Three mass models that fit the same curve

A spiral galaxy’s rotation curve is the single most cited piece of evidence for dark matter: the circular speed stays flat far beyond where the light ends, and the visible material cannot hold anything on such an orbit.

The argument is sound at large radii, where there is essentially no light and the entire rotation is dark. It is much weaker in the inner regions, where the stars and the halo both contribute — and that is where the interesting questions about the halo’s shape live.

A rotation curve that refuses to fall is the observation. This essay is about what happens when it is turned into a mass distribution, which requires deciding how much of it the stars supply — and nothing in the curve decides that.

The one number nobody measures #

The disc’s contribution to the rotation is proportional to the square root of its mass, and its mass is its luminosity times a mass-to-light ratio. That ratio is the free parameter, and it is genuinely unknown at the factor level.

Stellar population synthesis predicts it from the galaxy’s colour: a red population is old and has a high ratio, a blue one is young and has a low one. The prediction is good in shape and uncertain in normalisation, because it depends on the initial mass function — the low-mass stars that contribute mass and no light. Changing the assumed function between two common choices changes the ratio by a factor of two, which changes the disc’s rotational contribution by a factor of 1.4, which is the whole width of the degeneracy.

So the free parameter in a rotation-curve decomposition is not a fudge invented for the fit. It is a real physical quantity that another subfield cannot yet supply.

There is a second free parameter, and it is worth naming because it compounds the first. The halo’s density profile is also fitted, with two parameters of its own — a scale radius and a normalisation, or a concentration and a mass. So the decomposition has three free parameters and one curve, and the curve has perhaps ten independent points. The fit is not under-determined in the naive sense; it is that the parameters trade against each other along a direction the data barely constrain, which is the ordinary situation when a model has more freedom than the observation has structure.

It is worth putting a number on how badly the ratio is known. Two widely used initial mass functions differ in their low-mass slope, and the difference changes the stellar mass at fixed light by about a factor of 1.6. A third, favoured by some observations of massive ellipticals, adds another factor. The disc’s rotational contribution goes as the square root of its mass, so those factors become 1.26 and rather more in velocity — and since the halo makes up the remainder in quadrature, a twenty-five per cent change in the disc’s velocity contribution changes the halo’s inner density by a factor of two or three. A parameter known to a factor in one subfield propagates into a factor of several in another.

The maximum-disc convention #

Faced with a one-parameter family of acceptable models, the field adopted a convention: choose the largest disc consistent with the data — the one that supplies as much of the inner rotation as it can without exceeding the observed curve.

That is a defensible choice and it is not a measurement. It is a statement of the form “assume the visible matter does as much as possible”, which is a reasonable default and is not what the data say. The opposite convention — minimum disc — is equally consistent, and it implies a dark halo dominating everywhere including at the centre.

The two conventions give inner dark-matter densities differing by an order of magnitude, and the inner density is precisely the quantity that the cusp-core argument is about.

Be explicit about what the convention costs. Under maximum disc, the halo’s inner density is low and its profile is core-like; under minimum disc, it is high and cusp-like. Those two are the competing predictions of two different pictures of dark matter’s nature — and the observation that is supposed to distinguish them has a free parameter that moves the answer from one to the other. That is not a subtle contamination; it is the whole measurement. Every published statement about inner halo profiles from rotation curves carries an assumption about the mass-to-light ratio, and the assumptions are not always stated in the abstract.

There is an important asymmetry between the two components that is easy to lose. The disc’s shape is known — it follows from the light distribution, which is measured directly and to high precision — and only its amplitude is free. The halo’s shape is not known at all; it is assumed from simulations, and it is the thing under investigation. So the fit has one free amplitude on a known shape and two free parameters on an assumed one, and the freedom is unevenly distributed in a way that flatters the halo.

The two components are not independent #

There is a physical objection to the whole decomposition that is stronger than any of the fitting difficulties, and it is worth putting before the list of remedies.

