Ejected from a globular, merged in a nucleus

A hard binary in the core of a star cluster gains binding energy at the same average rate whatever its hardness, and in doing so heats the cluster around it. The population of such binaries in a core that is continually fed settles into a distribution flat in energy, from where binaries enter to where they leave, and the leaving is the one place the population’s shape is decided. The essay on that distribution assumed binaries leave by ejection: an encounter releases enough energy that the binary’s share of the recoil exceeds the cluster’s escape speed, and it goes.

There is a second way out. The binary’s two stars have sizes. As the pair hardens, its separation shrinks in inverse proportion to its binding energy, and at some point the stars fill the regions of space each can hold on to — their Roche lobes — and begin to exchange mass, and then to merge. A binary that reaches contact before an encounter throws it out does not leave the cluster; its two stars become one.

Which happens first is a race between two separations, and it is decided by how fast the cluster’s stars move.

Two separations, one of which moves #

The ejection separation follows from the escape condition. For equal masses the binary keeps a third of the energy an encounter releases; it escapes when that third exceeds its kinetic energy at the escape speed, and with Heggie’s typical fractional jump of 0.4 that happens near a binding energy of $7.5\,m v_\text{esc}^2$. A cluster’s escape speed from its core is about four times its velocity dispersion, so ejection comes near $x \approx 120$ in units of $m\sigma^2$, and the separation there is

$$a_\text{ej} = \frac{G m}{2\,x_\text{ej}\,\sigma^2} \;\propto\; \frac{1}{\sigma^2}.$$

A hotter cluster — one whose stars move faster — needs a harder binary to throw it out, because the binary’s recoil must beat a higher escape speed, and the harder binary is a smaller one.

The contact separation is set by the stars alone. For two equal stars, each fills its Roche lobe when the separation falls to about 2.6 stellar radii. For main-sequence stars of 0.7 solar masses, typical of an old globular cluster’s turn-off, with radii of 0.68 of the Sun’s, that is 0.0082 astronomical units — 1.8 solar radii — and it does not depend on the cluster at all.

The two lines cross at a dispersion of about 18 kilometres a second. Open clusters, with dispersions below a kilometre a second, eject their hard binaries while the separations are still tens of astronomical units; the binaries never come near touching. A typical globular cluster, at five to ten kilometres a second, ejects them at a few hundredths of an astronomical unit — eleven solar radii for a 7 km/s core, as the period distribution of the last essay showed. The most massive globular clusters, with central dispersions of fifteen to twenty, sit on the line. And the nuclear star clusters at the centres of galaxies, with dispersions of tens to a hundred kilometres a second, drive every hardening binary into contact long before the escape speed could be reached.

Who leaves, and how fast #

When an encounter ejects a binary, it ejects the passing star as well, and faster. The energy released is shared in inverse proportion to mass: the binary, of mass $2m$, takes a third and the single star, of mass $m$, takes two thirds. Their speeds follow: if the binary leaves at just above the escape speed, the single star leaves at twice it. In a globular cluster with a core dispersion of seven kilometres a second and an escape speed of about twenty-eight, the binary departs at a little over that and the single star at nearly sixty.

Those speeds are small against the Galaxy’s own — a globular cluster orbits at a couple of hundred kilometres a second — so the escapers do not leave the Galaxy; they leave the cluster, spread along its orbit, and join the stellar halo as stars that were once in a dense core and carry, in their binarity and in the tightness of their orbits, a record of having been processed there. The ejected binaries are short-period, tighter than almost any binary formed in the field at the same mass, and some of them will later come into contact by their own evolution as their stars swell, long after the cluster has let them go.

In a nucleus the same arithmetic would give escapers of hundreds of kilometres a second. It mostly does not happen, because the binaries merge first. The fastest stars that do leave galactic nuclei are made by a different three-body process: a binary passing close to the central massive black hole is separated, one star captured into a tight orbit and the other flung out at more than a thousand kilometres a second — fast enough to leave the Galaxy altogether. In a nucleus the central black hole, not the stellar binaries, is where three-body encounters release their largest energies.

A boundary blurred by large encounters #

The crossing in the figure uses the typical jump. Real encounters scatter: most add less than the mean, a few add several times it, and the largest can eject a binary from well below the typical ejection energy or carry it into contact in two or three steps.

