A galaxy born without a halo

Every disc galaxy whose rotation has been measured carefully obeys one relation. The acceleration measured from its rotation, $v^2/r$, is a fixed function of the acceleration its stars and gas alone would produce at the same radius — the same function for galaxies a thousand times apart in mass, with a scatter about as small as the measurement errors. That relation has a characteristic acceleration in it, about $1.2\times10^{-10}$ metres per second squared, below which the observed acceleration exceeds the baryonic one by more and more.

Two readings of it are possible, and they have been argued for forty years — since the first flat rotation curves made the gap between the mass and the light impossible to ignore. In one, the relation is a regularity: every galaxy formed inside a dark halo, and the processes that assembled the halo and the disc together tied the two so tightly that the dark matter’s contribution can be predicted from the light. In the other, the relation is a law: gravity itself departs from Newton’s at low accelerations, there is no halo, and the “missing mass” is the difference between two laws of motion. Both readings reproduce every ordinary galaxy, because ordinary galaxies are where the relation was measured.

The way to separate a regularity from a law is to find a galaxy whose history should have broken the regularity. A law has no history to be broken; a regularity does. And there is a class of galaxy whose history is unlike any other’s — formed not inside a halo but out of the debris of a collision, from material that started life in someone else’s disc.

Galaxies made from debris #

When two disc galaxies pass close to each other, their tides draw out long tails and bridges of stars and gas, tens of kiloparsecs in length. In some encounters the gas in a tail is dense enough to collapse under its own gravity, and clumps of a few hundred million to a few billion solar masses condense out of it — self-gravitating, rotating, forming stars, and indistinguishable at a glance from ordinary dwarf galaxies. They are tidal dwarf galaxies, and they are seen at the tips and along the lengths of tidal tails around many interacting systems.

What makes them a test is their composition. A tidal tail is made of material that was in the disc of a parent galaxy, and a disc is made of baryons. The parent’s dark halo is not in the disc. It is a spheroidal cloud whose particles move on orbits in every direction at speeds comparable to the parent’s rotation, and a tidal encounter that flings the disc’s cold, coherently moving gas into a tail does not fling the halo’s particles in the same way, because they were not moving coherently to begin with. So a dwarf assembled in a tail is assembled from baryons alone. If dark matter is a halo of particles, a tidal dwarf should have almost none.

How little can be computed. A condensing clump can only bind the particles whose velocity, relative to the clump, is below the clump’s escape speed. The tail’s gas and stars share the clump’s velocity almost exactly, which is why they are bound. The halo’s particles passing through the same volume have velocities spread over hundreds of kilometres a second in every direction, centred on the parent’s rest frame rather than on the tail’s motion.

The calculation is generous to the halo. It assumes every halo particle inside the right velocity window is captured, which over-counts, and it ignores that the clump forms gradually, so that its escape speed is lower while most of the capturing would have to happen. Simulations of interacting galaxies with full dark-matter haloes find the same result in more detail: tidal dwarfs form with dark-matter fractions of a few per cent at most. On the halo reading, a tidal dwarf’s rotation is set by its own gas and stars, with nothing to add.

Two places one dwarf could sit #

The hero figure puts that prediction on the radial acceleration relation. A tidal dwarf of a billion solar masses, mostly gas, in a disc with a scale length of a kiloparsec and a half, has a baryonic acceleration everywhere below the characteristic value — between about a twentieth and a fifth of it. That is the regime where the relation departs furthest from the line of equality.

If the dwarf carries no dark matter, its points sit on the line of equality: the observed acceleration is the baryonic one. If the relation is a law that every galaxy obeys regardless of history, they sit on the curve, and at six kiloparsecs the observed acceleration is 5.6 times the baryonic, which is a rotation speed 2.4 times higher. The two predictions do not overlap anywhere.

A smaller dwarf makes the gap wider, not narrower, because its accelerations sit further below the characteristic value.

The trend is the one the baryonic Tully–Fisher relation expresses for ordinary galaxies: the less massive and more diffuse a disc, the larger the share of its rotation that its visible matter cannot account for. In a dwarf of a few hundred million solar masses formed in its own halo, that share is most of the mass. A tidal dwarf of the same size has the same accelerations and, on the halo reading, none of the halo.

That is what makes the tidal dwarfs a clean case on paper. For an ordinary dwarf galaxy, the halo reading predicts that the dwarf sits on the relation, because that is where all the measured dwarfs are and the halo model is tuned to put them there. For a tidal dwarf, the same halo reading predicts that it sits off the relation, on the line of equality, because the tuning is a product of a formation history the tidal dwarf did not have — the slow assembly in which baryons settling into a halo drag its centre inward and tie the two together. The law reading predicts that it sits on the relation like every other galaxy. A regularity and a law, which agree everywhere else, disagree here by a factor of four in mass.