A rotation-curve fit treats the disc and the halo as two additive components whose parameters may be varied independently. They are not independent. The baryons that formed the disc were once spread through the halo, and as they cooled and sank inward they dragged the dark matter with them — the halo’s orbits contract in response to the deepening potential. So a galaxy with a heavier disc has a denser inner halo than the same galaxy with a lighter one, not the same halo with less added.

That is called adiabatic contraction, and including it changes the direction of the degeneracy rather than its width. Without it, more disc means less halo and the total is preserved. With it, more disc means more halo too, so the fit that reproduces the observed curve requires a less concentrated halo before contraction — and the quantity simulations predict is the profile before contraction. The comparison between an observed decomposition and a simulated halo therefore passes through a correction that is itself uncertain by a factor.

Feedback runs the other way. Gas driven out of the centre by supernovae or by a nucleus removes mass from the inner potential quickly enough that the dark matter orbits do not follow adiabatically, and repeated cycles of that flatten the cusp into a core. So the inner halo profile is not a property of dark matter at all but a record of the baryons’ history — which means the measurement this essay is about is trying to extract a fundamental parameter from a quantity that astrophysics has been modifying.

That is the deepest reason the decomposition is hard, and it is not a fitting problem. Two components that interact cannot be separated by assuming they add, and the observation constrains only the sum.

What breaks it #

Four observations constrain the disc’s mass independently of the rotation, and none of them is easy.

The vertical velocity dispersion. The thickness of a disc at a given vertical velocity dispersion measures its surface density directly, because the vertical structure is set by the balance between the disc’s own gravity and the stars’ vertical motions. Measuring the dispersion requires a face-on galaxy, and measuring the rotation requires an edge-on one, so the technique needs a sample with a range of inclinations and a statistical argument.

Spiral structure. A disc that dominates its own rotation is dynamically responsive and develops strong spiral arms and bars; one embedded in a dominant halo is stabilised. The morphology therefore constrains the disc’s fraction, and the argument runs in the direction of maximal discs for barred galaxies.

Gravitational lensing. For a galaxy acting as a lens, the total mass within the Einstein radius is measured geometrically, with no dynamics. Combining that with the rotation curve constrains the decomposition inside that radius — and the radius is a few kiloparsecs, which is where the degeneracy is.

And the baryonic Tully–Fisher relation. The relation between a galaxy’s total baryonic mass and its flat rotation speed is remarkably tight, and its tightness constrains how much the mass-to-light ratio can vary from galaxy to galaxy. It does not fix the normalisation, but it says the answer is nearly the same everywhere.

There is a fifth constraint that has become important and is worth naming because it comes from an unexpected direction: the dynamics of the disc itself. A disc that supplies most of its own rotation is gravitationally responsive, and its response shows up in the amplitude and pitch angle of its spiral pattern, in whether it has a bar, and in the vertical structure of its outer regions. Each of those is a dynamical measurement of the disc’s surface density that does not use the rotation curve at all. They are individually weak and they point consistently towards discs that are sub-maximal, and their consistency with the vertical-dispersion result is the main reason the field’s default has drifted away from maximum disc over the last two decades — quietly, and without any single decisive measurement.

What was actually measured #

Three of them, and the third has changed the framing.

The vertical dispersion measurements. A survey of face-on discs measuring the vertical velocity dispersion of their stars found surface densities implying sub-maximal discs — the stars supply perhaps sixty per cent of the rotation at their peak rather than the ninety-plus of the maximum-disc convention. That is currently the best direct constraint.

The lensing plus dynamics combination. For the handful of spiral galaxies that act as strong lenses, the combination gives disc fractions consistent with sub-maximal as well, though with large uncertainties.

And the tightness of the scaling relations. The baryonic Tully–Fisher relation has scatter smaller than the range of disc fractions the degeneracy permits, which means the decomposition cannot vary freely from galaxy to galaxy. Whatever the answer is, it is nearly universal — and a universal answer is a much stronger statement than a per-galaxy one, because it points at a mechanism rather than at an accident.

Where the picture stops #

Three of them, and the second is the one that makes the inner regions so hard.

