A drag with nothing to drag against

A star cluster contains no air. A galaxy halo contains, for dynamical purposes, nothing at all — its constituents are separated by parsecs and never touch. And yet a massive object moving through either one slows down, on a timescale that can be computed and that decides the fate of every satellite galaxy in the Local Group.

The mechanism has no medium in it. It is gravity acting on gravity’s own consequences: the moving body pulls the background towards its own path, the background arrives behind it because the body has moved on, and the resulting overdensity pulls backwards. The body is decelerated by a structure it made itself, in a system that is collisionless in every other respect.

Why the deflections do not cancel #

The obvious objection is symmetry. A particle passing in front is pulled backwards, one passing behind is pulled forwards; a particle on the left and one on the right cancel sideways. In a uniform background, why should anything be left over?

The answer is that the background is uniform before the encounter and not after. Each particle that passes is deflected towards the line of motion, and the deflection takes time to develop — by the moment a particle has been pulled inward, the perturber has moved forward. What accumulates behind is therefore not a symmetric enhancement but a trailing one, and it is the trailing part that has an unbalanced pull.

Seen from the other side the argument is even shorter, and it is the version worth keeping. Each encounter is a gravity assist, and a gravity assist exchanges momentum: the light body gains, the heavy body loses. The light bodies gain a little each and there are very many of them; the heavy body loses a little each time and there is one of it. Dynamical friction is a gravity assist run backwards, summed over a population, and the energy that leaves the perturber is exactly the energy that appears as heating of the background.

The formula, and the three surprises in it #

Chandrasekhar wrote the result down in 1943 for a background with a Maxwellian distribution of speeds:

$$\frac{d\mathbf{v}}{dt} = -\frac{4\pi G^2 M \rho \ln\Lambda}{v^3}\left[\operatorname{erf}(X) - \frac{2X e^{-X^2}}{\sqrt{\pi}}\right]\mathbf{v},\qquad X = \frac{v}{\sqrt{2}\,\sigma}.$$

Three features of it are worth pulling out separately, because each is counterintuitive on its own.

The background’s individual masses are absent. Only $\rho$ appears — the mass density. A perturber ploughing through a sea of solar-mass stars feels exactly the same drag as one ploughing through a sea of dark-matter particles of $10^{-60}$ solar masses at the same density. That is not an approximation; it survives the full derivation, because the momentum given to each background body scales as its mass and the number of them at a given density scales as the inverse.

The force goes as $M^2$. One power of $M$ because the wake’s amplitude is proportional to the perturber’s mass, and a second because the wake pulls on the perturber. The deceleration therefore goes as $M$: heavy things slow down, light things effectively do not, and the effect sorts a population by mass.

It is not monotonic in speed. The bracket is the fraction of the background moving slower than the perturber, and only those contribute; the $v^{-3}$ prefactor pulls the other way.

How the sum is actually done #

The derivation is worth a paragraph because it explains where the logarithm comes from and why the answer has the shape it has.

Take one background body of mass $m$ passing the perturber at impact parameter $b$ with relative speed $v$. In the impulse approximation — the encounter is fast enough that the paths are nearly straight — the transverse velocity it picks up is $\Delta v_\perp = 2GM/(bv)$, and the momentum taken from the perturber along its own direction of motion follows from conserving both energy and momentum in the two-body encounter:

$$\Delta v_\parallel = -\frac{2m v}{M+m}\left[1+\frac{b^2v^4}{G^2(M+m)^2}\right]^{-1}.$$

Now count encounters. In time $dt$ the perturber sweeps a cylindrical shell between $b$ and $b+db$ containing $n\,v\,2\pi b\,db\,dt$ background bodies, so the deceleration is an integral over $b$ of $\Delta v_\parallel$ times that number. The integrand behaves as $db/b$ over most of the range, which is where the logarithm appears: every decade of impact parameter contributes equally, so an encounter at a parsec matters as much as one at a kiloparsec, and the answer depends on where the range is cut off rather than on where the integrand peaks.

That is an unusual property and it is why the effect is so hard to localise. There is no characteristic distance at which dynamical friction happens. The wake in the opening figure is drawn at three focusing radii because a figure needs a scale, and the real wake extends outward until the background stops being uniform, with each decade pulling as hard as the last.

