Barred discs should be the small ones

A disc galaxy is the size its halo was born with. The gas that settles into a disc keeps the specific angular momentum it acquired, along with the dark matter, from the tidal torques of its neighbours, and a halo’s angular momentum is summarised by one dimensionless number, the spin parameter λ. The disc’s scale length comes out proportional to λ times the halo’s radius. Haloes of a given mass have a wide spread of spins — a lognormal distribution with a median near 0.035 and a spread of about a factor of 1.6 either way — and why that median is the same for haloes of every mass is one of the cleaner results of the theory of how structure forms. The spread in spin is why galaxies of the same mass differ in size.

The same number decides whether a disc can hold together. A disc too compact for its mass is dominated by its own gravity and is unstable to forming a bar, and in the model in which discs take their size from their haloes, a disc bars if its halo spins too slowly: the classic stability criterion becomes $\lambda < 1.16\,m_d$, where $m_d$ is the share of the halo’s mass in the disc. With a disc share of a few per cent and a median spin of 0.035, that threshold cuts through the middle of the spin distribution, and it predicted a barred fraction of between a third and two-thirds — about what is seen.

The criterion predicts something more specific than a fraction, and this essay works it out. If spin decides both a disc’s size and whether it bars, then the barred discs are the small ones. How much smaller is the measurement: a sharp threshold makes them half the size, and every effect other than spin that influences whether a disc bars shrinks the contrast. A measured size difference is therefore a measurement of how decisive the spin is.

Barred discs, if the spin decides, are half the size #

Take haloes all of the same mass, holding discs that are each three per cent of their halo’s mass. The threshold spin is then 0.035, which is the median of the distribution, so exactly half the discs are below it and bar. Their spins average about 0.022, and the spins of the stable half average about 0.050. Since the disc scale length is proportional to the spin, the barred discs are on average 0.45 the size of the unbarred ones — a factor of more than two.

For a lognormal distribution split at its median the ratio has a closed form: it is $\Phi(-\sigma)/\Phi(\sigma)$, the ratio of the normal distribution’s tails one standard deviation from the centre, which for σ = 0.5 is 0.446. Moving the threshold away from the median changes the barred fraction a great deal but the size ratio surprisingly little. With a disc share of five per cent the threshold is at a spin of 0.058, eighty-five per cent of discs bar, and the ratio is 0.42; with two per cent, fewer than a fifth bar and the ratio is 0.43. Wherever both kinds of disc are common, a sharp spin threshold makes barred discs between two-fifths and a half the size of their unbarred neighbours of the same mass.

That is a large effect, larger than almost any other systematic difference between barred and unbarred galaxies that has been looked for. It is exactly the kind of prediction that should be easy to test, and its size is itself the first thing worth noticing: a model in which spin decides bars predicts a difference that could hardly be missed.

Two populations of disc size in one kind of halo #

The distribution of sizes shows the starkness of the prediction more directly than its means. With a sharp threshold, every disc smaller than the median size bars and every disc larger does not. The two distributions do not overlap at all; they meet at a single size. A survey of galaxies of one halo mass, sorted by disc size, would find bars in all of the smaller half and none of the larger.

Nothing in nature divides that cleanly, and no one expects it to. A disc’s stability depends on more than its own share of its halo’s gravity: on how concentrated the halo is, on whether there is a bulge whose own gravity stabilises the centre, on how much of the disc is cold gas rather than stars, on whether a passing companion has given it a tidal kick that a marginally stable disc would amplify into a bar. Each of these varies from galaxy to galaxy independently of the spin, and each blurs the threshold. The question is how much.

The same split when the spin only leans on the outcome #

A simple way to represent everything else is to blur the threshold: instead of a disc barring if and only if its spin is below the critical value, the probability that it bars falls smoothly from nearly one well below the threshold to nearly zero well above it, over a width w in the logarithm of the spin. A width of zero is the sharp threshold. A width of 0.5 means that a disc with half the critical spin is still not certain to bar and one with twice the critical spin is not certain to be stable.

