An arm that cannot be made of stars

A spiral galaxy has two obvious properties and they are almost impossible to reconcile.

The first is that it rotates differentially. The rotation curve is flat, so the angular speed falls as $1/R$: material at four kiloparsecs goes round twice for every once at eight. The second is that it has spiral arms, which are open — a pitch angle of ten to twenty-five degrees is typical — and which are seen in essentially every disc galaxy at every epoch that can be observed.

Put those together and something has to give, because differential rotation is a very efficient destroyer of open structures.

The problem as a number #

The intuition is easy to state and easy to under-rate, so it is worth reducing to an expression.

Suppose an arm is a fixed set of stars, arranged at $t=0$ along a radial line. At time $t$ the star at radius $R$ has turned through $\Omega(R)t$, so the arm’s shape is $\theta(R) = \Omega(R)t$. The pitch angle — the angle between the arm and the local circle — satisfies

$$\tan i = \frac{1}{R\,t\,|d\Omega/dR|}.$$

For a flat rotation curve, $\Omega \propto 1/R$, so $R|d\Omega/dR| = \Omega$ and the expression collapses to

$$\tan i = \frac{1}{\Omega t}.$$

The pitch angle falls as the reciprocal of the angle the disc has turned through, in radians. That is a strong statement: it does not depend on the galaxy’s size, mass or rotation speed except through how many times it has gone round.

What the alternatives are #

If an arm cannot be a fixed set of stars, it has to be one of three things, and the argument since 1964 has been about which.

A pattern. The stars move through the arm rather than with it: the arm is a region of enhanced density that rotates at its own speed, and any given star spends most of its time outside one. This is the density-wave picture of Lin and Shu, and it is the one the figures below draw.

A transient. Arms form, live for a few hundred million years, and dissolve — with new ones forming continually, so that at any moment a galaxy has arms while no individual arm is old. Modern simulations produce this behaviour readily, and it is now the more widely held view for flocculent and multi-armed galaxies.

A response to something else. A bar, or a companion galaxy, drives a two-armed pattern directly — and a passing companion demonstrably reshapes a disc on exactly the timescale required. Many of the most photogenic grand-design spirals have a companion, and that is not a coincidence.

The three are not exclusive and the honest modern position is that different galaxies do different ones. What all three share is the abandonment of the material arm, and that abandonment is forced by the arithmetic above.

A pattern made of orbits #

The density wave has a construction that makes it visible without any wave mechanics at all, and it is the most satisfying picture in this field.

Take a set of closed elliptical orbits, nested, with each one’s major axis rotated slightly relative to the next. Where neighbouring orbits come close together, the density of stars is higher, simply because the same number of stars occupies less area. If the orientation advances smoothly with radius, the crowded regions line up into a spiral.

The crucial property is what happens when the whole configuration is allowed to rotate. The pattern — the set of crowded places — can turn at a single speed even though the stars on the orbits turn at speeds that vary with radius, because the pattern is not made of stars. It is made of the relative orientation of orbits, and that can be made to rotate rigidly by construction.

A rigidly rotating pattern does not wind up. The winding problem is solved not by finding a way for material to resist differential rotation but by noticing that the thing which needs to persist is not material.

What the stars do as they pass through #

If the arm is a place, a star should enter it, cross it, and leave. That is a testable statement, and its test is one of the most direct pieces of evidence in the subject.

Inside the corotation radius the stars go round faster than the pattern, so they overtake each arm from behind. Gas being overtaken is compressed on entering — a shock, in fact, since the relative velocity is supersonic — and collapses into stars. The most massive of those stars live only a few million years, but the disc has carried them some distance downstream by then.

The result is a colour gradient across the arm: the shock and the dust lane on the trailing edge, the young blue stars just downstream of it, and the older redder population further ahead. The gradient reverses on the other side of corotation, where the stars are being overtaken by the pattern instead. That reversal is the signature the picture makes and a transient arm does not, and it is seen in some galaxies and not others — which is the empirical form of the “different galaxies do different things” position above.

The observation behind the number #

The pitch angles that the shaded band in the second figure represents are measured, and measuring them is less straightforward than it sounds.

