The binaries a core keeps are spread evenly in energy

A binary star in the core of a dense cluster is not an isolated pair. Every so often a third star passes close enough to exchange energy with it, and which way the energy flows is settled by one comparison: a pair bound more tightly than the passing stars’ kinetic energy — a hard binary — gets harder, and the energy it gives up goes into the motion of the star that leaves. Hard binaries are a heat source, and after a cluster’s core has collapsed they are the heat source that stops it collapsing further. How much heat the core makes is set by the cluster around it, not by the binaries, and the core adjusts its density until its binaries supply what the halo carries away.

That account treats the binaries as a population with a total output and asks nothing about the individuals. The question here is the one it left open: what does the population look like? A core is fed — new hard binaries form in close encounters of three single stars, or are freed from the primordial population — and it is drained, as binaries harden until a single encounter throws them out. Between the feeding and the draining the binaries settle into some distribution of binding energies, and that distribution is what a census of a cluster’s binaries would measure.

The answer turns out to depend on almost nothing, and the reason is the same law that made the heating rate independent of the binaries in the first place.

A drift that does not care how hard the binary is #

Write a binary’s binding energy as $x$ in units of $m\sigma^2$, the kinetic energy scale of a star of mass $m$ in a cluster of velocity dispersion $\sigma$. A binary is hard when $x$ is larger than one or two. Below that boundary the same encounters do the opposite: a soft binary, bound more weakly than the passing stars are moving, is loosened a little by each and eventually pulled apart. The boundary is a thermodynamic one, the point at which the pair’s binding equals the kinetic energy of the stars around it, and it is why a system that gets hotter as it loses energy can use its hard binaries as a fuel: energy flows out of them, into stars, as heat flows from hot to cold.

Two facts about its encounters set everything. The first is that an encounter changes $x$ by a fraction of itself: Douglas Heggie showed in 1975 that for hard binaries the fractional change $\Delta = \delta x/x$ has a distribution independent of $x$, falling off as $(1 + \Delta)^{-4.5}$ with a mean of 0.4. A harder binary does not get a smaller kick in proportion; it gets the same proportional kick. The second is that harder binaries are hit less often. A binary’s cross-section for a strong encounter is set by its separation, enlarged by gravitational focusing, and for a hard binary it is proportional to the separation itself — and the separation is proportional to $1/x$.

Multiply the two: encounters come at a rate proportional to $1/x$ and each adds energy proportional to $x$. The average rate at which a binary gains binding energy is

$$\frac{\mathrm{d}x}{\mathrm{d}t} = \text{constant},$$

independent of how hard it is. This is Heggie’s law, and it is the reason the heating rate per binary in the core does not depend on the binaries’ hardness.

The paths in the figure are individual, and individually they look nothing like a constant drift. Early on, at small $x$, a binary is hit often and gains a little each time; later it is hit rarely and gains a lot each time, so the path becomes a staircase of long flat stretches and tall steps. But averaged over thousands of binaries, the energy grows at 0.394 per unit time below $x = 20$ and 0.396 between 40 and 100, the same to within the sampling noise. The staircase gets coarser with $x$ and its slope stays the same.

The path ends on a step. Each encounter gives the binary a recoil: the energy released is shared between the binary and the escaping star in inverse proportion to their masses, so for equal masses the binary keeps a third. When one encounter’s third exceeds what the binary needs to escape the cluster — $\Delta x > 3(v_\text{esc}/\sigma)^2$ in these units — the binary leaves and takes its future heating with it. For an escape speed four times the dispersion that is $\Delta x > 48$, and with Heggie’s mean jump of 0.4 it typically happens near $x \approx 120$.

A steady flux through energy #

Now feed the core. Binaries enter near some energy $x_0$, drift upwards at a constant average rate, and leave near $x_\text{ej}$. If they enter at a steady rate $S$, the core reaches a steady population in which the number of binaries per unit energy, $n(x)$, is unchanged from moment to moment.

That requires the flux through every energy between entry and ejection to equal the feeding rate — every binary that enters must pass through every energy on its way out. And the flux is the number per unit energy times the speed of the drift: $n(x)\,\dot x = S$. With $\dot x$ constant, $n(x)$ is constant. The population is flat in energy: the same number of binaries per unit binding energy everywhere between where they are born and where they leave.

