A spin too low to keep a disc from barring

The size of a disc galaxy is set by a number its halo was born with: the spin parameter $\lambda$, a dimensionless measure of how fast the dark-matter halo turns compared with how fast it would have to turn to be held up by rotation. Gas falling into the halo shares its angular momentum, settles into a disc where that angular momentum puts it on circular orbits, and the disc’s scale length comes out as about $\lambda$ times the halo’s radius divided by $\sqrt{2}$. Haloes spin slowly — $\lambda$ is typically three or four hundredths — so discs are small compared with their haloes, and their spread of sizes at a given mass follows the spread of $\lambda$.

That account treats the disc as a passive tracer of the halo’s spin, and for most discs it is. But a disc is also a self-gravitating system with dynamics of its own, and a small disc is a dense one. If the halo turned slowly enough, the disc that forms inside it is compact enough for its own gravity to dominate the rotation in its inner parts, and a disc in that state is not stable. It rearranges itself — not into spiral arms that come and go, but into a bar, a long-lived elongated structure of stars rotating as a rigid pattern through the centre of the galaxy.

This essay connects the two facts. The condition for a disc to bar can be written in terms of the halo’s spin and the disc’s share of the halo’s mass, and the scatter of halo spins then predicts what fraction of discs should be barred. The prediction depends on one number that is hard to measure directly, and the count of barred galaxies measures it.

When a disc’s own gravity wins #

A disc of stars supported by rotation is held in balance by the combined gravity of the disc and the halo. Its stability against large-scale distortions depends on which of the two dominates. If most of the inward pull comes from the halo, which does not respond to the disc’s perturbations, then a small distortion of the disc produces a small change in the force and the disc settles back. If most comes from the disc itself, a distortion changes the force that holds the disc up, and a bar-shaped distortion grows.

That was the insight of Ostriker and Peebles in 1973, who found in early numerical experiments that isolated discs barred within a few rotations and proposed that real discs were stabilised by unseen haloes — one of the first arguments for dark matter. Efstathiou, Lake and Negroponte turned it into a usable criterion in 1982: a disc of mass $M_d$ and scale length $R_d$, with peak rotation speed $V_{\max}$, is stable if

$$\epsilon = \frac{V_{\max}}{\sqrt{G M_d / R_d}} \gtrsim 1.1 .$$

The denominator is the speed the disc’s own gravity would supply at about one scale length; the ratio is a measure of how much of the rotation the disc carries. A disc in a halo with a flat rotation curve of speed $V_{200}$ has $GM_d/R_d = \sqrt{2}\, m_d V_{200}^2/\lambda$, where $m_d$ is the disc’s share of the halo’s mass. Adding the disc’s own contribution to the peak speed and squaring:

$$\epsilon^2 = \frac{\lambda}{\sqrt{2}\,m_d} + 0.39,$$

and the disc is stable only if $\lambda > 1.16\, m_d$.

The condition is a comparison of two small numbers: the spin of the halo against the fraction of its mass in the disc. The same disc share that is safe in a fast-spinning halo bars in a slow one, because the slow halo makes the disc smaller and the smaller disc is more self-gravitating. A disc holding five per cent of its halo’s mass needs $\lambda$ above about six hundredths, which is a high spin; a disc holding one per cent is safe at a third of that.

The haloes that fall below the line #

Halo spins are not a single value. In simulations they follow a lognormal distribution with a median near 0.035 and a width of about one half in $\ln\lambda$, the same for haloes of every mass — a scatter that the tidal torques acting before collapse put there.

The critical spin moves in proportion to the disc share, and it sweeps across the distribution. For a disc share of one per cent it sits deep in the low tail, and only one halo in a hundred would bar its disc. For three per cent it sits at the median, and half the haloes would. For five per cent it sits on the high side and most haloes would. The broadness of the spin distribution means that the predicted fraction is a smooth function of the disc share, without any sharp threshold.

The curve can be read in reverse. Bars are common: about a third of nearby disc galaxies show a strong bar in optical images, and in the near-infrared, where dust is less of a disguise, about two-thirds show a bar of some strength. The Milky Way has one. If those bars were made by the instability described here, the disc share that produces them is between about two and a half and four per cent — and the answer hardly depends on the width of the spin distribution, because all three curves cross the observed band at nearly the same place.

