A drag that stops where every orbit keeps time

The Fornax dwarf galaxy, a faint satellite of the Milky Way about 140 kiloparsecs away, has five globular clusters. Nearly every dwarf of its size has none. The clusters are massive, between a few times ten thousand and a few times a hundred thousand solar masses each, they are old, and they orbit within a kiloparsec or so of the galaxy’s centre.

They should not be there. A massive body moving through a sea of lighter ones is slowed by the wake it raises behind itself, loses angular momentum and spirals inward, and Fornax is small enough and dense enough that its clusters should have spiralled to the centre long ago and merged there into a nucleus. There is no nucleus. In 1976 the problem was noticed and stated as a puzzle about the clusters’ history; it is now usually stated as a piece of evidence about the halo they orbit in.

The resolution most often given is that Fornax’s dark matter has a constant-density core rather than a density that keeps rising towards the centre, and that in a core dynamical friction stops. The second half of that sentence is the subject of this essay, because it is not what Chandrasekhar’s formula says. The formula weakens the drag in a core. Simulations show it switching off altogether, and the reason is something the formula cannot see by construction.

Two haloes the stars cannot tell apart #

Fornax’s dark-matter halo is measured through the motions of its stars: thousands of line-of-sight velocities, a dispersion of about twelve kilometres a second, nearly flat with radius. A dispersion profile constrains the mass inside a radius very well near one particular radius, where the unknown shapes of the stars’ orbits stop mattering, and constrains the density at the centre poorly. Two families of halo fit the stars: the cusped profile that simulations of cold dark matter produce, whose density rises inward as the inverse of the radius, and a cored profile whose density levels off inside some radius.

These are models chosen to bracket the question, not fits. The real argument about Fornax’s centre is the argument about the innermost kiloparsec that runs through every dwarf galaxy: the cusps that simulations of collisionless dark matter predict, against the cores that the stellar kinematics of many dwarfs seem to prefer, with energy injected by bursts of star formation as one proposed way of turning one into the other. The clusters are interesting because they are a different kind of witness. Stars measure the potential through their velocities; the clusters measure it through how fast it would have dragged them in.

The formula’s answer #

The drag on a body of mass $M$ moving at speed $v$ through stars of density $\rho$ with a Maxwellian distribution of dispersion $\sigma$ is

$$\frac{dv}{dt} = -\frac{4\pi G^2 M \rho \ln\Lambda}{v^2}\left[\operatorname{erf}(X) - \frac{2X}{\sqrt\pi}e^{-X^2}\right], \qquad X = \frac{v}{\sqrt2\,\sigma}.$$

The bracket is the fraction of the stars moving more slowly than the body — only those contribute, because a star that overtakes the body is focused into a wake ahead of it as much as behind. For a cluster on a circular orbit, the drag removes angular momentum at a rate the formula fixes, and the orbit shrinks.

Two features of the curves deserve attention before anything is said about the core. The first is how fast the cusp works: a cluster that begins at a kiloparsec, the outer edge of where Fornax’s clusters are now, is gone in about two-thirds of the age of the universe. The second is that the two haloes agree for the first few billion years. Outside the core the two have similar densities, and outside it the cored halo is in fact slightly the denser of the two, because it was normalised to the same enclosed mass.

The Coulomb logarithm of three is low, because the system is small: the logarithm of the ratio of the largest to the smallest impact parameter is only a few when the orbit is a kiloparsec and the cluster tens of parsecs across. Every time in the figures scales inversely with it. That is a factor of two of uncertainty on all of them, which is less than the factors the argument turns on.

Five clusters and one age #

Running the formula for each of Fornax’s clusters gives the timing problem in its usual form.

The cusp’s verdict is clear. The two innermost clusters, at a quarter and nearly half a kiloparsec in projection, would reach the centre in about a billion years; a third, at a kiloparsec, in eight. The clusters are at least ten billion years old. For them to be where they are, they would have had to be born much further out and arrive at their present radii by coincidence in the last billion years, all of them at once, and Fornax would have had to keep no record of any clusters that arrived earlier. Each of those is possible; together they are a coincidence of the kind that makes a model suspect.

The core’s verdict on the formula is not as clean as it is usually made to sound. The three outer clusters are saved: their sinking times exceed fourteen billion years. But the two inner ones are not — the formula, run in a cored halo, still brings them to the centre within their ages. A core helps, through the formula, and does not by itself resolve the problem.

What resolves it is that inside a core the formula is not what happens.

What the formula does in a core #

The formula’s drag is a product of the local density, the fraction of stars moving more slowly than the body, and a factor of one over the body’s speed squared. A core changes all three.

