One acceleration, and every galaxy's missing mass

A rotation curve that refuses to fall is the classic evidence for dark matter: outside the visible disc of a spiral the orbital speed stays flat, where the visible mass alone would make it fall as the inverse square root of radius, and the difference has to be made up by mass that is not seen. Stated that way the discrepancy is a property of each galaxy separately, measured in each galaxy’s own units — a radius outside which the curve does not fall, a mass that has to be added.

Stated another way, it is a single relation obeyed by all of them. At every radius of a rotation curve two accelerations can be computed. One is observed: the centripetal acceleration $v^2/r$ of gas moving in a circle at the measured speed. The other is predicted: the acceleration that the galaxy’s stars and gas alone would produce at that radius, computed from the distribution of their light and their twenty-one-centimetre emission with Newton’s law. Plot one against the other — every radius, every galaxy, from dwarfs of ten million solar masses to giant spirals a thousand times heavier — and the points fall on one curve.

What the curve is made of #

The relation was assembled in 2016 by McGaugh, Lelli and Schombert from a sample of 175 disc galaxies with both high-quality rotation curves and near-infrared photometry from the Spitzer Space Telescope — the SPARC sample — and it contains about 2,700 individual points. The near-infrared matters: at 3.6 microns the light of a galaxy is dominated by old stars and is barely affected by dust or by young ones, so the stellar mass-to-light ratio is nearly the same from galaxy to galaxy and can be taken as a single number rather than fitted galaxy by galaxy. That removes the freedom a decomposition normally has — the disc–halo degeneracy that lets three mass models fit one curve — by fixing the stars in advance.

With the stars fixed and the gas measured directly, the baryonic acceleration at each radius is computed without reference to the rotation curve at all. The observed acceleration is measured without reference to the light. The two are independent measurements, and the relation between them is therefore not built in.

The scatter is the striking part. The points lie within about 0.13 dex of the curve — a third of a factor of two — and the estimated observational errors in distance, inclination and mass-to-light ratio account for most of that. What is left for intrinsic scatter is at most a few hundredths of a dex. Galaxies with very different histories, sizes, gas fractions and surface brightnesses, measured at very different radii, all obey one function of one variable to within their errors.

Which errors move a point, and which way #

A relation this tight invites the question of whether it could be manufactured by the way the points are made, and working through the errors shows that most of them move a point along the curve rather than off it.

The distance to a galaxy enters both accelerations. A radius in kiloparsecs is an angle times the distance, so an error in distance scales every radius by the same factor. The observed acceleration $v^2/r$ falls as the distance rises. The baryonic acceleration is a mass over a squared radius, and the mass from the light scales as the distance squared, so $GM/r^2$ is independent of distance for the stars — but the whole radial profile moves. The net effect of a distance error is to slide a galaxy’s points diagonally, and on the high-acceleration branch, where the relation is the line of equality, a diagonal slide keeps them on it.

The inclination of the disc enters only the observed acceleration: a rotation speed is measured along the line of sight and divided by the sine of the inclination, so a nearly face-on galaxy with an uncertain inclination has an uncertain speed. That moves points vertically, and it is the largest single contribution to the observed scatter, which is why the sample excludes galaxies closer to face-on than about thirty degrees. The mass-to-light ratio enters only the baryonic acceleration, and moves points horizontally.

So the scatter the errors produce has a direction, and the observed scatter has the directions the errors predict. What is left over, after the errors are modelled, is consistent with zero. That does not prove the relation has no intrinsic scatter; it means the intrinsic scatter is smaller than a measurement of this quality could reveal, which is a small number — a few hundredths of a dex, a few per cent in acceleration.

The same discrepancy, at different radii #

The significance of the relation is clearest when it is turned back into radii, galaxy by galaxy.

That is the empirical fact that a scale in radius cannot describe and a scale in acceleration can. A giant spiral has high surface density in its inner disc; its stars there produce accelerations well above $a_0$, and its rotation curve there is explained by its stars, with a maximum-disc fit. A low-surface-brightness galaxy of the same total luminosity spreads its stars thinly; its accelerations are below $a_0$ everywhere, and its rotation curve is dominated by the discrepancy from the centre outward. The amount of dark matter a disc galaxy appears to need, at every radius, is predicted by the surface density of its visible matter.

This was noticed first as a pattern among low-surface-brightness galaxies in the 1990s, which were predicted by some halo models to lie off the Tully–Fisher relation and turned out to lie on it; and before that it was predicted, in 1983, by Mordehai Milgrom, whose modified Newtonian dynamics proposed that gravity itself departs from Newton’s law below an acceleration of about $10^{-10}\,\mathrm{m\,s^{-2}}$. Whatever the explanation, the relation is a measured regularity, and it has to be accounted for by any theory of galaxies.

