Resonant Dynamical Friction Around a Super-Massive Black Hole: Analytical Description

TL;DR

The authors derive an analytical framework for resonant dynamical friction in a thin stellar disc around a super-massive black hole, showing that a massive inclined perturber induces a rapid, boundary-layer alignment of the disc stars' nodal angles with a 90-degree phase offset relative to the perturber. A double-averaged Hamiltonian with a Laplace-Lagrange disc component and a disc–perturber interaction term reveals two time scales: a nodal precession $\nu_p$ and a fast alignment scale $b_{pn}$, leading to a condition $|\nu_p|\lesssim|b_{pn}|$ for alignment and a derived RDF time-scale $τ_{\rm RDF}$. The resulting inclination damping of the perturber occurs on a timescale shorter than standard Chandrasekhar dynamical friction, scaling as $τ_{\rm RDF}\propto M_\bullet/(m_p M_{d,\mathrm{loc}})^{1/2} t_{\rm orb}$ and matching a wide range of $N$-body simulations. The mechanism remains robust across initial conditions, disc thickness, and even in simulations including a live spherical component, with potential relevance to the Milky Way's center and possibly to planetary systems. Overall, the work provides a concrete analytical description of a fast, resonant collective effect that aligns a perturber with a stellar disc through the synchronized nodal dynamics of the system.

Abstract

We derive an analytical model for the so-called phenomenon of `resonant dynamical friction', where a disc of stars around a super-massive black hole interacts with a massive perturber, so as to align its inclination with the disc's orientation. We show that it stems from a singular behaviour of the orbit-averaged equations of motion, which leads to a rapid alignment of the argument of the ascending node $Ω$ of each of the disc stars, with that of the perturber, $Ω_{\rm p}$, with a phase-difference of $90^\circ$. This phenomenon occurs for all stars whose maximum possible $\dotΩ$ (maximised over all values of $Ω$ for all the disc stars), is greater than $\dotΩ_{\rm p}$; this corresponds approximately to all stars whose semi-major axes are less than twice that of the perturber. The rate at which the perturber's inclination decreases with time is proportional to its mass and is shown to be much faster than Chandrasekhar's dynamical friction. We find that the total alignment time is inversely proportional to the root of the perturber's mass. This persists until the perturber enters the disc. The predictions of this model agree with a suite of numerical $N$-body simulations which we perform to explore this phenomenon, for a wide range of initial conditions, masses, \emph{etc.}, and are an instance of a general phenomenon. Similar effects could occur in the context of planetary systems, too.

Figures (11)

• Figure 1: The relevant times-scales associated with the problem, in units of the orbital time of the perturber, $t_{\rm orb} = 524$ yr, versus the stars' semi-major axes, plotted for the same parameters as in figure \ref{['fig:inclination histogram']}, i.e. for the first row of table \ref{['tab:runs']}. The time-scales are: the nodal alignment time-scale, $b_{{\rm p}n}(0)^{-1}$ (equation \ref{['eqn:b_pn definition']}), in yellow asterisks, the nodal precession time-scale $\nu_{\rm p}^{-1}$ in orange, and the inclination change time-scale, $\tau_{\rm RDF}$ (equation \ref{['eqn:tau RDF']}), in blue.
• Figure 2: A scatter-plot of the inclinations of the stars as a function of their semi-major axis, after $1$ Myr. The SMBH mass is $4\times 10^6~M_\odot$, the total disc mass is $2000~M_\odot$, the perturber's mass is $250~M_\odot$, and we add a Plummer model sphere of mass $2\times 10^5~M_\odot$ to the numerical simulation to regularise the precession of the argument of pericentre. Left: the analytical prediction of equation \ref{['eqn:i_n of t']}. Right: the result from the simulation with $N = 999$, corresponding to the first row of table \ref{['tab:runs']}. These plots show the state of the system after $1~\textrm{Myr}$.
• Figure 3: Like figure \ref{['fig:inclination histogram']}, but for the arguments of the ascending node. Left: the analytical prediction of equation \ref{['eqn: Omega_n alignment']}. Right: result from the simulation.
• Figure 4: Top panels: time evolution of the longitudes of the ascending nodes for the IMBH and stars in the disc. The thick red line shows $\Omega$ for the IMBH, the blue line shows mean $\Omega$ of the middle stars (stars with overlapping orbits with the IMBH), the shaded region is the area between 25 and 75% quantiles for $\left<\Omega\right>$ of the middle stars, the lightly-shaded region is the area between 10 and 90% quantiles of the same stars. The orange and green lines show the mean $\Omega$ for the inner and outer stars, respectively. Bottom panels: the time evolution of inclination angles of the IMBH with respect to the middle stars (blue) and the whole stellar disc (black). Right: the same, but in linear scale.
• Figure 5: The predictions of equation \ref{["eqn:perturber's inclination equation of motion"]} in blue, compared with the numerical simulation (orange). The set-up and initial conditions are the same as figure \ref{['fig:inclination histogram']}.