An arm that is undone by the work it does

A spiral arm cannot be a fixed set of stars, because differential rotation would wind such a thing into invisibility in a couple of turns. It is a wave — a pattern that stars move through — and the pattern’s single rotation rate decides where in the disc it can exist, between an inner and an outer Lindblad resonance.

This essay asks what the wave does while it exists. The answer is that it moves angular momentum, in exactly the way an accretion disc has to and by a completely different mechanism, and that the moving has a cost the pattern itself pays.

A wave with negative angular momentum #

The key idea has an odd sound to it and is not a trick of bookkeeping.

Inside corotation, stars orbit faster than the pattern turns. A wave there is a perturbation on material that is overtaking it. Working out the wave’s angular momentum content properly — as the second-order change to the total angular momentum of the disc when the wave is present — gives a negative number inside corotation and a positive number outside.

A negative-angular-momentum wave has a consequence that seems backwards until it is stated carefully: to make the wave larger inside corotation, angular momentum has to be removed from the material there. And to damp it, angular momentum has to be put back. So a wave excited at the inner Lindblad resonance and propagating outwards takes angular momentum out of the inner disc; carrying it across corotation converts it into a positive-angular-momentum wave; and depositing it at the outer Lindblad resonance gives angular momentum to the outer disc.

Net effect: angular momentum flows outwards, mass flows inwards, and the total is unchanged.

The requirement that the total be unchanged is not an assumption in this picture but a check on it, and it is the check the opening figure makes: the amount deposited at the outer resonance has to equal the amount removed at the inner one, to whatever precision the drawing is computed to. A pattern that appeared to create angular momentum would be a pattern drawn wrongly, because the only thing in the system that could supply any is the disc itself.

What the pattern does keep is a small amount, in its own wave energy, and that is why the wave grows. But the amount stored in a wave of realistic amplitude is a tiny fraction of what passes through it — the wave is a conveyor rather than a reservoir, and the figure’s flat central section is the conveyor running with nothing being added or removed along its length.

Why the exchange only happens at resonances #

A star passing through a spiral arm is pulled forwards as it approaches and backwards as it leaves. Over one passage the two nearly cancel, and over many passages with random phases they cancel exactly. Nothing accumulates.

The cancellation fails where the star’s own radial oscillation is commensurate with its encounters with the pattern. A star on a nearly circular orbit oscillates radially at the epicyclic frequency $\kappa$, and it meets a two-armed pattern at a frequency $2(\Omega - \Omega_p)$. Where those are equal — where $\Omega - \Omega_p = \pm\kappa/2$ — the star meets the arm at the same phase of its epicycle every time, the pulls add instead of cancelling, and a secular exchange occurs.

Those two radii are the Lindblad resonances, and they are the only places in the disc where the pattern and the stars exchange anything.

The cost #

Every resonant exchange changes a star’s orbit, and it does not change it symmetrically. A star that gains angular momentum at the outer Lindblad resonance moves outwards, and it also gains epicyclic energy — its orbit becomes more eccentric. The same at the inner resonance, in reverse.

So the transport heats the disc. The velocity dispersion of the stars rises, and it rises monotonically, because there is no cooling mechanism for a stellar disc: stars do not collide and cannot radiate their random motion away. A hot disc cannot carry a strong spiral wave. The criterion is Toomre’s: a disc is stable against the growth of such waves when its random motions are large compared with a threshold set by its surface density and its epicyclic frequency. A disc that starts marginally unstable, grows a pattern, transports angular momentum and heats itself will cross that threshold and stop supporting the pattern.

The arithmetic is unforgiving. A star’s radial excursion is set by how much epicyclic energy it carries, and a resonant kick changes that energy by an amount proportional to the square of the pattern’s amplitude — the same quantity the torque is proportional to. So the heating and the transport are not two effects that happen to accompany each other; they are one effect read against two different books, and a disc cannot buy more of the transport without paying exactly the matching amount of the heating. There is no parameter to tune that makes a wave efficient at moving angular momentum and gentle about raising the dispersion.

