A cluster's radius that records when its halo's centre softened

A massive body moving through a sea of lighter ones raises a wake behind it, and the wake’s gravity slows it. Dynamical friction is how a satellite galaxy spirals into its host and how a globular cluster sinks towards the centre of its galaxy, and in the Fornax dwarf spheroidal it poses a puzzle. Fornax has five globular clusters, each of tens to hundreds of thousands of solar masses, at radii from a quarter of a kiloparsec to a kiloparsec and a half, and in a halo whose dark-matter density rises as a cusp towards the centre the drag on them would have brought most of them to the middle within a few billion years. They are ten or more billion years old, and they are still out there.

The resolution most often offered is that the halo does not have a cusp. Inside a core of constant density, every orbit has the same frequency, the differential motion from which a wake is drawn vanishes, and the drag stops where every orbit keeps time: simulations of clusters sinking into cored haloes find them parked near the core’s edge indefinitely. A core in Fornax would explain the clusters, and it would be one more piece of evidence in the argument about the innermost kiloparsec of small galaxies.

But a core in a halo of cold dark matter has to be made, and the leading mechanism takes time. Repeated bursts of star formation drive gas out of the centre and let it fall back, and each cycle shakes the dark matter, which gains energy and spreads out. The core grows over many such cycles, over billions of years. Until it forms, the clusters are sinking through a cusp. This essay follows them through that history and finds that their present radii are a record of when the centre softened.

How a core is made, and why it takes time #

The idea that gas can reshape dark matter is a reversal of the more familiar effect, in which gas cooling into the centre of a halo deepens the potential and drags the dark matter inward, steepening the cusp. That happens when the gas settles slowly and stays. The core-making mechanism needs the opposite: gas that falls in and is then expelled suddenly, faster than the dark matter’s orbits can adjust. When the potential deepens slowly, the dark-matter orbits shrink adiabatically and give back nothing; when it then shallows abruptly, the orbits are left with more energy than the new potential can hold, and they expand. Each cycle of slow inflow and sudden blow-out ratchets energy into the dark matter.

A single cycle does little. Simulations that produce cores in dwarf galaxies need tens of bursts, each expelling a substantial fraction of the central gas, over periods of billions of years, and the energy needed scales with the core’s size and the halo’s central density. That makes core formation a process with a history, and a galaxy’s star-formation history becomes a candidate record of its core’s growth. Fornax formed stars over most of cosmic time, with its oldest populations more than ten billion years old and its youngest under a billion; its globular clusters, by contrast, are mostly old. The clusters were in place before most of the bursts that could have built the core had happened.

That ordering is the heart of the problem. If the clusters formed first and the core later, the clusters spent some interval in a cusp, and the length of that interval is what their present positions measure. It is the same logic by which a rotation curve absorbs a galaxy’s history — the present profile is the outcome of a sequence, and different sequences leave different marks — applied to a handful of point masses rather than to the whole curve.

One cluster, four histories of its halo’s centre #

Take the halo used to model Fornax — a cusp following the standard profile, normalised to the circular speed of eighteen kilometres a second that the stars’ motions give at a kiloparsec — and the alternative of the same mass with a core of 0.7 kiloparsecs. Release the most massive of Fornax’s clusters, about 360,000 solar masses, on a circular orbit at 1.2 kiloparsecs, and let Chandrasekhar’s formula for the drag decide its fate.

If the halo is cuspy throughout, the cluster reaches the centre in about five billion years. If the core is there from the start, the cluster sinks to the core’s edge and stops. In between, the history depends on when the core arrives: a cluster that has sunk partway through the cusp is caught wherever it happens to be — or, if it is already inside the future core’s radius, it is frozen there — and from then on it stays. A core arriving after about five billion years arrives too late for this cluster; it is already at the centre.

So the question is not whether a core can stop the clusters — it can — but whether it formed early enough to stop them where they are now.

Where a cluster ends up #

Running the same calculation for every birth radius gives a map from where a cluster was born to where it is now, one curve for each moment of core formation. A cluster of 180,000 solar masses born far out sinks slowly and is little affected by what happens at the centre; born close in, it either reaches the centre before the core forms or is frozen near the core’s edge. The curves for late cores bend down towards zero at small birth radii, marking the clusters that would have been lost.