What a low-spin disc looks like #

The difference between a disc that bars and one that does not is visible in its rotation curve.

In a halo with $\lambda = 0.06$ the disc is spread over a scale length of four per cent of the halo’s radius, and its own contribution to the rotation, the dashed curve, peaks at two-thirds of the halo’s speed. The total rotation curve rises gently and is flat — the kind of curve that refuses to fall and is dominated by the halo almost everywhere. In a halo with $\lambda = 0.02$ the same disc is a third the size, its own gravity supplies more speed at its peak than the halo does, and the total rotation curve has a hump in the inner few per cent of the halo’s radius where the disc dominates. That hump is the signature of a disc that carries its own rotation, and it is the configuration that bars.

Written this way the criterion becomes a statement about a single observable ratio, independent of the disc share and the spin separately: a disc whose own speed at its peak exceeds about 57 per cent of the total is expected to bar. That is less than the 85 per cent usually taken to define a “maximal” disc, the largest disc compatible with a rotation curve. The question of how much of a galaxy’s inner rotation its disc supplies — the division of a rotation curve between disc and halo that admits more than one answer from the curve alone — therefore has a dynamical consequence. Barred galaxies should, on this argument, have discs that are at least moderately heavy, and unbarred ones lighter discs.

Measurements of disc mass that do not use the rotation curve — from the vertical motions of the disc’s stars, which measure the disc’s surface density directly — have tended to find discs supplying less of the inner rotation than maximal, with typical shares near the threshold. The two lines of argument are consistent, and they point the same way: real discs sit near the edge of stability, which is where a population with scattered spins and a common disc share would put them.

Where bars should be commonest #

The disc share is not the same in every galaxy. The fraction of a halo’s mass that ends up as stars is small everywhere and largest in haloes of about $10^{12}$ solar masses — the mass of the Milky Way’s — falling steeply in smaller haloes, where supernova-driven winds blow the gas out, and in larger ones, where active nuclei keep it hot. That relation is measured by matching the counts of galaxies to the counts of haloes.

Taking each halo’s disc share from that relation turns the bar criterion into a prediction across the galaxy population. Barred discs should be commonest in galaxies of roughly the Milky Way’s mass, where the stellar share reaches three or four per cent, and rare in dwarfs, where it falls below one per cent, and in the most massive systems. Surveys of bars across galaxy mass find broadly that shape: bars are uncommon in the smallest discs and common at intermediate masses. The decline at high mass is harder to test, because the most massive haloes host few discs at all.

The prediction for dwarfs is the most uncertain, because dwarf discs are often dominated by gas rather than stars, and gas belongs in the disc share; adding it raises their share and their predicted bar fraction. But the gas in dwarfs is also turbulent and thick, which stabilises it in ways the criterion does not include, and dwarf discs are known to be dominated by their haloes in their rotation curves. That both effects push towards fewer bars in dwarfs is part of why the trend is robust.

The Milky Way as one test #

The Galaxy is the one disc whose mass and halo are measured from the inside, and it makes a fair test of the arithmetic. Its stars and gas amount to about six times $10^{10}$ solar masses, and its dark halo to about $10^{12}$, so its disc share is five or six per cent — at the high end of what the abundance-matching relation gives for its mass. By the criterion, a disc that heavy is stable only in a halo spinning faster than about $\lambda = 0.065$, which about one halo in nine does. The other eight would bar such a disc, and the Milky Way has a bar, a structure about five kiloparsecs long whose pattern turns once every two hundred million years or so and whose resonances shape the motions of stars near the Sun.

One galaxy cannot test a statistical prediction, but it can test consistency, and the Galaxy passes in a second way too. Its inner rotation curve is dominated by its disc: the stars within the Sun’s orbit supply most of the circular speed there, which is the “heavy disc” condition the criterion turns on. A Milky Way without a bar would have been the surprise.

Andromeda, the other large disc in the Local Group, has a heavier halo and a disc share near the Galaxy’s, and it too has a bar, weaker and seen nearly end-on. The Large Magellanic Cloud, a dwarf with a disc share below one per cent in stars but a substantial gas fraction, has an off-centre bar that is almost certainly the product of its interaction with the Small Cloud rather than of instability in isolation — a reminder that tidal encounters make bars by a separate route.

