Astronomy & Celestial Mechanics
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Galaxies

A halo weighed by counting its oldest clusters

The globular clusters round a galaxy add up to about four parts in a hundred thousand of the mass of its dark-matter halo — in a dwarf, in the Milky Way, in a giant elliptical and in a cluster of galaxies, across five decades of halo mass. The galaxy's own stars follow no such rule. So a count of a few dozen old star clusters weighs a halo that its stars cannot, and the strangest thing about the rule may be that it is less strange than it looks.

Assumes Dark matter, Halo profiles and Galaxy populations.

15 min read 6 figures Scales that defeat intuitionMeasuring the unreachable

The mass of a galaxy’s dark-matter halo is the quantity every theory of galaxy formation starts from and the one hardest to measure. Rotation curves reach only as far as there is gas to rotate, and three mass models can fit the same curve. Weak lensing of background galaxies measures the halo directly, but only averaged over thousands of foreground galaxies at once. The motions of satellites and of stars in the outer halo measure the enclosed mass for the few galaxies near enough to have them. For most individual galaxies the halo mass is inferred from the stars: the more stars, the bigger the halo, according to a relation calibrated on the statistics of large surveys.

That relation is not a straight line, and the reason it bends is the most important fact about galaxy formation. Halos of about a trillion solar masses, like the Milky Way’s, have turned a few per cent of their mass into stars, the most of any. Smaller halos have turned much less, because supernovae blow their gas out of their shallow potential wells, and larger ones have turned much less too, because their gas is too hot to cool and the black holes at their centres keep it hot. The mass that is not the light is not a fixed multiple of the light anywhere.

Globular clusters are another matter. They are the oldest bound groups of stars in any galaxy, their ages read off the bend where their stars leave the main sequence, each a few hundred thousand suns packed into a few parsecs, formed in the first two or three billion years and since orbiting in the halo. And the total mass in a galaxy’s clusters turns out to be a fixed fraction of the halo’s mass — the same fraction, to within a factor of two, from dwarf galaxies to the giant ellipticals at the centres of clusters of galaxies.

A straight line beside a bent one

The fraction was found by comparing the globular-cluster systems of galaxies with halo masses measured independently, from lensing, from the X-ray-emitting gas in groups and clusters, and from the kinematics of satellites. Its value is about four parts in a hundred thousand.

The clusters scale with the halo and the stars do not. The total mass in a galaxy's globular clusters (lower, with its ±0.3 dex scatter shaded) and its stellar mass (upper), against the mass of its dark-matter halo, all logarithmic. The clusters follow a straight line of slope one, 3.9×10⁻⁵ of the halo, across five and a half decades. The stars follow a curve that bends at both ends round a peak near 5.8×10¹¹ M☉, because star formation is suppressed in small halos and in large ones. Stars outweigh clusters by 12 times in a halo of 10¹⁰ M☉, 879 times at 10¹², 31 times at 3.2×10¹⁴.
Fig. 1 The total mass of a galaxy’s globular clusters (lower, with its ±0.3 dex scatter shaded) and its stellar mass (upper) against halo mass. The clusters follow a straight line of slope one, 3.9×10⁻⁵ of the halo; the stars bend round a peak near 6×10¹¹ M☉. The Milky Way’s hundred and sixty clusters sit on the line.

The two curves in the figure carry the whole argument. The globular clusters follow a straight line of slope one: double the halo, double the cluster mass. The stars do not. Below the peak they rise steeply with halo mass, about as its second power, and above it they rise slowly, as about its half power. In a dwarf with a halo of ten billion solar masses the stars outweigh the clusters by about twelve times; in the Milky Way, by nearly nine hundred; in the giant galaxies at the centres of groups, by thirty again. The ratio of stars to clusters changes by a factor of seventy across the range, and all of that change is in the stars.

The Milky Way’s numbers make the relation concrete. Its hundred and sixty or so globular clusters, at a typical mass of a quarter of a million suns, add up to about forty million solar masses. Divided by four parts in a hundred thousand, that is a halo of about a trillion solar masses — the value that the age of the universe and Andromeda’s approach and the motions of the Milky Way’s outer stars give independently. A census of old star clusters, made with a telescope and nothing else, lands within the range of halo masses that the dynamics allow.

A U-shape that is only a ratio

Long before the halo masses were known, the globular clusters were compared with the light of their galaxies, and the comparison seemed to demand explanation. The number of clusters per unit of the galaxy’s luminosity, called the specific frequency, is high in dwarf galaxies, low in spiral galaxies like the Milky Way, and high again in giant ellipticals. Galaxies at both ends appeared to have formed clusters with a special efficiency that spirals lacked.

