The bar leaves its gas where the orbits turn

A spiral pattern or a bar in a disc galaxy is a wave in the stars and gas, turning at its own angular speed — its pattern speed — while the material it is made of orbits at speeds that vary with radius. Where the material and the pattern turn together, at corotation, nothing special happens to either; where the material meets the pattern at the frequency of its own natural radial oscillation, at the Lindblad resonances, the pattern can exchange energy and angular momentum with it. An arm undoes itself by the work it does: the angular momentum a spiral pattern carries is deposited at its resonances, and inside corotation the gas and stars lose angular momentum to it and move inwards.

A bar does the same thing more strongly. Roughly two-thirds of disc galaxies have one, including the Milky Way, and a bar is the most effective engine a galaxy has for moving gas from its disc towards its centre. Gas falling freely towards a galaxy’s centre, with nothing to stop it, would feed the central black hole or pile up in a dense nucleus. In many barred galaxies it does neither. It stops in a ring — a nuclear ring, a few hundred parsecs to a couple of kiloparsecs across, bright with young star clusters and molecular gas, forming stars at a rate per unit area higher than anywhere else in the galaxy. The ring is where the bar leaves its gas, and the question is why it leaves it there.

An orbit that answers a rotating bar #

The answer is in how orbits respond to a bar. A star or a cloud of gas on a nearly circular orbit, seen in the frame that turns with the bar, feels the bar’s gravity pulling at it twice per circuit, because a bar has two ends. That periodic pull drives an oscillation in the orbit’s radius, and the response is that of any driven oscillator: in step with the forcing if the forcing is slower than the natural frequency, opposite to it if faster, and very large if the two are nearly equal.

The natural frequency is the epicyclic frequency, and the forcing frequency, for an orbit going round faster than the bar, is twice the difference between its orbital speed and the bar’s. In the drawn disc — a rotation curve that rises within a few hundred parsecs and is flat beyond, and a bar three kiloparsecs long — the two match at two radii inside corotation: the inner and outer inner Lindblad resonances, at 0.37 and 0.78 kiloparsecs. Between the outer resonance and corotation the orbits are elongated along the bar; this is the family of orbits, called x1, whose stars make up the bar itself. Between the two resonances the response has the opposite sign, and the orbits are elongated across the bar — the x2 family. Inside the inner resonance they align with the bar again.

For stars, which pass through one another without colliding, the two families simply coexist, each populated by the stars that happen to be on it. For gas they cannot. Gas on neighbouring orbits that are elongated in perpendicular directions would have to cross, and gas clouds that cross collide, shock, and lose energy. The solid curve includes a damping term that stands for that dissipation, and it shows what the gas does instead of flipping: its orbits turn through ninety degrees gradually, over a band of radius around the resonance, and in that band the gas’s streamlines crowd and cross.

Orbits drawn in the bar’s frame #

Drawn in the bar’s frame, the orbits outside the outer resonance are ovals aligned with the bar and those between the resonances are turned across it, with the orientation twisting smoothly between. The twist is what makes the flow visible in real galaxies. Where gas on bar-aligned orbits meets gas on turned ones, it shocks along two nearly straight lines running along the leading edges of the bar, from its ends towards the centre. Those shocks compress the gas into narrow lanes of dust, conspicuous in images of barred galaxies as dark streaks offset from the bar’s axis. They are where the gas loses the energy and angular momentum that let it fall inwards, and they end at the ring.

Where the gas falls out of step #

A bar can take angular momentum from the gas only if the gas’s response is out of step with the bar’s pull — if the gas’s density peaks lag or lead the bar’s axis, so that the bar’s gravity pulls backward on the gas more than forward. A response exactly in step or exactly opposite is pulled symmetrically and feels no net torque. The damped response is almost in step over most of the region between corotation and the resonance, and swings out of step sharply near each resonance, passing a quarter-turn from alignment at the resonances themselves.

The out-of-step response is the bar’s grip. Inside corotation the gas outruns the bar, and the lagging gas is pulled back, loses angular momentum and falls inward; the grip is strongest where the phase swings, near the resonances, and in a strong bar it is concentrated in the shocks along the bar’s edges. The gas arrives at the outer inner Lindblad resonance with little angular momentum left to lose, and there the orbits that would carry it further in are the x2 orbits, turned across the bar, which the bar’s pull no longer drives inward. The inflow stalls and the gas accumulates: a ring.

