A pair that heats what is trying to cool

A self-gravitating system gets hotter as it loses energy. That single fact makes a star cluster unlike anything with a positive heat capacity: heat flows from its core to its halo, the core loses energy, the core therefore heats up, and the flow accelerates. There is no equilibrium at the end of that road. The core contracts without limit, on a timescale that is finite, and the process is called core collapse. It is the same runaway that makes a cluster evaporate from the outside while its centre tightens, read from the inside instead.

Something stops it, because clusters are still here. What stops it is that a cluster is not made only of single stars, and a pair of stars in a tight orbit is a reservoir of energy far deeper than anything the cluster’s own motions can supply.

The crossing at about eight astronomical units is the hard–soft boundary, and everything in this essay is a consequence of the arrows on either side of it pointing outward.

Heggie’s law, stated as thermodynamics #

The rule is usually quoted as four words: hard binaries harden, soft binaries soften. What makes it more than a mnemonic is that it is a statement about which way heat flows between two systems at different temperatures.

A binary’s temperature is its orbital energy per unit mass, which for a circular pair is fixed by its separation and nothing else — one equation gives the speed anywhere on the orbit and the energy with it. The field of single stars has a temperature too, set by the velocity dispersion, and the virial theorem ties that dispersion to the cluster’s total mass. Heat flows from hot to cold, as always. A binary tighter than the boundary is hotter than the field and gives energy up; a binary looser than the boundary is colder and takes energy in.

The twist is the sign of the heat capacity. Giving energy to the field makes the tight binary tighter, therefore hotter, therefore more willing to give. Taking energy from the field makes the loose binary looser, therefore colder, therefore more willing to take, until it is not a binary at all. The boundary is a watershed and not an equilibrium: nothing settles there.

Measured rather than argued #

The thermodynamic account is persuasive and it is not a proof. It relies on averaging over encounters that are individually chaotic — three bodies interacting gravitationally have no closed-form solution and their outcomes are exquisitely sensitive to the starting conditions — so the sign of the mean is a claim about a distribution rather than about any one event.

That distribution can be measured. The way to measure it is to integrate several hundred encounters and average what they did.

The rejection step is not a nicety. A close triple approach can cost an integrator several digits, and the apparent energy change of a failed integration is of order the binding energy itself — so a handful of failures out of eighty will dominate the mean and produce a beautifully wrong figure. Discarding them is what makes the mean a mean of the physics rather than of the arithmetic.

The scatter between neighbouring points in that figure is real and it is the reason the claim asserted is about the sign and the number of sign changes rather than about the shape of the curve. Most individual encounters do something other than what the average says.

How often it happens #

An encounter has to occur before it can matter, and the rate is set by a cross-section that is not the geometric one.

The consequence runs through the whole subject. Encounters are not rare events that happen to be interesting; they are common events in the places where the speeds are low — which is exactly where binaries survive long enough to have them. A binary in the field of the Galaxy is essentially never disturbed. A binary in the core of a globular cluster is disturbed repeatedly.

What a sequence of encounters does #

Now run the process. Each encounter with a hard binary takes a roughly fixed fraction of its binding energy, not a fixed amount, so the reciprocal of the separation grows geometrically and the separation itself falls away.

That ending is the part worth carrying. The process does not run down gently. It terminates, at a computable separation, by removing the heat source from the system entirely — and by removing the single star it last interacted with as well, at a higher speed still. A cluster’s core does not merely lose energy to binary heating; it loses stars.

Where the energy is #

The numbers are worth a moment because they are counterintuitive. A globular cluster of a million stars with a dispersion of ten kilometres a second has a total kinetic energy of order the number of stars times the mean stellar kinetic energy. A single binary with a separation of a hundredth of an astronomical unit has a binding energy comparable to the kinetic energy of ten thousand of those stars.

So a handful of hard binaries carries as much energy as a substantial fraction of the whole cluster’s motions. That is why binary heating can halt core collapse using a population that is a per cent of the stars: the reservoir is deep, not wide.

