A core that makes what the cluster asks for

A hard binary in a star cluster is an energy source. Encounters with passing stars tighten it, and every time it tightens, the energy it gives up leaves with the intruder as extra speed, which the cluster’s core then shares out as heat. That is how core collapse is halted: the gravothermal catastrophe drives the core’s density upward until the encounter rate is high enough for binaries to pay for the heat the core is losing.

That account settles which way the energy flows. It says nothing about how much. A binary population could in principle heat a core far faster than the core needs, blowing it apart, or far slower, letting it collapse regardless. What actually happens is neither, and the reason is one of the cleanest arguments in stellar dynamics, due to Michel Hénon in the early 1960s: after collapse the core does not decide its own output. The rest of the cluster does, and the core adjusts its structure until it produces exactly what is demanded.

A heat source whose output does not depend on its fuel #

Start with a single hard binary and ask how much energy it releases per unit time.

Each hardening encounter removes a roughly fixed fraction of the binary’s binding energy, and that energy is proportional to $1/a$, where $a$ is the separation. So a tighter binary gives up more per encounter. But the tighter binary is also a smaller target. The encounter rate is the density of field stars times the cross-section times the relative speed, and for a hard binary in a cluster the cross-section is dominated by gravitational focusing: a star passing at the dispersion $\sigma$ is deflected into the pair from an impact parameter much larger than $a$, and the focused cross-section grows only in proportion to $a$ itself,

$$\Sigma \approx \pi a \,\frac{2 G m_{\text{tot}}}{\sigma^2}.$$

Multiply the energy per encounter, proportional to $G m^2/a$, by the rate, proportional to $n \sigma \cdot a\, G m / \sigma^2$, and the separation cancels:

$$\dot E_{\text{bin}} \sim \frac{G^2 m^3 n}{\sigma}.$$

A hard binary heats its surroundings at a rate that does not depend on how hard it is. It hardens at a constant rate in $1/a$, releasing energy at a steady clip until it recoils out of the core or its stars touch. That is the first ingredient, and it already has a surprising consequence: the total heating a core receives depends on how many binaries it holds and how dense it is, and not at all on the distribution of their separations, as long as they are hard.

Where the demand comes from #

The second ingredient is on the other side of the ledger. A cluster relaxes on a timescale fixed by its number of stars and its size, and relaxation carries energy outward: stars in the core, scattered by distant encounters, drift into the halo, taking energy with them, while the core loses it. At the half-mass radius, which encloses half the cluster and is where most of the cluster’s structure lives, that outward flux is set by the cluster’s bulk alone:

$$\frac{dE}{dt} = \zeta\,\frac{|E|}{t_{\text{rh}}},$$

with $|E|$ the cluster’s binding energy, $t_{\text{rh}}$ the relaxation time at the half-mass radius, and $\zeta$ a dimensionless number close to one-tenth that Hénon found from the structure of the self-similar solution and that simulations have since confirmed.

Nothing in that expression refers to the core. The half-mass region conducts heat outward at a rate its own relaxation sets, and the core is simply the reservoir at the bottom of the gradient. If the core supplies less than this, it loses energy and — having a negative heat capacity — contracts and heats up. If it supplies more, it expands and cools. Either way it moves, and the only place it can stop is where supply meets demand.

The square-root dependence needs one more step. The heating per unit volume from binaries is proportional to the number density of binaries times the density of stars, so to $f_b n^2$, where $f_b$ is the fraction of stars in hard binaries. The core’s volume is $r_c^3$, and its radius is tied to its density at a fixed velocity dispersion by the relation King derived for isothermal cores, $r_c^2 = 9\sigma^2/4\pi G \rho_c$, so $r_c^3 \propto \rho_c^{-3/2}$. Total heating is then $f_b \rho_c^2 \cdot \rho_c^{-3/2} = f_b\,\rho_c^{1/2}$. Denser cores generate more heat, but only slowly — which is what makes the balance stable rather than explosive. A core that overshoots in density generates more than the demand and pushes itself back; one that undershoots generates less and falls in.

A bigger supply makes a gentler core, not a hotter one #

Setting $f_b\,\rho_c^{1/2}$ equal to a demand that does not depend on the core gives the core’s structure directly. The central density is proportional to $f_b^{-2}$, and the core radius, as its inverse square root, is proportional to $f_b$ itself.

That inverts the intuition a reader brings from engines. More fuel does not mean more power here: the power is fixed by the load, and more fuel means the engine can idle. A cluster with a quarter of its core stars in hard binaries needs only a mild, extended core to meet its demand. A cluster with almost none has to drive its core to extreme density before the handful of binaries it has can keep up — and if it has none at all, it collapses until the density is high enough to make them in three-body encounters, whose rate goes as the cube of the density. The dynamically formed binaries then supply the heat, and the same balance sets the core at whatever size they can sustain.