Simulating the binaries one encounter at a time, with jumps drawn from Heggie’s distribution and the ejection condition applied at every step, turns the crossing into a transition. Below about ten kilometres a second not one binary in four thousand reaches contact. Half merge at about twenty, almost all by forty, all by sixty. The transition is a little higher and a little broader than the crossing of the two lines, because ejection is the outcome of single large encounters, which can come early, while contact requires the binary to climb all the way up — and at dispersions just above the crossing, some binaries are thrown out by a lucky encounter before they get there.

Less heat from each binary, so more binaries burnt #

The ending changes what the binary has given the cluster. A binary ejected near $x \approx 120$ has delivered all the energy it gained from entry to ejection, plus the third of its last jump that went into its own escape. A binary that merges stops wherever contact falls, and in a fast cluster contact comes long before the ejection energy.

At twenty kilometres a second a binary delivers 89 per cent of what it would in a slow cluster; at forty, 18 per cent; at sixty, five. The cluster’s demand for heat is set by the cluster, not by its binaries, so a cluster whose binaries each deliver less has to consume more of them. In a fast cluster the core contracts until enough binaries are being hardened to meet the demand — and every one of them ends as a merger.

At the highest dispersions the calculation reaches a limit of its own. In a cluster moving at sixty kilometres a second, the separation at which a binary becomes hard at all — where its binding equals the passing stars’ kinetic energy — is only a few times the contact separation, so the hard population is squeezed into a narrow range of energies between the hard–soft boundary and the stars’ surfaces. At about two hundred kilometres a second the two coincide for main-sequence stars, and there are no hard main-sequence binaries in the sense the argument uses: any pair bound tightly enough to be hard is already touching. In the densest galactic nuclei, binaries of ordinary stars cannot heat the system at all, and whatever holds the nucleus up must be something else.

The eccentricity that moves the boundary down #

The separations in the figures are semi-major axes, and the stars touch at the closest point of their orbit, not the average. Binaries hardened by encounters are not circular: each encounter kicks the eccentricity as well as the energy, and a population processed by many encounters approaches the thermal distribution, in which the fraction with eccentricity below $e$ is $e^2$. Half such binaries have eccentricities above 0.71 and pericentres less than 0.29 of their semi-major axes.

For those, contact comes at a semi-major axis 3.4 times larger than the circular figure assumes. Since the crossing dispersion goes as the inverse square root of the contact separation, the boundary for the more eccentric half of the population falls from 18 kilometres a second to about ten — into the range of ordinary globular clusters. A strongly eccentric pair does not necessarily merge at its first close passage; tides raised at pericentre drain orbital energy and circularise the orbit, sometimes leaving a tight, circular, detached binary that later comes into contact by its own evolution. But the effect runs in one direction: eccentricity brings contact forward, and the boundary between ejection and merger is lower for real binaries than for the circular ones drawn.

That correction is the largest single change to the calculation, and it is the one that most affects the comparison with observed blue stragglers. Ordinary globular clusters, which the circular figure says should almost never merge their hardening binaries, are brought by the eccentric half of their binaries close to the boundary, which is consistent with the correlation between blue stragglers and encounter rate that observations find in them.

What kind of star decides where the boundary falls #

The contact separation scales with the stars’ radii, so the dispersion at which a pair switches from ejection to merger scales as the inverse square root of the radius.

Red giants, with radii of ten suns, touch at a separation fourteen times larger than the main-sequence pair, and switch to merging above 5.2 kilometres a second — inside the range of ordinary globular clusters. A giant in a binary in a globular core is therefore likely to be driven into contact by encounters, and the outcome is not a clean merger but a common envelope, in which the giant’s envelope engulfs its companion and is partly ejected, leaving a tighter pair of a white dwarf and the companion. That is one route to the cataclysmic variables and the progenitors of the millisecond pulsars that globular cores hold in excess.

White dwarfs are a thousand times smaller than giants and switch only above about 124 kilometres a second, faster than any globular cluster and than most nuclear clusters. A pair of white dwarfs hardens until it is ejected or until it is so close that gravitational radiation finishes the work the encounters started — the same ending the black-hole pairs of the earlier essays come to, and for the same reason: compact objects never touch until gravity alone has brought them together.

Main-sequence stars are the case that divides the clusters. Ejected in globulars, merged in nuclei, and in the most massive globulars, sometimes one and sometimes the other.

What each kind of cluster makes #

Put the endings together with the population that feeds them and each kind of stellar system has a characteristic product.

