A falling curve seen from one side of the Galaxy

The flat rotation curve is the founding observation of dark matter in galaxies. Stars and gas far beyond the visible disc keep moving at the speed of those further in, so the mass within their orbits must go on rising with radius. For the Milky Way itself, the same statement has always been harder to make, because the observer is inside the disc, and the method that works in the inner Galaxy stops at the Sun’s orbit. The space astrometry mission Gaia changed that. Its third data release, in 2022, gave positions and motions for red giants well beyond twenty kiloparsecs, and when these were converted into a rotation curve, the curve fell.

Several analyses published since then find something of the sort. Yongjun Jiao and collaborators reported a decline they called Keplerian, beginning near 19 kiloparsecs and continuing to 26.5, and rejected a flat curve there at three standard deviations. It implied a mass of about $2\times10^{11}$ solar masses, a fifth or less of what had been commonly quoted. Xiaowei Ou and collaborators, using over thirty thousand giants with spectroscopic distances, found a gradual fall from about 230 kilometres a second at 5 kiloparsecs to 170 at 28. A mass that low would make the Milky Way’s dark halo surprisingly small and would leave little room for the dark matter that its disc would ordinarily have collected.

None of the measurements is careless, and the star velocities are among the best ever obtained in astronomy. What the argument over them concerns is the step between a measured velocity and a rotation curve. In the outer disc that step has a considerable size, and the things it depends on are the same things that are least known at those radii. Working through it shows that a claimed fall of about thirty kilometres a second has to be weighed against corrections of about the same size, in either direction.

From a mean velocity to a circular speed #

A rotation curve is defined as the speed of a body on a circular orbit, and no star in the outer Galaxy is guaranteed to be on one. A star population with random motions rotates more slowly than the circular speed, because random motion supplies part of the support against gravity. This lag has a name, the asymmetric drift, and it can be corrected for under some assumptions. If a tracer population is in a steady state and axisymmetric, the Jeans equation, a moment of the collisionless equation of motion, gives the correction in terms of quantities that can be measured:

$$v_c^2 = \bar{v}_\phi^2 + \sigma_\phi^2 - \sigma_R^2\left(1 + \frac{d\ln\nu}{d\ln R} + \frac{d\ln\sigma_R^2}{d\ln R}\right).$$

The velocity $\bar{v}_\phi$ is the direct measurement. The rest supplies the drift. A reader’s attention usually falls on the first term, but the bracket includes a logarithmic gradient: $d\ln\nu/d\ln R$ is minus the radius divided by the scale length of the tracer’s density. In an exponential disc that number grows without limit in the outer parts. At twenty-five kiloparsecs, with a scale length of three, it is more than eight.

So the circular speed at 25 kiloparsecs depends on a scale length, a quantity that describes how many stars there are and not how they move. A choice between two reasonable values, two and five kiloparsecs, moves the answer by nineteen kilometres a second. The whole decline in the outer part of the curve, from 19 to 26.5 kiloparsecs, is only a little larger than that. And the correction is a large share of the signal: the mean rotation sits a long way below the curve it is converted to.

What the correction is made of #

The terms are worth separating, because each has its own weakness.

So over a quarter of the mass inferred at the edge of the sample comes from the correction, and most of that from a term which is a star count. Star counts in the outer disc are exactly what is difficult. A catalogue of bright giants has a selection function that varies with position and with extinction, as the disc is seen through the dust of the plane. Its parent population has been shaped by whatever cuts were used to isolate the disc from the halo. The scale length of the sample has little to do with the scale length of the true disc unless the selection is known in detail, and an analysis usually imports its value from elsewhere.

The second term carries a gradient of the velocity dispersion. It is measured directly and decently, though at these distances the dispersion itself can be inflated by errors in distance, since a tangential velocity is the product of a proper motion and a distance. The third, the anisotropy between the azimuthal and radial dispersions, is small and measured. There is also a term omitted entirely, the cross term that couples radial and vertical motion, which vanishes only under an assumption about the shape of the velocity ellipsoid. In the sample here it is expected to be modest, and it is one more thing the steady-state equation takes on trust.

