A galaxy measured from inside it

Every other galaxy is a picture. Its centre is obvious, its inclination can be measured from the shape of its outline, and a radius on the sky converts into a radius in kiloparsecs as soon as its distance is known.

The Milky Way is none of those things. It is a band of light around the sky, seen from a point eight kiloparsecs off centre and buried in its own dust, and the quantity a telescope reports is a velocity along a line of sight with no distance attached. The astonishing thing is that this is enough — that a rotation curve for the inner Galaxy can be extracted from it without measuring the distance to anything at all.

The construction #

Set the Sun at radius $R_0$ from the Galactic centre and measure galactic longitude $l$ from the direction of the centre. A line of sight at longitude $l$ passes through gas at many radii; the closest it comes to the centre is

$$R_{\text{tangent}} = R_0 \sin l,$$

which follows from the right-angled triangle formed by the centre, the Sun, and the point of closest approach.

At that point — and nowhere else along the line — the circular velocity is directed exactly along the line of sight. Everywhere else, only a component of it is, so the Doppler shift is smaller. The largest velocity in the spectrum therefore belongs to the tangent point, and it belongs to a radius the geometry fixes.

Subtracting the Sun’s own motion gives the terminal velocity:

$$v_{\text{term}}(l) = v_c(R_0 \sin l) - v_c(R_0)\sin l.$$

One spectrum, one longitude, one point of the rotation curve. Sweep the longitude from zero to ninety degrees and the whole inner curve appears, sampled at whatever spacing the observer likes.

The elegance is that the two things a distance would normally be needed for — where the gas is, and how fast it is going in its own orbit — are supplied by the geometry and by the extremum of the line profile respectively. What is measured is a shift and an angle.

Why it stops at the Sun’s orbit #

Beyond $R_0$ the construction fails, and the failure is not a technical limitation but a geometrical fact: a line of sight at any longitude never comes closer to the centre than $R_0 \sin l$, and for outward-looking longitudes there is no tangent point at all. Every point on such a line is at a radius greater than the Sun’s, and the radial velocity increases monotonically outwards rather than reaching a maximum somewhere useful.

So the outer rotation curve of the Milky Way — the half that carries the whole dark-matter argument — needs distances, one object at a time, and each one has to be a star or a cluster whose intrinsic brightness is known.

That irony persists in the modern data. NGC 3198’s curve is smooth and well determined out to eleven disc scale lengths; the Milky Way’s outer curve is a scatter of points from different tracers with systematic offsets between them, and the value of $R_0$ itself — which every point depends on — was uncertain at the ten per cent level until very recently.

The observation behind the number #

The line that makes all of this possible was predicted before it was found. Hendrik van de Hulst worked out in 1944, during the occupation of the Netherlands and with almost no equipment available, that neutral hydrogen should emit at twenty-one centimetres through the hyperfine transition, and that the line ought to be detectable despite the transition being forbidden — because there is so very much hydrogen.

Ewen and Purcell detected it at Harvard in 1951 with a horn antenna built out of plywood and copper sheet, and Dutch and Australian groups confirmed it within weeks. Within two years the first maps of Galactic structure from twenty-one centimetre spectra had been published.

The dust that ruins optical work in the plane is irrelevant at twenty-one centimetres — the wavelength is a million times the size of a grain, and the extinction is negligible. That single fact is why the structure of the Galaxy is a radio subject.

The two things the Sun’s own motion does #

Every velocity in the construction is measured relative to a moving observer, and the corrections are not small.

The first is the Sun’s peculiar motion, some twenty kilometres per second relative to the average of the stars around it. The reference frame that removes it is the local standard of rest — the frame of a hypothetical object on a circular orbit at the Sun’s radius — and it is defined by averaging the motions of many nearby stars, which means it is measured rather than given.

The second is $v_c(R_0)$ itself, which enters the terminal velocity directly. The best modern value comes from a measurement that would have astonished the radio astronomers of the 1950s: the apparent motion of Sagittarius A*, the radio source at the Galactic centre, across the sky. It moves at about six milliarcseconds per year, almost entirely reflex — the source itself is nearly at rest — so the Sun’s total velocity around the centre follows from that angular rate and $R_0$ with no dynamics at all.

The local measurement, which needs no radio at all #

There is a second, older route to the same physics, and it works entirely within a few hundred parsecs of the Sun.

