Not all cores are equal: Phase-space origins of dynamical friction, stalling and buoyancy

TL;DR

This work addresses how dynamical friction operates in galaxies and halos with central cores, where the classic Chandrasekhar formula fails. By combining high-resolution $N$-body simulations with kinetic-theory insights, it shows that the fate of embedded massive bodies is governed by the host's distribution function $f(E)$, not solely the inner density slope. Core stalling occurs at a DF plateau $(\nabla f)_{ m CR}=0$, while buoyancy arises when the DF has an inflection that seeds a growing dipole mode. The results reveal that double power-law density profiles with rapid outer-to-inner transitions naturally produce the DF features that drive these phenomena, linking phase-space structure to the dynamics of black holes, nuclear clusters, and off-centered nuclei across cosmic time and affecting SMBH merger rates relevant for LISA and GW backgrounds.

Abstract

Dynamical friction governs the orbital decay of massive perturbers within galaxies and dark matter halos, yet its standard Chandrasekhar formulation fails in systems with cores of (roughly) constant density, where inspiral can halt or even reverse, phenomena known respectively as core stalling and dynamical buoyancy. Although these effects have been observed in simulations, the conditions under which they arise remain unclear. Using high-resolution N-body simulations and analytic insights from kinetic theory, we systematically explore the physical origin of these effects. We demonstrate that the overall distribution function (DF) of the host, not just its central density gradient, determines the efficiency and direction of dynamical friction. Core stalling arises when the perturber encounters a plateau in the DF, either pre-existing or dynamically created through its own inspiral, while buoyancy emerges in systems whose DFs possess an inflection that drives an unstable dipole mode. We show that double power-law density profiles with rapid outer-to-inner slope transitions naturally produce such DF features, which is why structurally similar cores can yield radically different dynamical outcomes. Our results provide a unified framework linking the phase-space structure of galaxies to the fate of embedded massive objects, with direct implications for off-center AGN, the dynamics of nuclear star clusters, and the stalled coalescence of black holes in dwarf galaxies and massive ellipticals.

Figures (8)

• Figure 1: Parameter space of stable, unstable, and unphysical $(\alpha, \beta, \gamma)$ systems in the $\alpha-\gamma$ plane, for fixed $\beta=4$. The red stars indicate the systems that constitute our Fiducial suite of simulations.
• Figure 1: Convergence tests to verify the robustness of our results. In all cases, we plot the trajectory of a BH of mass $M_{\rm BH}=10^{-3}$ in the RapidCore system, i.e. $(\alpha, \beta, \gamma)=(4,4,0.1)$. The horizontal black dashed lines denote the stalling radius $r_{\rm stall}\approx 0.65$ in the Fiducial simulation. Top left panel: effect of varying the BH's initial radius $r_{\rm init}$. The trajectories are shifted along the time axis such that $r_{\rm BH}(T=0)=1$ in all cases. Top right panel: trajectories with varying initial eccentricity. From these two panels, we see that the stalling radius is independent of the BH's initial radius and eccentricity. Bottom panels: effect of simulation resolution on stalling (left) and buoyancy (right). A well-resolved trajectory is seen only when $N \geq 10^6$ particles.
• Figure 2: Left: density profiles $\rho(r)$ vs $r$ for four $(\alpha, \beta, \gamma)$ systems, where the total mass is normalized to $1$ across all profiles. We fix $\beta=4$ and $\gamma=0.1$, and vary the value of $\alpha$ between 1 and 4. Right: the isotropic DFs corresponding to these systems. Despite having similar densities, these DFs differ drastically.
• Figure 3: Inspiral of a BH of mass $M_{\rm BH}=10^{-3}$ in systems with different density profiles. We fix $\beta=4$ for all cases, and each panel corresponds to a different value of $\gamma$. The dashed lines denote the prediction from the Chandrasekhar prescription (equation [\ref{['eq:c43']}]), and the horizontal arrows mark $r_{\rm stall}$, where the trajectory deviates from the prediction. Note how the different $\alpha$ values result in drastically different trajectories.
• Figure 4: Inspiral of BHs of varying masses, ranging from $M_{\rm BH}=10^{-4}$ to $10^{-2}$, in the RapidCore system. The solid lines show the N-body simulation results, and the dashed lines show the corresponding predictions from the Chandrasekhar prescription (equation [\ref{['eq:c43']}]) We see that regardless of mass, all BHs stall at $r_{\rm BH} \approx 0.65$. However, the more massive BHs are able to break through the stalling phase earlier, and sink in (slower than predictions from equation [\ref{['eq:c43']}]) down to $r_{\rm BH}\approx 0.1$. The lightest BHs stall perfectly at $r_{\rm BH} \approx 0.6$ and never sink further.