Bars arrive late #

If bars form whenever a disc is heavy enough, the bar fraction should also change with cosmic time, and it does. Among disc galaxies ten billion years ago, bars were much rarer than today; the fraction rises steadily towards the present. The disc share does not change that much over the same period, so something else must have been stabilising the young discs.

The likeliest candidate is their temperature in the dynamical sense. Discs at high redshift are gas-rich and turbulent, with random velocities a substantial fraction of their rotation — a stirring that has to be paid for by gravity and feedback together. A disc whose stars move randomly at a third of the rotation speed resists the coherent rearrangement a bar requires, in the way the criterion’s simple form does not include. As the gas is consumed and the discs cool and settle, they cross into the regime where the halo’s spin and the disc’s share decide the outcome, and the bars appear.

Space telescope images in the infrared have pushed the earliest bars back to within three billion years of the Big Bang, in discs that were evidently already settled. That is consistent with the picture: the bar fraction at any epoch is the fraction of discs that are both cold enough to bar and dense enough for their own gravity to win, and the second condition — the one the halo’s spin controls — is the one that sets the fraction once the first is met.

A bar is a way of moving angular momentum #

A bar is not just a shape. Once formed, it transfers angular momentum outward through the disc and into the halo, and the transfer reshapes the galaxy: gas inside the bar’s corotation radius loses angular momentum and flows inward, feeding star formation in a central concentration, while the bar itself slows down as it gives its angular momentum to the halo’s dark matter. The pattern speed of the bar decides where its resonances fall in the disc, and those resonances are where rings and the ends of spiral arms are found.

That connects the bar criterion to the question where the slowest gas went. A halo holds more material at low angular momentum than an exponential disc can accommodate, and one way to dispose of it is internal: a bar drives the slow gas inward into a central, rotating, flattened concentration — a pseudo-bulge — rather than a classical spheroidal bulge. Low-spin haloes have more slow material and make denser discs; denser discs bar; bars build pseudo-bulges. The same initial condition, a halo that happened to turn slowly, leads through two separate arguments to the same end, a disc galaxy with a bar and a disc-like bulge.

And it connects to the arms that undo themselves. Spiral arms in a disc that is not unstable to a bar are transient, regenerated by perturbations and dissolved by the angular momentum they transport. A barred disc has a persistent driver, and its arms are often attached to the bar’s ends. The halo’s spin decides which of the two kinds of spiral structure a galaxy has.

What the criterion leaves out #

The criterion is calibrated on numerical experiments with simple discs in rigid or live haloes, and it is a rough guide rather than a law. More recent simulations show that bar formation depends on the disc’s velocity dispersion, its thickness and the shape of the halo’s density profile at the centre; a hot disc resists bar formation, a live halo can both feed and stabilise a bar by exchanging angular momentum, and a central bulge can prevent a bar from forming even in a disc that would otherwise be unstable. The threshold of 1.1 is commonly quoted with a range from about 0.9 to 1.2.

The halo is treated as having a flat rotation curve everywhere, which overstates its pull at the centre for a cuspy halo and understates it for a cored one. The disc forms with exactly its halo’s specific angular momentum, which is the assumption of the size relation and the one known to fail at low angular momentum. Spins are drawn from a single lognormal for every halo mass; the disc share is the stellar mass alone, from one abundance-matching relation, with no scatter and no gas. And bars can also form or be destroyed by interactions with other galaxies, which this account of isolated discs leaves out entirely.

Still open: whether bars record their haloes’ spins #

The argument predicts more than a fraction. It predicts that among galaxies of the same mass, the barred ones should have formed in slower-spinning haloes and should therefore have smaller discs for their mass, at least when the bar formed. Galaxies’ disc sizes at fixed mass do scatter in the way the spin distribution predicts, and comparing barred and unbarred galaxies of equal mass should show the barred ones systematically more compact.

The comparison is complicated because a bar changes the disc it lives in — moving mass inward and outward, thickening the centre, building rings — so a disc that barred a few billion years ago no longer has the profile it had when the bar formed. Surveys that measure bars, disc sizes and stellar ages together in large samples are now able to make the comparison for galaxies whose bars are young, which are found at higher redshift as well as nearby. If barred discs at fixed mass prove more compact, the halo’s spin will have been read in a galaxy’s shape twice: once in the size of its disc, and again in whether that disc could hold together.