A population that cannot share the cluster's temperature

A dense star cluster is a gas of stars, and like a gas it relaxes toward equilibrium through collisions — not physical collisions but gravitational encounters, each deflecting a star by a tiny angle, which over a relaxation time of hundreds of millions to billions of years add up to a complete exchange of energy. One of the things relaxation does is push the cluster’s populations toward equal kinetic energy. A star twice the average mass, moving at the average speed, carries twice the average kinetic energy; encounters drain the excess, it slows, and it sinks toward the centre. The heavy members of a cluster end up concentrated in its middle, moving slowly.

That is mass segregation, and the usual picture of it is a smooth sorting by mass with the heaviest bodies at the bottom of the well. Lyman Spitzer showed in 1969 that the picture breaks down. A heavy population can only reach equal kinetic energy with the light one if it is a small enough share of the mass. Beyond a limit, the heavy bodies sink, gather in the centre, and become dense enough that their own gravity dominates there — and once their own gravity dominates, slowing down makes them sink further and speed up, and equal kinetic energy can never be reached. The heavy population decouples from the rest of the cluster and evolves as a cluster of its own.

For most of the twentieth century that was a theorem about idealised two-component models. It became one of the central facts about globular clusters when it was recognised that the clusters are full of stellar black holes.

A limit that has no physics of the heavy bodies in it #

Spitzer’s argument is a short one. A heavy population in equipartition with the stars moves slower than they do, by the square root of the mass ratio, so it occupies a smaller region in the centre — its spread in radius is set by its spread in speed. Shrink a population’s spread in radius while keeping its mass and its density rises as the cube. Eventually the heavy bodies’ own density in the centre exceeds the stars’, and from that point the heavy bodies sit in a potential well of their own making. In their own well, losing kinetic energy to the stars makes them fall further in, where the well is deeper and they move faster: a self-gravitating system gets hotter as it loses energy. Equipartition becomes impossible because the heavy population’s temperature rises as it tries to cool.

Working the virial balance through gives a condition on two numbers only: the heavy population’s share of the total mass, $M_2/M_1$, and its members’ mass relative to the stars’, $m_2/m_1$. Equipartition is possible only if

$$\frac{M_2}{M_1}\left(\frac{m_2}{m_1}\right)^{3/2} < 0.16.$$

The numerical constant comes from the shape of the cluster’s density profile and changes by tens of per cent with it; later work with more realistic models replaces the sharp line with a band. The dependence on the two ratios is robust, and it has a striking feature. Nothing about the heavy bodies except their mass enters. Whether they are black holes or binaries or massive stars, a population that is heavy enough and plentiful enough cannot share the cluster’s temperature.

The figure places the remnants of a globular cluster’s stellar population on that plane. Neutron stars are heavy but rare, and white dwarfs are plentiful but barely heavier than the stars that remain; both sit below the line. Binaries, which behave in encounters like single bodies of twice a star’s mass, sit just beyond it if they make up a tenth of the mass, and that is part of why binaries are always found concentrated toward clusters’ centres. Black holes are in a class of their own: ten times beyond the limit, with no realistic choice of numbers that brings them back.

How few it takes #

The heavy population does not have to be large to be beyond the limit, because the mass ratio enters raised to the power three halves.

A population of ten-solar-mass black holes breaks equipartition when it makes up eighteen hundredths of a per cent of the cluster’s mass, about one black hole for every eleven thousand stars. A cluster’s massive stars make roughly one black hole for every four hundred solar masses of stars formed — a few per cent of the mass — so a cluster that kept even a tenth of its black holes is beyond the limit, and it stays beyond it until all but a few dozen have gone. The subsystem stops being a subsystem only when it has almost ceased to exist.

The heaviest sink first #

Before the black holes can decouple, they have to reach the centre, and that is quick. A body sinks through a cluster by dynamical friction — the drag a moving mass feels from the wake it raises in the stars around it — at a rate proportional to its own mass, so the sinking time is the cluster’s relaxation time scaled down by the ratio of the average star’s mass to the body’s.

Black holes, twenty times an average star’s mass, sink in a twentieth of a relaxation time times a few — under half a billion years even in a slowly relaxing cluster. Neutron stars and white dwarfs take ten times longer. In every globular cluster older than a billion years, which is every globular cluster, the black holes have long since gathered at the centre and decoupled from the stars, while the lighter remnants are still on their way down in the slowest clusters.

