Dynamical Traction in Rotating Stellar Systems

• Dynamical traction is a mechanism in rotating, anisotropic stellar systems that transfers angular momentum to black holes through the induced stellar wake.
• It leverages gravitational focusing to create a non-zero orthogonal force, reversing the classical inward drag and promoting outward orbital migration.
• Numerical models reveal that in both warm and fragmented cold discs, ordered stellar streaming can delay central settling and drive significant angular momentum gain over hundreds of Myr.

Dynamical traction is the angular-momentum-gaining counterpart of Chandrasekhar dynamical friction in a rotating, anisotropic stellar background. In the formulation introduced for black-hole orbital migration, a massive black hole moving through an odd distribution function of the schematic form $f(E,L_z)$ does not experience only the usual drag antiparallel to its velocity. The induced stellar wake can also generate a force component aligned with the bulk stellar stream, so that angular momentum is transferred from the stars to the black hole and the black hole’s azimuthal speed and $L_z$ increase. In this sense, the same gravitational-focusing physics that produces dynamical friction in isotropic systems can, in a rotating field, delay central settling or even drive outward migration instead of monotonic inspiral (Boily et al., 13 Jul 2025).

1. Definition and relation to dynamical friction

Standard Chandrasekhar dynamical friction is written for a black hole of mass $M_\bullet$ and velocity $\mathbf v_\bullet$ moving through a stellar background of phase-space density $f(\mathbf v)$ as

$$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$

with $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$. In an isotropic background, this acceleration is parallel and opposite to $\mathbf v_\bullet$: it removes orbital energy and tends to bring the black hole to rest at the barycentre.

Dynamical traction arises when that symmetry is broken. In a strongly anisotropic rotating field, the force decomposition includes not only the usual parallel component $a_\parallel$, but also a non-zero orthogonal component $a_\perp$ associated with net stellar streaming. That orthogonal term is the defining feature of dynamical traction: it steadily increases the perturber’s azimuthal speed and hence its $L_z$0. The underlying physics is unchanged—gravitational focusing, wake formation, and orbit polarization—but the background symmetry is different. An isotropic $L_z$1 yields net drag and energy loss; an odd, rotating $L_z$2 yields net torque and angular-momentum gain.

The distinction is therefore not between two unrelated mechanisms, but between two manifestations of the same collisional stellar-dynamical response. In the limiting picture emphasized in the paper, if the black hole is imagined at rest while stars stream past it, “dynamical friction becomes instead dynamical traction.” A direct consequence is that traction can increase the black hole’s kinetic energy and can remain non-zero even when the black hole is momentarily at rest in an inertial frame, provided the background velocity field is anisotropic.

2. Stellar-dynamical setting and distribution-function structure

The mechanism is developed for a black hole already bound inside the central region of a galaxy after a merger, moving in a stellar system that retains net rotation. The black hole is treated as a low-mass perturbation to the host: $L_z$3 of the disc, about $L_z$4 of the total galaxy mass, and still much larger than individual stars. A possible initial displacement from the centre is modeled as an $L_z$5 flyby impulse,

$L_z$6

with $L_z$7, $L_z$8, and $L_z$9 the perturber mass, impact parameter, and speed.

The background potential is separated into two components. The first is a hot, nearly isotropic spherical component represented by an isochrone (Hénon) potential. The second is a cooler rotating disc represented by a Miyamoto–Nagai potential. This split is the stellar-dynamical environment in which dynamical traction is meant to operate: orbit families embedded in an isotropic isochrone background, but coupled to a live anisotropic disc that carries net $M_\bullet$0. The isochrone component provides a cored central potential and an approximately isotropic bulge/halo-like background; the disc supplies the ordered streaming needed to torque the black hole.

To isolate the streaming loop-orbit population, the analytic treatment uses a local distribution function

$M_\bullet$1

with

$M_\bullet$2

This construction encodes three features: oddness in $M_\bullet$3, azimuthal streaming locked to the local bulk speed $M_\bullet$4, and isotropic random motions in the meridional plane $M_\bullet$5. The mechanism therefore does not require an exactly cold razor-thin disc. It requires a stellar system in which ordered rotational kinetic energy exceeds the isotropic velocity-dispersion component strongly enough for the streaming response to dominate.