The flat curve and the fourth power #

The relation contains the flat rotation curve and the baryonic Tully–Fisher relation as special cases, and seeing how makes the single curve less mysterious.

Far outside a galaxy’s baryons, the baryonic acceleration falls as $GM/r^2$ for a total baryonic mass $M$. On the low branch of the relation the observed acceleration is the geometric mean of that and $a_0$: $g_{\rm obs} = \sqrt{GMa_0}/r$. A circular orbit has $g_{\rm obs} = v^2/r$, so $v^4 = GMa_0$ — independent of radius. The flat rotation curve is the low-acceleration branch of the relation, and its level is fixed by the galaxy’s baryonic mass alone, raised to the quarter power. That is the baryonic Tully–Fisher relation, with its slope of four and its normalisation both set by $a_0$, and the tightness of that relation when it is measured with the flat outer speed is the global version of the local tightness here.

The local relation says more than the global one. The Tully–Fisher relation constrains one number per galaxy; the radial acceleration relation constrains the whole shape of every rotation curve from the distribution of its baryons. A feature in the light — a bump in the surface brightness where a spiral arm or a ring concentrates stars — should produce a corresponding feature in the rotation curve at the same radius, and the relation says how large. Such features are seen, and they are the observations Renzo Sancisi summarised in a rule of thumb: for any feature in the luminosity profile there is a corresponding feature in the rotation curve, and vice versa. In a galaxy whose inner rotation curve was set by a dominant, smooth dark halo, the features in the light would be washed out of the curve.

What a halo has to do to reproduce it #

In the standard cosmology each galaxy sits in a halo of cold dark matter whose mass and concentration were set by the growth of structure, with a scatter from one halo to the next that reflects its individual history. Nothing in the dark matter knows about $a_0$ or about the galaxy inside. The relation therefore has to emerge from how the baryons settled into the halos — and it is fair to ask how hard that is.

The figure is a demonstration of what the input has to be. Given the observed Tully–Fisher relation as a constraint on each halo’s outer speed, and a halo core that scales with its disc, the shape of the relation follows roughly. What the demonstration does not show is why nature tuned the haloes that way. A halo’s formation history gives it a mass and a concentration independent of the galaxy that later forms inside it, and the baryons’ contribution to the inner acceleration depends on how compact that galaxy became, which depends on its angular momentum, its feedback and its mergers.

Hydrodynamical simulations of galaxy formation in cold dark matter reproduce the relation with a scatter comparable to the observed one. Different groups have argued that it emerges naturally from the way baryons and haloes co-evolve, or that it emerges only because the simulations’ feedback parameters were tuned to reproduce galaxy sizes and luminosities that themselves carry the relation. The disagreement is not about whether the simulations match; it is about whether they explain.

Where the model and the relation part #

The dwarfs are where the demand on a halo is sharpest, and where a simple halo model fails most visibly.

Real dwarf galaxies of the same outer speed have rotation curves of strikingly different inner shapes — some rising steeply, some slowly — and the difference tracks the distribution of their baryons, exactly as the relation requires. A dark-matter halo sets the inner curve by its own density profile, and haloes of the same outer speed have similar profiles, so a population of haloes predicts much less diversity in inner shapes than is observed at fixed outer speed, and the wrong correlation with the baryons. That is known as the diversity problem, and it is the cusp–core argument seen from the side of the baryons: the inner rotation curves of dwarfs are predicted by their visible matter better than by their dark matter.

The proposed solution within cold dark matter is feedback — repeated bursts of star formation that drive gas out of the centre and pull the dark matter outward with it, converting a cusp into a core by an amount that depends on how many stars the galaxy formed. That couples the halo’s inner shape to the baryons, as the relation demands, through a mechanism whose strength depends on details of star formation that the simulations model at resolutions far coarser than the processes themselves.

A family of relations with one scale in them #

The radial acceleration relation is the most general of several regularities in galaxy dynamics that share its acceleration scale, and the others are older.

Ken Freeman noticed in 1970 that the central surface brightness of spiral discs clusters near a single value, about 21.7 magnitudes per square arcsecond in the blue, and the surface density corresponding to that value, divided into Newton’s constant, gives an acceleration near $a_0$. Discs brighter than that limit are rare; discs fainter than it turned out to be common but had been missed, because a threshold in surface brightness is also a threshold in what a survey can see. The central surface density of the dynamical mass, measured from the inner slope of the rotation curve, follows the central surface density of the baryons in the way the relation predicts: equal to it for high-surface-density discs, and approaching a floor set by $a_0$ for the diffuse ones.

For pressure-supported systems the analogue is the Faber–Jackson relation between an elliptical galaxy’s luminosity and the fourth power of its velocity dispersion, which is the Tully–Fisher relation’s fourth power for a system supported by random motions rather than rotation. Its normalisation for the most diffuse pressure-supported systems — dwarf spheroidal satellites of the Milky Way, whose internal accelerations are far below $a_0$ — is again close to what the acceleration scale sets.