The U-shape that is only the ratio of two relations. The specific frequency — globular clusters per unit V-band luminosity, normalised to a galaxy of absolute magnitude −15 — against stellar mass, both logarithmic, predicted with no free parameter from a cluster mass proportional to halo mass and the stellar-to-halo relation. It falls from 19 in dwarfs of 3.2×10⁷ M☉ to a minimum of 0.7 at 2×10¹⁰ M☉ and rises again to 17 in the most massive ellipticals, of 3.2×10¹¹ M☉ in stars. The shape long read as evidence that dwarfs and giant ellipticals formed clusters unusually efficiently says instead that they formed stars inefficiently.
Fig. 2 The specific frequency of globular clusters — clusters per unit V-band luminosity — against stellar mass, predicted with no free parameter from a cluster mass proportional to halo mass and the bent stellar-to-halo relation. It falls from about 19 in dwarfs to 0.7 near 2×10¹⁰ M☉ and rises to about 17 in the most massive ellipticals.

The figure reproduces the U-shape with nothing in it but the two relations of the first figure. Clusters proportional to the halo, stars bent round the peak: divide one by the other and the specific frequency is lowest where stars are made most efficiently and highest wherever they are not. The dwarfs and the giant ellipticals did not make clusters unusually well. They made stars unusually badly, and the clusters, which track the halo, stand out against a dim galaxy. The minimum of the curve sits at the stellar mass where the stellar-to-halo ratio peaks, a little below the Milky Way’s, which is where the observed minimum of the specific frequency also lies.

That is a satisfying result because it replaces a property of clusters that needed explaining with a property of star formation that had been measured independently. It is also a warning about the specific frequency as a quantity: a ratio of a tracer of the halo to a tracer of star formation will show structure wherever star formation does, and the structure says nothing about the clusters.

When a count weighs better than a census

If the clusters track the halo and the stars do not, the clusters ought to be the better scale wherever the stellar relation is flat. The uncertainty in a halo mass inferred from either depends on two things: the scatter of the relation, and its slope, since a flat relation turns a small error in the measured quantity into a large error in the inferred one.

Which scale weighs a halo better. The uncertainty in a halo's mass, in dex, inferred from its stellar mass (with 0.15 dex of scatter about the stellar-to-halo relation, divided by that relation's logarithmic slope) and from its globular clusters (0.3 dex of intrinsic scatter, plus the counting noise of 2.5×10⁵ M☉ clusters), against halo mass. Below the peak the stellar relation is steep and stars weigh the halo to 0.06 dex at 3.2×10¹⁰ M☉; above it the relation flattens and a 0.15 dex error in stars becomes 0.38 dex in halo at 3.2×10¹³ and 0.38 at 2.5×10¹⁴. The clusters give 0.36, 0.30 and 0.30 dex: worse than the stars below about 1.7×10¹² M☉, better above it, and steady across the whole range because their relation never bends.
Fig. 3 The uncertainty in a halo mass inferred from the stellar mass (0.15 dex of scatter, divided by the relation’s slope) and from the globular clusters (0.3 dex of intrinsic scatter plus counting noise), against halo mass. Stars weigh dwarfs to 0.06 dex but only to 0.38 dex above 10¹³ M☉; the clusters give about 0.3 dex throughout, better above about 2×10¹² M☉.

Below the peak the stars are excellent: the stellar mass rises so steeply with the halo that even a scatter of a factor of 1.4 in stellar mass pins the halo to fifteen per cent. Above the peak the stellar relation flattens and the same scatter becomes a factor of two and a half in the halo. A giant elliptical’s stellar mass, however precisely measured, says little more than that its halo is somewhere between a few trillion and a few tens of trillions of solar masses — which is why the halos of such galaxies are weighed instead from the random motions that hold them up, from hot gas, or from lensing, wherever any of those is available. The clusters’ scatter, about a factor of two at fixed halo mass, is worse than the stars’ below the peak, and never improves. But it never gets worse either, because the relation never flattens. In groups and clusters of galaxies, where the stars are a poor scale, the clusters weigh the halo a little better than the stars do — and do so from a count of point-like sources that can be made at distances where neither rotation curves nor satellite kinematics are available.

The scatter itself is the important uncertainty, and it is not well known. Estimates range from about 0.2 dex in the cleanest samples to 0.3 or more when the halo masses are themselves uncertain, and a measurement of scatter is only as good as the independent halo masses it is measured against.

