A halo with too much slow gas for a disc

The size of a galactic disc is set by a single number describing how fast its dark halo turns. Gas falling into the halo shares its spin, conserves its angular momentum as it falls, and settles into a disc at the radius where its angular momentum puts it in a circular orbit; the disc’s scale length then comes out as the spin parameter times the halo’s radius, divided by about the square root of two. That simple account gets the sizes of disc galaxies right to within their scatter, across a factor of a thousand in mass. It is one of the most successful predictions in the theory of galaxy formation.

It uses only the mean. A disc is a distribution of material over angular momentum — a little at the centre, where the orbits are small, and more further out — and so is a halo. If the gas keeps its angular momentum element by element, as the simple account assumes, the disc’s distribution is the halo’s, and the question of whether the account works is a question about the whole distribution, not its average. This essay compares the two whole distributions, and finds that they disagree in a way that says something definite about what happened to the gas on its way in.

Two distributions with the same mean #

The disc’s distribution is easy to write down, because a disc is well described. The light of a disc galaxy falls off exponentially with radius over four or five scale lengths, and its rotation curve is close to flat over most of that range. Material at radius $R$ in a flat rotation curve of speed $V$ has specific angular momentum $j = RV$, and the mass inside $R$ of an exponential disc of scale length $R_d$ is $1 - (1 + R/R_d)\,e^{-R/R_d}$ of the total. The mean is $\bar{j} = 2R_dV$. Substituting, the fraction of a disc’s mass with angular momentum below $j$ is

$$M(<j) = 1 - (1 + x)\,e^{-x}, \qquad x = 2j/\bar{j}.$$

It starts flat — very little of an exponential disc’s mass has small angular momentum, because very little of a disc’s area is near its centre — and rises through most of the mass between a third of the mean and twice it.

The halo’s distribution comes from simulations. Bullock and collaborators measured how a halo’s mass is distributed over specific angular momentum in a large set of simulated haloes and found that a single two-parameter form fits almost all of them:

$$M(<j) = \frac{\mu\, s}{\mu - 1 + s}, \qquad s = j/j_{\max}.$$

The halo has a largest angular momentum, $j_{\max}$, and a shape parameter $\mu$ that sets how much of its mass sits near zero. Small $\mu$, near one, means a halo with a great deal of slowly rotating material; large $\mu$ means less. The simulated haloes had $\mu$ near 1.25 on average, with most between about 1.05 and 1.4.

The hero figure draws both, scaled to the same mean, as a disc that kept its halo’s angular momentum would be. The halo’s curve rises steeply from the origin where the disc’s starts flat. Below a tenth of the mean, the disc has two per cent of its mass and the typical halo eleven. The halo also runs out — at four times its mean, for $\mu = 1.25$ — where the disc has a thin tail extending further. Two distributions with the same mean have different shapes.

Why the two curves start differently #

The disagreement is concentrated at the bottom of the distribution, and the reason is a matter of how each curve starts. Near zero the disc’s distribution rises as the square of the angular momentum — expand $1 - (1+x)e^{-x}$ for small $x$ and the first term is $x^2/2$ — because the material with small angular momentum is the material at small radius, and the area of a disc inside a small radius grows as the radius squared. A disc simply has very little room near its centre.

The halo’s distribution rises linearly: for small $s$ it is $\mu s/(\mu - 1)$, a constant amount of mass for each equal step in angular momentum, all the way down to zero. A halo’s slowly rotating material is not confined to a small region. It is spread through the whole halo — mass elements on nearly radial orbits at every radius, elements in the inner shells assembled before the tidal torques had much time to act, and the debris of small haloes that merged with their angular momentum pointing against the host’s. A linear start against a quadratic one gives the halo far more mass at low angular momentum, and the smaller the shape parameter, the steeper the linear start.

The comparison is between a three-dimensional object, whose low-angular-momentum material fills a volume, and a two-dimensional one, whose low-angular-momentum material has to fit into the middle of a plane. Some mismatch at the bottom is unavoidable for any distribution that rises linearly from zero, and the size of the mismatch is what the shape parameter sets.

The disc a halo would build #

The difference is clearer when turned back into a disc.

Gas that kept the halo’s distribution exactly and settled on circular orbits would make a disc with a spike at its centre — six times the exponential’s density at a tenth of a scale length — then a body thinner than an exponential’s through the radii where most of a real disc’s light is, then a shallow outer skirt, and a sharp edge at eight scale lengths where the halo’s angular momentum runs out. Nothing about that resembles a disc galaxy. The central spike in particular would be a dense, compact component holding about a sixth of the mass, which in a real galaxy would have to be a bulge.

A surplus of slow gas #

The useful single number is the surplus: the largest amount by which the halo’s cumulative distribution exceeds the disc’s. It is the mass with too little angular momentum to sit anywhere an exponential disc would put it.

