A rotation curve that refuses to fall

Every orbit in this collection so far has been drawn around a mass that was, for the purposes of the drawing, a point. A planet around the Sun, a moon around a planet, a star around the centre of a binary: in each case the whole of the mass sits inside the orbit, and the speed of anything going round it falls as the inverse square root of distance.

A galaxy is the first object here whose mass is not a number but a function of radius. And when that function is measured, by the only method that reaches far enough, it does not do what the light says it should.

What the light predicts #

A disc galaxy’s brightness falls off exponentially with radius. That is an observation rather than a model: Ken Freeman established it in 1970 across a large sample, and the exponential is good over several factors of e in surface brightness, with a scale length of a couple of kiloparsecs for a galaxy of this size.

An exponential disc has a finite total mass, and nearly all of it is inside four scale lengths. So outside four scale lengths a test particle should be orbiting what is, to a good approximation, a point — and a sphere, or a sufficiently distant flattened distribution, pulls like a point.

That gives an immediate and sharp prediction: outside the light, $v = \sqrt{GM/r}$, so the speed falls as $r^{-1/2}$. This is not a modelling assumption. It is the same statement as the harmonic law, which is verified to eight digits in the solar system, and which every spacecraft in this collection is navigated by.

What is measured instead #

The measured curve is flat. Not slowly declining, not falling with a shallower exponent than expected — flat, from a few kiloparsecs out to the last radius at which there is any gas left to measure, which in NGC 3198 is about thirty kiloparsecs, more than eleven disc scale lengths.

The consequence is arithmetic and it is stark. If $v$ is constant then

$$M(<r) = \frac{v^2 r}{G} \propto r,$$

so the enclosed mass grows linearly, without limit, for as far as the measurement reaches. That straightness is the finding. A galaxy with more mass than its light suggested would be unremarkable — stellar mass-to-light ratios are uncertain at the tens of per cent level, and there is dust, and there are faint stars. A galaxy in which the enclosed mass rises linearly through the region where the light has already stopped is a different kind of statement, because there is no amount of unaccounted-for starlight that produces it. Doubling the assumed mass-to-light ratio of the disc raises the luminous curve and leaves it just as flat-topped as before.

The observation behind the number #

Everything above depends on being able to measure a rotation speed at radii where there are essentially no stars. That is possible for one reason: neutral hydrogen extends far beyond the stellar disc and it emits at twenty-one centimetres.

The line comes from a hyperfine transition — the electron’s spin flipping relative to the proton’s — with a transition probability so low that a given atom waits some ten million years for it. That would be hopeless for a laboratory and is ideal for a galaxy, because there are enough atoms that the emission is detectable, and because a line that faint is never optically thick, so what a radio telescope receives is proportional to the amount of gas rather than to the surface of a cloud.

For a galaxy that can be resolved, the observation is a velocity per position on the sky, and the rotation curve is built by fitting tilted rings — annuli each with their own inclination and position angle — to the velocity field. Two things have to be got right and neither is free. The inclination enters every velocity as $\sin i$, and it is estimated from the shape of the outer isophotes, which is why a galaxy seen nearly face-on is nearly useless for this. And the beam of a radio telescope is not a point: it averages over a region, which smooths the curve and, in the steeply rising inner part, biases it low.

The classic case is NGC 3198, whose curve van Albada and colleagues worked out in 1985 and which is still the textbook example, because the flat part extends further in units of the disc scale length than almost anywhere else. Vera Rubin and Kent Ford’s optical work through the 1970s had already found flat curves in dozens of spirals; the radio measurements extended them past the light, which is where the argument becomes unanswerable.

Two further things about that measurement deserve to be said, because they are the reason the curve can be trusted at all.

The velocity comes from a Doppler shift, and a shift measures only the component of motion along the line of sight. For a rotating disc that is exactly the useful component — the approaching side is blueshifted, the receding side redshifted — but it means the whole curve is measured in units of $\sin i$ and nothing observed in one galaxy can remove that factor.

And the radius is an angle. Converting it to kiloparsecs needs a distance, so every rotation curve inherits whichever rung of the distance ladder its galaxy sits on. A distance error rescales both axes: radii move in proportion, speeds do not move at all, and so the inferred enclosed mass moves in proportion too. The shape — flat or falling — survives it untouched, which is why the argument here is made from the shape.