The rotation curve that would decide it #

In terms of what a telescope measures, the difference is a rotation curve.

The baryonic curve rises to about thirty-three kilometres a second and then falls slowly, the shape of an exponential disc with nothing around it. The law’s curve keeps rising and flattens near seventy. Both are well within what a radio interferometer measuring the twenty-one-centimetre line of the dwarf’s hydrogen can resolve: the gas extends to several kiloparsecs, its rotation is a few tens of kilometres a second, and the beam of a modern array is a kiloparsec or less at the distances of the nearby interacting systems.

The baryonic mass is measured independently: the gas from the strength of the hydrogen line, plus a correction for helium, plus the molecular gas from carbon monoxide; the stars from their light and colours. For a tidal dwarf the stars are a small fraction of the total, so the uncertainty of stellar mass-to-light ratios — the dominant uncertainty for ordinary galaxies, where it makes the disc–halo decomposition degenerate — barely matters. On paper the test is simple: measure the rotation, measure the gas, divide.

The measurement, and what it found #

The best-studied tidal dwarfs are three condensations in a ring of gas around the galaxy NGC 5291, a system in which a high-speed collision some three hundred and sixty million years ago threw a ring of neutral hydrogen a hundred and fifty kiloparsecs across. Three clumps in the ring are massive, gas-rich and rotating.

The first analysis of their rotation, published in 2007, found dynamical masses two to three times their visible baryonic masses. That was a surprise on both readings. For the halo reading it meant tidal dwarfs contain dark matter they should not have been able to capture, and the proposal made at the time was that the extra mass was baryonic after all: cold molecular gas that emits no carbon monoxide and is invisible to the usual tracers, roughly doubling the gas mass. For the law reading the result was welcome: modified-gravity fits to the same three rotation curves, made shortly afterwards, matched them without free parameters beyond the inclination.

A reanalysis in 2015, with new interferometric data at higher resolution, reached a different conclusion: dynamical masses consistent with the baryonic masses, no dark matter required and none excluded at the level the law predicts. The difference between the two analyses lay largely in one number, and it is the number every rotation curve of a small, irregular disc depends on most.

An inclination worth a factor of three #

A rotation curve is measured along the line of sight. A disc tilted at inclination $i$ shows only $v\sin i$ of its rotation, and the true speed is the projected amplitude divided by $\sin i$. The dynamical mass goes as the square of the speed, so it goes as $1/\sin^2 i$.

For a large spiral the inclination is measured from the shape of its outline: a circular disc tilted at $i$ looks like an ellipse with axis ratio $\cos i$, and with a smooth, extended disc that ratio is measured to a few degrees. A tidal dwarf is not a smooth disc. It is young — a few hundred million years old, a handful of rotations — lumpy, star-forming in knots, embedded in the tail it condensed from and often blended with it. Its outline does not reliably trace a circle, and the alternative, fitting the inclination from the velocity field itself as one of the parameters of a tilted rotating disc, is poorly constrained when the disc spans only a few beams and its rotation is barely larger than its turbulent motions.

The figure shows what that does to the test. The projected amplitude chosen for it is one that a law-obeying dwarf would show at 28°, and a halo-free dwarf at 71°. Both inclinations are plausible for a small irregular disc; both are values that have been assigned to tidal dwarfs. The measurement does not fail because the rotation is hard to measure. It fails because the conversion from the rotation measured to the rotation present has an unknown in it as large as the effect.

This is the general weakness of the dwarf-galaxy tests of gravity, but in the tidal dwarfs it is sharper, because the predicted difference has to be read from a single small system rather than averaged over a population. For ordinary galaxies — where the harder problem is which stars a velocity was measured on — inclination errors scatter points around the relation in both directions and wash out; for a test that turns on whether three particular dwarfs sit on the line or on the curve, each one’s inclination is the measurement.

The parent’s pull #

There is a second effect, and it belongs to the law reading. If the radial acceleration relation is a law of gravity of the kind proposed in modified Newtonian dynamics, it is nonlinear: the response of a system to its own mass depends on the total acceleration it experiences, including the acceleration of the whole system in the field of its surroundings. A dwarf falling in the field of a large galaxy is not isolated, and the boost the law gives to its internal dynamics is reduced when the external field is comparable to or larger than its internal one. This is the external field effect, and it has no counterpart in Newtonian gravity, where a uniform external field has no effect on internal motions at all.