A ring needs a concentrated centre #

The resonances exist only if the orbital frequency minus half the epicyclic frequency rises above the bar’s pattern speed somewhere inside corotation, and that quantity is large only where the rotation curve rises steeply — where there is a lot of mass near the centre. In a disc whose rotation curve rises within a couple of hundred parsecs, the two resonances are well separated and there is a broad band of x2 orbits. As the rise is spread over a larger radius — as the centre becomes less concentrated — the two resonances move together and merge, and beyond a certain softness the disc has no inner Lindblad resonance at all. Then there are no x2 orbits, the gas the bar drives inward has no radius at which its orbits turn, and it flows all the way to the centre.

That makes a nuclear ring a statement about the centre. Galaxies with rings have bulges or dense central concentrations; galaxies with soft centres, dominated by a disc, tend not to. And the relation feeds back on itself. Gas driven into the centre by a bar adds mass there, steepening the rotation curve, strengthening and widening the resonances, and so building the structure that will catch the next gas. A bar can create the conditions for its own ring.

A ring whose size records the bar’s speed #

Where the ring sits depends on the pattern speed. A faster bar has a pattern speed closer to the peak of the orbital frequency minus half the epicyclic frequency, so the resonances lie close together and close to the centre; a slower bar has them further out. Bars are known to slow down over time — a bar turning inside a massive dark halo loses angular momentum to the halo through the same dynamical friction that drags a satellite inward — so the ring should move outward as the bar ages. The figure shows the ring growing from 0.8 to 1.4 kiloparsecs as corotation moves from the bar’s end to sixty per cent beyond it, and more for less concentrated centres. The observed rings span a hundred parsecs to a couple of kiloparsecs, and among them the rings in galaxies with stronger bars tend to be smaller relative to the bar, a trend that full orbit calculations for strong bars reproduce and the weak-bar theory drawn here does not capture.

Reading the bar’s speed off its ring #

The relation between a ring and a pattern speed can be run the other way. A bar’s pattern speed is one of the hardest quantities in galaxy dynamics to measure, because a pattern is not an object: nothing in it moves at the pattern speed, and no Doppler shift reports it directly. The standard method combines the surface brightness and line-of-sight velocities of the stars along slits parallel to the bar, using the continuity of the stellar distribution in the bar’s rotating frame, and it is demanding of data and sensitive to the orientation of the galaxy on the sky.

A ring offers an independent estimate. Its radius is measured from an image; the rotation curve, and with it the orbital and epicyclic frequencies at every radius, from the gas velocities; and the pattern speed is then the value for which the outer inner Lindblad resonance falls at the ring. The estimate inherits the weaknesses of the linear theory — for strong bars the ring sits somewhat inside the linear resonance — but it needs no assumption about the stars’ continuity and it can be made for galaxies seen at orientations where the standard method fails. Where both have been applied, they broadly agree that bars rotate fast, with corotation not far beyond the bar’s end, which is also what the survival of bars against their haloes suggests.

Other resonances leave rings too. Gas collects in an inner ring near the bar’s end, close to corotation, and in an outer ring near the outer Lindblad resonance, well beyond the bar; galaxies with all three rings — nuclear, inner and outer — have ratios of ring radii that match the ratios of their resonance radii in a flat rotation curve, a regularity that is one of the strongest pieces of evidence that the rings are resonant at all, rather than accidents of where gas happened to be.

Why the stars make a disc and the gas a ring #

Stars respond to the bar’s forcing exactly as gas does, orbit for orbit, and populate the same x2 orbits between the resonances. But stars do not collide and do not dissipate, so they do not shock where orbits cross, and they do not pile up at one radius: the stars on x2 orbits fill the band between the resonances as a flattened, rapidly rotating nuclear disc. Many barred galaxies show exactly that in their stellar kinematics — a region at the centre where the stars rotate fast and have a low velocity dispersion, a disc cooler than the bulge around it, often coincident with the gas ring and made of stars that the ring itself formed.