What a flat curve does to an orbit #

A curve that is flat rather than Keplerian changes the character of the orbits inside it, and the change is worth seeing because it separates a galaxy from every other system in this collection.

If $v$ is constant, the orbital period at radius $r$ is $2\pi r/v$, so it grows linearly with radius rather than as $r^{3/2}$. At the Sun’s distance from the Galactic centre that period is about 230 million years; at twice the distance it is 460 million rather than the 650 million a Keplerian galaxy would give. The Sun has gone round some twenty times since it formed, and a star at 30 kiloparsecs perhaps six.

More consequentially, the orbits do not close. In a Keplerian potential the radial oscillation frequency and the angular frequency are equal — which is exactly why an orbit is a closed ellipse and why the whole of celestial mechanics has such a convenient starting point. In a flat-rotation-curve potential the radial frequency is $\sqrt{2}$ times the angular one, so a star that is displaced from a circular orbit traces a rosette that never repeats. Every galactic orbit is precessing, all the time, and the amount of the precession is fixed by the shape of the curve.

Why the inner part agrees, and why that matters #

It would be easy to read the discrepancy as a failure of the theory of gravitation at large distances, and that reading has been seriously pursued. But note what the figures show inside a few kiloparsecs: the luminous prediction and the observation agree there, and they agree without any adjustment.

That is a strong constraint on any explanation. Whatever produces the outer speeds contributes almost nothing at the centre, so it cannot be a modification that scales with distance in a simple way and it cannot be extra mass concentrated where the stars are. It has to be a component that is diffuse, extended, and — in the inner regions — subdominant. The Milky Way shows the same shape, measured by an entirely different route, and its curve is worth drawing because the numbers are the ones a reader is most likely to have heard.

The coincidence in the middle of the figure #

There is a feature of the decomposition that ought to be more remarkable than it usually seems, and it has a name.

The disc’s contribution rises, peaks at about two and a quarter scale lengths, and falls. The halo’s rises monotonically. For the total to be flat, the halo’s rise has to compensate the disc’s fall — not roughly, but closely enough that no bump or dip appears in the observed curve at the crossover, and the observed curves are smooth to a few per cent.

That is a coincidence, because the two components have nothing to do with one another. The disc’s scale length is set by how the baryons cooled and settled; the halo’s core radius and asymptotic speed are set by how the dark matter collapsed. There is no reason for the second to know about the first, and yet the sum has no feature where they exchange dominance.

The problem has been called the disc–halo conspiracy since the 1980s, and it is the strongest argument against reading the decomposition too literally. Three responses are on offer and none is decisive.

The first is that the conspiracy is manufactured by the fitting. If the disc’s mass-to-light ratio is a free parameter chosen to make the total smooth, then smoothness is an output of the procedure rather than a property of the galaxy — which is precisely the disc–halo degeneracy stated as a criticism.

The second is that it is physical: the baryons settling into the disc pull the halo inwards, so the halo’s inner profile is shaped by the disc that formed inside it and the two are not independent after all. That is adiabatic contraction, it is calculable, and it produces some of the required correlation.

The third is that it is not a coincidence but a clue — that the acceleration scale at which the discrepancy appears is a constant across galaxies, which is the empirical claim underlying modified dynamics.

It is worth noticing that the three responses are not mutually exclusive and that the first one is testable on its own. If the smoothness were manufactured by the fit, then galaxies whose disc mass-to-light ratio is constrained independently — by stellar population modelling, or by the vertical velocity dispersion of the disc’s own stars — should show a residual bump at the crossover where the free fit produces none. Such measurements exist for a handful of nearby galaxies, and they do not show a bump, which weakens the first explanation without settling between the other two.

A feature that requires an explanation in one framework and is definitional in another is the sharpest kind of evidence available, and this one has not yet been made to decide between them.

What the picture cannot show #

A rotation curve is a statement about mass inside a radius, and nothing else. It cannot distinguish a spherical halo from a thick, flattened one, because the enclosed mass is what enters and the shape is not. Flattening is constrained by other things — the vertical structure of the disc, the shapes of tidal streams — but not by the curve.

