order resonances are generically unstable; (ii) for $M/ M_\odot\lesssim 10^{6.5}$, stable MMRs are always present; (iii) for $10^{6.5}\lesssim M / M_\odot \lesssim 10^{7.5}$, stability depends on disk mass flux, the summed mass of the orbiters, and the nuclear-cusp slope. When present, stable MMRs commonly occur between an inner anti-trap and an outer trap set by thermal torque, a region where embedded objects migrate outward in the disk. These results imply that high-mass AGN allow convergent migration to proceed to LVK-band mergers largely without resonant chains, whereas low/intermediate-mass AGN can host MMRs, with the potential to reshape merger pathways.

Figures (7)

• Figure 1: The boundary $\dot{\omega}_{\text{MMR}}/ \Delta\dot{\omega}_{\text{GR}} \approx 1$, which separates the interior parameter space where GR dominates the dynamics (and the resonance is broken) from the exterior region where GR precession is sufficiently weak that the MMR can exist. Colors denote different masses for the inner object participating in the resonance, and line-styles denote the harmonic $k$ in the $k+1:k$ first-order resonance. Higher harmonics shift the GR disruption boundary to smaller semimajor axes $a$ by reducing the differential relativistic apsidal precession $\Delta \dot{\omega}_{\rm GR}$ between a resonant pair.
• Figure 2: Colored contours show the migration timescale $\tau$ as a function of distance from the MBH $(a/R_g)$ and disk parameters. The red curve marks $\tau=10^8$ yr, the maximum AGN lifetime. Top panel:$\tau_{\rm I}$ assuming Type I migration only, with $\alpha_{\rm disk}=0.01$, $\dot m_{\rm disk}=0.1$ (the disk mass flux in units of the Eddington rate), and a migrating mass $m=10\,M_\odot$. Middle panel:$\tau_{\rm I/II}$ computed from Equation (\ref{['eq:typeI-II']}), i.e., the Type-I rate rescaled by the local surface density; both top and middle panels vary the central mass $M$. Bottom panel:$\tau_{\rm I/II}$ as a function of $\dot m_{\rm disk}$ and $a/R_g$ for fixed $M=10^7\,M_\odot$. The orange line indicates the minimum radius where marginal stability is imposed; to the right of this line the sirko2003 models assume stellar feedback maintains $Q_{\rm T}=1$.
• Figure 3: Thermal torque migration maps, analogous to Figure \ref{['fig:migration_typeI/II']} but with the additional inclusion of thermal torques 2017masset Colored contours show the migration timescale $\tau$ versus distance from the MBH $(a[R_g])$; the red curve marks $\tau=10^8$ yr. As in Figure \ref{['fig:migration_typeI/II']}, the top two panels use $M$ on the y–axis, and the bottom panel uses $\dot m_{\rm disk}$ (all three panels adopt the same parameters as their Figure \ref{['fig:migration_typeI/II']} counterparts). These maps differ in the torque used to compute $\tau$: Top panel: we exclude the possibility of gap opening and show $\tau_{\rm I/thermal}\propto (\Gamma_{\rm I}+\Gamma_{\rm GW}+\Gamma_{\rm thermal})^{-1}$, with $\Gamma_{\rm thermal}$ given by Equation \ref{['eq:gamma_therm_final']}. Middle and bottom panels: gap opening is now accounted for and we show $\tau_{\rm I/II/thermal} \propto \Gamma_{\rm{tot}}^{-1}$, where $\Gamma_{\rm tot}=\Gamma_{\rm I/II}+\Gamma_{\rm GW}+\Gamma_{\rm thermal}$, and $\Gamma_{\rm I/II}$ follows Equation (\ref{['eq:typeI-II']}). Cyan (orange) lines mark migration traps (anti-traps) defined by zero total torque and white contours indicate $\Gamma_{\rm thermal}=0$.
• Figure 4: Iso-diffusion contours where $\mathcal{D}_{\rm scat} = \mathcal{D}_{\rm turb}$ in the ($M/M_\odot$, $a/R_g$) plane. The dark purple contour shows the fiducial model ($\alpha = 0.01$, $m = 10 ~\rm{M}_\odot$, $\dot{m}_{\rm disk} = 0.1$, $\gamma = 7/4$). Dashed and dotted curves vary one disk parameter relative to fiducial: dashed uses $m = 250 ~\rm{M}_\odot$; dotted uses $\alpha = 0.1$. Solid magenta, orange and aquamarine curves adopt a core profile for the NSC stars ($\gamma = 1$) with $\dot{m}_{\rm disk} = 0.01, ~0.5$ and 0.1, respectively. $\mathcal{D}_{\rm scat}$ obeys a simple power law with semimajor axis $a$, but $\mathcal{D}_{\rm turb}$ has a more complex evolution driven by the AGN disk model. Scattering achieves greater importance at the smallest and largest radii, where disk turbulence is weakened by declines in $\Sigma$ arising from radiation pressure dominance and $Q_{\rm T}=1$ self-regulation, respectively.
• Figure 5: Inverse effective diffusion, $\mathcal{D}_{e,\mathrm{eff}}^{-1}$, across the $(\log_{10} (M\,[M_\odot]),\ \log_{10} (a\, [R_g])$ plane. Filled contours show $\mathcal{D}_{e,\mathrm{eff}}^{-1}$ [yr] computed from turbulent and scattering contributions added in quadrature. white contours indicates where $\mathcal{D}_{e,\rm scat} = \mathcal{D}_{e,\rm turb}$. Top: scattering term evaluated for a stellar cusp with slope $\gamma=7/4$. Black contours show where the product of migration timescale and diffusion, $\tau_{\rm I/II},\mathcal{D}_{e,\mathrm{eff}}=10^x$ and contour labels show $x\in[-4,3]$. Green contours assume only Type-I migration Bottom:$\gamma=1$ and $x\in[-6,3]$.