An arm is not a curve; it is a broad region with fuzzy edges, and its apparent shape depends on the wavelength observed. In blue light an arm is dominated by the young stars and looks narrow and high-contrast; in the near infrared it is dominated by the old disc population and is far broader and fainter. The infrared arm is the one that traces the mass, and its amplitude is the number the density-wave theory actually predicts.

That number is small. The surface density contrast between arm and interarm regions in the old stellar population is typically ten to twenty per cent — a modest ripple. The contrast in blue light is a factor of several, because star formation responds to compression far more steeply than linearly.

So the arms are mostly a lighting effect. The mass distribution of a spiral galaxy is much smoother than the photograph suggests, and the photograph is dominated by the small fraction of stars that are young and bright. This is the single most useful correction to make to the mental image of a spiral galaxy.

Forty years of the problem being known and unsolved #

The history here is unusually clean, because the difficulty was recognised long before anything could be done about it.

Bertil Lindblad was working on the dynamics of the disc through the 1920s and 1930s, and he understood the winding problem in its modern form: he knew that differential rotation destroys material arms, and he spent decades looking for a description in which the arms were kinematic rather than material. His work on the epicyclic frequency and on what are now called Lindblad resonances is exactly this search, and it supplies the machinery the next rung uses.

What was missing was the idea of a quasi-stationary pattern, and the wave mechanics to make it work. C. C. Lin and Frank Shu supplied both in 1964: a density wave in a differentially rotating disc, with a dispersion relation connecting its pattern speed to its wavelength and to the disc’s own surface density. The picture immediately explained the open arms, the shocks and the star formation in them, and the association of dust lanes with the inner edges.

It has one durable difficulty, which was noticed almost at once. A wave in a disc propagates, radially, and where it reaches a Lindblad resonance it is absorbed. A pattern that is being absorbed needs replenishing, and a genuinely stationary arm pattern therefore needs a continuous driver. That is the loose end from which the transient-arm picture was eventually pulled.

How the alternatives are told apart #

Three tests separate the accounts, and none of them is easy.

The age gradient across an arm. If the pattern rotates rigidly and the stars overtake it, the offset between the gas shock, the young stars and the older stars should vary systematically with radius and should reverse at corotation. Measured in some galaxies; not found in others.

The pattern’s rigidity. A density wave has one pattern speed for the whole disc; a transient arm has a pattern speed that tracks the local rotation, because it is being sheared. Distinguishing the two needs pattern speeds measured at several radii independently, which is at the edge of what can be done.

The number and rag of the arms. Grand-design two-armed spirals are strongly associated with bars and companions; flocculent galaxies with neither. That correlation is the strongest single piece of evidence that some arms are driven and some are not.

What the picture cannot show #

Neither of these figures is a simulation. The winding figure is kinematics — stars carried round at fixed speeds — and the density-wave figure is a static construction of nested ellipses. Neither includes the self-gravity of the pattern, which is what makes a real density wave a wave rather than a drawing.

The kinematic spiral is not stable on its own. Left alone, the ellipses’ orientations would drift apart, because the rate at which an orbit’s apsides precess depends on radius. Making the pattern persist needs the self-gravity of the density enhancement to hold it together, and that is the substance of the Lin–Shu theory rather than an ornament on it.

And a two-armed pattern is drawn here because two is the easiest case. Real galaxies have one, two, three, or a dozen ragged arm segments. The kinematic construction produces $m$ arms if the orientation advances by $\pi/m$ across the disc, so the number of arms in this picture is a parameter and not a prediction.

What an arm is made of, in the end #

Putting the pieces together gives an account that is worth stating in one place, because it is not what a photograph suggests.

A spiral arm is a ten to twenty per cent enhancement in the surface density of the old stellar disc; a compression of the gas passing through it, strong enough to be a shock; a burst of star formation triggered by that compression; and a population of very bright, very short-lived stars that make the whole structure conspicuous in blue light and negligible in mass. The stars that make the arm visible are not the stars that make it exist, and neither set stays in it.