The clock the whole process runs on #

None of this happens quickly, and how quickly is set by relaxation: the time it takes for the accumulated small deflections from distant encounters to change a star’s velocity by of order itself.

That last entry is why this essay is about clusters and not about galaxies. The mechanism exists everywhere; it has had time to act only in the smallest systems.

What was actually observed #

The observational case for binary heating is indirect, and it is worth being explicit about the chain.

Nobody has watched a binary harden. The separations involved are milliarcseconds at best and the timescales are millions of years. What is observed is a population.

First, the binaries are there. Globular clusters have binary fractions of a few per cent in their cores, measured by looking for stars sitting above the main sequence in a colour–magnitude diagram — an unresolved pair of equal stars is twice as bright and the same colour, so it appears exactly three-quarters of a magnitude above the sequence.

Second, the cores stopped collapsing. About a fifth of Galactic globulars have the steep central surface-brightness profile expected of a collapsed core, and four fifths do not, despite relaxation times short enough that most of them should have collapsed by now. Something is holding them up.

Third, the products are visible. Clusters contain blue stragglers — stars above the main-sequence turnoff, which single-star evolution forbids because the more massive stars burn out first — at rates that correlate with the encounter rate rather than with the cluster’s mass. Those are the mergers and mass transfers that hard binary encounters produce.

Where the binaries come from #

The account so far has assumed a supply of binaries. Where they come from is a question with two answers, and which one dominates decides how the process behaves.

Primordial binaries formed when the cluster did, from the fragmentation of the same collapsing cloud, with whatever distribution of separations star formation produces. Most of them are soft by the standards of a cluster core and are destroyed within a relaxation time; the hard tail survives and is the fuel.

Dynamically formed binaries are made in the cluster itself, by three-body encounters. Two stars passing close cannot become bound on their own — a two-body encounter is time-reversible and leaves both on the hyperbolic orbits they arrived on — but a third star present at the right moment can carry away the excess energy and leave the first two bound. The rate for that depends on the cube of the stellar density, because three bodies have to be in the same place, so it is negligible everywhere except in a core that has already contracted a long way.

That density dependence is what makes the whole process self-limiting in a satisfying way. A cluster with no hard primordial binaries collapses until its core is dense enough to manufacture them, at which point the heating switches on and the collapse halts. A cluster with a healthy primordial population never gets that far, and its core stops contracting at a much lower density.

So the two supplies produce different outcomes from the same physics, and the observable difference is the central density at which a cluster settles. Systems with high binary fractions should have less concentrated cores, and the correlation is seen — though weakly, because the binary fraction measured photometrically is the fraction of near-equal-mass pairs and the fraction that matters dynamically is the fraction of hard pairs, and the two need not track each other.

A mechanism that manufactures its own fuel when it runs out is difficult to switch off, which is the deeper reason clusters survive rather than collapsing: the halt does not depend on the initial conditions supplying anything in particular.

Where this stops being clean #

Two things the account above has assumed are not safe.

The first is that the intruder is a single star. In a dense core it often is not: binary–binary encounters are more common than the binary fraction suggests, because a binary is a bigger target, and their outcomes include the destruction of one pair to tighten the other. The scattering experiment above measures the wrong process for the densest cores.

The second is that the stars are point masses. When the separation has fallen far enough, the stars touch, and the outcome is a merger or a mass transfer rather than a further hardening. That puts a floor on how hard a binary can become, which is a floor on how much energy it can deliver, and it is a floor set by stellar radii rather than by dynamics.

One more consequence of the sign is worth drawing out, because it explains a fact about clusters that looks unrelated. A cluster whose core is being heated by binaries expands, and an expanding core has a lower density and therefore a longer relaxation time — so the heating slows the very process that made it necessary. That is a negative feedback wrapped around a positive one, and the result is not a steady state but an oscillation: the core contracts until binary heating becomes strong enough to reverse it, expands until the heating shuts off, and contracts again. Those gravothermal oscillations are seen in simulations with a clarity that has never been matched observationally, because the period is comparable to the relaxation time and no cluster has been watched for a hundred million years. What is observed instead is a distribution of core densities across the Galactic globular population, and the width of that distribution is broadly what an oscillating population would produce and also what a spread of ages and masses would produce.