3. Energy exchange, torque, and the traction transition

The kinetic description is written in Fokker–Planck form by decomposing velocity diffusion into components parallel and perpendicular to the black hole motion,

$M_\bullet$6

The specific kinetic-energy change is

$M_\bullet$7

In isotropic backgrounds the first-order orthogonal term vanishes by symmetry. In the rotating case, the relevant combination

$M_\bullet$8

shows that the first-order azimuthal response changes sign with $M_\bullet$9: a black hole that lags the local stream is accelerated forward, whereas one that leads it is decelerated. The tendency is toward corotation, but the approach to corotation is accompanied by substantial $\mathbf v_\bullet$0 exchange.

The torque estimate is

$\mathbf v_\bullet$1

so over a time interval $\mathbf v_\bullet$2,

$\mathbf v_\bullet$3

This is the explicit transfer statement: the anisotropic wake exerts a net torque that pumps angular momentum into the black-hole orbit.

A central result is that traction is not immediate. The transition from stochastic diffusion to systematic angular-momentum gain is defined by the inequality

$\mathbf v_\bullet$4

Once this condition is satisfied, the black hole no longer random-walks in velocity space; it begins to gain azimuthal kinetic energy systematically and can be transformed from a low-$\mathbf v_\bullet$5 box or radial orbit into a high-$\mathbf v_\bullet$6 loop orbit. For $\mathbf v_\bullet$7 and $\mathbf v_\bullet$8 at $\mathbf v_\bullet$9 kpc, the paper gives two example thresholds: if $f(\mathbf v)$0, traction dominates for $f(\mathbf v)$1; if $f(\mathbf v)$2, the crossover is at $f(\mathbf v)$3. For $f(\mathbf v)$4, the threshold shifts lower. The corresponding “critical line of stability” in the $f(\mathbf v)$5 plane has an asymptotic low-$f(\mathbf v)$6 trend $f(\mathbf v)$7 (Boily et al., 13 Jul 2025).

The timescale is likewise explicit. Using

$f(\mathbf v)$8

the paper finds $f(\mathbf v)$9 in its applications, but the Fokker–Planck integration shows that for a $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$0 black hole at $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$1 kpc in a streaming field, the azimuthal component overtakes the random components after about $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$2 Myr, and by about $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$3 Myr the streaming kinetic energy exceeds the sum of the dispersive components. The practical implication is that the same few-$$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$4 yr interval often associated with inward sinking can instead become a period of outward angular-momentum pumping, delaying central settling by several hundred Myr (Boily et al., 13 Jul 2025).

4. Warm-disc and cold-disc orbital evolution

In a dynamically warm rotating disc, where substantial substructure does not form, the mechanism is comparatively clean. A black hole released from rest at $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$5 kpc on a radial or low-$$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$6 orbit first sinks inward by dynamical friction. As the radial velocity is damped, however, the orthogonal traction term continues to act. The numerical evolution shows decreasing $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$7 for about $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$8 Myr, damping of $$\Gamma M_\bullet \frac{d\mathbf v_\bullet}{dt} = (\Delta \mathbf v_\bullet)_I = - F M_\bullet \int d^3v\, f(\mathbf v)\, \frac{\mathbf v_\bullet-\mathbf v}{|\mathbf v_\bullet-\mathbf v|^3},$$9, steady growth of $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$0 and $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$1, and eventual reversal into outward migration; the black hole does not settle at the centre. By contrast, a black hole started on a circular orbit remains approximately trapped near its original radius, with only perturbations from strong local scattering and global disc response. High-$\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$2 orbits are therefore more resistant to systematic migration than radial ones.

The cold-disc case is more nonlinear because the disc is Jeans/Toomre unstable and fragments into spirals and clumps. The paper quotes a Jeans length $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$3 with $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$4 pc for its reference model, and a mean Toomre parameter $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$5, peaking at $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$6 near the centre and falling to $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$7 by $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$8 kpc. In this regime a black hole started from rest at $\Gamma = 4\pi G^2 m_\star^2 \ln\Lambda$9 kpc excites large density waves, acquires angular momentum early, and is often driven inward first, reaching the centre on about $\mathbf v_\bullet$0 Myr timescales. Near the centre, however, non-axisymmetry changes the outcome: a clump or large $\mathbf v_\bullet$1 asymmetry can kick the black hole off-centre, after which dynamical traction becomes dominant and drives outward migration. In the fiducial simulations the black hole can then migrate outward to kiloparsec scales over about $\mathbf v_\bullet$2–$\mathbf v_\bullet$3 Gyr.