A cluster keeps the remnants its escape speed can hold #

Whether the black holes are there at all depends on what happened at their birth. A supernova that makes a neutron star usually kicks it: the explosion is slightly asymmetric, and the neutron star recoils at hundreds of kilometres a second — the speeds measured for young pulsars. A globular cluster’s escape speed from its centre is a few tens of kilometres a second.

The neutron stars are almost all lost. At a fifty-kilometre-a-second escape speed a cluster keeps two in a thousand of the neutron stars it made — the rest are thrown out in the first few tens of millions of years, which is itself a puzzle, because globular clusters are rich in millisecond pulsars and must have kept more than this. Black holes are kicked far less. If the kick carries the same momentum as a neutron star’s, a ten-solar-mass black hole moves seven times slower, and a cluster keeps a third or more of them; if, as models of the explosions suggest, much of the stellar envelope falls back onto the black hole and cuts the kick further, the cluster keeps nearly all of them. So the remnant population that dominates a globular cluster’s centre is the black holes, not the neutron stars that outnumber them at birth.

The energy the cluster asks for #

Once decoupled, the black-hole subsystem evolves fast. It is a small, dense cluster with a short relaxation time of its own, and it core-collapses within a few of its own relaxation times, forming black-hole binaries by three-body encounters at its centre. Those binaries are hard — their binding energy exceeds a black hole’s typical kinetic energy — and a hard binary heats what is trying to cool: each encounter with a third body tightens it and throws the third body out faster than it came in.

What makes the subsystem central to the whole cluster is Hénon’s principle, which governs a core powered by binaries whatever they are made of. A cluster’s relaxation carries energy outward at a rate fixed by its global structure — about a tenth of its binding energy per half-mass relaxation time — and the centre must supply that energy or collapse. With a black-hole subsystem present, the black-hole binaries supply it. The stars never need to make binaries of their own or collapse their core; the black holes do the work, and the stellar core stays large.

The cost is the black holes themselves. Each hardening encounter that pays the cluster’s energy bill eventually ejects something, and in a black-hole subsystem the bodies in the encounters are black holes. A binary hardens until the recoil from its next encounter exceeds the escape speed, and then it leaves; the single black holes it flings out leave too.

The arithmetic is a budget. The cluster demands energy at a rate proportional to its binding energy divided by its relaxation time. Each black hole ejected supplies energy of order a black hole’s mass times the square of the escape speed, a fixed multiple of its kinetic energy in the cluster. Dividing one by the other gives an ejection rate inversely proportional to the relaxation time, and the subsystem lasts a fixed number of relaxation times. With the multiple set at sixteen the number is a little under five; the multiple is uncertain by a factor of two either way, because the energy carried off per ejection depends on how hard the binaries get before they leave.

Which clusters have spent theirs #

The depletion time is proportional to the relaxation time, and the relaxation times of the Milky Way’s globular clusters span more than two decades, from tens of millions of years in the densest to tens of billions in the most diffuse. So the clusters divide.

Clusters whose relaxation time is shorter than two or three billion years have spent their black holes within the age of the Galaxy. Once the subsystem is gone, nothing powers the centre but whatever stellar binaries the cluster has, and the cluster goes on toward the deep core collapse that the stellar binaries then halt. Clusters relaxing more slowly still hold black holes and keep large stellar cores, puffed up by the subsystem’s heating.

That is the observed division. About a fifth of the Milky Way’s globular clusters have the cusped, core-collapsed profiles of clusters that have run their centres down; they are, as a class, the clusters with the shortest relaxation times. The rest have large cores, a fact that for decades needed explaining, since every cluster’s relaxation time is shorter than its age and core collapse was expected to be common. The black-hole subsystem explains the large cores, and its depletion explains why the collapsed clusters are the fast ones. Detailed simulations of individual clusters that follow their black holes reproduce the division cluster by cluster.

The division also explains a puzzle about the clusters’ sizes. Clusters with large cores are, on average, larger overall, with longer half-mass radii, than core-collapsed ones of the same mass, and for a long time this was read as a difference in how they formed. The black-hole picture reads it as a difference in what they have been doing since: a cluster heated by a black-hole subsystem expands as the energy flows outward, and keeps expanding until the subsystem is spent. Its size today records how much heating its black holes supplied, and so how many of them it kept.