For the typical halo the surplus is seventeen per cent; for the haloes richest in slow material, a quarter. It falls below a tenth only for haloes with $\mu$ above 1.75, which are rare. If the gas keeps its angular momentum and nothing else happens, every disc galaxy would sit on a central component holding a sixth of its baryonic mass or more.

Many disc galaxies do have bulges, and the classical account was that the surplus was the bulge — the slow-rotating gas of the halo’s interior collapsing into a dense central component while the faster gas made the disc. The trouble is that a large fraction of disc galaxies have no such component, or have one far smaller than a sixth of their mass. Among the giant spirals within about ten megaparsecs, more than half have no classical bulge, and those that have a central component often have a pseudo-bulge — a flattened, rotating concentration that looks like a product of the disc’s own internal evolution rather than of early collapse. Among the dwarf discs, pure exponentials with no central component at all are the rule.

Telling a classical bulge from a pseudo-bulge is itself a measurement with its own uncertainties. A classical bulge is round, supported by random motions, with a light profile that falls off steeply and a Sérsic index above two; a pseudo-bulge is flattened, rotating, often star-forming, with a profile close to exponential. The surplus computed here is a mass of slowly rotating gas, and gas that collapses in the first billion years would form something like the first; gas that settles into the disc and is driven inward later would form something like the second. Measuring the mass in either needs the disc’s light separated from the bulge’s, and the stellar mass-to-light ratio of each — the same freedom that lets a rotation curve be divided between components in more than one way. The census of bulgeless discs is robust to those uncertainties, because a galaxy with no central component at all needs no decomposition.

This is a different problem from the one that first made angular momentum the difficulty of galaxy formation. The early simulations with gas produced discs ten times too small, because the gas cooled in small clumps, which sank to the centres of their haloes by dynamical friction and handed their orbital angular momentum to the dark matter on the way — the angular momentum catastrophe, a failure of the mean. Simulations with feedback that keeps gas hot until it has settled solved that, and their discs now have about the right sizes. The distribution problem survives the solution: a halo whose gas keeps its mean exactly still has too much of it at the bottom.

Twists make it worse #

The universal profile describes the magnitude of each mass element’s angular momentum, and implicitly assumes all of it points the same way. It does not. The angular momentum of a halo’s inner shells and its outer shells points in directions that differ by tens of degrees, because the shells were assembled at different times from different parts of the tidal field that spun them, and the gas and dark matter of a single halo are often misaligned by a similar amount.

What a disc inherits is not the magnitude of each element’s angular momentum but its component along the disc’s axis. An element tilted away from the axis contributes less than its magnitude; one tilted past ninety degrees contributes against the disc’s rotation. A modest twist of thirty degrees changes little. A twist of sixty, which simulations find in some haloes between the gas and the dark matter, more than doubles the material below a tenth of the mean. Any process that mixes the directions of the incoming material deepens the surplus, and mergers do exactly that — the same vector cancellation that turns two discs into a slower spheroid acts, at lower amplitude, within every halo as it accretes.

How much of the slowest gas has to go #

If the surplus does not become a bulge, it must have gone somewhere. The candidate is feedback: winds driven by supernovae and young stars, blowing gas out of the galaxy’s centre. That kind of feedback is selective in angular momentum without being designed to be. Gas with the least angular momentum falls furthest in, reaches the highest densities, forms stars first and fastest, and sits where the energy from those stars is released; the outflows that follow carry it away preferentially.

The figure removes the slowest gas first, renormalises what remains to its own mean, and compares it with an exponential. Each curve has a minimum. Too little removed leaves the surplus; too much leaves the remainder short of the low-angular-momentum gas an exponential’s centre does need. For the typical halo the best match comes from removing a quarter of the gas, and for haloes rich in slow material nearly half.

That is a large amount of gas to blow out of a galaxy, but it is in line with what is inferred independently. The baryons in a dwarf galaxy’s disc are a small fraction of what its halo should have collected — most of them are missing — and the metals a galaxy keeps say that much of what it made was thrown out. Simulations that include strong enough outflows produce bulgeless dwarfs with exponential discs, and they do it by exactly this mechanism: the outflows are launched from the centre during the first bursts of star formation, they carry the slow gas, and the gas that settles later has more angular momentum. The same outflows, rearranging the gas at the centre fast enough to stir the dark matter, are one of the proposed answers to the argument about the innermost kiloparsec of dwarf haloes.

The removal never gives a perfect exponential. A residual remains — about six per cent of the mass for the typical halo — and it is at the other end, at high angular momentum, where the halo runs out and the exponential’s tail does not. Removing gas from the bottom cannot fix a shortage at the top. The outer parts of disc galaxies do have long, faint tails, often steeper or shallower than the inner exponential, and the mismatch at high angular momentum may be visible there: in the second component of a galaxy that has two, or in discs that break to a steeper fall-off at several scale lengths.

