A disc that turns slower than its mass requires

A disc galaxy today is a cold system. The gas in it moves in nearly circular orbits, with random motions of about ten kilometres a second on top of rotation of a hundred to three hundred. The ratio of rotation to random motion is ten or twenty or more, and every speed read off such a disc — its outer flat speed, its line width, its speed at some radius — is overwhelmingly a speed of rotation.

The discs that formed most of the stars in the universe, around ten billion years ago at redshifts of one to three, were different. Spectrographs that map the velocity field across a galaxy found them rotating, but with random motions of their ionised gas typically several tens of kilometres a second, and in the smaller ones comparable to the rotation itself. They were thick, clumpy and stirred, forming stars at many times the rate of a present-day spiral.

Measuring the Tully–Fisher relation for those galaxies ran into a problem that looked, at first, like evolution. The relation between rotation speed and mass was far looser than in nearby galaxies, and a population of objects lay well off it, turning much too slowly for their mass. The resolution is not that these galaxies obeyed different physics, and not necessarily that they had different amounts of dark matter. It is that their random motions carry part of their weight, and a rotation speed measures only the part that rotation carries.

Pressure holds up part of a disc #

In a galaxy that barely rotates the random motions of its stars hold it up entirely, and the virial theorem converts their dispersion into a mass. A disc is the opposite extreme, held up almost entirely by rotation. A turbulent disc is in between, and the balance in it can be written for each radius.

The random motions act as a pressure, equal to the density times the square of the dispersion. Where the density falls outward — and in an exponential disc it falls everywhere — that pressure pushes outward, doing some of the work gravity’s inward pull otherwise demands of rotation. The gas needs less centripetal acceleration from its orbital motion, and it turns more slowly than the circular speed its enclosed mass sets.

For an exponential disc of scale length $R_d$, with a dispersion σ that does not change with radius and a thickness set by its own gravity, the balance is

$$V_{\rm rot}^2 = V_c^2 - 2\sigma^2\,\frac{R}{R_d}.$$

The factor of two comes from the thickness: a disc held vertically by its own gravity is thinner where its surface density is higher, so its midplane density falls twice as fast with radius as its surface density does, and the outward pressure gradient is correspondingly steeper. The correction is sometimes called asymmetric drift, a name inherited from the stars of the Milky Way, where the older and more disordered populations lag behind the rotation of the young ones for the same reason.

This is the same balance that makes a flat rotation curve a measurement of mass in a cold disc: there, the dispersion term is negligible and the rotation speed is the circular speed. In a turbulent disc the rotation speed is only one of two terms, and a mass read from it alone is too small.

The figure’s most important feature is the shape. A cold disc with a flat circular speed has a flat rotation curve. The same disc with 70 km/s of turbulence has a rotation curve that falls steeply outward, not because the mass stops increasing but because the pressure correction grows with radius. Falling outer rotation curves were reported in massive star-forming galaxies at redshift two, and read at face value they would mean galaxies with far less dark matter within their discs than their present-day counterparts — a decomposition that is ambiguous even for a cold disc with a perfectly measured curve. How much of the fall is instead pressure support is one of the central arguments in interpreting them, and the answer depends on exactly the quantities in the equation above: the dispersion, whether it is constant, and how the disc’s thickness varies.

The same lag in the Galaxy’s own stars #

The correction was not invented for distant galaxies. It was found in the Milky Way, among its stars, long before anyone measured a disc at redshift two.

Stars near the Sun belong to populations of different ages, and the older a population is, the larger the random velocities its stars have acquired from a few billion years of gravitational encounters with clouds and spiral arms. When the average motion of each population around the Galactic centre is measured, the populations with larger random velocities lag further behind the circular speed. The lag grows roughly as the square of the dispersion, as the balance above requires, and extrapolating it to zero dispersion is one way the Sun’s own motion relative to a truly circular orbit is found. The relation between lag and dispersion in the Sun’s neighbourhood was worked out in the 1920s and gave the correction its name.

The Galaxy’s stars and the gas of a distant turbulent disc are held up by the same kind of pressure, and they lag by the same kind of amount. The difference is only in which component is measured. A star’s orbit is traced one star at a time, from inside the Galaxy; a distant disc’s gas is seen all at once, blurred, in a single spectrum per position. In both cases the circular speed — the quantity the mass determines — is recovered only after the random motions have been accounted for, and in both cases a speed read without that correction describes the population rather than the potential it moves in.

