Velocity dispersion — where it appears
Named by 32 essays across 5 fields — each of them below, with the objects they name alongside it.
An average that weighs what cannot be watched
A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.
A drag with nothing to drag against
A massive body moving through a sea of light ones raises an overdensity behind itself and is pulled back by it. The force does not depend on the masses of the background bodies at all — and it is strongest at one particular speed.
A galaxy held up by disorder
An elliptical galaxy barely rotates. What holds it up against its own gravity is the randomness of its stars' motions, and the virial theorem turns that randomness into a mass by exactly the reasoning that turns a rotation speed into one.
A cluster weighed three ways
The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.
The mass at the centre that is not stars
One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole.
A distance measured with a stopwatch
The images of a lensed quasar sit at the stationary points of an arrival-time surface, and the height between two of them is a delay in days. That delay is proportional to a distance, and the distance is proportional to one over the Hubble constant — so a flickering quasar gives H₀ with no calibrated distance under it.
A map stretched by the thing it measures
A redshift survey plots galaxies at distances derived from their redshifts, and a galaxy's redshift contains its own motion as well as the expansion. So the map is systematically distorted — squashed on large scales, drawn out into radial spikes on small ones — and both distortions are caused by the gravity the survey exists to measure.
A drag that is strongest in the middle
Dynamical friction is not like air resistance. Only the stars moving slower than the body contribute at all, so the drag rises from nothing, peaks when the body has slowed to about the speed of everything around it, and falls away again — which is why a fast satellite is never captured and a slow one sinks in a hurry.
Red, gas-poor, and still spiral-shaped
A galaxy falling into a cluster meets a headwind of hot gas at two thousand kilometres a second. Where that ram pressure exceeds the disc's own grip on its gas, the gas goes; the stars, which are not a fluid and feel no pressure, do not. The result is a spiral with no fuel, and it is the commonest kind of galaxy in a cluster.
Whether it merges is a ratio of two times
Two galaxies passing each other either merge or do not, and what decides it is not how close they come. It is whether the encounter lasts long enough for the internal motions to respond — a slow, prograde passage transfers orbital energy into stellar orbits and the pair is bound; a fast one leaves both galaxies heated and still moving.
An arm that is undone by the work it does
A spiral pattern takes angular momentum from the inner disc and delivers it to the outer, which lets mass move inwards without violating anything. It is paid for in random motion, and a disc with too much random motion cannot carry a wave — so the pattern destroys the conditions it needs to exist.
A dispersion inflated by orbits nobody resolved
A velocity dispersion measured from single spectra is not the dispersion of the system's centres of mass. Every star in a binary carries its own orbital velocity of a kilometre or so, and for a dwarf galaxy whose real dispersion is smaller than that, the measured value — and the dark-matter content computed from its square — is mostly binaries.
Three mass models that fit the same curve
A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.
The factor that multiplies every quasar mass
A reverberation lag and a line width give a length times a velocity squared, which is a mass multiplied by an unknown number of order one. That number is fixed by making a few dozen nearby active galaxies lie on a correlation measured in quiescent ones — so every black-hole mass at redshift two rests on a fit performed at redshift zero.
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.
The speed a line width stands in for
A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.
A disc that turns slower than its mass requires
The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.
Stars that weigh and do not shine
Most of the mass in an old stellar population is in stars too faint to contribute to its light. How many of them there are is set by the mass function stars are born with, and the mass-to-light ratio every dark-matter measurement divides by can be wrong by a factor of two without a single colour changing.
A black hole thrown out by its own merger
When two black holes merge, the gravitational waves they radiate carry momentum as well as energy, and unless the two are identical and spinning in line the waves leave more in one direction than the other. The remnant recoils, at anything from a few tens of kilometres a second to several thousand — and whether a galaxy keeps its central black hole comes down to how its spins were pointing in the last orbit before the merger.
The black holes that can be weighed are the heavy ones
A black hole's mass is measured from the motions of stars or gas only inside its sphere of influence, the region where its gravity beats the galaxy's, and that region has to be resolved. At a given galaxy, a heavier hole rules a larger sphere and can be weighed further away, in a much larger volume — so the sample of weighed holes is drawn from the top of the distribution, and the relations built from it describe the heavy end of every galaxy rather than the typical one.
The one radius where the orbits stop mattering
A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.
A census of visitors is a census of the survey
Three bodies from other stars have passed through the inner solar system in eight years of looking. Turning that count into a number of such bodies per cubic astronomical unit needs the volume in which they could have been seen, and that volume is set by the Sun's gravity, by each body's size and by how many weeks a survey had to catch it — so the first number the census gives is a measurement of the survey.
A baseline that grows four astronomical units a year
The Earth's orbit gives a parallax a baseline of two astronomical units and never more. The Sun's own motion through the nearby stars gives one that lengthens by four a year without limit — but the stars move too, by as much, so the long baseline yields a distance only for a group whose motions average away, and for a century it was the only way to reach past a few hundred parsecs.
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.
A dispersion that weighs the sheet only if the orbits are known
A lens's images cannot tell how much of its mass is in the galaxy and how much is spread through a sheet around it, and the time delay — and so the Hubble constant — scales with the answer. The stars' velocity dispersion can tell, because orbits feel mass in three dimensions where light only integrates along a line. One dispersion through one aperture weighs the sheet only to sixteen per cent, because the shape of the orbits moves it as much; the same dispersion measured in rings weighs it to four.
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.
Motions across the sky measure the orbits first
A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.
A stirring supernovae cannot pay for
The gas in a disc forming stars at redshift two moves at fifty kilometres a second in random directions, five times faster than in a spiral today, and turbulence that fast decays in a few tens of millions of years. Supernovae are the obvious source of the energy, and they run out at about thirteen kilometres a second however fast the disc forms stars. What holds the discs up is gravity — gas stirring itself by falling inward — and a disc stirred that way is also a disc pouring its gas into its own centre.
A galaxy born without a halo
When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference.
A drag that stops where every orbit keeps time
Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.
What Jupiter kept of the Sun's siblings
The Sun was born among a few thousand other stars, and while their planets formed they flung out small bodies at a few kilometres a second — exactly the speeds at which Jupiter can capture a passing body. Counted honestly, the birth cluster supplied almost every capture Jupiter has ever made: billions of bodies in thirty million years, against a few million in the four and a half billion since. And billions is still a hundred-thousandth of the Sun's own comets. Whatever the siblings gave the Sun, Jupiter was not how it arrived.
A galaxy that grows faster by eating small
Two equal elliptical galaxies merge and the result has twice the mass and twice the size. A galaxy that takes in the same mass as thirty small companions ends up nearly four times the size. The difference is one line of the virial theorem, in which companions bring mass without much energy, and it may answer a puzzle thrown up by the first deep infrared surveys: the most massive galaxies at redshift two are a fifth of the size of anything like them today.
Named alongside it
The objects these essays reach for when they reach for this one.
Dark matterMass-to-light ratioCoulomb logarithmDynamical frictionM sigma relationVirial theoremDwarf spheroidalDynamical massGravitational focusingRotation curveThe baryonic Tully–Fisher relationThe core–cusp problem