An equilibrium reached without a single encounter

The virial theorem turns a photograph into a mass. For a bound system of stars in equilibrium, twice the kinetic energy equals minus the potential energy, $2K + W = 0$, and since the kinetic energy is measured by the spread of the stars’ velocities and the potential energy depends on the mass and the size, a velocity dispersion and a radius give a mass with no orbit followed. Applied to the same system as a statement about energy, it says that a self-gravitating body heats up as it loses energy.

Both uses assume the system has reached the theorem. For a star cluster that is not an assumption but a history: stars in a cluster pass close enough to one another often enough to exchange energy, and a cluster forgets its initial state on a relaxation time of a few hundred million years. For a galaxy it is a puzzle. The relaxation time of a galaxy’s hundred billion stars is a million times the age of the universe; its stars have never exchanged energy with one another in any meaningful way; and yet galaxies held up by the disorder of their stars are in virial equilibrium to the precision anyone has measured. Something brought them there without encounters.

A system that cannot have encounters #

The figure is computed with the simplest system that has the answer in it and nothing else. It is a sphere of four hundred concentric shells of equal mass. By Newton’s shell theorem, each shell is pulled only by the mass inside it and feels nothing from the shells outside, so a shell’s motion depends on one number, the mass enclosed, and nothing about where that mass is. The shells can pass through one another freely, and when they do the enclosed masses change. No shell ever collides with or scatters off another; there is no such thing as an encounter in this system.

That makes it a sharper test than a simulation of stars would be. An N-body calculation with a few thousand particles has encounters whether it wants them or not, because its particles are too few and too massive, and part of any relaxation it shows is numerical graininess rather than physics. A system of shells cannot relax that way at all; if it reaches equilibrium, the collective potential did it.

The sphere starts almost at rest, with a small random radial velocity for each shell — enough that twice the kinetic energy is a tenth of the potential energy — and with its density falling as one over the radius, concentrated towards the centre as a real protogalaxy’s would be. Time is measured in units of the free-fall time: the time a cold sphere of the same mass and radius takes to collapse to its centre.

The virial ratio $2K/|W|$ starts near zero, because almost nothing is moving. As the sphere falls, potential energy becomes kinetic, and the ratio rises past one — past equilibrium — to 1.7 when the inner shells, which have the shortest free-fall times, reach the centre and turn round. The system rebounds and the ratio swings low; it swings high again, with a smaller amplitude; and by five free-fall times it stays within ten per cent of one. The system has reached the virial theorem in a few crossing times. Nothing in it has touched anything else.

Half the size it fell from #

Where it ends up is fixed before it moves, by two facts together.

The total energy $E = K + W$ is conserved, because nothing enters or leaves. At equilibrium the virial theorem makes $K = -W/2$, so $E = W/2$. So the final potential energy is exactly twice the total energy, and the size the system settles at — measured by the gravitational radius $r_g = GM^2/|W|$ — is

$$r_{g,\,\rm final} = \frac{GM^2}{2|E|}.$$

For a sphere starting exactly at rest, $E$ is its initial potential energy, and the final gravitational radius is exactly half the initial one. A cold self-gravitating cloud settles at half the size it fell from, and it does so whatever the details of the collapse, because the answer uses only conservation of energy and the endpoint. The computed sphere started slightly warm, which moves the prediction to 0.79 of the initial radius; it settles at 0.86, the difference being the few shells flung out unbound and the softening that keeps gravity finite at the very centre.

The factor of two is the reason for a number every cosmologist quotes. A region of the early universe slightly denser than average expands with everything else, slows, stops at a turnaround radius and falls back. At turnaround its density is $9\pi^2/16 \approx 5.6$ times the background’s. It settles at half that radius, eight times denser; and it takes as long again to collapse and settle as it took to reach turnaround, during which time the background’s density falls by a further factor of four. The product, $5.6 \times 8 \times 4 = 178$, is the density contrast of a collapsed halo — the reason a dark-matter halo is conventionally defined as the region within which the mean density is about two hundred times the background’s. The number comes from the same two lines of algebra as the figure.

The half-mass radius has no such rule, because the collapse rearranges the mass, and it shrinks by more than the gravitational radius: from 0.71 to 0.32 of the initial radius. The inner shells end deeper in the potential than they started and the outer ones further out, and the smooth initial profile becomes a dense core surrounded by an extended envelope.