A density that does not stay exponential #

The sharpest form of the problem has to do with the shape of the tracer’s profile, and it does not have a preferred sign. Galactic discs seen from outside do not usually have one exponential all the way out. Many have a break, beyond which the light drops much faster, and others flatten into an outer extension. The Milky Way, seen from within, has been reported to have a break as well, near the outer disc, and it has a substantial warp and flare by twenty kiloparsecs. A flare moves stars away from the plane and so changes what a sample near the plane counts as its radial density.

Read one way, this is a warning, and read another, it is a diagnosis. If the true disc steepens where an analysis assumes it continues exponentially, then the analysis has under-corrected, and a real, nearly flat curve will appear to fall. A recent test of the method, by Koop and collaborators, using the same catalogue and simulated galaxies, reached that conclusion with some force: steady-state axisymmetric modelling becomes less robust at large radii, and a tracer density not as exponential as assumed can produce the appearance of a steeply declining rotation curve. A flare would act in the other direction, and a curve that looks only modestly declining might be declining more steeply than reported. So nobody should read the figure as saying that the fall is an artefact. What it says is that a measurement of the outer curve needs the outer density as well, with a precision comparable to that of the velocities, and that the two are not equally well measured.

A few stars that are not going round #

Two further sources of apparent decline do not involve the equation at all. The first is contamination. Red giants at large distance are selected as disc stars by their positions, their metallicities or their motions, and any sample will include some members of the stellar halo, which as a population has essentially no net rotation.

The arithmetic is simple, and the danger lies in its scaling: the disc’s density falls exponentially with radius while the halo’s falls as a modest power law, so a contamination fraction that is negligible at 15 kiloparsecs can become substantial at 28. Even a ten per cent admixture is an apparent fall of twenty kilometres a second. Selecting by motion, as many analyses do, cleans out the halo stars but imposes a velocity cut that truncates the disc’s own velocity distribution, which biases both the mean and the dispersion. Selecting by metallicity needs a metallicity scale that holds at low values in the outer disc, where the disc and the halo overlap. The authors of the analyses know this and treat it, and it adds a term to the error budget that cannot be calculated from Gaia’s own uncertainties.

A one-sided view of a disc that is not round #

The second has to do with where the sample lies. Stars at 25 kiloparsecs from the Galactic centre on the far side of the disc are too distant and too reddened for the catalogue’s precise astrometry. So an outer curve in practice comes mainly from a wedge on the Sun’s side of the Galaxy. For an axisymmetric disc a wedge would be enough, and the Jeans equation assumes this. But the outer Milky Way is not axisymmetric: it has a warp, ripples in velocity and density, and large-scale patterns that are visible in the catalogue itself.

A mode with one arm is what an encounter tends to leave. The Galaxy is being encountered: the Large Magellanic Cloud is on its first passage, with a mass that is a large fraction of the Milky Way’s, and the Galaxy’s inner parts have been displaced towards it, leaving the outer stars with a reflex motion measured at tens of kilometres a second beyond about 30 kiloparsecs. In a real sense the Galactic centre is not a fixed origin for those motions, and a steady axisymmetric equation does not describe it. The perturbation at 25 kiloparsecs is smaller than the one in the outer halo, and it has not been separately measured. A disc also responds to stellar streams and passing satellites, so a coherent 5 per cent pattern in rotation at the disc’s edge, though not established, is not implausible.

This matters for the interpretation in another way. The Jeans equation has a term in the time derivative of the mean radial velocity that has been dropped, because steady state sets it to zero. No measurement of the disc’s motions can test that assumption, because a catalogue records only one moment. Comparison with simulations that have realistic perturbers is the only route, and those tests find the recovery of the true circular speed worsening at these radii. Mass goes as the square of the speed, so even a ten per cent uncertainty in the speed would be a twenty per cent uncertainty in the mass, and that would be enough to matter for the conclusion.

What it would mean if the mass did stop #

There is a test from outside the disc, which uses objects that have no need for the Jeans equation to be accurate in the outer disc. Globular clusters and satellite galaxies orbit at much larger distances, and they respond to the mass within their orbits.