A population of bulges that is not there #

Put the halo statistics together and the problem is quantitative.

If gas kept its angular momentum in every halo, the median galaxy would have a central component of a sixth of its mass and almost none would have less than a tenth. The observed population has a large fraction with less than a tenth and many with none. The two distributions hardly overlap. Either the gas did not keep its angular momentum, or it did not keep all of its mass, or both, and in every case the simple account of disc sizes — which works — works because it uses the mean, which the processes that fix the distribution mostly preserve.

That last point needs care. Removing the slowest quarter of the gas raises the mean angular momentum of what remains, by about thirty per cent for the typical halo, and a disc formed from the remainder is larger than the simple account predicts. The success of the simple account in matching disc sizes is therefore partly a coincidence of two errors: haloes lose some angular momentum to the dark matter during assembly, which shrinks discs, and outflows remove the slowest gas, which enlarges them. The sizes come out right while both effects are of order tens of per cent.

Exchanges inside the disc #

There is a second escape, which does not need any gas to leave. A disc does not keep its angular momentum distribution after it forms. Spiral arms and bars transfer angular momentum outward through the disc, moving material inward at the centre and outward at the edge, and over billions of years that redistribution can turn a profile of one shape into another. Viscous discs, in which angular momentum is transported locally, evolve towards exponential profiles under fairly general conditions, which is one long-standing explanation for why discs are exponential at all.

That route makes the opposite prediction about the centre. A disc that started with a surplus at low angular momentum and redistributed it internally would pile the surplus into a central concentration that rotates — a pseudo-bulge — rather than eject it. The nearby giant spirals with pseudo-bulges and no classical bulges fit this; the bulgeless dwarfs, whose central densities are low, do not. The two escapes are not exclusive, and the balance between them probably changes with mass: outflows dominant in dwarfs, where shallow potentials let winds escape, and internal redistribution dominant in giants, where the gas cannot get out.

The two escapes leave different records in the stars. The stars in a disc keep the angular momentum their gas had when they formed, apart from the slow stirring of spiral arms, so the oldest stars in a disc carry the angular momentum distribution of the first gas to settle. If outflows removed the slowest gas early, the oldest disc stars should already be missing the low-angular-momentum tail; if internal redistribution moved it inward later, the oldest stars should show the tail and the youngest should not. The distribution of angular momentum among stars of a given age is not easy to measure — it needs ages for individual regions of a disc and velocities good enough to separate rotation from random motion — but it is exactly what the combination of a velocity map and a stellar-population fit gives, spaxel by spaxel, for thousands of nearby galaxies.

What the comparison assumes #

The halo profile is the universal fit to simulated haloes of dark matter, and the gas is assumed to start with the same distribution. In simulations with gas, the gas and dark matter acquire similar angular momentum during the linear phase but diverge afterwards: gas shocks and loses energy without losing angular momentum, while the dark matter passes through itself. The direction of the difference depends on the halo’s history, and measured gas distributions in simulations with hydrodynamics differ from the dark matter’s by tens of per cent in the low-angular-momentum tail.

The disc is an exponential with a perfectly flat rotation curve. Real rotation curves rise in the inner scale length or two, so the gas at small radii has less angular momentum per unit radius than a flat curve gives, and the disc’s own distribution has a little more low-angular-momentum material than drawn. That reduces the surplus somewhat — by a few per cent of the mass for a rising inner curve — but does not remove it.

The twist model tilts each mass element by an independent random angle, which mixes directions more evenly than a real halo, whose misalignment is between coherent shells. The removal model takes away gas strictly in order of angular momentum, which no real outflow does; it gives the minimum amount that has to be removed, and a less selective outflow has to remove more.

And the census treats the surplus as if it would end up in a bulge of that mass, with no star-formation efficiency, no time delay and no internal evolution. The comparison with observed bulge-to-total ratios is therefore a comparison of masses with light, and of a population at the end of its history with one modelled at its start.

Still open: where the slow gas went #

The angular momentum distribution of a disc is a record of its formation, and it says that the gas that made the disc was not a fair sample of its halo. Something removed or rearranged a fifth to a quarter of the slowest gas, and it did so in most galaxies, because most discs are closer to exponential than their haloes would allow.

What remains unknown is the split. Outflows that eject the slow gas should leave their mark in the gas around galaxies — in the metals carried out, and in the circumgalactic medium that the ejected gas joins — and should be strongest in dwarfs. Internal redistribution should leave pseudo-bulges and rings and should be strongest in giants with bars. A survey that measured the angular momentum distributions of disc stars by age, separating the stars that formed early, when the slow gas was still present, from those formed later, would show which happened when. Integral-field maps of nearby discs, which give both stellar ages and velocities, now make that measurement possible; the answer will say whether the barrier that angular momentum builds against collapse was removed by violence from outside or by patience from within.