Where a pattern is allowed to turn

The previous rung left a pattern rotating rigidly at some speed, and said nothing about what that speed can be or where in a disc such a thing is allowed to exist. Both questions have the same answer, and it comes out of the rotation curve without any new physics.

The second frequency a disc has #

A star on a circular orbit has one frequency: $\Omega = v_c/R$, the rate at which it goes round. Displace it slightly outwards and it acquires a second: it falls back, overshoots, and oscillates radially about the radius it belongs at. That radial oscillation has its own frequency, called the epicyclic frequency, and it follows from the rotation curve alone:

$$\kappa^2 = \frac{2\Omega}{R}\frac{d(R^2\Omega)}{dR}.$$

Two special cases make the expression legible.

For a Keplerian potential — everything at the centre — $\Omega \propto R^{-3/2}$ and the algebra gives $\kappa = \Omega$ exactly. The radial oscillation takes exactly as long as one circuit, so the star returns to its maximum radius at the same point in the orbit every time, and the orbit closes. That is the ellipse, arrived at from a completely different direction: an ellipse is what a rosette becomes when the two frequencies happen to be equal.

For a flat rotation curve, $\Omega \propto 1/R$ and $\kappa = \sqrt{2}\,\Omega$. The radial oscillation is now faster than the circuit by a factor of $\sqrt 2$, which is irrational, so the orbit never closes and every star traces a rosette that fills an annulus.

The epicycle, and the orbit as an ellipse again #

There is a way of seeing the two frequencies together that makes the whole construction of the previous rung obvious rather than clever.

Write a star’s position as a guiding centre — a point on a circular orbit at radius $R_g$ — plus a small displacement about it. To first order in that displacement, the star traces an ellipse about the guiding centre: an epicycle, with axis ratio $2\Omega/\kappa$ and traversed in the retrograde sense, while the guiding centre goes round at $\Omega(R_g)$.

For a flat rotation curve, $2\Omega/\kappa = \sqrt 2$, so the epicycle is an ellipse twice as long tangentially as it is radially, whatever the galaxy. The star is therefore never far from its guiding radius, and the whole orbit is a slow drift with a small rocking superimposed on it.

Now watch it from a frame rotating at $\Omega_p$. The guiding centre drifts at $\Omega - \Omega_p$, which is slow near corotation and fast far from it, and the epicycle carries on at $\kappa$. When the two are commensurate the star’s path in that frame closes, and a set of stars sharing the resonance forms a closed pattern rather than smearing out.

That is exactly the nested-ellipse picture from the previous rung, now derived rather than drawn: the closed orbits in the rotating frame are the epicyclic ellipses at the resonant radii, and their systematic change of orientation with radius is the spiral.

The frame that makes a pattern stationary #

A rotating frame is needed at all because a disc has no single rotation rate to sit in. If the arms were made of the same stars for ever, differential rotation would wrap them: at the Sun’s radius the disc turns once in 230 million years and at half that radius roughly twice as often, so an initially straight spoke is wound into a tight coil within a couple of galactic rotations, and after ten billion years it would be wrapped hundreds of times. Spiral arms are open and there are galaxies far older than that, so the arms cannot be a fixed set of stars. What can persist is a pattern that individual stars pass through.

Now put an observer in the frame that rotates with the pattern, at $\Omega_p$. In that frame a star at radius $R$ drifts around at $\Omega(R) - \Omega_p$, and it also rocks radially at $\kappa(R)$, which is unaffected by the choice of frame.

An $m$-armed pattern presents $m$ arms per circuit, so the star encounters the pattern at frequency $m|\Omega - \Omega_p|$. Resonance occurs when that forcing frequency matches the star’s own radial frequency:

$$m\,|\Omega(R) - \Omega_p| = \kappa(R).$$

For $m = 2$ that is $\Omega \pm \kappa/2 = \Omega_p$, which is precisely the pair of curves the first figure draws. The inner solution is the inner Lindblad resonance, the outer is the outer Lindblad resonance, and between them lies corotation, where $\Omega = \Omega_p$ and the star and the pattern go round together.

The three radii are not decorations on the theory. A wave propagating radially through the disc is absorbed at a Lindblad resonance, so the region between the two resonances is where a wave of a given pattern speed can exist — a statement about the extent of spiral structure, made from the rotation curve and one number.

Why this is the same mechanism as a Kirkwood gap #

The condition above is a resonance condition of exactly the kind the asteroid belt already supplied, and recognising that is worth doing carefully, because the two look nothing alike.