Out of step #

The oscillation damps without any friction, and the mechanism is visible in the individual shells.

Each shell, once the collapse is over, oscillates in and out through the centre on an orbit with a period set by its own energy. Shells with different energies have different periods, and so they drift out of step: a group of shells that passes through the centre together will, a few periods later, be spread around their orbits with no memory of having been together. That is phase mixing. The total density at each radius, which is the sum over all the shells there, stops oscillating once the phases are spread, even though no individual shell has stopped moving or lost any energy. Equilibrium, for a collisionless system, is not rest; it is the statistics of many orbits no longer changing.

Phase mixing alone, however, cannot explain the settled state, because it preserves each shell’s energy. A system whose shells kept their initial energies would mix to a steady state that remembered them exactly. The collapse did something more.

A potential that moves the energy #

A particle moving in a potential that does not change keeps its energy. A particle moving in a potential that changes while it moves does not: its energy changes at the rate

$$\frac{dE}{dt} = m\,\frac{\partial\Phi}{\partial t}.$$

During the collapse the potential is changing violently — deepening as the mass falls in, becoming shallower as it rebounds — on the same timescale as the orbits themselves. A shell that happens to be falling in while the potential is deepening gains binding; one climbing out while the potential is relaxing loses it. The outcome depends on where each shell was in its orbit during the fluctuations, which is different for different shells.

The figure shows the result. Before the collapse each shell’s energy was nearly its potential energy at its starting radius, a smooth curve. After it the energies are scattered by amounts comparable to themselves. On average the shells that started near the centre have become much more bound, and those that started near the edge less bound, and some of the outermost have gained enough to escape altogether. The total is exactly the same as it was: the collapse has moved energy outward, from the core to the envelope. That redistribution is what Donald Lynden-Bell called violent relaxation in 1967. It is fast, because the potential fluctuates on the dynamical time; it is irreversible in practice, because the fluctuations damp as phase mixing smooths the density that causes them; and it stops when they do.

It has one signature that encounters do not share. The rate $\partial\Phi/\partial t$ is the same for every particle at a given place, whatever its mass, so violent relaxation changes the energy per unit mass and not the energy per particle. It therefore produces no mass segregation. A star cluster relaxed by encounters sinks its heavy stars to the centre, as equipartition requires; a galaxy relaxed violently leaves its heavy and light stars mixed in the same proportions everywhere — which is what is observed, and one of the ways the history of an elliptical galaxy can be read from the fact that its stars have not sorted themselves.

A limit the collapse cannot break #

Violent relaxation is powerful, but there is one thing it cannot do, and the thing it cannot do has been used to weigh a particle.

A collisionless system obeys Liouville’s theorem: the density of its stars in the six-dimensional space of positions and velocities is carried along unchanged by every orbit. The collapse stirs that density into ever finer filaments, and what an observer sees — averaged over any finite volume of phase space — is a coarse-grained density that can only fall as the stirring proceeds, never rise. The densest a relaxed system can be, in positions and velocities together, is the densest its initial state was. The shells in the figure have concentrated into a core in space, but they have done it by spreading out in velocity, and the product of the two has not gone up.

For dark matter this becomes a statement about the particle. If the dark matter is a fermion — a massive neutrino, or something like one — the exclusion principle caps its phase-space density at a value set by its mass, and the relaxed core of a halo can be no denser in phase space than that cap. The smallest, densest dark-matter-dominated galaxies, the dwarf spheroidals around the Milky Way, have measured velocity dispersions of ten kilometres a second in cores a few hundred parsecs across. Requiring their phase-space density to fit under the cap gives a lower bound on the particle’s mass of a few hundred electronvolts. The argument, from 1979, was the first to rule out ordinary neutrinos as the galaxies’ dark matter — the particles whose mass cosmology now weighs by the structure they stopped from forming — and it rests on nothing more than the fact that a collapse can stir phase space but not compress it.

A collapse the oldest stars remember #

The question of whether the Milky Way itself was assembled by a collapse like this is the oldest argument in the study of galaxy formation, and violent relaxation sits in the middle of it.