The simulation weights each binary by the time it spends at each energy, so the histogram is a snapshot of a steady population rather than a count of encounters. Between twice the entry energy and a third of the ejection energy it is flat to within five per cent, as the flux argument requires. The edges are soft for two reasons that the simple argument leaves out. Near entry, binaries are born over a spread of energies, so the plateau ramps up rather than starting abruptly. Near ejection, the heavy tail of Heggie’s distribution means a single large encounter can eject a binary from well below the typical ejection energy — at $x = 24$ a jump of two, which happens about one encounter in fifty, is enough — so the population thins out over a range of energies rather than ending at a wall.

Most binaries are nearly as hard as they get #

A distribution flat in energy looks unremarkable on a linear axis. Its meaning is clearer counted per factor of energy, which is how binaries are usually described: by the logarithm of their separation or period.

The number per unit logarithm of energy is $x\,n(x)$, which rises in proportion to $x$. Half the binaries are harder than $x = 78$, and 94 per cent are within a factor of ten of the typical ejection energy. A core’s binaries are not spread over the hard range; they are piled up near its top.

That does not mean the hardest binaries do most of the heating. Each binary heats at the same rate whatever its hardness — that is Heggie’s law again — so a binary at $x = 100$ contributes no more than one at $x = 10$. There are simply many more at $x = 100$, because a binary spends equal time on equal intervals of energy: the stretch from 12 to 120, a factor of ten, takes it eleven times as long as the stretch from 5 to 15, a factor of three. The heating is shared equally among all of them; the binaries are counted unequally only because the logarithm compresses the soft end.

The practical consequence is for the arithmetic of the previous essay. A core’s heating rate per binary is $\dot x$ times $m\sigma^2$, and the population a core needs to meet the cluster’s demand is that demand divided by $\dot x\, m\sigma^2$. The distribution says what that population looks like: a few binaries recently formed and still nearly soft, and most of them tight, close to the hardness at which the next strong encounter will remove them.

A shape that forgets how the binaries were born #

The flatness was derived from two things — a constant drift and a steady feed — and nothing about how binaries form or exactly how each encounter goes. That is a strong claim, and it can be tested by changing both.

Entering at three, ten or thirty times $m\sigma^2$ changes where the plateau begins and nothing else: above its entry, each population sits at the same level and falls away at the same energy. Replacing Heggie’s broad distribution of jumps with a fixed jump of exactly 0.4 changes only the shape of the fall-off near ejection — it becomes sharper, because no single encounter can throw a binary out early — and leaves the plateau untouched.

That universality is the population-level version of Hénon’s principle. The cluster’s demand fixes the total heating rate regardless of the heat source; Heggie’s law fixes the shape of the population that supplies it regardless of how the binaries were made. The entry energy is a property of the formation mechanism and of the primordial population, both poorly known. The exit energy is set by the cluster’s escape speed, which is measured. A theory of a core’s binaries needs the second and can very nearly ignore the first.

Where the binaries come from, and where the soft ones go #

The feed has two sources, and the steady population does not need to know which dominates. The first is the primordial population: binaries formed with their stars, spread across every separation, of which the hard ones in the core are already part of the population and the soft ones are being taken apart. The second is the formation of new binaries in the core itself, when three single stars meet closely enough that one carries off the energy that leaves the other two bound. That process is extremely sensitive to the core’s state — its rate goes as the cube of the density divided by the ninth power of the dispersion — so it is negligible in most cores and dominant at the deepest point of a collapse, which is why it was the mechanism first invoked to stop one.

Three-body binaries are born only just hard, at a few times $m\sigma^2$, because the third star can carry off only an energy of order its own kinetic energy. Primordial binaries enter the hard population wherever they happen to sit, mostly far above the boundary. The figure that tested the entry energy is the answer to the question of which source matters: above their entry, both feed the same plateau.

The soft binaries form the other half of the picture and are not drawn. They drift downwards in energy, at a rate that grows as they loosen, and are destroyed; their steady population is concentrated near the boundary, a balance between the binaries that three-body encounters form there and the ones that ordinary encounters take apart. The boundary itself moves as the cluster evolves: a cluster that expands and cools lowers its dispersion, so binaries that were soft become hard without changing at all, and the hard population is fed from below by the cluster’s own cooling.

The same law where it fails #

The flatness depends on one condition that a cluster’s core meets and a galaxy’s nucleus does not: a steady supply of single stars to encounter. In a cluster, two-body relaxation refills the core’s population of stars on orbits that pass close to the binaries faster than the binaries use them up, and Heggie’s constant drift holds.