In the belt, the perturber is Jupiter, and the resonance is between an asteroid’s orbital period and Jupiter’s: at the 3:1 radius, the asteroid receives the same kick at the same phase of its orbit every three circuits, and the kicks accumulate rather than cancel. The result is a gap.

In the disc, the perturber is a wave rather than a body, and the resonance is between the frequency at which a star meets the arms and the frequency at which it rocks radially. The mechanism is identical: a forcing at the natural frequency, delivered at the same phase every time, so the response accumulates. What differs is the outcome. An asteroid at resonance is removed. A star at a Lindblad resonance is not removed; the wave is absorbed there, and the star gains the angular momentum the wave loses. That exchange is the engine of the disc’s slow evolution, and it is what makes spiral structure something more than an ornament — the same slow exchange that circularises a close binary or a hot Jupiter, operating between a wave and a disc rather than between two bodies.

What the pattern does to the disc #

A density wave transports angular momentum outwards. Inside corotation the pattern takes angular momentum from the stars; outside it, gives. The net effect is to move mass inwards in the inner disc and outwards in the outer one, which is the same thing viscosity would do and is far more effective than any real viscosity in a galaxy.

Two consequences follow, and both are observed.

A bar forms readily and then survives. A rotating bar is the same phenomenon at $m=2$ with a large amplitude, and it drives gas inwards very efficiently — into the region inside the inner Lindblad resonance, where the wave cannot follow. Rings of star formation at the inner resonance radius of barred galaxies are one of the more striking confirmations that the resonance radii are real places.

The disc heats. Every scattering off a spiral arm gives a star a little more random velocity, so the support of an old population drifts from rotation towards disorder, so the disc’s velocity dispersion rises with the age of the population. Old stars in the solar neighbourhood move faster relative to their neighbours than young ones, in a relation that is measured and that this mechanism explains.

The numbers, for one galaxy #

It is worth putting the Milky Way’s own values in, because they are what the first figure draws and they land somewhere provocative.

At the Sun’s radius the circular speed is about 230 kilometres per second, so $\Omega_0 \approx 28$ kilometres per second per kiloparsec. The curve is close to flat there, so $\kappa_0 \approx \sqrt2\,\Omega_0 \approx 40$ in the same units — a figure that can be checked independently, since $\kappa$ is also what the Oort constants measure locally, through $\kappa^2 = -4B(A-B)$.

A two-armed pattern turning at 25 kilometres per second per kiloparsec then corotates at about 8.7 kiloparsecs, which is very close to where the Sun is.

If that is right — and the pattern speed is the contested quantity, so the “if” is doing work — then the Sun sits near corotation, where it overtakes the arms only slowly and crosses one every few hundred million years rather than every few tens. There is a small industry attempting to connect that crossing rate to the terrestrial record of ice ages and extinctions, and it is a good example of a calculation that is easy to do and very hard to do convincingly.

What corotation does #

Corotation is the one radius where the pattern exerts no systematic torque, because a star there is not overtaking the arms at all. It is also, for that same reason, the radius at which stars can be moved furthest.

A star near corotation stays close to an arm for a long time, and a long, slow interaction can change its angular momentum substantially without heating it — that is, it can move a star to a different radius while leaving it on a nearly circular orbit. This is radial migration, and it was recognised as a major effect only in the 2000s.

Its consequence is uncomfortable for a whole line of argument: a star’s present radius is not where it formed. The Sun’s chemical composition is a slightly awkward fit to the solar neighbourhood, and one plausible reading is that it was born a kiloparsec or two further in and has since migrated out. Any attempt to reconstruct the history of the disc from the chemistry of stars at a given radius has to contend with this, and the machinery that does the mixing is the same spiral pattern that this essay has spent its length bounding.

The observation behind the number #

A pattern speed is hard to measure, and it is the quantity everything here depends on.

The standard method is due to Tremaine and Weinberg and is elegant enough to state: for a tracer that obeys the continuity equation — which the old stars do, since they are neither created nor destroyed — an integral of the surface brightness weighted by velocity along a line parallel to the major axis, divided by the same integral weighted by position, gives $\Omega_p$ directly. No model of the arms is needed.

Applied to barred galaxies, it finds bars that rotate fast, with corotation just outside the end of the bar — a result that recurs and appears to be a genuine regularity. Applied to spiral arms it is harder, because the young stars that mark the arms most clearly are exactly the tracer that violates the continuity assumption, being created and destroyed all the time.