The second effect is the larger one and it is worth stating plainly. In a core, the circular speed rises linearly from the centre, because the enclosed mass grows as the cube of the radius; the stars’ velocity dispersion stays at its measured twelve kilometres a second. A cluster deep in the core is therefore moving slowly through a sea of fast stars, and the formula’s bracket — the fraction of stars slower than the cluster — goes as the cube of the ratio of the two speeds. That is a steep suppression, and it is real: a body that is slow relative to its surroundings raises a weak wake.

But a suppression is not a stop. The formula at every radius gives a positive drag, and a body under a positive drag keeps losing angular momentum, however slowly. In the formula’s world, a cluster that reaches the core creeps inward for ever, reaching the centre in a time that is long but finite. That is what the core curves in the figures do.

What simulations find instead #

N-body simulations of clusters sinking into haloes with constant-density cores find something qualitatively different. Outside the core, the cluster sinks as the formula predicts, to within the usual uncertainty in the logarithm. On reaching the region of the core, the sinking stops. The cluster settles onto an orbit at a fraction of the core radius and stays there, for as long as the simulations are run. Several studies have found the cluster pushed slightly outward after stalling — a buoyancy, where the formula admits only drag.

The stall was published in 2006 under a title that called it a failure of the Chandrasekhar formula, and the description is accurate. No setting of the Coulomb logarithm reproduces it, because the logarithm multiplies the drag and the drag needs to vanish. No adjustment of the velocity distribution does either. The formula has a structural assumption that fails in a core, and the stall is what happens when it does.

The assumption is that the stars passing the cluster are on orbits unrelated to its own. Chandrasekhar derived the drag for a body moving in a straight line through an infinite uniform medium: each star comes in from infinity, is deflected once, and leaves. In a galaxy, the stars are on orbits, and the cluster is on one too, and a star deflected by the cluster comes back round and meets it again. The drag that survives repeated encounters is the drag from stars whose orbits are out of step with the cluster’s, so that the encounters do not repeat in the same configuration and undo each other. In the language of the resonant formulations of dynamical friction, the drag comes from resonances between the cluster’s orbital frequency and the stars’ — and resonances require a spread of frequencies to resonate with.

Every orbit with the same period #

That spread is what a core removes.

In a region of constant density, the gravitational force grows linearly with the distance from the centre, exactly like the force of a spring. A particle in such a potential oscillates harmonically: every orbit, whatever its size, shape or orientation, has the same period. An orbit grazing the centre and a circular orbit at the core’s edge take the same time to go round.

A cluster in a cusp is surrounded by stars whose orbits turn faster inside its own and slower outside, so the material it disturbs is continually sheared away from it and replaced with fresh material: the wake trails, and pulls back. A cluster inside a harmonic core is surrounded by stars that all turn at its own rate. The disturbance it raises does not shear away. Stars that it deflects come back in step and are deflected back, and over an orbit or two the net torque averages to nothing. The more careful analyses describe the cluster as sitting at a corotation resonance with the whole core, where the torques from stars on either side balance.

That is why the local formula gets the wrong answer and not merely an imprecise one. It reads the density and the velocity dispersion, which in a core are perfectly ordinary, and it does not read the frequency structure, which is degenerate. The information the drag depends on is not local.

The innermost cluster #

The figure for Fornax’s most massive cluster makes the difference concrete.

On the formula alone, even the cored halo cannot keep this cluster. With the stall, it can, and so can every other cluster in Fornax, at radii comparable to the ones observed. The full argument — clusters of these masses, at these radii, in a halo of this mass — goes through only for a core of a few hundred parsecs or more and only because the formula fails inside it.

There is a second dwarf with the same problem: a single cluster near the centre of the faint satellite Eridanus II, a galaxy so small that its dark-matter density is set by very little else. Its cluster sits where friction in a cusp would have brought it to the centre quickly, and the same argument, made with one cluster rather than five, favours a core there too — in a galaxy with so few stars that bursts of star formation are hard to invoke as the thing that made the core. If the stall argument holds, that is a constraint on the dark matter itself rather than on how the galaxy’s stars rearranged it.

What the clusters measure #

The timing argument is a measurement of the halo’s central density, made with clocks rather than with velocities, and it is worth being explicit about what it can and cannot say.

It can say that a cusp as steep as the cold-dark-matter simulations predict, normalised to the mass Fornax’s stars require, would have brought at least two of the five clusters to the centre within their ages, with no fine-tuning of anything but their birth radii. It can say that a core of a few hundred parsecs would let all five survive — through the formula’s weakening for the outer three, and through the stall for the inner two.

It cannot say that the core is made of dark matter that is anything other than cold. The cores of dwarf galaxies are argued over in the essay on the innermost kiloparsec, and one proposal is that repeated bursts of star formation drive gas in and out of the centre fast enough to heat the dark matter and flatten the cusp. Fornax has had an extended star-formation history, and such a mechanism is not excluded there; the clusters would then be telling the history of the galaxy’s gas rather than the nature of its dark matter.