Which scale weighs a halo better. The uncertainty in a halo's mass, in dex, inferred from its stellar mass (with 0.15 dex of scatter about the stellar-to-halo relation, divided by that relation's logarithmic slope) and from its globular clusters (0.2 dex of intrinsic scatter, plus the counting noise of 2.5×10⁵ M☉ clusters), against halo mass. Below the peak the stellar relation is steep and stars weigh the halo to 0.06 dex at 3.2×10¹⁰ M☉; above it the relation flattens and a 0.15 dex error in stars becomes 0.38 dex in halo at 3.2×10¹³ and 0.38 at 2.5×10¹⁴. The clusters give 0.28, 0.20 and 0.20 dex: worse than the stars below about 8.7×10¹¹ M☉, better above it, and steady across the whole range because their relation never bends.
Fig. 4 The same comparison if the clusters’ intrinsic scatter is 0.2 dex. The cluster count then beats the stars above about 9×10¹¹ M☉ and approaches the stars’ precision in dwarfs, limited there by counting only a handful of clusters.

At 0.2 dex the clusters overtake the stars from about a trillion solar masses upward, and in the smallest dwarfs the limit becomes the counting itself: a dwarf with a halo of ten billion suns has one or two globular clusters, and Poisson noise on two is a factor of two. Below that mass the method has nothing to count. The cluster relation is a scale for galaxies with at least a dozen clusters, which means halos above about a hundred billion solar masses.

The galaxies with too few stars to weigh

One kind of galaxy shows the method at its most useful: the ultra-diffuse galaxies, as large as the Milky Way but with a hundredth or a thousandth of its stars. Their stellar masses, a hundred million suns, would put them in dwarf-sized halos on the stellar relation. Their sizes suggest larger ones. And the stellar relation cannot settle which, because it was calibrated on ordinary galaxies, and an ultra-diffuse galaxy is by definition not ordinary.

Their globular clusters can. Some ultra-diffuse galaxies in the Coma cluster carry tens of globular clusters — as many as galaxies a hundred times brighter — which on the cluster relation implies halos of hundreds of billions of solar masses, approaching the Milky Way’s: galaxies that failed to make stars in an otherwise ordinary halo. Others carry a handful, consistent with dwarf halos puffed up by some internal process. The clusters divide the population into two kinds that the stars alone cannot separate. And at least two galaxies near NGC 1052 with few stars and unusually luminous clusters appear to have almost no halo at all, which is either a striking failure of the relation or evidence that they formed in a collision rather than in a halo of their own.

A census of clusters in an ultra-diffuse galaxy is a hard measurement — they are faint, a few per galaxy, mixed with background sources and with the clusters of the surrounding cluster of galaxies — but it is an image and a count, made with a single telescope, against a halo mass that would otherwise need spectroscopy of the faint stars themselves.

Why should old clusters know about dark matter?

That a population of star clusters formed in the first few billion years should know, today, the mass of the halo round it to a factor of two is puzzling at first sight. Clusters form in dense gas, their formation depends on the pressure of the interstellar medium, and many are destroyed by tides in the first billion years. Nothing in that list mentions dark matter.

One family of explanations says the clusters formed very early, before the feedback that bends the stellar relation had begun to act, when the gas in every halo was a fixed fraction of its mass and cluster formation tracked the gas. On this view the linear relation is a fossil of a time when everything scaled with the halo, preserved because the clusters, once formed, were rarely added to.

The other family says the linearity is not formation physics at all but arithmetic. A massive halo is assembled from many smaller ones, and its clusters are the sum of theirs. Whatever relation the small halos had, summing many of them averages it.

Merging averages the clusters into a straight line. Halos built by merging progenitors of 3.2×10⁹ M☉, each carrying a globular-cluster mass that scatters by 0.6 dex about 3.9×10⁻⁵ of its halo, 60 seeded realisations at each final mass: the typical ratio, averaged in the logarithm (line), and its ±1σ spread (band), in dex relative to the mean ratio, against final halo mass. A single progenitor scatters by 0.51 dex; a halo of 10¹² M☉ built from 316 of them by 0.048; one of 10¹⁴ by 0.008. A lone small halo typically sits below the mean, because a few rich ones carry it; summing many independent ratios forces the total towards their average, so a tight linear relation in massive halos is what assembly predicts whatever the scatter in the small ones — and the relation's tightness above 10¹² M☉ is therefore weaker evidence for any formation law than it looks.
Fig. 5 Halos built by merging progenitors of 3×10⁹ M☉, each with a cluster-to-halo ratio scattered by 0.6 dex, sixty seeded realisations at each final mass: the typical ratio (line) and its spread (band). One progenitor scatters by half a dex; a halo of 10¹² M☉ built from three hundred of them by 0.05; one of 10¹⁴ by 0.02.