The final step is to average over the background’s own velocities, and this is where only the slower bodies survive: a body moving faster than the perturber in the same direction contributes with the opposite sign, and in a Maxwellian the two populations partly cancel. What is left is the bracket in Chandrasekhar’s formula, which is exactly the fraction of the distribution inside speed $v$.

The logarithm, and what it hides #

$\ln\Lambda$ is the Coulomb logarithm, and it is where the derivation admits that it does not know the answer. The integral over impact parameters diverges logarithmically at both ends: at small $b$ because the impulse approximation fails, at large $b$ because the background is not actually infinite. $\Lambda$ is the ratio of the largest useful impact parameter to the smallest, and in practice it is between about 3 and 30 — so $\ln\Lambda$ is between 1 and 3.5 and the whole result is uncertain by a factor of a few.

That is worth stating plainly rather than hiding: the drag is known to a factor of about three and no better from theory alone, and every quantitative statement below inherits that. What pins it down is calibration against N-body simulations, in which the same quantity is measured rather than derived.

What it does, at three scales #

Satellites sink. A satellite galaxy orbiting inside a host halo loses angular momentum to the halo and spirals in. In a singular isothermal halo the arithmetic closes in a line — the drag goes as $M^2/r^2$, the angular momentum as $Mr$, and $r^2$ falls linearly with time.

Massive stars sink too. The same $1/M$ dependence inside a cluster whose mass is known from its dispersion produces mass segregation: the heaviest stars drift towards the centre on a timescale shorter than the cluster’s relaxation time by the ratio of their mass to the average. In an old globular cluster the blue stragglers and the neutron stars are concentrated centrally, and that concentration is a measurement of this process rather than of where those objects formed.

Black holes pair up. When two galaxies merge, each brings a central black hole, and the two must reach the centre of the merged system before they can do anything else. Dynamical friction is what brings them from tens of kiloparsecs to about a parsec.

The energy has to go somewhere #

A drag dissipates energy, and in an ordinary fluid the energy becomes heat in a medium that can radiate it away. Here there is no medium and nothing radiates. So where does it go?

It goes into the background’s own random motion. Every light body that gains speed in an encounter keeps it, and the sum of those gains is exactly the perturber’s loss. The background gets hotter — its velocity dispersion rises — and since a self-gravitating system that gains energy expands and cools, the halo responds by puffing up slightly. That response is small for a single satellite and is not small for the accumulated history of a galaxy’s mergers, and it is one of the mechanisms invoked for turning a cuspy dark-matter profile into a cored one.

The bookkeeping is worth stating because it distinguishes this from friction in the ordinary sense. No energy is lost anywhere in this process. The perturber’s orbital energy becomes the background’s kinetic energy, and the total is conserved to the precision of the integration. What is lost is only the ordered character of the perturber’s motion — which is the same thing that is lost when a disc is heated by an encounter and never recovers its original thinness.

Where the sign reverses #

The formula predicts a deceleration, always, and it is derived for a background that is uniform. In a background with a core — a central region of nearly constant density — the prediction fails in a way that took twenty years to be accepted, because it fails in the wrong direction.

The mechanism is the same wake seen from a different starting point. In a cored profile the stars near the centre are not on radial orbits falling through; they are on nearly harmonic orbits in a nearly uniform density, all with nearly the same period. A perturber moving among them can find itself in a resonance with that common period, and a resonance exchanges angular momentum in a direction set by the resonance rather than by the wake’s geometry.

What is observed in simulations is that a satellite sinking into a cored halo slows, stalls at about the core radius, and in some cases moves outward — the background transfers angular momentum to it rather than taking it. The effect has been called dynamical buoyancy, and it is the same physics as the classical drag with the sign of one term changed by the presence of a resonance the uniform derivation cannot contain.

Whether it is a general result or a feature of particular configurations is still argued about, and the argument matters beyond the dynamics: cores and cusps are the observable difference between several proposals about what dark matter is, and the sinking of satellites is one of the few dynamical measurements sensitive to which of them a halo has. A prediction that a satellite should stall at a particular radius, and a population of satellites observed to sit at such radii, is a test — provided the stalling is not being produced by something else in the simulation.

The general lesson is worth extracting from the specific one. A formula derived by summing independent encounters in a uniform medium cannot represent a coherent response, and a coherent response is what a resonance is. Wherever the background has a characteristic frequency — a core, a bar, a disc, a rotating halo — the classical calculation is describing something other than what is happening, even where it happens to give the right magnitude.