In 1962 the orbits of the Galaxy’s oldest, most metal-poor stars were found to be highly eccentric, plunging through the disc on nearly radial paths, while more metal-rich stars moved on rounder orbits. The interpretation offered was a single rapid collapse of the protogalaxy, a few hundred million years long, during which the first stars formed while the gas was still falling and kept the radial orbits of the collapse — the shells of the figure, frozen into stars before they could settle. In 1978 a different reading of the halo’s globular clusters proposed the opposite: the halo was assembled slowly from independent fragments, each with its own history, falling in over billions of years.

Both pictures need the same physics to end in the equilibrium observed, and the modern answer is that both happened. The inner halo is dominated by stars from one massive early merger, whose debris is on exactly the radial orbits a violent collapse leaves; the outer halo is threaded with streams from small systems still being taken apart, whose relaxation is incomplete and whose phases have not mixed. The Galaxy is a collisionless system at every stage of the process at once, relaxed in its centre and still collapsing at its edge.

Two clocks for one equilibrium #

The difference between the two routes to equilibrium is not one of degree.

Two-body relaxation, in which stars exchange energy through the cumulative effect of many weak encounters, takes about $N/(8\ln N)$ crossing times, where $N$ is the number of stars. An open cluster of a thousand stars relaxes in eighteen, well within its lifetime; a globular cluster of a hundred thousand in about a thousand, comparable to its age; a galaxy of a hundred billion in five hundred million crossing times, where its age is about a hundred. Violent relaxation takes a few crossing times regardless of the number of stars, because it depends on the potential and not on the graininess of the mass.

Collapse simulations made with many thousands of particles in the 1980s found that a cold, lumpy collapse settles into a profile whose surface brightness falls as the exponential of the quarter power of the radius — the law that describes the light of elliptical galaxies, found empirically in 1948 and never derived. The agreement is one of the reasons elliptical galaxies are thought to owe their structure to violent relaxation; the law’s own origin is part of what is still not understood.

So the virial equilibrium assumed when a galaxy is weighed, when a cluster of galaxies is weighed three ways, or when a dark-matter halo’s mass is inferred from its satellites’ velocities, is an equilibrium reached collisionlessly, in the collapse that made the system — and for a cluster of galaxies still accreting, one reached only in its inner parts. A cluster observed while a subcluster falls in has a virial ratio oscillating like the first swings of the figure, and a mass computed from its velocities in that state is wrong by tens of per cent in whichever direction the swing happens to be. The theorem is exact; the moment at which it applies is not guaranteed.

What the shells cannot do #

The shell model is spherical by construction, and a spherical collapse is the least violent one there is. Real collapses are lumpy, with sub-clumps falling in along filaments, and the potential fluctuates in shape as well as in depth; that makes the relaxation faster and more complete than the figure shows. Spherical shells also cannot exchange angular momentum, so the model has no rotation, no flattening and none of the instability by which a collapse of stars on nearly radial orbits becomes a bar or a prolate spheroid. Gravity is softened at the centre, at a fiftieth of the initial radius, so that shells can pass through the middle with a finite speed; it slightly alters the equilibrium and is why the virial ratio is computed from the forces rather than from the potential energy. The initial profile and the initial warmth are choices: a perfectly uniform, perfectly cold sphere collapses homologously, every shell reaching the centre at the same instant, and then stays in step for many oscillations — the one initial condition that does not relax quickly, and a warning against reasoning about collapse from the most symmetric case. And four hundred shells are enough to show the mechanism and too few to measure the final profile’s shape precisely.

Still open: why the end state has the shape it has #

Lynden-Bell’s theory of violent relaxation predicted not only that it happens but what it produces: a most-probable distribution of energies, derived by statistical mechanics, which for a collisionless system obeying phase-space conservation takes a form like the Fermi–Dirac distribution. The prediction has never matched simulations well. Real collapses stop relaxing before they reach the most probable state, because the fluctuations that drive the process die away as it proceeds — relaxation is incomplete, and how incomplete depends on the details. Meanwhile, simulated dark-matter haloes from every kind of initial condition converge on nearly the same density profile, a shallow cusp inside and a steeper fall-off outside, whose innermost kiloparsec is still argued about. That universality is the most striking regularity in the structure of collisionless systems, it is what violent relaxation leaves behind, and no theory derives it from first principles. The collapse computed here reaches the virial theorem exactly and a profile nobody can yet predict.