A cluster that boils itself away

A globular cluster looks like the most permanent object in the sky. A hundred thousand stars, bound, twelve billion years old, sitting in the halo of a galaxy where nothing much happens.

It is dissolving. Not from any external cause — although those exist and matter — but because a self-gravitating system of point masses cannot be in equilibrium in the sense a gas in a box can. It has a tail of stars moving faster than escape speed, it loses them, and the loss makes the situation worse rather than better.

The word “boils” is doing real work in the title and is not a metaphor for something else. The escape of stars from a cluster is evaporation in the ordinary sense: a thermal distribution against a finite escape energy, with the tail leaving and being replenished. What makes it unlike a puddle is that the container is made of the same stuff as the contents, so removing some changes the container.

Relaxation, and why it is slow but not slow enough #

Two stars passing at a distance exchange a small velocity kick. One such encounter is negligible; the effect accumulates as a random walk in velocity, and the time for the accumulated kicks to matter is the relaxation time.

The scaling is worth carrying, because it is what divides the whole subject in two. Each encounter changes the velocity by a small amount; the changes add in quadrature; and summing over all impact parameters gives a logarithmic factor. The result is that the relaxation time is longer than the crossing time by roughly the number of stars divided by the logarithm of it. For a galaxy with $10^{11}$ stars that is $10^{10}$ crossing times — vastly longer than the age of the universe, so a galaxy’s stars move in a smooth potential and remember nothing.

For a globular cluster with $10^5$ stars it is a few thousand crossing times, which is a few hundred million years. The dividing line between the two regimes is not a matter of size or of mass but of number, and the number required is large — which is why a galaxy with a thousand times the mass of a cluster is collisionless and a cluster is not. Over twelve billion years such a cluster has relaxed many times over, and its present state is not a memory of how it formed but a consequence of what relaxation does.

Where the escapers come from #

Relaxation drives the velocity distribution towards a Maxwellian. A Maxwellian has a tail extending to arbitrarily high speed, and a cluster has an escape speed — so at any moment a small fraction of the stars are above it, and they leave.

The fraction above escape speed in a Maxwellian at the relevant temperature is around one per cent, and it is repopulated on a relaxation time. So the cluster loses of order one per cent of its stars per relaxation time, and the evaporation time is roughly a hundred relaxation times.

The estimate has a subtlety in it that is characteristic of the subject. A star does not need to be above escape speed at some instant to leave; it needs to be above it in the outer parts of the cluster, where the escape speed is lowest, and the population there is not the same as the population in the core. Doing the calculation properly requires solving for the distribution function everywhere at once, and the answers differ from the crude one by factors of a few rather than by orders of magnitude.

For a typical globular cluster that is comfortably longer than the age of the universe, which is why globular clusters still exist. For an open cluster in the disc it is a few hundred million years, which is why open clusters do not: nearly all of them dissolve into the field, and the Sun’s own birth cluster is long gone and unidentifiable. The stripped stars leave along the tidal field’s axes and form tidal tails — thin streams stretching for tens of degrees across the sky, which are among the best tracers of the galactic potential available, because every star in one was on very nearly the same orbit. A stream is a set of test particles with a known common origin, which is exactly what measuring the galaxy from inside it otherwise lacks.

The part that is genuinely strange #

Losing stars ought to relieve the situation. It does the opposite, and the reason is that a self-gravitating system has negative heat capacity.

So: stars escape, carrying away energy. The remaining cluster is more tightly bound, contracts, and its velocity dispersion rises. A higher dispersion means a shorter relaxation time and a more populated escape tail, so the loss accelerates. There is no equilibrium to settle into and no thermostat anywhere in the system.

The same instability, elsewhere #

Negative heat capacity is not a peculiarity of star clusters. It belongs to anything held together by its own gravity, and it accounts for several things in this collection that otherwise look unrelated.

A star contracts as it radiates, and contracting makes its centre hotter — which is why a star’s interior is a thermostat running backwards and why a star that loses energy faster burns fuel faster rather than cooling down. A cluster of galaxies whose gas cools in the centre has the same problem: the cooling gas is compressed by the weight above it and heats, so the flow does not simply drain. And a satellite whose orbit decays through drag speeds up, for the identical reason at the level of a single orbit.