This is the quantitative form of a correlation the essay on binary heating could only name: that clusters with more binaries should have less concentrated cores. The proportionality is now definite, and in simulations the core-to-half-mass ratio of post-collapse clusters does rise with the primordial binary fraction, from about a hundredth or less with no primordial binaries to several hundredths and more with a ten-per-cent population. The exact normalisation depends on the binaries’ mass ratios and the details of three- and four-body encounters, and the figure’s anchor at five per cent is a choice made to match those simulations rather than a derivation. The slope does not depend on it.

It also changes what an observer’s category means. A Galactic globular cluster is classified as core-collapsed when its surface brightness keeps rising into the centre rather than flattening; about a fifth are. The other four-fifths are usually described as not yet collapsed. On Hénon’s argument many of them may well have been through collapse and be sitting in the post-collapse state with a large core, because their binaries — or, as it now appears, their black holes — are plentiful enough that the balance never demanded a small one. A large core is not evidence that a cluster is dynamically young; it may be evidence that its heat source is abundant.

An analogy the argument was built on #

Hénon’s own framing was stellar, and it is worth drawing out because the parallel is exact in structure.

A main-sequence star’s luminosity is not set by how vigorously its core burns hydrogen. It is set by how fast radiation can diffuse out through the envelope, which depends on the star’s mass, radius and opacity — which is why luminosity follows mass by a steep power and the nuclear rates barely enter. The core’s reaction rate is exquisitely sensitive to temperature, so the core simply sits at whatever temperature makes the reactions produce what the envelope conducts away. Raise the reaction rate at fixed temperature, and the core cools a little and the luminosity barely changes. The furnace is thermostatted by the chimney.

A post-collapse cluster is the same object in different materials. The envelope is the half-mass region; its opacity is the relaxation time; the conducted flux is $\zeta |E|/t_{\text{rh}}$. The furnace is the core’s binaries, and their rate is sensitive to density rather than temperature. The thermostat works the same way, and it has the same consequence: the details of the energy source do not affect the rate of energy production. They affect only the conditions in the core where it is produced.

The analogy breaks at one point, and the break is the interesting part. A star’s core has a positive heat capacity where it matters: when it loses energy, it contracts, heats and burns faster, restoring the balance smoothly. A cluster’s core has the negative heat capacity of any self-gravitating system, and the stabilising feedback has to come entirely from the binaries’ dependence on density. When that feedback is too slow compared with the core’s own relaxation — in clusters of more than about seven thousand stars, as Jeremy Goodman showed — the balance is not approached monotonically but overshot, repeatedly: the core bounces in the gravothermal oscillations that simulations show. Those oscillations are a fluctuation about Hénon’s balance rather than a departure from it. Averaged over a few cycles, the core still produces what the halo asks for.

Expansion at a rate that forgets its beginning #

The demand side has a consequence for the whole cluster, not just its core. Energy is carried outward and deposited in the halo, so the cluster’s binding energy falls and it expands. With the binding energy $|E| \approx 0.2\,GM^2/r_h$ and a relaxation time growing as $r_h^{3/2}$, the flux equation becomes a statement about the half-mass radius alone:

$$\frac{1}{r_h}\frac{dr_h}{dt} = \frac{\zeta}{t_{\text{rh}}} \propto r_h^{-3/2},$$

and so $r_h^{3/2}$ grows linearly in time. At late times $r_h \propto t^{2/3}$, and the starting radius has dropped out of the answer.

This is where the principle becomes testable against a population rather than a simulation. A compact cluster expands fast because its relaxation time is short; a diffuse one expands slowly because its relaxation time is long. They converge. The cluster’s present half-mass radius is set, to within a modest factor, by its mass and its age, and its initial size is erased — and the Galactic globulars do show half-light radii of a few parsecs across three decades in mass, with a scatter much smaller than any reasonable spread of initial conditions would produce.

The same integration makes a prediction about mass that is less often drawn out. Solving for the late-time radius gives $r_h^{3/2} \propto t\,\ln\Lambda/\sqrt{N}$, so at a fixed age a heavier cluster should be smaller, as $M^{-1/3}$, because its longer relaxation time slows its expansion. Integrated for a million solar masses rather than a hundred thousand, the same four starting radii end at three to seven parsecs rather than six to ten.