The simulation in the figure takes small halos with a wildly scattered ratio of cluster mass to halo mass — a factor of four either way — and builds larger halos by adding them together. The spread in the ratio falls as the square root of the number of progenitors, and by a trillion solar masses it has fallen to a tenth of a dex; by a hundred trillion, to a twentieth. The mean ratio converges on the average of the progenitors’ ratios, which makes the relation linear by construction. A lone small halo typically sits below that average, because the average is carried by the few progenitors that happen to be rich in clusters, which is also why the smallest galaxies show the largest scatter in the real relation.

Merging averages the clusters into a straight line. Halos built by merging progenitors of 3.2×10⁹ M☉, each carrying a globular-cluster mass that scatters by 1.2 dex about 3.9×10⁻⁵ of its halo, 60 seeded realisations at each final mass: the typical ratio, averaged in the logarithm (line), and its ±1σ spread (band), in dex relative to the mean ratio, against final halo mass. A single progenitor scatters by 1.03 dex; a halo of 10¹² M☉ built from 316 of them by 0.176; one of 10¹⁴ by 0.078. A lone small halo typically sits below the mean, because a few rich ones carry it; summing many independent ratios forces the total towards their average, so a tight linear relation in massive halos is what assembly predicts whatever the scatter in the small ones — and the relation's tightness above 10¹² M☉ is therefore weaker evidence for any formation law than it looks.
Fig. 6 The same assembly from progenitors whose ratio scatters by 1.2 dex, a factor of sixteen either way. The typical small halo now sits far below the mean, but by 10¹² M☉ the scatter has still fallen below a fifth of a dex and the relation is linear.

Even with an absurdly scattered starting point, a factor of sixteen either way, the assembled halos converge on a tight linear relation by the Milky Way’s mass. That is the uncomfortable part of the argument for anyone who wants the relation to be a clue to how clusters formed: above a trillion solar masses, almost any formation law gives a straight line, because the halos there are built from hundreds of pieces. The relation in massive halos is evidence that they were assembled hierarchically, which was not in doubt. The information about how globular clusters formed is in the dwarfs, where the halos have had few mergers and the scatter has not been averaged away — and the dwarfs are where the counting is noisiest.

The arithmetic version leaves one real question standing: why the ratio in the small progenitors averages to four parts in a hundred thousand and not some other number. That number is set by how efficiently the earliest, smallest halos turned gas into dense, long-lived clusters and how many survived the tides of their first billion years, which is formation physics. The straight line is assembly; its height is not.

What the clusters are not telling

The cluster count is a statement about a galaxy’s past as much as about its present. A galaxy that has accreted many satellites has accreted their clusters, which is part of why giant ellipticals have so many, and why the outer halo of the Milky Way contains clusters that arrived with dwarf galaxies now dispersed into streams. A galaxy whose clusters have been stripped by a neighbour, or dragged into its centre by the drag a massive object feels moving through the halo, has fewer than its halo would predict. The clusters also evolve internally: two-body encounters slowly evaporate them, a system that gets hotter as it loses energy losing its lightest stars to the halo. The mass in clusters today is less than at formation by an uncertain factor, probably two to several, which is absorbed into the fraction without being explained.

Each of these processes moves an individual galaxy off the relation, and their sum is part of the scatter. What is striking is that, together, they do not destroy the relation. Whatever the clusters have done since they formed, they have done it in proportion to the halo.

A relation that any theory of dark matter has to keep

The cluster relation is also a constraint on alternatives to dark matter, and a peculiar one. Theories that replace the halo with a modification of gravity tie the dynamics of a galaxy to its visible mass — one acceleration scale that accounts for every galaxy’s missing mass in rotation curves. In such a theory there is no halo for the clusters to track; there is only the baryonic mass, stars and gas, which follows the bent relation of the first figure. A fixed ratio of cluster mass to an inferred dynamical mass would then be a coincidence that has to be manufactured separately in every mass range: in dwarfs, where the gas dominates the baryons; in spirals; and in groups and clusters of galaxies, where the dynamical masses inferred from modified gravity still fall short and need some unseen matter anyway.