For the Milky Way the pattern speed is contested, with estimates spanning a factor of two, precisely because the whole disc cannot be seen at once and the tracers available are the ones with the shortest lives.

The pattern is being slowed down #

A pattern speed is treated above as a given, and it is not constant: a bar rotating inside a massive halo loses angular momentum to it and slows.

The mechanism is dynamical friction, of exactly the kind a massive body moving through a sea of lighter ones experiences. The bar is not a body, but it is a rotating asymmetry in the gravitational field, and the halo particles whose orbits resonate with it exchange angular momentum with it preferentially — taking more than they give, because there are more particles just below the resonant condition than just above it.

The rate depends on how much halo there is where the bar is. A bar sitting inside a dense, massive halo slows quickly; one inside a disc-dominated inner galaxy, with little halo mass to talk to, barely slows at all.

That makes the pattern speed a probe of the halo, and the measurement is made through a dimensionless ratio: the corotation radius divided by the bar’s own length. A bar that has not slowed has corotation just outside its end, at a ratio between about 1.0 and 1.4 — a fast bar. One that has been braked has corotation much further out, and a ratio above about 1.4.

The measurements overwhelmingly find fast bars. Taken at face value that says the inner regions of barred galaxies are dominated by their discs rather than by their halos, which is one of the stronger arguments on the maximal-disc side of the decomposition debate — and it is an argument that comes from a rate of change rather than from a mass measurement.

The tension is that simulations of bars in cosmological halos slow them down considerably faster than the observations allow, and reconciling that has been an open problem for two decades. A pattern speed is therefore not a parameter of a galaxy but a measurement of its history, and the histories the simulations produce and the ones the galaxies show do not yet match.

The disagreement has produced the usual three candidate resolutions — that the halos are less dense in the centre than simulated, that the bars are younger than assumed, or that the measured ratios are biased by the difficulty of defining where a bar ends — and the third is the one the observers take most seriously, since the bar length is measured by eye far more often than by any objective criterion.

A quantity defined by where a visual feature stops is a quantity with an unquantified systematic in it, and the ratio being argued about is a factor of 1.4 against measurement scatter of comparable size.

Attempts to replace the eye with a photometric definition exist and give systematically different lengths, which shifts the whole distribution of ratios without settling the argument.

What the picture cannot show #

Everything here is linear theory. The resonance conditions come from treating the pattern as a small perturbation on circular orbits. A bar is not a small perturbation, and the orbits in a strong bar are qualitatively different — families of closed orbits aligned with and perpendicular to it, which no epicyclic treatment produces.

The figure draws Ω ± κ/2 and nothing about amplitude. Whether a wave of a given pattern speed actually exists, and how strong it is, is a question the dispersion relation answers and this diagram does not. The diagram says only where such a wave is permitted.

And the inner resonance may not exist. For a rotation curve that rises steeply enough in the centre, $\Omega - \kappa/2$ may never reach the pattern speed at all, in which case there is no inner Lindblad resonance and a wave can propagate right through the middle. That case is not exotic: it is thought to be what allows a bar to form and to persist.

The generalisation #

The move being made here is one this collection has now made in several fields, and it is worth naming as a method rather than as a result: when a system has two natural frequencies, the interesting behaviour lives where their ratio is rational.

In the solar system the two frequencies are two orbital periods, and the consequence is gaps, locks and the Laplace resonance among Jupiter’s moons. In an exoplanetary system they are the periods of two planets, and the consequence is a resonant chain that records how the system migrated. In a galactic disc they are the orbital and epicyclic frequencies of one star, and the consequence is a boundary on where a wave can live.

The commonality is not superficial. In each case a forcing arrives at the same phase repeatedly, so the response adds rather than averages away — and a system with no rational frequency ratio anywhere in it would be a system nothing interesting ever happened to.

Where the ladder goes next #

The resonances bound a pattern; they do not create one. The mechanism that generates spiral structure in the first place — swing amplification, in which a leading disturbance is sheared into a trailing one and amplified enormously in the process — is the next rung.

Later rungs on this anchor: swing amplification and the shearing sheet; bar formation and what stops a bar growing; angular-momentum transport and radial migration, which moves stars several kiloparsecs from where they were born; the Tremaine–Weinberg method applied to real galaxies; the inner Lindblad resonance rings in barred spirals; and warps as a bending wave rather than a density wave, which uses the same machinery with the vertical frequency in place of the epicyclic one.