Jan Oort showed in 1927 that if the Galaxy rotates differentially — inner material going round faster in angle than outer material — then the radial velocities and proper motions of nearby stars must show a double sinusoid in galactic longitude. The amplitudes of those two sinusoids are the constants now named after him:

$$A = \tfrac{1}{2}\left(\frac{v_c}{R} - \frac{dv_c}{dR}\right)_{R_0}, \qquad B = -\tfrac{1}{2}\left(\frac{v_c}{R} + \frac{dv_c}{dR}\right)_{R_0}.$$

Their sum and difference are worth more than either separately. $A - B$ is the angular speed at the Sun’s radius, and $A + B$ is minus the local slope of the rotation curve. So two numbers, extracted from the motions of stars none of which is more than a per cent of the way to the centre, give both the local rotation rate and whether the curve is locally rising or falling.

The measured values are close to $A = 15$ and $B = -12$ kilometres per second per kiloparsec. Their sum is small, which is the local statement of the whole finding of this field: the rotation curve at the Sun is nearly flat. Oort had that result in 1927, before there was any way to measure the curve at large radii, and before anybody knew what a galaxy was made of.

Nothing in that measurement depends on hydrogen, on radio astronomy, or on a distance to the centre. It depends on being able to measure proper motions well — which is why the modern values come from a satellite that does nothing else, and why they moved when it reported.

The Galaxy from outside, inferred #

An essay about a galaxy nobody can photograph should be explicit about what is nevertheless known, and how.

The Milky Way is a barred spiral of moderate size: a disc scale length of about 2.6 kiloparsecs, a stellar mass around six times ten to the tenth solar masses, a bar some three kiloparsecs long, and four or so arm segments. Each of those numbers comes from a different and indirect route. The scale length comes from infrared star counts, which see through the dust. The bar was established from the asymmetry of those same counts — one side of the inner Galaxy is systematically brighter than the other, which a symmetric disc cannot produce — and confirmed by the non-circular gas velocities it drives.

The comparison that then becomes possible is worth making, because it is the closest thing to a photograph available: the Milky Way’s rotation curve, luminosity and scale length place it beside galaxies that can be photographed, and its properties are unremarkable among them. Its position on the Tully–Fisher relation is normal. It is, from outside, an ordinary large spiral, and there is a certain satisfaction in having established that from within.

The two numbers everything is divided by #

Every quantity in this essay is expressed in units of the Sun’s distance from the centre and its circular speed, so those two numbers propagate into every radius and every velocity on the curve. For most of the twentieth century they were the weakest links, and both have now been fixed by measurements that owe nothing to the gas.

The distance came from tracking individual stars on orbits around the compact source at the Galactic centre. One of them has a period of about sixteen years and passes within a hundred and twenty astronomical units of the centre at closest approach, which is close enough that its orbit shows relativistic precession. Because the orbit is measured both astrometrically — as an ellipse on the sky, in angular units — and spectroscopically — as a radial velocity, in kilometres per second — the ratio of the two fixes the distance without any assumption at all. It is a moving-cluster measurement applied to one star, and it now gives the Galactic centre distance to a few tenths of a per cent.

The circular speed came from the reflex proper motion described above, combined with that distance. The compact source itself is very nearly at rest at the centre of mass, so its apparent motion across the sky is the Sun’s own orbital motion seen from outside, and the product of that angular rate and the distance is a speed.

Two things follow that are worth stating. First, both numbers are now known better than the rotation curve they scale, which reverses a situation that held for fifty years — the curve’s uncertainty used to be dominated by them and is now dominated by the tracers. Second, the two measurements are not independent of one another, since the circular speed uses the distance; so a revision to the distance moves every radius and every velocity together, in a correlated way that has to be propagated rather than added in quadrature.

There is a satisfying closure in the method. The distance to the centre of the Galaxy — the quantity that the whole of Galactic structure is measured against, and that Shapley first estimated from globular clusters in 1918 — is now obtained from a single star’s ellipse, by the same geometry that fixes a binary’s orbit anywhere else in this collection.

The ambiguity the method leaves behind #

There is a second use of these velocities, and it comes with a defect that shaped decades of work on Galactic structure.

Once a rotation curve is known, the process can be run backwards: a cloud at longitude $l$ with radial velocity $v$ must be at the radius that produces $v$, and hence at a distance from the Sun that the geometry gives. That is a kinematic distance, and it is how most of the Milky Way’s molecular clouds and star-forming regions are placed.