And it cannot rule out the coincidence entirely. Clusters born further out, at the right radii, could have sunk to where they are just now. The timing argument rests on that being improbable for five clusters at once, which it is, and on Fornax having no remnant of earlier arrivals at its centre, which it does not.

The same stall in other places #

Fornax is the cleanest case because its clusters are clocks with known ages, but the frequency argument is general, and it turns up wherever something massive sinks into something with a flat centre.

Bars in disc galaxies. A rotating bar in a disc galaxy loses angular momentum to the dark halo it turns inside — the same exchange, through the same resonances, with a rigid pattern in place of a cluster. The slowdown can be fast: in simulations with a dense, cusped halo a bar is braked within a few rotations to the point where its corotation radius lies far beyond its own end. Most measured bars are fast, with corotation just beyond the bar’s end, and that has been used as an argument that the haloes of barred galaxies are not dense at their centres, or are dominated by the disc there — the argument this essay makes with clusters, made with a pattern whose speed sets where its resonances fall. It is contested for the same reason: a bar can also be spun up by gas, and the halo’s density is one of several things that set the slowdown.

Black holes in cored galaxies. The most massive elliptical galaxies have cores in their stellar light, flat centres hundreds of parsecs across, usually attributed to a pair of supermassive black holes that ejected the stars at the centre as their orbit shrank. A third black hole arriving later, or one displaced by the recoil of a merger, sinks back through a core that its predecessors made. Simulations of that return find it slowing in the core, oscillating there for longer than the local formula predicts — the same stall, applied to a body ten thousand times heavier in a galaxy ten thousand times larger.

Buoyancy. In several of the cored simulations, a cluster that has stalled is not merely held but pushed gently outward. The formula has no way to produce this, since its drag always points against the motion. A resonant calculation does: once the stars near the cluster’s orbit have been rearranged by it, the torque they exert can change sign. The effect is small and depends on details of the core, but it is the clearest sign that the local picture of a wake pulling back on a body has stopped describing what happens.

Where the model stops #

Circular orbits only. The sinking curves follow circular orbits. Real clusters have eccentric orbits, on which the drag acts most strongly at pericentre, and eccentric orbits sink faster and also circularise as they sink. The times would shorten somewhat in both haloes; the comparison between them would not change.

A Maxwellian at one dispersion. The formula’s bracket assumes the stars and dark matter near the cluster have a Maxwellian distribution of velocities with the measured dispersion. In a cusp the dark matter’s dispersion falls towards the centre and in a core it is roughly constant, and the models here use one value throughout. That is closer to right for the core than for the cusp.

Projected radii. The present radii are projections onto the sky and are lower bounds. A cluster seen at a quarter of a kiloparsec could be at half a kiloparsec, which roughly triples its sinking time in the cusp — still short of its age.

The stall is drawn, not computed. The shaded regions mark where simulations find the sinking stops; the curves inside them are the formula’s, and the figures do not draw the stalled orbit. A calculation of the stall needs either an N-body simulation or the linear-response theory of the torque, and the claim here rests on the published simulations rather than on anything the figures compute.

The surprising connection #

The property that stops the drag is the same one that makes a harmonic oscillator’s period independent of its amplitude — the property that made pendulum clocks possible and that distinguishes Hooke’s law from every other force law in mechanics. Inside a uniform sphere of matter, gravity is Hooke’s law: it is the tunnel through a uniform planet, in which anything dropped from any height arrives at the far side in the same forty-two minutes. Every body inside such a sphere keeps the same time, and a body that keeps the same time as everything around it cannot be dragged by them, because nothing about its surroundings is moving relative to it in the way a drag requires.

A dwarf galaxy’s dark matter, if it has a core, is a clock in which every orbit ticks at once. The five clusters of Fornax have been ticking with it for ten billion years, and their survival is a measurement of how flat the centre of a galaxy is — taken not from any star’s velocity but from a drag that did not happen.

Still open: what a stalled cluster does next #

The simulations that show the stall run for a few billion years and find the clusters parked at a fraction of the core radius, sometimes pushed slightly outward. What happens over a Hubble time is less clear. A core is not perfectly harmonic at its edge, the cluster itself loses mass to the galaxy’s tide and so feels less drag, and the core may not have existed at all when the clusters formed — if bursts of star formation built it, it grew over billions of years under clusters that were already sinking. Whether the clusters could have sunk partway in an early cusp and then been frozen in place when the core formed, and whether their present radii record when that happened, is a question the clusters might answer if their orbits, rather than their projected positions, were measured.