The common statement is that a bound gravitational system responds to energy loss by becoming more tightly bound and faster-moving. It is the one thermodynamic fact about gravity that has no laboratory analogue, and it is why gravitating systems evolve rather than settling.

Core collapse #

The instability has a specific spatial form, and it is worth following.

A cluster is not uniform. Its core is denser than its halo, so its relaxation time is shorter there, and the core equilibrates faster than the whole. Heat — which here means random kinetic energy — flows from the hot core to the cooler halo, as it would in any system with a temperature gradient.

But losing energy makes the core hotter, not cooler. So the temperature difference driving the flow increases, so the flow increases, and the core contracts without limit. This is the gravothermal catastrophe, and left alone it drives the central density to infinity in a finite time — a few hundred relaxation times, which for many globular clusters is less than their age.

Roughly a fifth of the galaxy’s globular clusters show the signature: a surface-brightness profile that rises as a power law into the centre rather than flattening into a core. Reading that off an image is not trivial — surface brightness is the quantity distance cannot touch, which helps, but crowding in a collapsed core defeats photometry at exactly the radii the diagnostic lives at.

What stops it #

The catastrophe does not actually reach infinity, and what stops it is the smallest structure in the system.

A binary star is an energy reservoir. A hard binary — one whose orbital speed exceeds the cluster’s velocity dispersion — statistically hardens in encounters with single stars: it gives up orbital energy to the passing star, which departs faster, while the binary’s own orbit shrinks. Since a shrinking bound orbit becomes more negative in energy, the binary can supply an essentially unlimited amount of kinetic energy to its surroundings.

So as the core contracts and the encounter rate rises, binaries — either primordial ones, or new ones formed in three-body encounters at the extreme densities of a collapsed core — begin to deliver energy to the core faster than it is being conducted away. The collapse halts and reverses. The cluster then oscillates: it expands as the binaries pump energy in, the density falls, the binary heating shuts off, and it recollapses. These gravothermal oscillations are seen in simulations and are, as far as anybody can tell, what a post-collapse cluster is doing.

Mass segregation, which happens first #

One further consequence of relaxation runs ahead of all of this and is observed directly.

Encounters push the system towards equipartition of kinetic energy, which means heavy stars end up moving more slowly than light ones. Slower stars sink. So the massive stars concentrate in the core and the light ones are pushed outwards and are preferentially lost — which changes the cluster’s mass function over time, from the outside in.

The timescale for a star of mass $m$ to segregate is the relaxation time scaled by the ratio of the mean mass to $m$, so a ten-solar-mass object in a cluster of half-solar-mass stars segregates twenty times faster than a typical member. That is dynamical friction acting on an individual rather than on a satellite, and it is the same expression.

That last consequence is worth its own sentence, because it has turned out to matter far beyond cluster dynamics. A dense core full of black holes forms binaries by three-body encounters, those binaries harden through further encounters, and the hardest of them are ejected from the cluster or merge. Clusters are therefore a formation channel for merging black-hole binaries entirely distinct from the evolution of isolated massive binaries, and the two channels predict different distributions of spin and eccentricity. Which of them dominates is one of the questions the gravitational-wave catalogue is being read to answer.

Equipartition is never actually reached. A system in which heavy objects have sunk to the centre has concentrated its mass, which deepens the potential, which drives further segregation — so mass segregation is itself an instability, and in a cluster with a wide range of masses the heavy component decouples and undergoes its own collapse.

Two corrections to the boiling picture #

The evaporation estimate above — a Maxwellian tail, a fraction above escape speed, a loss per relaxation time — is the standard back-of-envelope, and two things about real clusters make it wrong in ways worth knowing.

A star above escape energy does not necessarily escape. In a cluster bounded by a tidal field rather than by its own potential alone, the escape route is through the two Lagrange points, which occupy a small solid angle. A star with more than the escape energy but with the wrong angular momentum simply orbits inside the cluster, energetically unbound and geometrically trapped, sometimes for hundreds of crossing times before its orbit is deflected into the aperture. These are the potential escapers, and at any moment a few per cent of a cluster’s stars are in that condition.