Near-circular orbits behave differently in cold discs. The black hole can open a gap and torque the stellar disc in a process compared with Type-II migration in protoplanetary discs. In that regime, inward motion is driven mainly by gravitational torques rather than classical friction or traction. The black hole can reach within about $\mathbf v_\bullet$4 pc of the centre by $\mathbf v_\bullet$5 Myr, yet fragmented central structure can still pull it back out, after which traction again spins it up and renews off-centre wandering. The eccentricity dependence is therefore explicit: nearly circular orbits favor gap-opening, torque-driven migration; sufficiently eccentric or low-$\mathbf v_\bullet$6 orbits evolve more through gravitational focusing, friction, and traction. The paper identifies the absence of a closed-form transition criterion in terms of $\mathbf v_\bullet$7, $\mathbf v_\bullet$8, and $\mathbf v_\bullet$9 as an open problem (Boily et al., 13 Jul 2025).

5. Two-stage orbital migration instability and the role of clumps

The paper’s central nonlinear claim is the existence of a two-stage orbital migration instability in fragmented cold environments. In the first stage, inward migration seeds or amplifies Jeans-unstable clumps and global non-axisymmetric structure. Once the black hole reaches or approaches the centre, a sufficiently massive or compact clump, or a global $a_\parallel$0 asymmetry, can scatter or pull it away from the barycentre. In the second stage, once displaced back into a region where ordered stellar streaming dominates, dynamical traction transfers angular momentum to the black hole, increases $a_\parallel$1 and $a_\parallel$2, and drives outward migration rather than prompt return.

The binding energy of the clumps controls the instability. Strongly bound clumps can survive and dislodge the black hole. Weakly bound clumps instead dissolve near the black hole or merge with it, suppressing long-lived ejection. This distinction also determines the final central morphology. If clumps are weakly bound, the end state can resemble a nuclear star cluster; the paper notes that such a remnant may still move at about $a_\parallel$3 relative to the barycentre.

The hot isotropic isochrone background acts as a stabilizer. If the rotating anisotropic component is embedded in a much more massive hot, nearly isotropic component, the clumps that form in the disc are less strongly bound and more easily disrupted by the black hole’s tidal field. Under those conditions the black hole can settle centrally on about $a_\parallel$4 Myr, fragmentation-driven ejection is suppressed, and a central cluster-like remnant is favored. The paper suggests that the isotropic-to-anisotropic mass ratio likely needs to be at least about $a_\parallel$5 to stabilize the centre against the clump-driven instability (Boily et al., 13 Jul 2025).

6. Numerical realization, astrophysical implications, and scope

The $a_\parallel$6-body calculations were performed with AMUSE, using a live Miyamoto–Nagai disc, a frozen or partially live isochrone background, direct Hermite PH4 integration for the black hole, and a Barnes–Hut tree for the stars. The fiducial setup uses $a_\parallel$7, $a_\parallel$8, $a_\parallel$9 kpc for both components, $a_\perp$0 for the disc, $a_\perp$1 disc particles, $a_\perp$2, and $a_\perp$3, with star-star softening $a_\perp$4 pc and star–black-hole softening $a_\perp$5 pc. Total angular momentum is conserved to about $a_\perp$6 over $a_\perp$7 Gyr, which the authors treat as sufficient for angular-momentum-transfer analysis.

The broader implications are direct. In warm anisotropic systems, low-$a_\perp$8 black holes can acquire angular momentum and avoid central settling. In cold fragmented systems, a black hole may reach the centre quickly but need not remain there; clump-induced displacement followed by traction can drive it back outward. In the fiducial $a_\perp$9, $L_z$00 kpc model, outward migration can reach about $L_z$01 kpc over about $L_z$02–$L_z$03 Gyr. More generally, settling to the galactic centre can be delayed by several $L_z$04 yr, and for a black hole to settle at the heart of a galaxy on timescales of about $L_z$05 Myr or less, either a large fraction of $L_z$06 must be dissipated or the black hole must grow in situ in an isotropic environment devoid of sub-structures.

The paper also states a concrete environmental contrast. In a nearly isotropic nuclear star cluster with $L_z$07, such as the Milky Way example discussed there, the black hole should not undergo the traction transition and should sink back rapidly. A rotating nuclear stellar disc is more favorable to traction. This suggests that dynamical traction is not a generic replacement for dynamical friction, but a symmetry-dependent orbital-migration channel whose efficiency is a strong function of streaming support, isotropic dispersion, and the fragmented state of the stellar density field (Boily et al., 13 Jul 2025).