That is also why arms are where a galaxy’s colour structure lives. Colour is a thermometer for a single star and an age indicator for a population, and an arm is a place where the population’s age is reset. A galaxy’s arms are blue for the same reason a firework is bright — not because there is more material there, but because a small part of the material has just been made very hot.

The generalisation #

The move made here — a pattern that persists in a medium whose constituents do not stay in it — is one of the most widely useful ideas in physics, and this is one of its cleanest instances.

A traffic jam is the standard analogy and it is a good one: the jam is a region of high density that can travel backwards along a motorway at a few kilometres an hour while every car in it is travelling forwards at eighty, and no car stays in the jam for long. The jam is nevertheless a real object with a location, a speed and a lifetime.

The same distinction separates a wave from the water it moves through, and the resonances that clear the Kirkwood gaps from the asteroids that are missing from them. It is worth noticing where the analogy stops: a traffic jam has a cause upstream of it, and one of the open questions about spiral arms is whether they need one.

The one galaxy whose arms are hardest to map #

There is a pleasing irony in the subject: the spiral galaxy about which least is known is the one the observer is inside.

Every other galaxy is seen from outside, so its arms are laid out on the sky and their pitch angles can be measured off an image. The Milky Way’s arms are seen edge-on through the disc, at every distance at once, with dust between. What is observed is a direction and a velocity, and turning those into a position requires a rotation curve and an assumption that the gas moves on circles — which is precisely the assumption the arms themselves violate.

That method, kinematic distance, also has a structural ambiguity. Inside the solar circle each line-of-sight velocity corresponds to two distances, one on each side of the tangent point, and choosing between them requires additional information that is often unavailable. Whole arm segments have been placed on the wrong side of the Galaxy and later moved.

The number of arms the Milky Way has has accordingly changed more than once, and the map has been redrawn every few decades. What settled the modern version was not a better rotation curve but geometry: very long baseline interferometry measures trigonometric parallaxes to the masers embedded in star-forming regions, which are exactly the objects that trace arms, and does so at distances of several kiloparsecs with no kinematic assumption anywhere in the chain.

The Galaxy’s own structure is therefore known by the one technique that also navigates spacecraft, and it is known for a few hundred sight lines rather than for the disc as a whole.

What an arm does to the stars that pass through #

The picture above has stars passing through an arm and coming out the other side unchanged, and that is not quite true — the exception has consequences for a subject that looks unrelated.

Near corotation a star and the pattern move together, so instead of a brief encounter the star sits in the arm’s potential for a long time. The arm can then exchange angular momentum with it substantially, and the star’s guiding-centre radius changes: it ends up on a different circular orbit, kiloparsecs from where it began.

The remarkable feature is that this happens without heating. An ordinary perturbation raises a star’s random velocity and puffs the disc up, and there is a limit to how much of that a thin disc can absorb before it stops looking thin. Corotation churning moves stars radially while leaving their orbits nearly circular, so the disc stays thin and the migration leaves no kinematic trace.

That is a problem for anyone trying to read a galaxy’s history from its chemistry. A star’s composition records the gas it formed from, and the gas at each radius is enriched at its own rate, so a metallicity ought to say where a star was born. Radial migration severs that link — a metal-rich star in the outer disc may be an interloper rather than evidence of enrichment there — and reconstructing the disc’s chemical evolution has had to be redone with migration included.

A pattern that no star stays in still rearranges the disc permanently, which is a stronger claim than the winding argument needed and is a consequence of the same construction.

Where the ladder goes next #

The construction above rotates the pattern at some speed without saying what that speed can be, or where in the disc a pattern is permitted to exist at all. Both are decided by the same rotation curve, through the epicyclic frequency, and the next rung takes them up.

Later rungs on this anchor: swing amplification and how a transient arm is made; the bar as a driver, and why bars are common; the colour gradient measured across an arm and what it says about corotation; flocculent against grand-design and what distinguishes them; arm-driven radial migration of stars, which scrambles the record of where a star was born; the Milky Way’s own arms and how many of them there are; and the arms of galaxies at high redshift, which are clumpier and shorter-lived.