Inside the solar circle it is two-valued. A line of sight crosses each circle of radius $R < R_0$ twice — once before the tangent point and once after — and both crossings have the same radial velocity. So a cloud with a given velocity is either at the near distance or the far one, and nothing in the spectrum says which.

That is the kinematic distance ambiguity, and it is resolved object by object, by absorption against a background continuum source, or by the cloud’s angular size, or by its height above the plane. Get it wrong and a piece of spiral arm is placed on the wrong side of the Galaxy.

The curve the Galaxy actually has #

The outer curve was described above as a scatter of points from different tracers, and that description has stopped being accurate. A survey that measures positions, proper motions and parallaxes for over a billion stars, combined with spectroscopic radial velocities for a large subset, supplies something the older work could not: the full six-dimensional phase-space coordinates of millions of individual disc stars, each with its own distance rather than a kinematic one.

From that, the circular speed can be computed directly as a function of radius, by asking what rotation the observed velocity field requires at each place — with corrections for the fact that stars, unlike gas, have velocity dispersions and therefore lag the circular speed by a computable amount.

The result has a feature nobody was looking for. Out to about fifteen kiloparsecs the curve is nearly flat, as it has always been reported. Beyond that it declines, and the decline is faster than the gentle fall a flat-curve halo would produce — steep enough to be consistent with the enclosed mass having stopped growing.

If that holds, it is a statement about the edge of the Milky Way’s dark halo, measured kinematically rather than assumed, and it revises the Galaxy’s total mass downward relative to the values that a flat extrapolation gives. The caution is the same one this essay has been repeating: the measurement depends on the asymmetric-drift correction, on the assumption that the outer disc is in equilibrium, and on distances that are least reliable exactly where the effect appears. The outer disc is warped and flared and is known to be responding to the passage of a satellite, none of which is equilibrium.

So the one galaxy that can be measured star by star is also the one where the assumption of a settled, axisymmetric disc is most obviously false, and the two advantages the Milky Way offers — proximity and perturbation — are not separable.

What the picture cannot show #

The tangent-point method assumes circular orbits. Gas in the inner Galaxy is not on circular orbits: the bar drives strongly non-circular streaming, and inside about three kiloparsecs the terminal velocities are dominated by it. The inner rotation curve derived this way is therefore not to be believed at small radii, and the figure above draws it only as a model.

It assumes the disc is flat and the Sun is in the plane. Neither is exactly true. The outer disc is warped by several degrees, and the Sun sits some twenty parsecs above the midplane, which is negligible for the tangent points and not negligible for tracing structure.

A terminal velocity is a choice of threshold. The edge of a line profile is soft, and the velocity assigned to it depends on the fraction of the peak at which the edge is called. Different authors have used different fractions, and the resulting curves differ by a few kilometres per second systematically.

And the picture is a plan view of something nobody has ever seen in plan. Every diagram of the Milky Way’s spiral structure, this one included, is a reconstruction from velocities and models. The Galaxy is the one object in this collection that is drawn rather than photographed, and each such drawing carries the assumptions of the decade it was made in.

The generalisation #

The method’s shape is more general than its subject: when a quantity cannot be measured directly, find the geometry in which one particular point along the line of sight is distinguished by an extremum, and measure the extremum.

The same reasoning gives the transit’s contact points, where a light curve’s corners fix a geometry that the depth alone cannot, where the ingress and egress are distinguished not by being observable in isolation but by being the moments at which something turns over. It gives the maximum elongation of an inner planet, which fixes an orbital radius from an angle and nothing else — the same trigonometry as $R_0\sin l$, three hundred years earlier and one scale down. And it gives the terminal velocity here.

In each case the extremum does two jobs at once: it is easy to identify in the data, and it corresponds to a geometrically special location. That coincidence is not luck. An extremum in an observable usually occurs precisely where a projection factor reaches one, and a projection factor reaching one is what a special geometry means.

The tangent-point construction and the line profile it is built from are worth reading at a second geometry each.

Where the ladder goes next #

The immediate next rung is the structure this method is used to map — the arms, the bar, and the argument about how many of the first there are.

Later rungs on this anchor: the warp and flare of the outer disc; the bar’s effect on the inner velocity field, and why the terminal-velocity curve inside three kiloparsecs is not a rotation curve; maser parallaxes and the arm map they have produced; the thick disc as a separate population with its own kinematics; the vertical structure of the disc and the Oort limit; the halo’s stellar populations traced by streams; and the Galactic centre itself, where one star’s orbit weighs the thing at the middle.