The tension is plain enough. Tracer masses at forty to fifty kiloparsecs come out at about five times $10^{11}$ solar masses, more than twice what a sharply truncated curve allows at any distance. The Large Magellanic Cloud has been estimated from the deflection of stellar streams at over $10^{11}$ solar masses, so that a Milky Way of $2\times10^{11}$ solar masses would have a companion nearly its own size, and it would be hard to describe what is orbiting what. The timing of the Local Group’s approach offers a total of a few times $10^{12}$ for the two big galaxies. These are not independent of modelling either. Satellites might not be bound or relaxed, and halo mass from globular clusters has its own systematic errors. But they point the same way, and nothing obvious favours a sharp truncation of the mass exactly where the disc’s stars run out.

Thus the plausible conclusion is not a choice between flat and Keplerian. A moderate decline in the curve is probably real, as several analyses agree and as is expected in a galaxy whose baryonic disc still makes a contribution to the force at those radii. Its size is uncertain. And the extrapolation to a total mass is the weakest step of all: a fall over a limited range does not imply a truncated halo at all, as a halo fit with a profile allowed to fall more steeply can reproduce the stars and still enclose much more mass further out. Three mass models that fit the same curve is a familiar difficulty here, and it recurs on a different scale.

A precise instrument and its weak link #

There is a general point here about what precision does to an argument. The data have improved so much that the statistical uncertainty on the mean rotation in each radial bin is about a kilometre a second, and this makes the statistical error bar on a plotted curve look tight. A figure showing a falling line with tiny error bars then reads as decisive. But the error bars describe only the velocities, whereas the conversion comes from quantities measured far less well, and it is much larger than the statistical term. So here improved velocities raise the significance of a difference without raising its reliability, a pattern this subject meets often. A rejection of a flat curve at three sigma is a statement within one model of the tracer, with one density, one treatment of the halo stars and one assumption of steady state.

A further point concerns the choice of tracer. Young disc stars, such as Cepheids, have small dispersions, so their correction is much less. A sample of Cepheids has a small asymmetric drift and a density that can be read directly from its distribution. Those samples are smaller and stop at a somewhat lesser radius, near twenty kiloparsecs, and the curve Przemek Mróz and collaborators drew from them in 2019 declines only gently. If a young cold population and an old warm one, moving in the same potential, yield different circular speeds at the same radius, the difference is a direct measure of how much the correction has gone wrong. That is the same logic that underlies the way disc mass is read from how far stars climb: the same potential must explain every population moving in it.

What the picture cannot show #

The velocities in these figures are not those of any actual catalogue. They were reconstructed so that a chosen analysis reproduces a straight-line approximation to one published curve. The dispersions have a representative size and an assumed exponential shape, and they are held fixed in each comparison. The real analyses fit dispersions and densities from the data with different choices of their own, and so these figures say how large each sensitivity is, not what any specific paper got wrong. No conclusion here depends on a particular dataset.

The one-armed pattern is a schematic form, and its amplitude has not been measured at these radii. The rising halo fraction is a linear ramp chosen for clarity, and the actual contamination depends on the selection. The dots in the mass figure come from different methods with their own systematic errors. The picture also treats the disc as a two-dimensional sheet, whereas the warp means that the plane itself is not defined at 25 kiloparsecs, so the separation between radial and vertical motion is itself a choice. Nor do these figures show that the curve is flat, since they show only that the present evidence cannot make it Keplerian without further assumptions.

Still open: a curve from stars on both sides #

The direct way past most of the difficulties is to see the far side. Near-infrared astrometry would penetrate the dust that hides the opposite half of the outer disc, and it would average the one-armed modes that a single wedge cannot, and allow the density to be counted in the same sample. A proposed infrared successor to the present mission would do this, and until then, wide-field surveys of variable stars in the infrared offer a thinner sample with good distances. If a curve measured all the way round a circle at twenty-five kiloparsecs falls by the same amount on both sides, the fall belongs to the potential. If it differs, the difference is a map of how far the outer Galaxy is from the steady state in which it is usually written down, and that is a measurement worth having whichever way the mass turns out.