The same accounting underlies the way a pressure-supported system is weighed. The virial theorem turns a dispersion into a mass for a system with no rotation at all, and the rotation-curve method turns a speed into a mass for a system with no dispersion. A turbulent disc needs both at once, and the balance above is how they are added.

The relation read two ways #

To separate what pressure support does from anything else, a model population can be built in which nothing else happens. Eighty model discs are spread in mass from 3 × 10⁹ to 2 × 10¹¹ solar masses of stars and gas, and every one has a circular speed that lies on the same baryonic relation as nearby galaxies, $M = 47\,V_c^4$, with 0.08 dex of scatter. None has evolved; the relation between mass and circular speed is identical for all of them. The only thing varied is the dispersion.

First, the present day: dispersions between 8 and 12 km/s, and rotation measured at 2.2 disc scale lengths.

Cold discs return the relation almost unaltered. The slight flattening, to 3.87 from four, is the pressure correction at its smallest: even ten kilometres a second of turbulence takes a few per cent off the rotation of the slowest discs, and nothing noticeable off the fastest.

Now the same discs with the dispersions measured at redshift two, between 25 and 90 km/s. A disc of a given circular speed cannot hold arbitrarily large random motions and remain a disc, so the dispersion is capped at 0.45 of the circular speed, where a disc measured at 2.2 scale lengths would have almost no rotation left. Twenty-two of the eighty reach the cap, all of them among the smaller discs.

The picture is the one early surveys at high redshift found. The fastest-turning, most massive discs lie near the local relation. Moving down in mass, the discs become more dominated by their dispersion and slide to the left, turning too slowly for their mass. The least ordered — turning at less than their own dispersion — form a group far off the relation at low speeds. A line fitted through all of them is much shallower than the relation any of them obey and has more than three times the scatter.

Read as a statement about galaxies, this would say that at redshift two the relation between mass and rotation was loose and differently sloped, and that many galaxies were anomalously slow rotators. Read correctly, it says the rotation speed is the wrong quantity to put on the axis when a galaxy is partly held up by something else.

The trend with mass is not an accident of the model. A fixed range of dispersion is a larger fraction of a smaller galaxy’s circular speed, so the small discs are the disordered ones, as they are observed to be. That correlation is what tilts the relation rather than merely scattering it: the correction is largest precisely at the low-mass end.

Adding the disorder back #

If random motion carries part of the weight, a speed that includes random motion should restore the relation. The simplest such speed combines the two in quadrature,

$$S_K = \sqrt{K\,V_{\rm rot}^2 + \sigma^2},$$

and the value $K = 0.5$ was proposed for it on a clean argument. A body supported entirely by rotation at speed $V_c$ has $S_{0.5}^2 = 0.5\,V_c^2$. A body supported entirely by random motion, an isothermal sphere with the same circular speed, has a one-dimensional dispersion of $V_c/\sqrt2$, and so also $S_{0.5}^2 = 0.5\,V_c^2$. With $K = 0.5$ the two extremes of support give the same combined speed for the same mass, and galaxies between them should land in between.

This is essentially what was found when the combined speed was applied to galaxies out to redshift one: the relation in $S_{0.5}$ was tight, and it showed little change in its zero point with time, where the relation in rotation alone had appeared to scatter and shift. The combined speed needs no model of any individual disc, only its rotation and its dispersion, and in integrated spectra of faint, poorly resolved galaxies that is often all that can be measured.

In this model population, though, the combined speed does not recover everything. The exponent is still well short of four, and the scatter is nearly twice that of the cold discs. Something in the argument for 0.5 does not match these discs.

Correcting each disc #

The alternative is to use the pressure correction itself. Given a disc’s measured rotation at radius $R$, its dispersion and its scale length, the circular speed follows from the balance above: $V_c^2 = V_{\rm rot}^2 + 2\sigma^2 R/R_d$.