The common element is not a mass or a size but a surface density, because an acceleration from a disc is Newton’s constant times a surface density, and a scale in one is a scale in the other. The regularities say that galaxies know about a characteristic surface density of about a hundred solar masses per square parsec, which is roughly the surface density of the Sun’s neighbourhood of the Milky Way’s disc and of the densest parts of the dwarfs. In a halo picture, that surface density must emerge from how discs acquire their sizes inside haloes and how feedback limits their central concentrations; in a modified-gravity picture it is the scale of the theory.

An acceleration among other accelerations #

Two consequences follow from where $a_0$ sits.

The first is that no measurement in the Solar System can test a departure from Newton’s law at $a_0$. The Sun’s pull at Neptune is fifty thousand times larger, and the outer planets and spacecraft have been tracked with a precision that sets tight limits on anything happening at their accelerations — and says nothing about $a_0$. The low-acceleration regime is a regime of galaxies, of the outer parts of star clusters, and of wide binary stars separated by thousands of astronomical units, whose orbital accelerations fall below $a_0$. The wide binaries measured by the Gaia spacecraft have become a test of exactly this, and the published analyses disagree about whether their orbits show any departure from Newton’s law at all.

The second consequence is a coincidence nobody has explained. $a_0$ is close to $cH_0/2\pi$, the acceleration scale set by the expansion rate of the universe, and close to the acceleration associated with the cosmological constant. In a picture in which galaxy dynamics is set by dark matter, the scale in the relation should come out of galaxy formation, and there is no reason for it to match the expansion rate today. In a picture in which gravity itself is modified, the match suggests a connection between the dynamics of galaxies and the cosmology they sit in — which no modified theory has yet derived convincingly. It may be an accident, and the history of physics has plenty of numerical coincidences that meant nothing. But it is the kind of accident that is worth noticing when one side of it is the least understood quantity in cosmology.

Where the relation is not obeyed so simply #

The relation was measured in disc galaxies, and its reach beyond them is a test as much as an extension.

Early-type galaxies measured with lensing and stellar dynamics fall on the same curve at the accelerations where both can be measured, to the extent their stellar masses are known — which depends on a mass function that may not be universal. Weak lensing, which measures the total mass round galaxies to hundreds of kiloparsecs, extends the relation to accelerations a hundred times below where rotation curves end, and the stacked lensing signal of isolated galaxies continues along the low branch further than any rotation curve reaches.

Galaxy clusters do not obey it. The accelerations in a cluster’s outer parts are near $a_0$, and the observed mass — from the hot gas’s temperature, the galaxies’ velocity dispersion and lensing — exceeds what the relation predicts from the cluster’s visible baryons by a factor of about two. A modified-gravity explanation of the galaxy relation therefore still needs unseen mass in clusters, while a dark-matter explanation needs to account for the tightness in galaxies. The relation is simple where it holds and not universal, and the boundary between where it holds and where it does not is itself a measurement neither explanation has fully absorbed.

What the figures leave out #

The relation is drawn as a fitting function. The curve and its band are the published fit to the SPARC sample, not a fit made here; the model galaxies are computed from the mass models used elsewhere in these figures, with their gas taken as a second exponential disc twice the stellar scale length.

The model haloes are cored isothermal spheres, not the cuspy profiles cold dark matter simulations produce without feedback, and their outer speeds are tied to the discs by construction. They show what a halo has to do; they do not show what cold dark matter predicts, which requires a simulation.

The accelerations assume circular orbits in a thin disc. Pressure support in the gas, non-circular motions from bars and warps, and the thickness of the disc all alter both accelerations at the level of the scatter, and they are largest in exactly the dwarfs whose inner curves make the sharpest test.

And the stellar mass-to-light ratio is held fixed. The relation’s tightness depends on it being nearly constant in the near infrared; a variation in the mass function between galaxies would add scatter, and its absence is itself weak evidence that the mass function does not vary much among disc galaxies.

Still open: whether the relation is a law or a regularity #

A law would hold in every system at the relevant accelerations; a regularity would hold where the formation histories that produce it apply, and fail elsewhere. The two differ most clearly in systems whose histories are unusual. Tidal dwarf galaxies, formed from debris pulled out of colliding galaxies, should contain almost no dark matter if dark matter is a halo, because debris flung from a disc does not carry the halo with it; if they nonetheless fall on the relation, the relation is about the baryons and their dynamics, not about haloes. The few measured seem to fall close to it, which is surprising for dark matter and expected in modified gravity, and their inclinations and equilibrium are uncertain enough to leave the question open. Ultra-diffuse galaxies with apparently little dark matter, galaxies with measured external fields, and the wide binaries in the solar neighbourhood are the other places the question is being asked — each of them a system in which a regularity would have a reason to break and a law would not.