That is what makes the difficulty structural rather than quantitative. A weak wave heats slowly and moves almost nothing; a strong one moves a great deal and heats itself out of existence in a few hundred million years. The observed dispersion of the solar neighbourhood — rising from about ten kilometres a second for the youngest stars to forty-odd for the oldest — is consistent with either a mild continuous heating over ten billion years or a handful of vigorous episodes, and the two histories are not distinguishable from the present-day numbers alone.

The wave therefore destroys the conditions that allow it. That is not a paradox but it is a real difficulty for the theory: a self-heating wave in a disc with no cooling should have a lifetime of a few rotations, and spiral galaxies have been spiral for ten billion years.

How much is actually moved #

An order of magnitude is worth having, because “the pattern transports angular momentum” is a statement that could describe a negligible effect as easily as a decisive one.

The torque exerted by a spiral pattern on the disc is proportional to the square of its amplitude — the fractional surface density contrast between arm and interarm. That contrast is a few tenths for a well-defined grand-design spiral and a few hundredths for a flocculent one, so the transport rate differs by two orders of magnitude between the two kinds of galaxy, and only the strong-armed ones move a significant amount.

For a grand-design spiral, the resulting inward mass flux is of order a solar mass a year through the mid-disc. Over ten billion years that is enough to rearrange a substantial fraction of the inner disc, which is why the effect is taken seriously as an evolutionary mechanism rather than as a curiosity. It is nevertheless small compared with what a bar does: a strong bar’s torque exceeds a spiral’s by an order of magnitude, and a barred galaxy funnels gas to its centre on a timescale of a rotation or two rather than of a Hubble time.

The resolutions #

Three answers are current and they are not exclusive.

Cold gas. Gas, unlike stars, does cool: it radiates. Every disc still forming stars is being fed from outside, and a disc continuously supplied with fresh gas has a cold component that can carry the wave even when the stellar component has been heated beyond it. That makes spiral structure a property of the gas-rich part of the disc, which matches the observation that spiral arms are traced most sharply by young stars and molecular gas rather than by the old stellar population. Transience. Rather than one long-lived wave, a disc may support a succession of shorter-lived patterns, each growing by swing amplification from a local disturbance, transporting for a few rotations and dissolving. Numerical simulations produce exactly this, and the resulting structure looks like a spiral galaxy in every statistical measure while having no single pattern speed at all.

Driving. A bar, or a companion galaxy, can drive a pattern continuously against the damping. Barred galaxies and interacting pairs do have stronger and more regular arms, which supports this for those cases and leaves the unbarred isolated spirals to the other two mechanisms.

Each resolution is really a claim about where the energy comes from, and stated that way they stop competing. Cold gas supplies a component that can be re-cooled, so the disc’s capacity to carry a wave is replenished rather than merely preserved. Transience abandons the requirement that any one wave last: a pattern that dies after four rotations never heats the disc past the threshold, and the next one grows out of whatever the last one left. Driving supplies the amplitude from outside, so the disc’s own thermal state stops being the limiting quantity. Only the first and third add anything to the disc; the second is an argument that nothing needs to be added, and it is the one that has gained ground as simulations have got large enough to resolve individual amplification events.

Distinguishing them observationally is harder than it looks, because the three make similar-looking pictures. The measurement that separates them is the pattern speed: a single long-lived wave has one, and a superposition of transients does not. Techniques for measuring it exist — the best known uses a continuity argument applied to a tracer whose total is conserved — and applied to real galaxies they tend to return a pattern speed that decreases outwards, which is what a set of overlapping transients would give and what a single wave would not.

How a pattern speed is actually measured #

Everything above turns on a single number that no image contains. A photograph shows where the arms are; it does not show how fast the pattern is turning, and the difference between a long-lived wave and a succession of transients is entirely a statement about that number.

There is one model-independent way to get it, and it is a piece of bookkeeping rather than a fit. If a tracer obeys a continuity equation — if the amount of it is conserved, so that it is neither created nor destroyed as the disc turns — then integrating the observed surface brightness and the observed line-of-sight velocity along a strip parallel to the disc’s major axis gives two numbers whose ratio is the pattern speed times the sine of the inclination. Two integrals over data, and the answer comes out.

