A spiral drawn by velocities alone

The Milky Way’s spiral arms have never been seen from outside. Every map of them is a reconstruction from inside the disc, and for half a century almost every map of the inner Galaxy was built by one method: measure the velocity of a cloud of gas along the line of sight, assume that the cloud moves on a circular orbit at the speed the rotation curve gives for its radius, and solve for the distance at which a circular orbit would produce the velocity measured. That is a kinematic distance, and it carries a famous ambiguity — two solutions inside the Sun’s orbit, one near and one far — that has been argued about since the first maps of neutral hydrogen.

The ambiguity is the famous problem. The quieter one is the assumption in the middle of the method, that the gas moves on circles. It does not, and the reason it does not is the spiral arms themselves. An arm in a disc galaxy is a density wave rather than a fixed set of stars, and gas flowing through one is pulled by the arm’s extra gravity: it slows as it approaches, is deflected inward or outward as it passes, and speeds up again as it leaves. The departures from circular motion are ten to twenty kilometres a second, a few per cent of the orbital speed, and they are not random. They have the arm pattern’s own shape.

So the errors a circular curve makes are not random either. A velocity disturbance shaped like a spiral turns into a distance error shaped like a spiral, and a distance error shaped like a spiral moves gas toward some places on the map and away from others. The hero figure is the whole argument in one picture. On the left is a disc of gas with no arms at all — clouds scattered uniformly — given only the motions a two-armed density wave would impose. On the right are the same clouds placed where the kinematic method puts them.

How much a kilometre a second is worth #

The size of the effect follows from one number: how fast the circular velocity changes with distance along a line of sight. If moving a cloud one kiloparsec further away changes its expected velocity by fifteen kilometres a second, then a fifteen-kilometre-a-second error in the velocity is a one-kiloparsec error in the distance. The conversion is the inverse of the slope of the velocity–distance curve, and it depends on where the line of sight points and how far along it the cloud lies.

Two features of the figure matter for everything after it.

The sensitivity is never small. Even where a line of sight is most favourable, a kilometre a second costs about sixty parsecs. A molecular cloud is a few tens of parsecs across, and the spacing between the Milky Way’s arms near the Sun is two or three kiloparsecs, so streaming of ten kilometres a second moves a cloud by about a quarter of the distance between two arms. That is not a correction to an arm’s position. It is enough to put a cloud in the wrong arm.

The sensitivity is worst exactly where the gas is most often measured. Every line of sight into the inner Galaxy reaches a tangent point, where it grazes a circle around the centre and the velocity reaches its extreme. There the velocity barely changes with distance, and any velocity error becomes an unbounded distance error. The tangent points are also where the gas is densest in the longitude–velocity data, because a long stretch of the sight line shares nearly the same velocity there, and where the terminal velocity used to measure the rotation curve from inside is read off. So the most heavily sampled directions are the ones where the method is least able to say where anything is.

Toward the outer Galaxy the problem is different but not milder. Beyond the Sun’s circle there is no ambiguity, because the velocity changes monotonically with distance; but it changes slowly, because a flat rotation curve makes distant outer orbits look almost like the Sun’s own. At a longitude of 120° the sensitivity climbs from a tenth of a kiloparsec per kilometre a second nearby to a quarter at fourteen kiloparsecs, and a fifteen-kilometre-a-second stream there moves a cloud by three or four kiloparsecs.

An error with a shape #

A random velocity error would scatter clouds along their lines of sight and blur any structure the gas has. Streaming does something else, because it is a field: every cloud at a given arm phase streams the same way. Along one line of sight the result is a distance error that oscillates as the sight line passes through successive arms.

The model behind the figure is the simplest that has the right ingredients. The rotation curve is flat at 233 kilometres a second. The spiral is logarithmic, two-armed, with a pitch angle of twelve degrees — close to what the Milky Way’s arms are measured to have. In the frame of each cloud the streaming has a radial component and an azimuthal one, both of the chosen amplitude and a quarter of a cycle apart in arm phase, which is the shape linear density-wave theory gives for gas on either side of an arm. Every cloud is placed on its correct side of the tangent point, so that the near–far ambiguity is resolved perfectly and everything in the figures is the streaming alone.