The tidal dwarfs are exactly the systems this matters for. They form in the tails of their parents, typically tens of kiloparsecs away, where the parent’s field is a substantial fraction of the characteristic acceleration — much larger than the dwarf’s own field at its outer edge. In a field of a tenth of the characteristic value the law’s predicted speed falls from 68 to 45 kilometres a second; in a field equal to it, to 35, barely above the baryonic 33.

So the law reading makes a prediction that depends on the dwarf’s distance from its parent, and that distance is measured only in projection. A tidal dwarf seen near the line of equality could be a dwarf without dark matter, or a law-obeying dwarf close to its parent; a dwarf seen on the relation could be a law-obeying dwarf far from its parent, or a dwarf whose inclination was misjudged. Each reading has a way to accommodate each answer, and the two ways have different free parameters.

That symmetry is not a stalemate. It says what the test needs: a sample of tidal dwarfs at a range of projected distances from their parents, with inclinations constrained by kinematics at high resolution, so that the external field and the inclination can be separated statistically rather than argued for each object. If the law holds, the inferred ratio should fall systematically with proximity to the parent. If the dwarfs carry no dark matter and gravity is Newtonian, it should be independent of proximity and equal to one.

A regularity, a law, and the systems that separate them #

The tidal dwarfs belong to a small class of systems that are valuable precisely because their histories are abnormal. Ultra-diffuse galaxies that appear to have little or no dark matter, satellite galaxies in strong external fields, wide binary stars in the solar neighbourhood whose orbital accelerations fall below the characteristic value — each is a place where a regularity produced by formation would have a reason to fail and a law would not, or where the law’s nonlinearity makes a prediction Newtonian dynamics does not.

None of them has given a decisive answer, and the reasons rhyme. Each depends on a geometric quantity measured in projection — an inclination, a three-dimensional distance, an orbital orientation — and the quantity enters the test at a power that makes a modest error as large as the effect. That is not a coincidence of bad luck. The systems that are dynamically clean enough to test gravity are, almost by definition, small, faint and rare, and those are the systems whose geometry is hardest to pin down.

The decomposition of an ordinary rotation curve is degenerate because two unknown mass distributions can trade off against each other. The tidal dwarf was supposed to escape that degeneracy by having only one mass distribution, the visible one. It does escape it. What replaces it is a degeneracy between a mass and an angle, which is less fundamental and more tractable — inclinations can be measured better, and populations can average them — but which for now carries the whole of the disagreement.

What the figures leave out #

The dwarf is in equilibrium. A tidal dwarf a few hundred million years old has completed only a few rotations, and may still be collapsing, rotating unevenly, or being stretched by its parent’s tide. Gas that is not on circular orbits does not trace the potential, and the rotation curve can over- or under-state the mass in ways that depend on the dwarf’s state.

The baryons are fully counted. Molecular gas without carbon monoxide, cold enough to be invisible in the usual tracers, was proposed as the extra mass in the first analysis and has never been excluded. It is measured in the Milky Way’s own clouds, where it adds a quarter to a half to the molecular mass; in a young, metal-poor tidal dwarf the proportion could be larger.

The capture is a velocity-window estimate. It sets an upper limit on the halo a clump can bind; simulations that follow the capture in detail find fractions of a few per cent at most, consistent with it.

The external field is uniform. The 1-D approximation used for it is standard but simplified; a full calculation in a specific modified-gravity theory gives somewhat different factors, and in some theories the external field effect is weaker or absent.

The relation and its curve are those fitted to ordinary galaxies. If the relation is a regularity, there is no reason the tidal dwarfs should follow it; if it is a law, the exact interpolating function used here is still a fit rather than a derivation.

Still open: a tidal dwarf with a measured inclination and a known distance #

What would end the argument is a tidal dwarf that is old enough to have settled, extended enough to have a well-measured rotation curve, inclined enough that $\sin i$ is not small but not so edge-on that the disc’s thickness matters, and far enough from its parent that the external field is weak. A few such systems are candidates — older tidal dwarfs found by their unusually high metal content, which they inherited from their parent’s disc rather than building themselves, and which marks them out from primordial dwarfs of the same size.

Their metal content is the strongest evidence that they are tidal: a dwarf of a few hundred million solar masses that formed in its own halo should be metal-poor, and one with the metallicity of a large spiral’s outer disc has borrowed it. If such a dwarf rotates at the speed of its own gas and stars, the radial acceleration relation is a regularity and the tidal dwarfs are its first clean exception. If it rotates at the speed the relation predicts, something other than a captured halo is supplying the extra pull — in a galaxy that had no way to acquire one.