In the model the correction is exact, because the model’s discs obey the balance they are corrected with. In real galaxies it is only as good as three assumptions: that the dispersion is constant with radius, that the disc’s thickness is set by its own gravity, and that the measured dispersion is the random motion of the gas and not something else. The third is the hardest. A spectrum taken through a telescope’s blurred beam mixes gas from neighbouring parts of the disc moving at different rotation speeds, and that unresolved spread of orbital motion inflates the measured dispersion exactly as unresolved binary orbits inflate the dispersion of a dwarf galaxy. The inflation is largest in the centre, where the rotation curve rises steeply, and in the smallest, most poorly resolved galaxies — the same galaxies the correction matters most for.

Nor is the disc’s gas the only thing with a pressure. The gas layer of the Milky Way itself is thicker than its thermal pressure can support, and turbulence, magnetic fields and cosmic rays share the work. Which of those a measured line width includes, and which it misses, is part of the correction’s uncertainty at every redshift.

Why one half, and when #

The mismatch between $S_{0.5}$ and the pressure correction has a precise origin, and it shows up if the weight $K$ is treated as a free parameter.

Substituting the balance into $S_K$ gives $S_K^2 = K V_c^2 + \sigma^2(1 - 2K R/R_d)$. When $K = R_d/2R$, the dispersion term vanishes and $S_K^2 = K V_c^2$ exactly: the combined speed is then a fixed fraction of the circular speed for every disc, whatever its turbulence. The best weight on rotation is therefore set by where the rotation was measured.

The two arguments for a weight agree only at one radius. The isothermal-sphere argument gives 0.5 and knows nothing about discs. The pressure correction gives $R_d/2R$, which equals 0.5 for rotation measured at one scale length and 0.23 at 2.2 scale lengths, where many rotation speeds are quoted. A survey that measures rotation further out and adds dispersion with a weight of one half is over-weighting the dispersion, and the relation it recovers is flatter and looser than the one its galaxies obey.

Real data soften the contrast, because measured dispersions and measured rotation speeds are not the idealised quantities of the model. Integrated dispersions include some unresolved rotation, which partly stands in for the missing rotation term, and a disc’s dispersion need not be constant with radius. The empirical success of 0.5 in faint, unresolved galaxies is real. The model says what that success depends on, and why a single universal weight should not be expected once rotation curves are resolved well enough to measure at a chosen radius.

What changed, and what did not #

The point of the model is a separation. Every one of its eighty discs has exactly the same relation between mass and circular speed as a nearby galaxy, and yet read one way the population looks radically different, and read another way it looks identical. Evidence that the Tully–Fisher relation itself evolved has to survive the correction for pressure support before it means anything.

Some evidence of change does survive, and it is about something else. The baryonic mass of a galaxy at redshift two is dominated by gas that cannot be measured directly for most galaxies. It is estimated from the rate of star formation, through the relation between gas and star formation that holds nearby, and that relation’s own timescale is one of the things that may have been different then. A shift in the relation’s zero point at high redshift is therefore a shift that could be in the speed, in the mass, or in the calibration that turns a measured rate of star formation into a gas mass. Pressure support is the correction that has to be made first, because it is the largest, and it is the one that looks most like evolution when it is left out.

There is also a check on the whole picture that does not depend on the relation. A disc whose thickness is set by its turbulence has a scale height of roughly its radius times the ratio of its dispersion to its circular speed, so a disc a few kiloparsecs across with a dispersion a quarter of its circular speed should be about a kiloparsec thick — several times the thickness of the thin disc of a present-day spiral. Star-forming galaxies at those redshifts that are seen edge-on do appear thick and clumpy rather than thin, which is what dispersions of that size require. The random motions are a real, structural property of those galaxies, and a relation between mass and speed that ignored them would be ignoring the shape of the galaxies it describes.

Still open: what kept the discs stirred? #

Turbulence decays. Random motions in gas dissipate their energy in shocks within about the time it takes to cross the disc’s thickness, a few tens of millions of years, so the dispersions measured at redshift two must have been continuously driven. Two sources are argued for. One is the energy released by young stars — winds, radiation and supernovae — which was abundant in galaxies forming stars so fast. The other is gravity: gas flowing inward through a disc that is marginally unstable releases orbital energy as random motion, and gas accreting from outside adds more. The two predict different relations between a disc’s dispersion and its rate of star formation and gas content, and measurements have been read as favouring each, or a mixture changing with galaxy mass and time. Which one sets the dispersion decides how much of a galaxy’s weight its own turbulence carries, and so how much correction the relation needs.