Within that model the error curve has two properties that the rest of the essay relies on. It is proportional to the streaming amplitude, which the three curves show directly. And it has a fixed shape for a given line of sight, so that the clouds at one distance are always pushed the same way. That second property is what turns an error into a structure.

A row of clumps where the gas is smooth #

Wherever the distance error decreases with distance, the method compresses: clouds from a longer stretch of the sight line are placed into a shorter stretch of the map, and the map’s density there goes up by the ratio. Wherever the error increases with distance, the map is stretched and thinned. A sight line through gas of perfectly constant density therefore comes out as a sequence of clumps and gaps.

The effect has a name in the older literature: velocity crowding. It was understood in the 1970s, by W. B. Burton among others, that the ridges seen in longitude–velocity diagrams of neutral hydrogen need not be ridges of density at all, because a shallow gradient of velocity with distance concentrates emission from a long path into a narrow range of velocity. The argument here is the same effect carried one step further, from the velocity data into the face-on map that the kinematic method builds from them. Crowding in velocity becomes crowding in the map, and it does so in the pattern of the streaming that produced it.

The size of the clumps is the point. A density contrast of three between arm and interarm gas is a large contrast — larger than molecular gas shows in many external spirals, where the arms can be seen from outside and no distance has to be inferred. A uniform disc, given realistic streaming and read through a circular rotation curve, produces contrasts of that size with no arm in the gas at all.

The same disc in longitude and velocity #

The longitude–velocity diagram is what a survey of the Galactic plane actually measures, before any distance is computed: at each direction, how much emission arrives at each velocity. It is the raw material of every kinematic map, and the place where streaming first shows.

On circular orbits the uniform disc fills its allowed region smoothly. The edges are the terminal velocities — the tangent points at each longitude — and the region narrows toward the centre and the anticentre, where the velocity projected along the line of sight vanishes and the method fails entirely. Nothing in the interior would be called an arm.

The diagram with streaming is no longer smooth. Bands appear where clouds along long stretches of a sight line have been brought to similar velocities, and gaps where they have been spread apart. Some of the bands run parallel to the terminal-velocity edge, the way the classical arm tracers do in real surveys. None of them corresponds to more gas.

That is an uncomfortable picture, because the identifications of the Milky Way’s arms in longitude–velocity data were made by tracing exactly such features: the Scutum arm, the Sagittarius arm, the Norma arm, each a ridge of emission that can be followed across tens of degrees of longitude. The real arms do have more gas in them; external galaxies leave no doubt of that. But the diagram alone cannot separate the part of a ridge made by extra gas from the part made by the streaming that extra gas produces, and both have the shape of the arm.

A contrast made from motion alone #

How large an arm can streaming draw? One way to measure it is to bin the kinematic map by arm phase — the position within the spiral pattern — and compare the densest phase with the emptiest.

The curve says that motions of the observed size are enough, by themselves, to produce arm contrasts of the size real galaxies have. That does not mean the Milky Way’s arms are illusions. It means that a kinematic map cannot measure how strong they are: the contrast it reports is the true contrast compounded with a contrast the method makes, and the two point the same way, because the streaming is caused by the arm. A strong arm produces strong streaming, which produces a strong artificial arm on top of the real one, at a slightly different place.

That last clause matters as much as the amplitude. The artificial arm is not in the same position as the real one. The streaming field is a quarter-cycle out of step with the density in linear theory, and the conversion from velocity error to distance error depends on the angle between the streaming and the line of sight, which changes across the map. So the kinematic arm is displaced from the density arm by an amount that varies along its length, and its pitch angle is not the real one. Measurements of the Milky Way’s pitch angles from kinematic maps have ranged from about five degrees to more than fifteen, and part of that spread is a property of the method rather than of the Galaxy.

What the parallaxes found #

The decisive test needs distances that do not come from velocities at all. Since the mid-2000s very-long-baseline interferometry has measured trigonometric parallaxes of water and methanol masers in star-forming regions across the Galaxy, with a precision of ten microarcseconds or better — enough to measure a distance of ten kiloparsecs to a few per cent, with no assumption about how anything moves. The same observations measure each maser’s proper motion, so its full three-dimensional velocity is known as well as its position.