The difference between the two is the dissipation, and it is also what makes the gas useful as a tracer. Stars on the two families of orbits overlap and wash out the transition; gas cannot overlap, marks the transition sharply, and lights it up with the star formation its own collisions trigger. The ring is where the orbital structure of the bar becomes visible, because it is where the one component that cannot tolerate crossing orbits is forced to reveal where they cross.

A ring as a star-forming engine #

The gas that reaches a ring does not stay as gas for long. A ring concentrates the inflow of several kiloparsecs of bar into a narrow annulus, and gas there reaches densities at which it collapses into stars quickly — nuclear rings are among the most intense sites of star formation in nearby galaxies, strings of young massive clusters around the galaxy’s centre, sometimes forming stars at a solar mass per year or more in an area a kiloparsec across. Because the gas enters the ring at particular points, where the dust lanes meet it, and then orbits round it, the clusters along the ring form an age sequence, youngest near the entry points, older further round.

A galaxy’s gas supply runs out faster than the galaxy ages, and a bar accelerates that for the inner disc: it sweeps the gas from within its corotation into the ring, where it is consumed. Barred galaxies show a deficit of gas and star formation in the region swept by the bar, between the ring and corotation — the bar desert — and a concentration of both in the ring. Over a few billion years the ring’s star formation builds a dense disc of stars at the centre, a pseudo-bulge, which in turn steepens the rotation curve and changes the resonances. The bar reorganises the centre of the galaxy it lives in.

Inside the ring #

The ring is not always the end of the inflow. Some galaxies have a second, smaller bar inside the ring, turning at its own speed — a bar within a bar — which can drive gas further inward from the ring towards the centre, and some have nuclear spirals of dust and gas inside the ring, which do the same more weakly. How gas finally reaches the central black hole, from a ring at a few hundred parsecs to the accretion disc at a fraction of a parsec, is a chain of such mechanisms, each handing the gas down a factor of ten in radius, and it is not known which links dominate in which galaxies. The chain ends at the last parsec, where the problem becomes removing angular momentum from a pair of black holes rather than from gas, and at the black hole whose mass the stars round it reveal.

The Milky Way has its version. The region within about two hundred parsecs of its centre, the Central Molecular Zone, holds several per cent of the Galaxy’s molecular gas in a ring-like stream, fed by gas flowing along the leading edges of the Galactic bar, visible as dust lanes in the distribution of molecular gas along the line of sight. Its star formation rate is, for the amount of dense gas it contains, surprisingly low — a puzzle about what stops the gas collapsing once the bar has put it there, which the linear picture of orbits and resonances does not address.

What the linear picture leaves out #

The figures use linear, epicyclic theory: a weak bar, nearly circular orbits, and a damping term standing for everything gas does that stars do not. Real bars are strong — they elongate orbits by tens of per cent, not a few — and for them the orbit families must be found by integrating orbits in the bar’s potential. The x1 and x2 families are then real, separate families of closed orbits with a gap between them, not the two sides of a sign flip, and the ring forms where the x2 family exists, a location that in strong bars moves inward from the linear resonance. The damping is a single constant; gas dissipates through shocks whose strength depends on the flow, and the ring’s width and exact position come out of hydrodynamic simulations rather than from a formula. And the bar is steady; real bars grow, slow, and may buckle vertically, and their rings evolve with them. None of this changes the logic: the ring forms where the orbits turn, the turn requires a resonance, and the resonance requires a concentrated centre.

Still open: how the gas gets from the ring to the black hole #

The bar solves the first two orders of magnitude of the problem of feeding a galaxy’s centre, bringing gas from kiloparsecs to hundreds of parsecs, and the nuclear ring is where its contribution ends. Below the ring, the angular momentum still to be removed is large, the mechanisms proposed are several and possibly different from galaxy to galaxy, and the timescales are short enough that what is seen in any one galaxy may be a transient. Whether active nuclei are fed preferentially in galaxies with bars within bars, with nuclear spirals, or by events unrelated to the disc altogether, is a statistical question that the high-resolution images of molecular gas now being made of nearby galactic centres, at tens of parsecs, are beginning to answer.