The conditions are where the difficulty lives. The tracer has to be conserved, which rules out the young stars and the ionised gas that make the arms conspicuous — they are created in the arm and destroyed downstream, which is the whole reason they trace it. The old stellar population is conserved and is the usual choice, and its arm contrast is the ten-to-twenty per cent ripple that makes the integrals small differences of large numbers.

The geometry has to cooperate too. A face-on disc has no line-of-sight velocity to integrate and an edge-on one has no arms to see, so the method needs an intermediate inclination and degrades at both ends.

And the disc has to have one pattern. Applying the method to a galaxy with a bar and a spiral turning at different rates returns some average of the two, weighted in a way that depends on which radii the strips crossed.

Applied where those conditions hold, the results are awkward for the simple picture. Pattern speeds measured over different radial ranges in the same galaxy frequently come out different, decreasing outwards — which is what a set of overlapping transients would give and what a single rigidly rotating wave would not.

That result is not decisive, because the same gradient would be produced by a single wave whose amplitude falls outward and whose integrals are therefore dominated by different radii in different strips. It is, however, the only measurement of the quantity the argument is about, and it does not favour the simple picture.

The disc that stayed thin #

There is a constraint on all of this that comes from a completely different direction, and it is the sharpest one available.

The Milky Way’s thin disc is thin: a scale height of a few hundred parsecs against a scale length of a few kiloparsecs, a ratio of about one to ten. It has been forming stars for ten billion years, and the heating described above has been running the whole time. A disc heated at a constant rate for that long would not look like this.

The velocity dispersion of solar-neighbourhood stars does rise with age, which is the heating happening. What is disputed is the shape of the rise. If it goes as a power of time with a small exponent, the disc heats quickly at first and then hardly at all, and there is no difficulty. If it goes as the square root of time, the oldest stars should be far hotter than they are.

The complication is that spiral arms are not the only thing doing the heating. A star scattering off a giant molecular cloud also gains random velocity, and the two mechanisms differ in an observable way: a spiral wave acts in the plane and heats the radial and azimuthal components, while a cloud is compact and scatters in all three directions. So the ratio of vertical to in-plane dispersion is a diagnostic of which mechanism dominates, and the observed ratio is intermediate — which is read as both operating, with clouds supplying most of the vertical heating and waves most of the radial.

The thinness of the disc is therefore a budget rather than an accident, and it bounds how much transport the arms can have done over the galaxy’s life more tightly than any measurement of the arms themselves.

Both of the constraints above are budgets rather than pictures, and that is the useful thing about them: a picture of a spiral galaxy is compatible with all three explanations at once, and a budget is not.

Where the transported material goes #

The mass that moves inwards has somewhere to go, and the consequences are visible.

Over a galaxy’s life the transport builds up a central concentration — a pseudobulge, distinguishable from a merger-built bulge by rotating rapidly and having a nearly exponential light profile rather than the steeper profile of a spheroid. That is secular evolution: a galaxy becoming more centrally concentrated without any external event. There is also an effect at corotation that has nothing to do with heating and is now thought to be substantial. A star sitting near the corotation radius, where it does not overtake the pattern, can exchange angular momentum with it while keeping its orbit nearly circular — moving radially by several kiloparsecs without becoming eccentric. That is radial migration, and it scrambles the relationship between where a star is now and where it was born, which is the central difficulty in reading a galaxy’s chemical history off a histogram of its present stars.

Where the ladder goes #

The same accounting, applied to a gas disc round a compact object, is the transport that lets anything accrete at all — and the mechanism there is turbulence rather than a global wave, which is a different way of solving an identical problem.

The other direction leads to the transport a galaxy does at its very centre, where a pattern’s inner Lindblad resonance is not the end of the story. A bar drives gas inwards to a nuclear ring; from there, a secondary bar or a nuclear spiral can drive it further; and at the smallest scales the problem becomes the removal of the last of the angular momentum from a pair of black holes, where the stars that would carry it away run out.