With a blur of 0.6, the two distributions of size overlap over most of their range. Small discs are still more often barred than large ones, and the barred mean is still smaller, but now by a factor of 0.60 rather than 0.45. The overlap is what makes a measured contrast informative: the contrast falls steadily as the blur grows, so the contrast measures the blur.

How much the size difference says about what makes a bar #

Drawn against the blur, the size ratio rises from 0.45 for a sharp threshold to 0.57 at a blur of half — comparable to the spread of the spins themselves — to 0.70 at a blur of one. The curves for different disc shares nearly coincide, so the relation between the contrast and the blur does not depend much on the uncertain disc share. That makes it a clean instrument: a measured ratio of sizes reads off a blur.

The interesting values are the small contrasts. If barred and unbarred discs of the same mass turn out to differ in size by a fifth, the blur must be about 1.7, which means the threshold is smeared over a factor of six in spin. At that point the spin is no longer deciding whether a disc bars; it is leaning on the decision. A contrast of a few per cent would mean the spin hardly matters at all, and the explanation for which discs bar would lie almost entirely elsewhere — in the haloes’ concentrations, the discs’ gas fractions or their histories of encounters. A factor of two would mean the spin decides almost everything.

That is the sense in which the size contrast is more useful than the barred fraction. The barred fraction depends on the disc share and the threshold’s position, both uncertain by tens of per cent, and many combinations give the observed third to two-thirds. The size contrast is almost independent of those, and depends on the one thing the fraction cannot say: how sharp the dependence on spin is.

What a survey binned by spin should see #

There is a second way to put the same question, which is closer to what surveys have done. Spins cannot be measured for individual galaxies directly, but a proxy can be built from what can be measured — the disc’s scale length and its rotation speed, which in the disc-formation model give back the spin of its halo — and galaxies can be sorted into bins of estimated spin and the barred fraction counted in each. A sharp threshold would give a step: all barred below, none above. A blur gives a ramp, falling over a factor in spin that grows exponentially with the blur.

Analyses of this kind, using samples of several thousand galaxies from the large spectroscopic surveys, have reported that barred galaxies are indeed found preferentially at lower estimated spin, with the barred fraction falling as the spin rises. That is the predicted sign. The ramp they find is gentle rather than steep, which in this language means a substantial blur. Part of it is real and part of it is the estimate: a spin inferred from a size and a speed carries errors of tens of per cent from inclinations, from the conversion of light to mass, and from the model that turns size into spin, which blurs any observed step by about 0.3 on its own. And the proxy is built from the disc size, which is the very quantity in question — sorting by spin estimated this way is sorting by size at fixed speed, so the two tests are the same test.

Where in mass the contrast should be sharpest #

Haloes of different mass hold different shares of their mass in their discs. The share of stars peaks at about three per cent in haloes like the Milky Way’s and falls steeply in both directions: dwarf galaxies are inefficient at making stars, and the largest haloes have their gas kept hot. With the disc share taken from that relation, the model predicts barred fractions that peak in Milky-Way-mass haloes and fall towards both ends, and size contrasts that do something less obvious.

Where bars are common, the contrast is near the value for a split at the median. In the smallest haloes, where the disc share is a fraction of a per cent, the threshold sits far down in the tail of the spin distribution; only the very slowest-spinning haloes produce discs that bar, and those discs are very small indeed compared with the typical stable disc, so the contrast is largest of all. It is also hardest to see, because the barred galaxies are rare there and the samples small.

The prediction for dwarfs has a complication worth stating. The disc share that matters for stability is the share of mass in the disc’s own gravity, gas included, and dwarf galaxies are gas-rich — their discs can hold several times more gas than stars. With the gas counted, a dwarf’s disc share rises and the predicted bars become more common, which is the direction surveys of nearby dwarfs suggest. A comparison done with stellar mass alone would put the threshold in the wrong place for exactly the galaxies where the contrast is strongest.

The nearest barred disc is a compact one #

The galaxy that can be measured best is a single data point, but it points the predicted way. The Milky Way is barred — its bar, about five kiloparsecs long, was established from the motions and distribution of stars towards the centre, as a galaxy has to be measured from inside it — and its disc is compact for its mass. Its stellar scale length, about 2.5 to 3 kiloparsecs, is smaller than that of most spirals of the same stellar mass and rotation speed: comparisons with large samples of external galaxies place it roughly one standard deviation below the typical size for its mass, which in the disc-formation model corresponds to a halo spin about two-thirds of the median.