Non-circular motions. A rotation curve assumes circular orbits. A bar, a spiral arm or a warp produces streaming motions of tens of kilometres a second, and the inner regions — where the degeneracy matters — are exactly where bars are. Correcting for them requires a two-dimensional velocity field rather than a single curve.

Beam smearing. A radio observation’s angular resolution is a few arcseconds, which at a typical distance is a kiloparsec, so the inner rotation curve is averaged over the region whose shape is in dispute. Optical spectroscopy has better resolution and worse coverage. The measurement that would settle the argument is a high-resolution two-dimensional velocity field, and it exists for a modest number of galaxies.

The inclination enters as a factor. The observed velocity is the true one times the sine of the inclination, and the inclination is measured from the apparent axis ratio of a disc assumed to be circular and thin. A five-degree error at sixty degrees of inclination is a five per cent error in every velocity and a ten per cent error in every mass — applied to the total, so it moves the disc and the halo together and does not change the decomposition, but it does change the comparison against any external calibration.

And the halo’s shape is assumed. Fitting a halo requires a parameterised profile, and the profile is chosen from simulations — which assume the halo is made of collisionless dark matter. Using the fit to test that assumption is circular unless the parameterisation is general enough to accommodate the alternative, which is not always the case.

A fourth sits alongside them, and it is about what a “rotation curve” is for a dwarf galaxy. The systems where the inner profile question is sharpest are the smallest, because they are the most dark-matter dominated — and they are also the ones with the slowest rotation, where a few kilometres a second of non-circular motion is a large fraction of the signal, and where the gas may not be in a settled disc at all. So the degeneracy is least severe exactly where every other systematic is worst, and the two effects have kept the question open together. A dispersion inflated by unresolved binaries is the same story for the pressure-supported systems.

Why a degeneracy at the centre and not at the edge #

The general shape here is worth naming, because it decides which conclusions from rotation curves are safe.

The evidence is strongest where one component dominates and weakest where two compete. At twenty kiloparsecs there is no light and no ambiguity: the rotation is dark matter, and the conclusion does not depend on any decomposition. At two kiloparsecs the two contributions are comparable and their sum is measured, and no rearrangement of the measurement separates them.

So the two claims usually made from a rotation curve have completely different standing. “There is dark matter” is a statement about the outer curve and is as secure as any in extragalactic astronomy. “The dark matter’s inner density profile is such-and-such” is a statement about the inner curve, and it is a statement about a decomposition that a mass-to-light ratio dominates.

That distinction is worth keeping because the two are often quoted together. The same curve for two galaxies makes the first argument beautifully and says nothing about the second, and it is the second that the last thirty years of the subject have been arguing about.

Say which other measurements share this structure, because the remedy is the same in each. Two parameters that lensing measures as one is a lens’s total mass traded against its concentration; a sheet of mass that changes nothing but the answer is the same lens’s mass traded against a uniform screen. In all three the observation is an integral of the mass distribution and the question is about its shape, and in all three the escape is a second observable with a different weighting — a velocity dispersion, a time delay, a vertical structure — rather than a better measurement of the first.

A last observation about the history, because it explains why the convention persisted. Maximum disc was adopted in the 1980s for a good reason: it is the conservative assumption for the question that was then being asked. If the aim is to establish that dark matter is necessary, then assuming the stars do as much as possible and finding that they still cannot account for the outer rotation is the strongest form of the argument. The convention was chosen to be conservative with respect to one conclusion, and it was then carried forward into a different question — the halo’s inner shape — where it is not conservative at all but simply one end of a range.

That is a general hazard with conventions. A default chosen to be safe for one purpose is not safe for another, and the reason for the choice is usually recorded in a paper twenty years earlier than the analysis using it. The maximum-disc assumption is not a measurement, and its status as an assumption is the single most useful thing to know when reading a decomposition.

Where the ladder goes next #

The natural next rung is the mass-to-light ratio itself: what stellar population synthesis predicts, why the initial mass function is the dominant uncertainty, and how a spectroscopic measurement of the low-mass stars in a distant galaxy is attempted. The rung after it is the inner profile — what the decomposition is being used to measure, and why the answer has been contested for three decades.