None of that rules the alternatives out. It does mean that the cluster relation is a test that the halo picture passes almost by construction — clusters formed in halos and halos merge — and that a theory without halos must explain separately. The relation’s very simplicity, which the assembly argument shows is partly arithmetic, is easy to reproduce with halos and hard without them.

The same simplicity gives a practical tool in the opposite direction. Where a halo mass is known from lensing or dynamics, the clusters it should contain are known too, and a galaxy with far more or far fewer clusters than its halo predicts is a galaxy with an unusual history — one that has been stripped, or one that was assembled from unusually cluster-rich pieces. Stars that weigh and do not shine in a galaxy’s halo are inferred from the gap between its light and its dynamics; clusters that should be there and are not are a gap of the same kind, read in a census rather than a spectrum.

A fitted relation, a straight line and an idealised merger tree

The stellar-to-halo relation used here is one published fit; others differ by a factor of two in the stellar mass at the high-mass end, which moves the bend in the specific frequency but not its shape. The cluster relation is drawn as exactly linear with a log-normal scatter, while the data at the lowest masses suggest a slight steepening and a scatter that grows. The merging simulation is the simplest possible version: equal-mass progenitors, no mass in the progenitors other than their halos, no clusters formed after the progenitors merged and none destroyed. Real halos grow from a spectrum of progenitor masses and accrete smooth matter as well as lumps, and the averaging is slower when one progenitor dominates. The argument survives these simplifications; the specific numbers in the figures do not depend on them very much, but they do depend on them.

Still open: whether the line holds in the smallest halos

The relation is measured best between ten billion and a thousand trillion solar masses, and most firmly above a hundred billion, where there are clusters enough to count. Below that, the dwarf galaxies with one or two clusters are where formation and assembly could be told apart — where a linear relation would mean that even the smallest halos made clusters in proportion to their mass, and a bent one would mean that the line in massive halos is an average and nothing more. The census of the faintest dwarfs round the Milky Way and Andromeda, and of isolated dwarfs in the Local Volume, is now reaching the point where the presence or absence of a single cluster in a halo of a billion solar masses can be set against that halo’s mass from the motions of its stars. If the smallest halos carry clusters in the same proportion, the relation is telling something about the first star clusters that no other observable records. If they do not, it is the cleanest demonstration there is that hierarchical assembly turns any law into a straight line.

Where this goes next

The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection. The mass that is not the light A galaxy's mass-to-light ratio is not a number, it is a curve, and it rises without turning over. What stellar populations can plausibly weigh sets a ceiling; the dynamics sit far above it, and the gap is a shape rather than a discrepancy. Two places one tidal dwarf could sit. The radial acceleration relation — observed acceleration against the acceleration the baryons alone produce, logarithmic, in m/s² — with the fitted relation for ordinary disc galaxies (solid) and the line of equality (dashed). The marks are a tidal dwarf galaxy of 10⁹ solar masses of gas and stars in an exponential disc of scale length 1.5 kpc, sampled every half kiloparsec to 6 kpc. If it carries no dark matter, as debris pulled from a disc should not, its points lie on the dashed line (open). If the relation is a law that every galaxy obeys, they lie on the curve (filled), and at the outermost radius the acceleration is 5.6 times the baryons' — a rotation speed 2.37 times higher. Every point is below a₀, where the two predictions are furthest apart. A galaxy born without a halo When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference. Three initial mass functions, normalised to the same light. The number of stars per logarithmic interval of mass, on a logarithmic axis, for three initial mass functions — Kroupa, Salpeter, bottom-heavy — each normalised to the same number of stars above one solar mass, so that the stars which make an old population's light are the same in each. They differ only below a solar mass. In a 10 billion year old population the stars below half a solar mass hold 62 per cent, 78 per cent, 91 per cent of the living stellar mass, and they give 15.8, 24.2, 41.7 per cent of the light of the main sequence, which is itself a third of the population's. The difference between them is almost entirely mass that emits almost nothing. Stars that weigh and do not shine Most of the mass in an old stellar population is in stars too faint to contribute to its light. How many of them there are is set by the mass function stars are born with, and the mass-to-light ratio every dark-matter measurement divides by can be wrong by a factor of two without a single colour changing.

The objects this essay names

Each one links to every other essay that touches it.

Dark matterDark matter haloGlobular clusterHierarchical assemblySpecific frequencyStellar to halo mass relationUltra-diffuse galaxyWeak lensing