At the centre of a galaxy after a merger, the binary is a pair of supermassive black holes and the single stars are the galaxy’s stars. The pair hardens by the same slingshot, each star that passes close carrying off energy and leaving the galaxy’s centre, but the galaxy is far too large to relax: nothing replaces the stars on orbits that come close enough. The drift slows as the supply runs out, the population of one binary does not settle into anything, and the pair can stall at the separation where the stars that would have hardened it are not there. The difference between a cluster core and a galactic nucleus is not the law of each encounter, which is the same, but whether the system relaxes quickly enough to forget what the binaries have used.

What a census would find #

A binding energy in units of $m\sigma^2$ becomes a separation and a period once the cluster’s dispersion and the stars’ masses are given.

Binding energy goes as the inverse of separation, and by Kepler’s third law separation goes as period to the two-thirds, so a distribution flat in energy is a distribution in the logarithm of period that falls as $P^{-2/3}$. For a globular-cluster core with a dispersion of 7 km/s, binaries entering near $x = 5$ have periods of about 440 days, and they are ejected near 3.7 days, when their separation has fallen to about 0.05 astronomical units — eleven solar radii, still wide enough for two stars of 0.7 solar masses not to touch. Between those limits the population is tilted towards short periods by a factor of $(440/3.7)^{2/3} \approx 24$ in the number per logarithmic interval.

That tilt is the prediction. Binaries formed with their stars and never processed — the binaries of the field around the Sun — are spread roughly evenly in the logarithm of period across this range, with no strong preference for short periods. A core that has been heating its cluster for billions of years should have reshaped its hard binaries into a population that rises towards the short end until the escape speed cuts it off. Surveys of globular-cluster binaries, by the variations they cause in the velocities of their stars and by the photometric signature of unresolved pairs, find binary fractions in the cores of a few to a few tens of per cent; the period distribution of the hard end is harder to measure, because the shortest-period systems are eclipsing or spectroscopic binaries in crowded fields, and the prediction is a target rather than a confirmation.

Two further populations carry the same signature in a different form, and they are the binaries whose masses and orbits are known best — the only stars whose masses are measured directly are in binaries, and in a cluster those binaries have been through the process drawn here. The millisecond pulsars of globular clusters, spun up by accreting from a companion, and the cataclysmic variables and X-ray binaries in their cores are all products of hard binaries. Their numbers correlate with the rate of stellar encounters in each cluster rather than with its mass, which says that encounters, not birth, made them — the same dynamics that shapes the distribution drawn here.

What the steady state leaves out #

The calculation treats the core as a fixed background: a dispersion, an escape speed, a supply of single stars to encounter. In a real cluster the core changes as it is heated, the escape speed falls as the cluster expands and boils its stars away, and the population never quite reaches a steady state, though its relaxation into one is fast compared with the cluster’s evolution. The binaries are equal-mass, and the encounters are with single stars of the same mass; real encounters include exchanges, in which the intruder replaces one of the binary’s members, and binary–binary encounters, which can form triples or destroy a binary outright. Exchanges preferentially put heavier stars into binaries, so the hard population drifts towards the heavy end of the mass function — the neutron stars, white dwarfs and black holes — which is exactly why the compact binaries above are over-represented in cluster cores.

And the binaries end only by ejection. Real binaries can also end by their stars touching. For the cluster drawn here they do not, with eleven solar radii to spare at the ejection energy; but the ejection separation shrinks as the square of the dispersion, and in a cluster a few times faster the stars would touch before the cluster could throw them out.

Still open: whether the binaries leave or merge #

The distribution’s upper edge is the one quantity that controls it, and it was taken here to be set by the escape speed. That is correct only while the ejection separation is wider than the stars. In clusters with larger dispersions — the most massive globulars, and the nuclear star clusters at the centres of galaxies, where stars move at tens or hundreds of kilometres a second — a binary hardening at the same average rate reaches contact before the next encounter can eject it, and its two stars become one. Whether a core’s binaries end as escaping pairs or as merged single stars, and how that boundary depends on the cluster and on the kind of star, decides what the heating leaves behind: high-velocity escapers in one kind of system, and in the other a population of stars that are more massive than they should be — the blue stragglers that an encounter-driven factory was already suspected of making.