Astronomy & Celestial Mechanics
the sky, computed
Generator

The binary-encounter generator

Hard below 8.3 astronomical units, soft above, and nothing settles at the line
Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium.

The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium.

6 essays call binary-encounter. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium. Gravitation

A pair that heats what is trying to cool

A star cluster has a negative heat capacity, so it cannot reach equilibrium — its core contracts and gets hotter without limit. What stops it is a binary, and which way a binary exchanges energy with the stars around it is settled by one comparison of two energies.

 The last parsec, priced in stars. The two timescales that shrink a pair of 10⁸-solar-mass black holes at the centre of a merged galaxy, against their separation in parsecs, both axes logarithmic, for a stellar velocity dispersion of 200 km/s and a central density of 500 solar masses per cubic parsec. Ejecting stars hardens the binary at a rate proportional to the separation, so that timescale grows as the orbit shrinks; gravitational radiation goes as the fourth power of the separation, so its timescale collapses. Neither alone finishes the job and the crossing of the two is where the answer is set. With the loss cone kept full, the crossing is at 2.9e-2 parsecs and the total is 4.2·10⁸ years, inside a Hubble time. With the supply of low-angular-momentum orbits emptied by a factor of 100 — which is what happens in a smooth spherical nucleus, because the stars that could interact have already been thrown out and two-body relaxation refills the orbits far too slowly — the crossing moves outward to 7.3e-2 parsecs and the total becomes 1.7·10¹⁰ years. That is the final-parsec problem, and it is not a problem about gravity: it is a problem about supply. Real nuclei are not spherical, and the figure cannot show what a triaxial potential does, which is to keep feeding the binary orbits it has not already used. Galaxies

The last parsec, and the stars that are not there

Two galaxies merge and their central black holes sink towards each other, and then stop. From about a parsec apart, friction no longer works and gravitational radiation is not yet strong enough — and the only mechanism in between throws away the stars it depends on faster than they can be replaced.

 Supply and demand in a cluster's core. The rate at which hard binaries heat a star cluster's core, against the core's central density, for binary fractions of 1%, 5%, 20%, both on logarithmic axes and in units that put the balance for five per cent at one. The heating rate per binary does not depend on how hard the binary is — the energy released per encounter is proportional to its binding, and the chance of an encounter to its size, so the two cancel — and summed over a core whose radius shrinks as its density rises, the total grows as the square root of the density. The horizontal line is the demand: the energy the rest of the cluster carries away by relaxation through its half-mass radius, which the core has no say in. Each supply curve meets it at one density: 25 for 1%, 1.0 for 5%, 0.063 for 20%. The core contracts or expands until it produces exactly what is demanded, and a core rich in binaries does so at a lower density. Gravitation

A core that makes what the cluster asks for

After core collapse, a star cluster's binaries heat it — but how much they heat it is not theirs to decide. The rest of the cluster carries energy outward at a rate fixed by its own size and mass, and the core contracts or swells until its binaries supply exactly that. So the core's structure follows from the halo's demand, and the cluster expands at a rate that forgets how it was born.

 The binaries in a fed core are spread evenly in energy. The steady distribution of binding energies among the hard binaries of a core where new binaries enter at x = 5 and leave by ejection from a cluster whose escape speed is 4 times its dispersion: the number per unit binding energy, against binding energy in units of mσ², from 20,000 simulated binaries weighted by the time each spends at each energy. Between twice the entry energy and a third of the typical ejection energy (about 120) it is flat to within 5 per cent, and it falls away smoothly near ejection because a single encounter can throw a binary out from anywhere above about 24. A flat distribution is what a population with a constant flux through energy and a constant drift must have: binaries pass every energy at the same rate, and spend the same time at each. Gravitation

The binaries a core keeps are spread evenly in energy

A hard binary in a star cluster gains binding energy at the same average rate however hard it already is. So in a core that is fed new binaries and loses hardened ones, every binding energy is passed through at the same speed, and the steady population is flat in energy — which, counted per factor of energy, means most binaries are within a factor of ten of being thrown out. Where they were born hardly matters; where they leave decides everything.

 Ejected before touching, or touching before ejected. The separation at which a hardening pair of 0.7-solar-mass main-sequence stars is typically ejected — where one encounter's recoil exceeds an escape speed 4 times the dispersion, at binding energy about 120 mσ² — against the cluster's velocity dispersion, both axes logarithmic, beside the separation at which the two stars fill their Roche lobes and begin to merge, 0.0082 au (1.8 solar radii). The ejection separation falls as the inverse square of the dispersion, because a hotter cluster needs a harder binary to throw it out. The lines cross at a dispersion of about 18 km/s. In clusters slower than that — open clusters and nearly all globulars — a binary is ejected while its stars are still apart. In faster ones — the most massive globulars and the nuclear star clusters at galaxies' centres — it is driven into contact first, and its two stars become one. Gravitation

Ejected from a globular, merged in a nucleus

A hard binary in a star cluster hardens until something ends it: an encounter that throws it out, or its own stars touching. Which comes first is decided by one comparison — the separation at which the cluster's escape speed is exceeded against the separation at which the stars meet — and the first shrinks as the square of the cluster's velocity dispersion. In a globular cluster binaries leave while their stars are still apart; in the nucleus of a galaxy they merge first, and the heating that holds the cluster up becomes a factory of stars more massive than they should be.

 Which populations can share a cluster's temperature. The share of a cluster's mass in a heavy population against the mass of its members relative to an average star of 0.5 solar masses, both on logarithmic scales, with Spitzer's limit: a heavy component can reach energy equipartition with the stars only below the line where its share of the mass times its members' mass ratio to the power three halves equals 0.16. Above it the heavy bodies sink, cannot slow down enough to match the stars' kinetic temperature, and decouple into a subsystem of their own. Stellar black holes, 10 solar masses and about 2% of the mass, sit at 1.8 — 11 times beyond the limit. Neutron stars sit at 0.05 and massive white dwarfs at 0.14, below it; binaries, at a tenth of the mass, sit at 0.28, just beyond it, which is part of why they gather toward the centre. Only the black holes are beyond it by an order of magnitude. Gravitation

A population that cannot share the cluster's temperature

Encounters between stars push a cluster toward equal kinetic energies, so heavy bodies slow down and sink. Spitzer showed in 1969 that past a certain point they cannot slow down enough: the heavy population falls away from the rest and becomes a cluster of its own. Stellar black holes are past that point by a factor of ten, in every globular cluster that kept them — and the subsystem they form, not the stars' binaries, is what holds most clusters' cores open until the black holes have been spent.

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