An open cluster ejects its hard binaries, at a few kilometres a second, and they join the field as ordinary binaries a little tighter than they started. A typical globular cluster ejects them too, at tens of kilometres a second; the escapers become fast-moving halo binaries, and the stars they last interacted with leave faster still. The most massive globular clusters split their binaries between the two endings, about a quarter merging in this model. A nuclear star cluster merges all of them.

A merger of two main-sequence stars is a single star of nearly twice the mass, burning hydrogen afresh in a stirred-up core. In an old population whose turn-off is 0.8 solar masses, it sits above and blueward of the turn-off, where single-star evolution forbids anything to be: a blue straggler. Blue stragglers are found in every globular cluster, concentrated towards the core, at numbers that correlate with the rate of stellar encounters, and they are one of the clearest confirmations that the cores are sorted by mass. The calculation here adds a prediction about which clusters should make them by this route: mostly the fastest ones, with the boundary near the most massive globulars, while in ordinary globulars the blue stragglers must come from other channels — mass transfer in primordial binaries, and direct collisions during the resonant encounters in which three stars are briefly bound together.

A merged star has one more property that marks it. The orbital angular momentum of the pair goes into the merged star’s spin, and a star formed that way rotates far faster than a single star of its mass and age. A giant turning at fifty kilometres a second has to have been given its angular momentum by something, and a merger in a dense nucleus is one of the things that can give it.

A cluster that merges its heat source #

A merger changes the cluster in two ways that an ejection does not. It removes the binary’s binding energy from the cluster’s accounts without giving it to anyone: the energy goes into the merging stars’ structure and is radiated away over the following thousands of years. And it leaves behind a star nearly twice as massive as its neighbours, which sinks towards the centre as relaxation shares energy between stars of different masses, and there becomes a heavy single star more likely than any other to capture a companion and form a new hard binary. A fast cluster’s core is therefore populated, over time, by its own merger products, and its binaries are increasingly made of them.

That feedback has an end. A star built by successive mergers is a more massive star, and a more massive star lives a shorter life: a merger product of twice the turn-off mass, in a population ten billion years old, has a remaining lifetime of a billion years or so before it evolves, swells and dies. A nucleus that merges its binaries is continually manufacturing short-lived massive stars out of long-lived small ones, and the white dwarfs, neutron stars and black holes those leave behind are the compact objects that do not touch, that harden until gravitational radiation takes over, and that may supply the heat the ordinary binaries could not.

There is a further limit at the very highest dispersions. The figures assume the stars meet only through their binaries; in the densest nuclei, single stars collide directly, two stars passing within a few radii of each other without ever having been bound, at a rate that rises with density and with the stars’ gravitational focusing. Where direct collisions dominate, the binaries are no longer the main route to mergers, and the race drawn here is one contribution among several to a nucleus’s population of rejuvenated stars.

What the race leaves out #

The model is a pair of equal stars hardened by single stars of the same mass, with contact at a fixed number of stellar radii and an escape speed a fixed multiple of the dispersion. Real binaries have unequal masses, so the recoil share and the contact condition both change; real encounters include exchanges, which replace a light member with a heavier intruder and so move the population towards the heaviest stars present; and real binaries are eccentric, so that the stars touch at pericentre while the mean separation is still several times larger, which moves every boundary in the figures to lower dispersions. Tides and the stars’ own evolution can bring a pair into contact without any encounters at all, and in the oldest clusters the turn-off stars are swelling as they leave the main sequence, so a binary that was safely detached can come into contact on its own.

What survives these complications is the scaling. The ejection separation falls as the square of the dispersion and the contact separation does not, so there is always a dispersion above which a given kind of pair merges before it leaves, and the boundary for ordinary stars lies between the globular clusters and the galactic nuclei.

Still open: what holds a nucleus up #

The argument ends with a gap. In a nuclear star cluster whose stars move at a hundred kilometres a second, hard binaries of ordinary stars either merge long before they can be ejected or do not exist as hard binaries at all, so they cannot be the thermostat that stops the nucleus collapsing. Something else must supply the heat its relaxation carries outward: the stellar black holes that segregate to the centre and form hard pairs that do not touch; the central massive black hole, which acts as a heat source by disrupting and swallowing stars that come too close; or the recurrent formation of new stars from gas that the nucleus accretes. Which of these holds up a given nucleus decides whether it is a factory of merged stars, of black-hole pairs heard by gravitational-wave detectors, or of tidal disruption flares, and the boundary between those regimes has not yet been drawn.