The consequence is that the dissolution rate is not the Maxwellian tail divided by a relaxation time. It depends on how long an unbound star lingers, which depends on the geometry of the tidal boundary, which depends on the cluster’s orbit through the galaxy. That is why the measured dissolution times of clusters correlate with their orbits and not merely with their masses, and why a simple evaporation timescale under-predicts the lifetime of a cluster on a wide orbit.

And equipartition has a threshold. Mass segregation was described above as an instability, and the condition under which it runs away is specific: the heavy component sinks and, if it carries enough of the total mass, its own self-gravity dominates the region it has collected into, and it decouples and collapses on its own. Below that threshold the light stars can absorb the energy the heavy ones give up, and a genuine near-equipartition is reached; above it, they cannot.

The threshold is a statement about the heavy component’s total mass rather than about the mass ratio alone, which is what makes it easy to state and hard to apply — a cluster’s heaviest members are its remnants, and how many it retains is precisely the quantity the last section listed as unsettled. The dynamical fate of a cluster turns on how many black holes it kept, and that number is inferred from the dynamics it is supposed to explain.

Both corrections push the same way, and it is worth noticing which way that is. A cluster survives longer than the crude evaporation estimate says, because its unbound stars leave slowly; and it collapses less predictably than the gravothermal argument says, because whether its heavy component decouples depends on an inventory nobody can take directly. The boiling picture is right about the mechanism and unreliable about the schedule, which is the usual relationship between a thermodynamic argument and the system it is applied to.

What a cluster is worth as a laboratory #

The reason any of this is studied so hard is that a star cluster is the only self-gravitating system whose evolution can be both computed exactly and observed in detail.

Computed exactly, because a hundred thousand point masses interacting through Newtonian gravity is a problem that can be integrated directly, star by star, for a Hubble time — which nothing about a galaxy or a cosmological volume permits. The integrations are expensive and they are not approximations.

Observed in detail, because the individual stars are resolved. A globular cluster’s colour–magnitude diagram, its density profile, its velocity dispersion as a function of radius, its binary fraction and its mass function are all directly measurable, which makes it the only place where a dynamical theory can be confronted with more observables than it has parameters.

That combination is rare enough to be worth naming. Most of this collection’s subjects are measured through one or two numbers and modelled with more. Here the ratio runs the other way, and the consequence is that the disagreements between theory and observation are informative rather than merely absorbable — a simulated cluster that reproduces the density profile and fails on the mass function has been caught doing something specific.

Where the picture stops #

Stellar evolution matters as much as dynamics early on. In the first hundred million years a cluster loses a third of its mass as its massive stars die and return their envelopes, which unbinds the cluster from the outside and can destroy it outright. Almost all clusters dissolve during that episode, before any of the relaxation physics has had time to act.

Black holes complicate the ending. A cluster retaining a population of stellar-mass black holes has a heavy component that segregates, forms binaries, and heats the rest of the cluster from within — which delays or prevents core collapse entirely. Whether globular clusters retain their black holes was assumed settled in the negative for decades and is now thought to be the opposite in many cases, which changes the expected dynamical state of the whole population.

The two-body approximation to relaxation is an approximation. The standard derivation treats a star’s encounters as a series of independent two-body events with an arbitrary cutoff at the size of the system, and the resulting Coulomb logarithm is fitted rather than derived. Correlated encounters and the fluctuating global potential contribute too, and how much is a question that direct integrations answer numerically rather than analytically.

And the tidal field is not static. A cluster on an eccentric orbit through a galaxy with a bar, spiral arms and giant molecular clouds is being shocked repeatedly, and the shocks inject energy and strip stars in ways that depend on the orbit rather than on the cluster. Disentangling internal evolution from external disruption is the central difficulty in interpreting any observed cluster.

The two processes the essay compares run at rates that depend on the same uncertain logarithm, so it is worth reading both at a second value of it.

Where this ladder goes next #

Later rungs on this anchor: the Fokker–Planck description, which is how relaxation is computed rather than estimated; binary heating in detail, and the statistical theorem that hard binaries harden; gravothermal oscillations and what a post-collapse cluster looks like; tidal tails as tracers of the galactic potential; and the retention of stellar-mass black holes, which is currently the largest open question about what is inside these systems.