The results confirmed both halves of the argument above. The masers lie in arms: placed by parallax, they trace spiral segments with pitch angles of roughly ten to twenty degrees, so the Milky Way’s arms are real and the density in them is real. And the masers do not move on circles. Their velocities depart from the circular rotation curve by typical amounts of five to fifteen kilometres a second, with a systematic tendency for star-forming regions to lag behind circular rotation — the azimuthal component of the streaming. Kinematic distances computed for the same sources differed from their parallax distances by tens of per cent, and the differences were correlated across neighbouring sources rather than scattered.

The comparison of the two has turned a method’s error into a measurement. The difference between where the kinematic method puts a source and where the parallax puts it is the streaming, measured. A few hundred such comparisons are now enough to calibrate the streaming field near the arms and correct kinematic distances for the thousands of clouds that have no maser to measure — which is the same architecture as the chain of cosmic distances, a few direct measurements calibrating many indirect ones, arriving inside the Galaxy.

Why the method’s errors and the arms are the same object #

There is a general shape to this failure that is worth stating, because it recurs. A measurement is converted into a position by a model of how the measured quantity depends on position. The model leaves out a perturbation. The perturbation is caused by the very structure the map is meant to find. Then the error in the map is not a blur over the structure; it is a second copy of the structure, displaced, with an amplitude set by the strength of the first.

The same shape appears in the redshift maps of galaxies, where peculiar velocities caused by clusters stretch the clusters along the line of sight into “fingers” pointing at the observer and compress the infall regions around them into walls — distortions that in that case are modelled and used, because their amplitude measures the growth of structure. It appears in the motions that are somebody else’s velocity in the local Hubble flow, where galaxies falling toward the same attractor share their errors. In each case the structure writes its own artefact into the map, and the artefact cannot be averaged away because it is not noise.

What makes the Milky Way’s version especially stubborn is that there is no second view. An external spiral can be photographed face-on, and its arms are seen in starlight and dust before any velocity is measured. The Milky Way’s disc is seen edge-on from inside, through dust that hides most of it in visible light, and the velocities are the only depth coordinate that reaches across it. Every arm traced in its gas before the parallaxes was traced partly in the gas and partly in the method, and the share was not known.

Where the model stops #

The spiral is a single two-armed wave of fixed pitch and amplitude. The Milky Way has at least four major arm segments, possibly with different pitches, a central bar whose own non-circular flow dominates the inner few kiloparsecs, and spurs and branches between the arms. A realistic streaming field is messier than the one drawn, and its artefacts would be correspondingly messier, though of the same size.

The streaming is linear. In real arms gas passes through a shock where the flow is abrupt, and the velocity jump is concentrated into a narrow layer rather than spread sinusoidally. A sharp jump produces sharper artefacts — a caustic, where the kinematic map folds — but the same overall amplitude.

The clouds are perfectly placed on their own side of the tangent point. Real analyses must resolve the near–far ambiguity, and a wrong choice moves a cloud by many kiloparsecs, far more than streaming does. Including it would add a second, larger error that the streaming makes worse near the tangent point, where the two candidate distances are close.

The rotation curve is known. It is measured from the terminal velocities, which are themselves perturbed by the same streaming at the tangent points. A curve fitted to streaming-perturbed terminal velocities carries a small wiggle with the arms’ spacing, and those wiggles have been seen in the Milky Way’s measured curve.

The disc is uniform. That is the point of the exercise, not a claim about the Galaxy. Real arms have real density contrasts, and the kinematic map shows the real contrast compounded with the artificial one.

Still open: the far side of the disc #

The maser parallaxes now reach beyond the Galactic centre in a few directions, and the far side of the disc — the half of the Galaxy behind the bar, twelve to twenty kiloparsecs away — is still mapped mostly by kinematic distances, with the near–far ambiguity resolved by statistical arguments rather than measurement. Its arms, as drawn, are continuations of the near side’s arms assumed to be symmetric.

Whether they are is a question the method cannot answer with confidence, because at those distances the sensitivity of distance to velocity is largest and the streaming is unmeasured. A far-side arm traced in velocity could be a real arm displaced by its streaming, a real arm at the right place, or a crowding artefact of an arm somewhere else along the same lines of sight. The parallaxes needed to settle it require masers bright enough to be detected at twenty kiloparsecs and baselines long enough to measure fifty microarcseconds to a few per cent; the arrays being extended southward and the space-based baselines being proposed are aimed at exactly that. Until they report, the Milky Way’s far side is a map of where gas is moving, labelled as a map of where it is.