For the heavier cluster the map is harsher. Drag grows in proportion to the cluster’s mass, so a cluster twice as heavy sinks twice as fast, and with a late core the whole inner kiloparsec is emptied of clusters this heavy. A cluster of that mass found today well inside a kiloparsec — which is where Fornax’s heaviest cluster is — either was caught early by the core or started far out and has sunk most of the way in.

A pile-up at the edge of the core #

The maps carry a feature worth drawing out, because it is a prediction no single cluster can test and a population can. For every core-formation time there is a wide band of birth radii that ends on the same plateau: clusters born anywhere from just outside the core to well beyond it sink until they reach its edge and stop there. A cluster born at 1.2 kiloparsecs and one born at 1.8 end within a few parsecs of each other, at the core radius. Only clusters born far out, which have not yet sunk that far, stay outside it; only clusters born inside it before it formed end up within it.

So a population of clusters in a halo that acquired a core should pile up at the core’s edge, with fewer inside and a tail outside. The pile-up marks the core radius directly, independently of the stars’ motions that are usually used to measure it. In Fornax, three of the five clusters are within about a kiloparsec in projection and the heaviest is at 0.43, which, allowing for projection, is consistent with a pile-up near a core of 0.6 to 0.8 kiloparsecs — but five clusters do not make a distribution. Dwarf galaxies with richer cluster systems, and the cluster systems of larger galaxies whose centres are cored, would show the pile-up if it is there.

The halo’s own mass profile at these radii is measured from the stars, using the one radius where the orbits stop mattering — near the half-light radius, where the mass inferred from the velocity dispersion is insensitive to whether the stars’ orbits are radial or circular. That is why both haloes in these figures are normalised at about a kiloparsec, and why the difference between them is confined to the inner region where the clusters’ fate is decided. The stars of a dwarf held up by disorder rather than rotation constrain the enclosed mass at their half-light radius well and the slope of the density inside it poorly; the clusters probe exactly the slope.

How far out a late core needs the clusters to have been born #

Turning the map round gives, for each real cluster, the birth radius it would need for each moment of core formation. The two outer clusters of moderate mass, at a kiloparsec and a kiloparsec and a half, barely notice: the drag on them is similar in either halo, they sink slowly, and they need to have been born only a few hundred parsecs further out than they are, whatever the core did.

The heaviest cluster, GC3, at 0.43 kiloparsecs in projection, is the one that notices. If the core formed within the first billion years, the cluster needs to have been born at about 0.7 kiloparsecs, close to where most of Fornax’s stars are. If it formed at six billion years, the cluster needs to have been born at about 1.4 kiloparsecs — twice the half-light radius of the stars, out in the galaxy’s thin outskirts — and to have sunk most of the way in before being caught.

Neither is impossible, but they are not equally likely. Globular clusters form in the densest gas, and the densest gas in a dwarf galaxy is near its centre. A model in which the most massive cluster formed far out in the galaxy’s periphery and drifted in has to explain why it formed there; a model in which the core formed early has to explain how feedback built a core within the first billion or two years. The cluster’s radius, read through the drag, trades one assumption for the other.

Partial sinking sorts clusters by mass #

There is a signature that does not depend on any one cluster’s birth place. Because the drag is proportional to mass, a halo that let the clusters sink for a while before stopping them leaves the heavier clusters further in than the lighter ones born at the same radius. A population of clusters born at similar radii and caught partway would show mass segregation: heavy clusters inside, light ones outside, with the separation growing with how long the sinking ran.

Fornax’s five clusters do arrange themselves somewhat this way. The lightest, GC1, at under forty thousand solar masses, is the outermost at 1.6 kiloparsecs; the heaviest, GC3, is the second innermost. But GC4, the innermost at 0.24 kiloparsecs, is of only moderate mass, and the radii are projected distances that understate the true ones by an unknown amount for each cluster. Five clusters with projected radii can suggest a pattern and cannot establish one. Larger systems of clusters in other dwarf galaxies, and three-dimensional positions from accurate distances, would turn the suggestion into a test.