It says nothing about what the mass is. Every figure here is consistent with a halo of faint stars, of cold gas, of compact objects, or of a particle that has never been detected. The exclusion of the ordinary possibilities came later and from elsewhere: counts of faint stars, the absence of enough microlensing events towards the Magellanic Clouds, and the amount of ordinary matter allowed by the light-element abundances.

The straight line in the enclosed-mass figure has a beginning and an end. It is drawn where there is gas to measure. Inside a kiloparsec the beam of a radio telescope averages over the steep rise and the curve there is not measured so much as reconstructed; outside the last measured point, the curve is not known to be flat, it is simply not known.

And the decomposition is not unique. The three components in the first figure add in quadrature to the total, and a heavier disc with a lighter halo fits nearly as well as a lighter disc with a heavier one — the disc–halo degeneracy, which is the standing frustration of the subject. What the data fix tightly is the total; how it is divided depends on a mass-to-light ratio for the stars that has to come from stellar population models rather than from the curve.

How long it took to be believed #

The history is instructive because the evidence arrived a long time before the conclusion did.

Jan Oort noticed in 1932 that the vertical motions of stars near the Sun implied more mass in the local disc than the counted stars supplied. Fritz Zwicky applied the virial theorem to the Coma cluster in 1933 and found a mass some two orders of magnitude above the light. Neither result changed anybody’s mind, and the reasons are ordinary: both were single measurements of hard quantities, both had error bars that a determined sceptic could widen, and neither had a shape to it — each was one number, larger than another number.

The rotation curves were different in kind. By the mid-1970s Vera Rubin and Kent Ford had optical curves for dozens of spirals, each one flat, each measured the same way, and each flat over a range of radius. Then the radio measurements pushed past the light and kept being flat. What made the case was not any single galaxy but the repetition of a shape across a population, which is the form of evidence this subject deals in more often than it deals in single crucial measurements — as the census of exoplanets also found.

It is worth being honest that the shape was not universally read as mass. Modified dynamics was proposed in 1983 precisely because a flat curve is what a modified force law would also produce, and it fits individual galaxy curves impressively well. What it does not do as easily is the cluster and the cosmological evidence, which is why the mass reading is the mainstream one — but the rotation curve alone does not settle it, and an essay that claimed otherwise would be overstating a genuinely good figure.

The same three readings for a second galaxy and a different viewing geometry are what turn one flat curve into a statement about galaxies.

The generalisation #

The pattern here recurs whenever a dynamical measurement is compared against a photometric one, and it is worth stating in its general form: a speed measures a mass, a brightness measures a luminosity, and the ratio of the two is a hypothesis.

The same reasoning weighs a binary star, where it is uncontroversial because both members are visible. It weighs a cluster of galaxies, where Fritz Zwicky applied it in 1933 and got an answer four hundred times the visible mass — four decades before the rotation curves, and largely ignored. It weighs a planetary system from a star’s reflex motion, and there the unseen component is a planet and nobody finds it surprising.

What makes the galactic case different is not the method but the shape of the answer. An unseen companion, an unseen planet, an underestimated stellar mass — all of those are corrections. A component whose enclosed mass rises linearly through and beyond the visible object is not a correction to the visible object; it is a larger object that the visible one is sitting inside.

And the twenty-one-centimetre profile at a steeper inclination, since the inclination is the one geometric quantity the whole measurement is divided by.

One more reading shows how much of the decomposition is actually determined by the curve.

The flatness is a measurement and the decomposition is a model, and the two are routinely quoted with the same confidence. Only the first of them is what the observations actually establish.

Where the ladder goes next #

The immediate question is what the ratio of the two curves looks like as a function of radius, and what it implies about how the mass is distributed rather than merely how much of it there is. The next essay takes that up.

Later rungs on this anchor: the disc–halo degeneracy and the maximum-disc hypothesis; the universal shapes fitted to halo profiles and the argument about whether their centres are cusped or cored; the baryonic Tully–Fisher relation and what it says about the coupling between the two components; the rotation curves of dwarf galaxies, where the halo dominates everywhere and the measurement is hardest; the outer curves traced by satellite galaxies and streams rather than by gas; the vertical structure of the disc as a second, independent dynamical constraint; and what modified dynamics gets right, which is more than its critics usually allow and less than a full account requires.