Andromeda, slightly more massive, has a disc whose scale length is about five and a half kiloparsecs, twice the Milky Way’s, and a bar that is weaker and hidden inside a large boxy bulge. The pair are about as far apart in disc size as the sharp-threshold prediction puts barred and unbarred galaxies, and the more compact of the two has the clearer bar. Two galaxies cannot measure a blur, and Andromeda’s bulge, much larger than the Milky Way’s, would on its own have stabilised its disc against barring strongly. But the comparison shows the kind of difference the prediction is about, in the two discs whose sizes are known best.

The Milky Way’s compactness has been noted for other reasons — it makes the Galaxy’s rotation curve fall more steeply in its outer parts than a typical spiral’s, and it makes its disc supply a larger share of its inner rotation than average. If the compactness reflects a slow-spinning halo, the bar, the steep outer curve and the dominant disc are three faces of one fact about how the Galaxy’s halo was torqued thirteen billion years ago.

A bar changes the disc it lives in #

The prediction is for discs at the moment they bar, and a bar does not leave its disc alone. A bar is a way of moving angular momentum: it takes angular momentum from the inner disc and gives it to the outer disc and to the halo, so the inner disc becomes more concentrated and the outer disc spreads. Over a few billion years a barred disc’s scale length can grow, by amounts that simulations put at tens of per cent depending on how strongly the bar couples to its halo. The halo absorbs angular momentum too, and in doing so slows the bar, as any rotating pattern is slowed by the material it drives at its resonances.

That cuts against the prediction. A disc that started small and barred would, a few billion years later, be less small than when the bar formed, and a comparison of barred and unbarred discs today would understate the contrast set at formation. The sign of the correction is known; its size depends on the bar’s age, which can be estimated from the ages of the stars in its central regions. The comparison that tests the prediction most directly is therefore between unbarred discs and discs whose bars are young — which is why the surveys now measuring bars, disc sizes and stellar ages together, nearby and at redshifts of one and more, are the ones able to make it.

There is a second, quieter complication. The disc size measured from its starlight is the size of its stars, and stars form where the gas was dense enough. A low-spin halo makes a compact gas disc, but how the light is distributed depends on where and when the stars formed, and a disc measured at one wavelength and one surface-brightness threshold can look more or less compact than its mass. Comparisons in the near-infrared, where the light follows the old stars and so the mass, are the ones to trust.

What the model leaves out #

The spin distribution is the lognormal fitted to simulated haloes, with no dependence on mass or environment. The disc scale length is taken as proportional to the spin at fixed halo mass, which ignores the factors in the full disc-formation model that depend on the halo’s concentration and the disc’s own gravity, and that make low-spin discs a little less compact than strict proportionality. The stability criterion is the classic one for a disc in a fixed halo, which simulations have found to be indicative rather than exact; a live halo that responds to the bar changes both the threshold and the growth. The blur is a single number standing for many separate effects with different dependences on mass and redshift, and a real survey would see a mixture. The disc share is the stellar share from the stellar-to-halo mass relation, without gas.

Still open: whether the contrast is large, small or absent #

The prediction is unusually sharp for a statement about galaxy populations: two numbers measurable in a survey — disc sizes of barred and unbarred galaxies of equal mass — and an interpretation that depends on almost nothing else. A factor near two would make the halo’s spin the dominant cause of bars, a factor near one would make it irrelevant, and anything in between measures the share.

What the surveys have found so far is the predicted sign at modest strength, with the complication that the bar has had time to change its disc. The measurement that would settle it is the contrast for galaxies whose bars formed recently, in near-infrared light, at fixed halo mass rather than fixed stellar mass — the last of these needs a weighing that does not come from the disc itself, from lensing or from satellites. If barred discs prove to have been much smaller when their bars formed, the spin of a halo will have been read twice in one galaxy’s shape: once in the size of its disc, and again in whether that disc could keep its symmetry.