GC4 is interesting for another reason. It is younger and more metal-rich than the other four, and it sits close to the centre: a cluster that formed later and closer in, perhaps after the core had formed, when nothing would drag it further. If its age could be tied to the timing of the star-formation bursts that are supposed to have built the core, it would be the one cluster whose radius was set after the change rather than before.

What a late core would have left at the centre #

The clusters that were not caught went somewhere. A globular cluster that sinks to the centre of a galaxy does not vanish; it arrives as a compact, dense ball of old stars, and several such arrivals merge into a nuclear star cluster, a bright knot at the galaxy’s centre. Many dwarf galaxies of Fornax’s mass and larger have one. Fornax does not.

That is a second clock, independent of the radii of the clusters that survived. For a population of clusters like Fornax’s, spread over the galaxy’s inner two kiloparsecs, a core from the start delivers none to the centre; a core at three billion years delivers about a sixth; no core at all delivers half. If Fornax began with more clusters than the five it has — which the metal-poor stars of its field population, some of which may be the debris of dissolved clusters, could indicate — the absence of a nucleus bounds how many were lost, and so how late the core can have formed. Both clocks, the radii of the survivors and the absence of the sunk, point the same way: early.

What the drag is computed from #

The drag is the local formula Chandrasekhar derived, with the density and velocity dispersion at the cluster’s position and a fixed Coulomb logarithm of three — a logarithm nobody can pin down to better than a factor of order two in a system this small. A larger logarithm sinks everything faster and makes the constraint on an early core tighter; a smaller one relaxes it. The halo models are those of the essay on the stall: a cusp with a scale radius of 1.5 kiloparsecs and a cored isothermal sphere with a core of 0.7, both normalised to the same circular speed at a kiloparsec, which is what the stars’ velocities measure.

The switch from cusp to core is instantaneous in these figures. A real core grows gradually, and its central density falls over billions of years, so a cluster in the inner region feels a drag that weakens progressively rather than switching off. The effect is to smear the core-formation time over the growth period: what the radii record is roughly the time by which the core had become shallow enough to stop the drag, not the time it began to form. The stall itself is imposed at the core’s edge, following the simulations, rather than computed, since the local formula cannot produce it.

What the figures leave out #

The clusters are on circular orbits throughout; real clusters are on eccentric orbits, which sink at different rates and circularise as they go. Their masses are fixed, whereas a cluster in a dwarf galaxy loses stars to the galaxy’s tide and to its own internal evolution, and was probably several times more massive when it formed — which would have made it sink faster early on, strengthening the argument. The radii compared are projected, and the true radii are larger by an unknown factor for each cluster. The galaxy’s own history — its accretion, any interaction with the Milky Way, the tidal field it orbits in — is ignored. And the population used for the delivered fraction is a uniform grid of masses and radii, not a model of how clusters form.

Still open: whether the core is old, young or absent #

The radii and the absent nucleus favour an early core, and an early core is hard to make. Feedback-driven cores in simulations of dwarf galaxies form over the period of bursty star formation, which in Fornax lasted for much of its history — its stars span ages from more than ten billion years to under a billion. A core in place within the first billion or two years needs most of the energy input to have come early, from the first generations of stars, which Fornax’s star-formation history does not obviously show.

The alternatives each move a different piece. If the dark matter is not cold — if it interacts with itself, or is light enough to have wave-like structure on kiloparsec scales — a core can exist from the start, with no feedback needed, and the clusters’ survival is natural. If the clusters were born much further out, perhaps in smaller systems that merged into Fornax, a cusp can be kept and the clusters arrive late. And if Fornax’s halo is less massive in the centre than the models here — which the stars’ motions across the sky, measured from space, may decide — the drag is weaker everywhere and the timing problem loosens. The clusters’ radii do not choose among these on their own. What they do is turn a qualitative puzzle — why the clusters survived — into a quantitative constraint on when the centre of a dwarf galaxy changed, which any account of the change has to meet.