The real population does not follow that trend cleanly, and the reason is a boundary the model leaves out. A cluster orbiting a galaxy is limited by the galaxy’s tide: beyond its tidal radius, stars are stripped away, and a cluster that expands to fill that radius stops expanding and starts losing mass instead. Mark Gieles, Douglas Heggie and Hongsheng Zhao described the result as balanced evolution in two stages. First, an expansion-dominated phase in which the half-mass radius grows as Hénon’s argument says and the tide does nothing. Then, once the cluster fills its tidal radius, an evaporation-dominated phase in which the same energy flux drives stars over the boundary, and the cluster shrinks in mass at a rate set by its relaxation. Low-mass clusters with short relaxation times reach the second stage early; the most massive globulars may still be in the first. In both stages the core is doing what Hénon said it would: supplying the flux, and not choosing it.

A demand that falls, and a core that follows it down #

As the cluster expands its demand falls, because both the binding energy and the inverse relaxation time decrease. So the core’s output must fall too, and the only way for it to fall is for the core to become less dense. A post-collapse cluster’s core therefore expands along with the whole, keeping roughly a fixed ratio to the half-mass radius set by the binary fraction, and its luminosity in the dynamical sense declines steadily through the cluster’s life.

The picture is one of a cluster running down in a controlled way. The core is never out of balance for long; the balance point drifts as the halo expands; and the binaries that sustain it are consumed at a rate the halo sets. Each binary, hardened until its recoil exceeds the escape speed, is eventually thrown out of the cluster or merges, and the population of hard pairs is replenished either from wider primordial ones sinking into the core or from new ones formed in three-body encounters. As long as that supply lasts, the heating continues at the rate required. When it fails, the core contracts until it has made more.

The heat source is interchangeable, which is the point #

Stated in full, Hénon’s principle is that the energy generation rate of a post-collapse cluster’s core is determined by the cluster’s global properties and not by the mechanism that generates it. The mechanism enters only by fixing the core’s density and size.

That generality is what gave the principle a second life. For decades the heat source in cluster cores was assumed to be binaries of ordinary stars. The stellar-mass black holes a cluster forms were expected to segregate to the centre and eject one another within the first few hundred million years, by the same hardening process. Simulations by Philip Breen and Douglas Heggie, and by several groups since, found instead that a black-hole subsystem at the centre of a cluster behaves as a heat source in exactly Hénon’s sense: it supplies the energy the cluster demands, and its own ejection rate is set by that demand. Because the demand is modest, the black holes are ejected slowly, and a substantial number can survive for a Hubble time.

The observable consequence follows directly from the core scaling. A cluster whose heat source is a population of black holes — each ten or twenty times heavier than a star, and hardening in encounters with the same focused cross-section — meets its demand at a much lower stellar density than a cluster relying on stellar binaries. Its visible core is large and its central density low. So the clusters that look the least evolved might be the ones hosting the most black holes, and the size of a globular’s core becomes a measurement, through Hénon’s balance, of a population no telescope can see.

That is a measurement with a check. The black-hole pairs a cluster ejects in hardening are the ones that later spiral together by radiating gravitational waves and are detected when they merge, and a cluster core’s ejection rate is fixed by the same demand that fixes its size. The merger rate from clusters inherits the halo’s thermostat.

What the principle leaves open #

The argument sets the rate and the scalings, and it does so cleanly because it avoids the core’s microphysics. That is also where it stops.

It does not say how quickly the core finds the balance, or whether it oscillates about it. The gravothermal oscillations are a real phenomenon in simulated clusters above a few thousand stars and have never been observed, because their period is a fraction of a relaxation time — millions of years — and what a survey sees is a snapshot of a population, not one cluster in time.

It does not fix the normalisation of the core radius, which depends on how efficiently binary–binary encounters convert binding energy into heat, how many of the recoiling stars stay in the core long enough to share their energy, and how the binaries’ mass ratios are distributed. Those enter as a constant, and the constant is currently known only from simulation.

And it assumes the heat goes where it is needed. A binary’s recoil energy is deposited as a single fast star, which may leave the core before sharing any of it, or leave the cluster altogether. The fraction retained is a detail that shifts the constant, and it is one of several details that make simulated clusters’ cores somewhat larger or smaller than the simple scaling suggests.

What survives all of those is the direction of causation, and that is the argument’s real content. In a system with a negative heat capacity, the energy source cannot set its own output, because any excess or deficit rearranges the source until it matches the load. The contrast is a massive black-hole pair at the centre of a galaxy, which hardens by the same slingshot but in a system far too large to relax: there nothing replenishes the stars it ejects, and the pair stalls for want of them rather than settling into a balance.

Still open: the binaries’ own statistics, and the black holes #

The next question is the one this essay has pointed towards without drawing: the binaries’ own statistics in a core that is being fed and drained at once — the distribution of binding energies that a steady supply of new pairs and a steady loss of hardened ones settles into, and whether that distribution is universal in the way the heating rate is. A second direction runs through the black holes: the retention of a black-hole subsystem, followed as a heat source whose own depletion Hénon’s demand sets, and what that predicts for the spins and masses of the pairs it ejects.