Astronomy & Celestial Mechanics
the sky, computed
Concept

Two-body relaxation — where it appears

The gradual randomisation of stellar velocities by the accumulated effect of distant encounters. Its timescale grows with the number of members, so a thousand-star cluster relaxes in tens of crossing times while a galaxy never does.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

7 decades of relaxation time, and a Hubble time between them. Crossing time and two-body relaxation time for 5 self-gravitating systems, against the number of bodies in each. The lower marks are the crossing time R/σ, which spans 3 decades; the upper ones are t_relax ≈ 0.1N/ln N times it, which spans 8. The horizontal rule is a Hubble time, and the same expression puts these systems on opposite sides of it. Two of them relax — an open cluster and a globular cluster — so their stars have exchanged energy, their heavy stars have sunk, and their present structure is not the one they were born with. The rest do not: a dwarf spheroidal, an elliptical galaxy, a cluster of galaxies, the elliptical missing it by a factor of 10⁶, which is why a galaxy is modelled as a smooth potential with no stars in it at all. The dependence is on N and hardly at all on anything else, because the Coulomb logarithm makes every decade of impact parameter contribute equally. A cluster that relaxes also evaporates: about a stellar mass in a hundred and forty leaves per relaxation time, so the globular cluster empties in 6.1·10¹¹ years.

How long a system takes to forget

A star crossing a galaxy is deflected by every other star, and the deflections add as a random walk. The time for that walk to change a star's energy by its own amount is a hundred million billion years for a galaxy and a billion for a globular cluster, and everything about how the two are modelled follows from which side of a Hubble time that falls on.

gravitation · Two-body relaxation 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.

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.

gravitation · Binary heating 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.

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.

galaxies · Galactic nuclei The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

gravitation · Impulse approximation Where the stars that are torn apart come from, round a 10⁶ solar-mass hole. The rate at which stars are delivered onto orbits reaching the tidal radius, per logarithmic interval of their orbital radius, for a hole of 10⁶ solar masses at the centre of a nucleus that is isothermal outside the hole's influence and a Bahcall–Wolf cusp within it, with a velocity dispersion of 54 km/s set by the black hole mass–dispersion relation. The radius is in units of the hole's influence radius, GM/σ² = 1.47 pc. Close in, a star's orbit is so short and its angular momentum changes so slowly that the loss cone empties every orbit, and the rate is limited by relaxation diffusing stars into it; far out, the angular momentum wanders across the whole cone in one orbit, the cone is full, and the rate is limited by how many stars there are and how long they take to come in. The flux peaks at 0.27 influence radii, near where the two regimes meet (0.19), and the total is 7.4·10⁻⁵ disruptions a year. The nucleus model is the simplest there is and real nuclei differ from it by factors of several; the shape — a narrow band of radii supplying most of the flares — is what survives.

The stars a black hole eats come from a narrow band

A star is torn apart only if its orbit happens to point almost exactly at the hole — within a millionth of the possible directions of its angular momentum. Stars on those orbits are gone within one orbit, and the only way new ones arrive is by the slow random walk of encounters with other stars. Close to the hole that walk is too slow; far from it the orbits take too long; and most of the stars a hole destroys come from a narrow band between.

galaxies · Tidal disruption 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.

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.

gravitation · Binary heating How fast a disc forgets, by the mass of the lumps it is made of. The two-body relaxation time of a star moving at 30 km/s among perturbers of mass m, against m, for a fixed mass density of 0.015 solar masses per cubic parsec — about the density of molecular gas in the plane near the Sun — with a Coulomb logarithm of 5. The time falls as one over the perturber's mass: the same mass in heavier lumps relaxes a population proportionally faster. The orange point is the stars themselves, 0.04 solar masses per cubic parsec in half-solar-mass pieces: 4.9·10⁹ million years, 3.5·10⁵ times the age of the universe (dashed). Molecular clouds of 5·10⁵ solar masses give 13 billion years as point masses — comparable to the age of the disc. Fewer than a hundred-thousandth of the bodies, carrying less than half the mass, relax the disc 3.8·10⁵ times faster than all the stars together.

A few thousand clouds outrun a hundred billion stars

Encounters between stars relax the Galactic disc in a time hundreds of thousands of times the age of the universe, so in practice they do nothing. Put a fraction of the same mass into a few thousand molecular clouds and the time falls to the age of the disc — because relaxation counts the mass of each perturber, not their number. The disc's present thickness is the record of that, and of a bound on anything heavier hiding in the halo.

gravitation · Two-body relaxation Five clocks around the black hole at the centre of the Galaxy. Timescales against distance from a black hole of 4.3·10⁶ solar masses at the centre of a cluster whose stellar mass inside r is 10⁶ (r/1 pc)^1.5 solar masses, made of stars of 1 solar mass. Ordinary two-body relaxation (orange) is slow everywhere, 8.4·10¹⁰ years at 0.01 pc. Orbits that close are nearly Keplerian and keep their orientation until they precess — through the stars' own mass farther out, through relativity close in — and over that coherence time the torques between them add rather than random-walk. That makes scalar resonant relaxation (blue) faster than two-body relaxation inside about 0.57 pc, and vector resonant relaxation (green), which turns orbital planes and is not stopped by precession within a plane, faster still: 3.5·10⁷ years at 0.1 pc, against the 6·10⁶-year age of the young stars there. Coefficients of order one are set to one; the slopes are the content.

Torques that add because the orbits hold still

Near a massive black hole every star's orbit is almost a fixed ellipse, repeated thousands of times before it precesses. The torques the other stars exert on it therefore point the same way orbit after orbit, and they add in proportion to time instead of cancelling in a random walk — so angular momentum near the hole relaxes hundreds of times faster than ordinary encounters allow, and orbital planes faster still.

gravitation · Two-body relaxation 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.

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.

gravitation · Binary heating An excess the hole cannot afford for long. The mass a 10⁶-solar-mass hole gains from disrupted stars, in units of its own mass, against how long an excess in the disruption rate lasts, for excesses of 30, 100, 300 over the relaxed nucleus's 7.4·10⁻⁵ a year, with half of each Sun-like star accreted. An excess of 100 doubles the hole in 2.7·10⁸ years, an excess of 300 in 9·10⁷. The post-starburst phase that selects the hosts lasts about a billion years, so an excess of a hundred sustained through it would have grown every such hole several times over, and the galaxies that were post-starbursts a few billion years ago would carry holes several times heavier than their stars' motions predict. A binary's spike of 0.1 a year for 3·10⁵ years feeds it only 1.5%.

A flare rate a hole cannot afford for long

Tidal disruption flares are found in galaxies that stopped forming stars within the last billion years far more often than those galaxies' numbers allow — by factors estimated between thirty and two hundred. Each proposed cause can be priced. A dense cluster from the burst would need ten times the hole's mass of new stars inside a parsec; a triaxial nucleus would give twenty times too much; and any excess of a hundred, kept for the length of the phase, would feed a million-solar-mass hole several times its own mass.

galaxies · Tidal disruption

Named alongside it

The objects these essays reach for when they reach for this one.

Relaxation timeGlobular clusterCore collapseCoulomb logarithmLoss coneBinary heatingAngular momentumCrossing timeEquipartitionHard binaryImpulse approximationMass segregation

All concepts