The binaries are beyond the line too #

The plane holds a smaller surprise beside the black holes. Primordial binaries — pairs of stars born bound together — behave in a cluster’s encounters as single bodies of about twice an average star’s mass, and if they make up a tenth of the mass they sit just beyond Spitzer’s limit. They too should sink, concentrate and fail to share the cluster’s temperature, and the ones that are hard should heat the centre exactly as the black-hole binaries do.

In a cluster that has black holes, the binaries cannot do this, because the black-hole subsystem has already taken the centre. It heats the stars around it, including the binaries, and keeps them from concentrating; the binaries wait in the outskirts of the core. Only when the black holes have been spent do the binaries take over — sinking, interacting, and becoming the energy source that halts the stellar core’s collapse. A cluster’s centre is powered by a succession of populations, each beyond the limit, each taking its turn in order of mass, and each spent before the next is needed. The gravitational waves from the first of them are what every pair that merges eventually sends out; the pulsars and X-ray binaries the second makes, when its neutron stars and white dwarfs finally reach the centre, are what makes core-collapsed clusters bright in X-rays.

A black hole seen in a cluster #

For most of this argument the black holes were inferred rather than seen, and through the 2000s the standard expectation was the opposite of the one here: that a cluster’s black holes would decouple, form a tiny subsystem, and eject one another within a billion years, leaving one or none. That conclusion came from the subsystem’s short internal relaxation time, and it missed that the subsystem’s rate of energy production is set by the whole cluster’s demand, not by its own relaxation. A subsystem that supplies only what is asked of it lasts as long as the cluster’s relaxation time allows.

The change of view was driven partly by detection. Radio and X-ray observations found candidate accreting black holes in several globular clusters from 2012 onward, and in 2018 radial-velocity monitoring of stars in NGC 3201 found a star orbiting an unseen companion of more than four solar masses — a black hole in a detached binary, inside a globular cluster. A few more have followed. Each is a black hole that happens to have captured a star, a small minority of what a cluster holds, and their existence says the subsystems were not all spent long ago.

The same physics applies elsewhere with different numbers. A young massive cluster in a star-forming galaxy today keeps its black holes and will spend them on the timescale its own relaxation sets; a nuclear star cluster at a galaxy’s centre, with escape speeds of hundreds of kilometres a second, keeps everything, and its black-hole binaries merge before they can be ejected. In both, Spitzer’s inequality guarantees the subsystem forms.

What the budget leaves out #

The ejection budget treats the cluster as fixed while the black holes are spent, and it is not. The subsystem’s heating expands the cluster, which lengthens its relaxation time and slows the demand, so the depletion stretches out toward the end. A cluster losing mass to the Galaxy’s tide shrinks instead, shortening its relaxation time and hastening the depletion. Both effects change the depletion time by tens of per cent and leave its proportionality to the relaxation time intact.

The black holes’ masses vary, from a few to perhaps forty solar masses, and the heaviest dominate the subsystem: they sink first, form binaries first, and are ejected last, because the lighter ones are flung out of the encounters that harden the heavy binaries. A subsystem therefore loses its light black holes first, and its last members are its heaviest — which is also what the cluster’s gravitational-wave sources would be.

And the retention fraction itself is uncertain by more than the depletion arithmetic. Whether a cluster kept a third of its black holes or nearly all of them depends on natal kicks that have been measured for neutron stars and only inferred for black holes, from a handful of black-hole binaries in the Galaxy whose space velocities are small.

Still open: how many each cluster still holds #

The inequality is certain and so is its consequence for black holes: wherever they are kept, they decouple and form a subsystem that powers the cluster until it is spent. What is not known for any individual cluster is how many black holes remain.

Several routes exist and none is decisive. The size of a cluster’s stellar core relative to its half-mass radius tracks the number of black holes in simulations, and the observed profiles can be inverted for it — though the inversion depends on the cluster’s history. The degree of mass segregation among the stars themselves is reduced when black holes are present, because the subsystem heats the stars’ centre; that is measurable with deep imaging. And the gravitational waves from black-hole mergers carry a population signature: binaries assembled and ejected from globular clusters should have spins oriented at random and include the heaviest black holes, while binaries from isolated stars should have spins aligned with their orbits. If a sizeable fraction of the observed mergers carry the cluster signature, the subsystems of the Galaxy’s own globular clusters have been producing them — and each merger seen from a distant cluster is one more black hole spent paying for a core that the stars could not have held open on their own.