The heaviest binaries, and what they do instead #

Everything above is about pairs of ordinary stars. A cluster also contains compact remnants, and because they are heavier they sink to the core and dominate the encounter statistics there — so the dynamics of a real core is largely the dynamics of its black holes.

That population’s fate was assumed settled for decades in one direction and is now thought to go the other way. The old argument was that a subsystem of black holes, being much heavier than the stars around it, would segregate, decouple, collapse on its own, and eject itself within a few hundred million years by exactly the mechanism this essay describes — leaving a cluster with at most one or two.

Two things changed that. Detailed simulations found that the ejection is slower than the simple argument allows, because the black-hole subsystem’s own heating puffs up the cluster around it and lowers the density that drives the ejection. And a handful of black-hole candidates were identified in Galactic globular clusters, by radial-velocity variations of a visible companion rather than by any accretion signature.

If clusters do retain substantial black-hole populations, several things follow. The core is heated by black-hole binaries rather than by stellar ones, so the halting of core collapse is a statement about a population nobody can see directly. The cluster’s observed structure — a large core, a low central density — is a symptom of that population rather than of its initial conditions. And the binaries ejected by the process are pairs of black holes on unbound orbits, which merge somewhere in intergalactic space and are detectable when they do.

That last point is what turned an argument about cluster structure into a question the gravitational-wave catalogue is being asked. Pairs assembled dynamically should have spins oriented at random relative to their orbit, because nothing correlated them; pairs assembled from a binary star system should have spins roughly aligned, because they inherited the orientation of the orbit they formed in. The distribution of spin orientations among merging black holes is therefore a measurement of how many of them were assembled in cluster cores — a statement about the interiors of globular clusters, made by an instrument on the ground measuring the length of a four-kilometre arm.

What survives to be seen #

A process that hardens binaries until they merge or eject themselves ought to leave products, and the products are what the observational case for the mechanism mostly rests on.

The clearest are the close binaries that a cluster has far too many of. Globular clusters contain cataclysmic variables, low-mass X-ray binaries and millisecond pulsars at rates per unit mass hundreds of times higher than the Galactic field. A field binary tight enough to transfer mass had to be born tight; a cluster binary of the same kind can be manufactured, by exactly the hardening sequence this essay describes, from a pair that started far wider — or by an exchange, in which a passing compact object displaces the lighter member of an existing binary and takes its place.

The overabundance scales with the encounter rate rather than with the cluster’s mass, which is the discriminating test. Two clusters of equal mass and different central density contain different numbers of these objects, in proportion to the rate at which their cores produce encounters, and the correlation holds across the Galactic population over two decades in rate.

That is as direct a confirmation as the subject offers. The mechanism cannot be watched and its consequences can be counted, and the count follows the quantity the mechanism depends on rather than the quantity anything else would depend on.

A cluster core is a factory, and the products are the objects the rest of astronomy studies for their own sake: the millisecond pulsars that time gravitational waves, the X-ray binaries that weigh neutron stars, and the blue stragglers that should not exist.

Where the ladder goes #

The natural next rung is the collapse this essay’s binaries are halting, followed properly: the self-similar contraction of a cluster’s core, the finite time in which it completes, and the gravothermal oscillations that follow when heating and cooling alternate rather than balance.

The other direction leads out of clusters entirely. The same energy argument applied to a binary of black holes at the centre of a merged galaxy gives the same conclusion — the pair hardens by ejecting stars — and there it runs into a genuine problem, because the supply of stars on orbits that come close enough runs out before the pair is tight enough for gravitational radiation to finish the job. A hardening rate that depends on a population it is itself destroying is the shape of every problem in this anchor, and at a hundred million solar masses it is the shape of an unsolved one.