Where it stops working #

The descent of a black hole pair stalls. Once the two are close enough that the stars whose orbits pass between them have all been ejected, the reservoir of slow background material is exhausted, and dynamical friction has nothing left to act on. The pair sits at about a parsec, with a separation that shrinks only as fast as new stars are scattered into the region — which is slow.

That is the “final parsec problem”, and it is a genuine open question rather than a rhetorical one: the gap between where friction gives out and where gravitational radiation takes over is about two orders of magnitude in separation, and closing it requires either gas or a non-spherical potential to keep refilling the loss cone.

What is actually observed #

Nothing on this page has been watched happening. The timescales are $10^8$ to $10^{10}$ years and the effect is not visible in any single system in real time, so the evidence is entirely circumstantial and worth setting out as such — this is a subject in which the observation behind the number is a population rather than an event.

The strongest piece is the Sagittarius dwarf galaxy, which is currently being disrupted while passing through the Galactic disc, on an orbit whose apocentre has demonstrably decayed — its stream of debris traces several previous passages at larger radii than the present one, and the spacing of those passages is a direct measurement of how fast the orbit is shrinking. The second is mass segregation in globular clusters, which is measured as a radial gradient in the mass function and matches what the $1/M$ scaling predicts. The third is the absence of a population that should exist: a halo full of satellites at every radius, undecayed, is what a universe without dynamical friction would show, and the observed radial distribution of surviving satellites is depleted at small radii by about the right amount. The wake and the sinking time are the two ends of the same calculation, and both are worth reading at other settings.

The same integral, elsewhere #

The name of the awkward factor gives the game away, and it is worth following, because the identical calculation was done first in a completely different subject.

A charged particle moving through a plasma is slowed by the same argument with one substitution: an inverse-square attraction becomes an inverse-square attraction or repulsion, the wake becomes a polarisation cloud of opposite charge trailing behind, and the sum over impact parameters diverges logarithmically at both ends for the same reason. The result is the stopping power of a fast particle in matter, its logarithm is the Coulomb logarithm, and the cutoffs are set by the Debye length above and the distance of closest approach below.

The two subjects differ in one respect that is instructive. A plasma screens: charges of both signs are present, so the polarisation cloud has a finite extent and the outer cutoff is a physical length the medium supplies. Gravity has one sign, so nothing screens, and the outer cutoff has to be imposed by hand at the size of the system. That is the whole reason the astronomical version of the constant is ambiguous and the laboratory version is not — a gravitational wake has no Debye length, and the missing quantity is exactly the one that would make the integral converge.

The correspondence is close enough that results have moved between the two fields in both directions. The velocity dependence that gives the peak in the gravitational case is the same function that gives the Bragg peak in the stopping power of an ion in matter, and for the same reason: only the medium’s slower constituents contribute, and the prefactor falls as the inverse square of the speed. A physicist who has computed one has computed the other, with a substitution of constants and a change of sign in the interaction that turns out not to matter, because the force enters squared.

What does not transfer is the geometry of the application. A charged particle in matter is stopped over a distance short compared with anything else in the problem, so the stopping power is used to compute a range. A satellite in a halo is decelerated over a distance comparable with the system it is moving through, so the same quantity is used to compute an orbit — and the background it is moving through changes along that orbit, which is the difficulty the previous section is about.

It is also worth putting a number on how negligible the effect is for anything small. The Sun moving through the solar neighbourhood, at a local density of about 0.1 solar masses per cubic parsec and a speed comparable with the local dispersion, is decelerated by dynamical friction on a timescale of order $10^{17}$ years — some seven orders of magnitude longer than the age of the universe. The quadratic dependence on mass is what does it: a body a million times heavier than the Sun feels a drag a million times stronger per unit mass, which turns an irrelevance into the process that assembles galaxies.

And the relaxation time at the other value of the logarithm, which is the number nobody can pin down.

One more reading shows the other thing the same wake does to the body that made it.

Where the ladder goes next #

Two rungs follow naturally. One is the resonant version of the same idea: when the perturber is on a circular orbit inside a disc rather than moving through a uniform sea, the drag comes from discrete Lindblad resonances rather than from a trailing wake, and the result is planetary migration in a protoplanetary disc — the same physics with a different geometry and a much larger consequence. The other is the reverse: what the background does with the energy it has absorbed, which is the heating that drives a cluster’s core towards collapse.