The same curve, two galaxies

The rung below measured a rotation curve that does not fall where the light runs out, and concluded that mass keeps arriving at a rate the light does not account for.

That conclusion is safe. What does not follow from it — and is very often assumed to — is that the split between the stars and the dark component is measurable. It is not, and the reason is a degeneracy so exact that two mass models differing by a factor of two in stellar mass produce curves indistinguishable at the precision of any real measurement.

Where the freedom is #

A rotation curve gives one number at each radius: the circular speed, which is a statement about the total enclosed mass through $v^2 = GM(<r)/r$ for a spherical distribution, and something slightly more complicated for a flattened one.

A mass model has at least two components with quite different radial shapes. The disc’s contribution rises, peaks at about 2.2 scale lengths and falls; the halo’s rises monotonically and flattens. Adding them in quadrature is a fit with the disc’s normalisation and the halo’s two or three parameters free.

The photometry supplies the disc’s shape — the surface brightness profile is measured directly and does not depend on the distance — but not its normalisation, because converting light to mass needs a mass-to-light ratio, and that is not an observable of the kinematics.

What the light does not say #

The mass-to-light ratio of a stellar population is computed from a stellar population synthesis model: assume an initial mass function, an age, a metallicity and a star-formation history, evolve the population, and sum the light and the mass.

Every one of those inputs is uncertain and one of them is uncertain in a way that matters enormously. The consequence is that population synthesis returns a stellar mass with an uncertainty of a factor of two, and the factor of two is a systematic rather than a statistical one — it is the same in every galaxy analysed with the same assumptions, so it does not average away.

Colour helps, but only somewhat. Redder populations are older and have higher mass-to-light ratios, and relations between colour and $M/L$ are calibrated and widely used. They tighten the estimate to perhaps 0.1 dex for a given IMF, and they do nothing at all about the IMF itself. Working in the near-infrared helps for a different reason — the light there is dominated by the old low-mass stars that carry the mass, rather than by the few hot young ones that dominate the blue — so the ratio is both larger and less variable, which is why disc decompositions are done at 3.6 microns wherever the imaging exists.

The maximum disc #

There is exactly one thing a rotation curve says about the mass-to-light ratio on its own, and it is an upper limit.

Scale the disc up far enough and its own contribution alone exceeds the measured curve somewhere. Past that point no halo can bring the total back down, because a halo cannot have negative mass. The largest scaling that still fits is the maximum disc solution.

Where a galaxy sits between those bounds is a genuine and long-running dispute. The maximum-disc position holds that the near-coincidence between the inner rotation curve’s shape and the disc’s own photometric shape is unlikely to be an accident, so the disc probably does dominate inside a few scale lengths. The opposing position notes that the coincidence is expected anyway if the halo responds to the baryons during formation, and that dynamical arguments about disc stability limit how heavy a disc can be.

The bound is worth stating carefully because it is often mis-stated. Maximum disc is not a claim that the disc is maximal; it is the largest disc the data permit, and quoting it as the stellar mass is quoting an upper limit as a measurement. The literature convention is to fit at some fraction of it — 0.85 of the peak speed is a common choice, chosen because it is roughly where disc stability arguments put the limit — and that fraction is a modelling decision that propagates silently into every halo parameter downstream.

Why the two components trade so exactly #

The degeneracy is not an accident of the particular halo profile used, and it is worth seeing why it is so nearly perfect.

Over the range where a rotation curve is measured, the disc’s contribution and the halo’s have similar shapes. The disc’s rises to a peak and falls slowly; the halo’s rises and flattens. Neither is anything like a delta function or a step, so their difference over a limited radial range is small and smooth, and adding a little of one while removing a little of the other changes the total by a residual that is smaller than the measurement error almost everywhere.

The technical statement is that the two basis functions are highly correlated over the interval sampled, so the fit’s normal equations are ill-conditioned. What the data determine well is the sum; what they determine badly is the difference, and the ratio of the two eigenvalues is the degree of degeneracy.

Two things make it worse than it needs to be. Rotation curves are measured over a limited range of radius, typically a few disc scale lengths, and it is at large radius that the halo’s shape becomes distinctive. And they are measured with points every kiloparsec or so, correlated by the beam, so the effective number of independent constraints is smaller than the number of points plotted.

Extending the curve outwards is worth more than measuring it better, which is why the most useful rotation curves are the 21-centimetre ones that reach two or three times the optical radius rather than the optical ones that are far more precise.

A number, to fix the range #

For NGC 3198 — a galaxy whose curve has been a test case for four decades — the two bounds are worth writing down.

The maximum-disc solution puts the stellar mass-to-light ratio at about four in solar units in the near-infrared, which for its luminosity is a stellar mass of a few times $10^{10}$ solar masses and a disc that supplies almost the whole rotation speed inside about eight kiloparsecs.

The minimal-disc end is bounded only by plausibility: a mass-to-light ratio of one, or half, produces a fit just as good with a halo that dominates everywhere.

Population synthesis with a Milky Way-like initial mass function puts the value somewhere in the middle, with an uncertainty that comfortably spans both ends.

So the honest statement about NGC 3198 is that its stellar mass is known to a factor of about three, its total mass inside the last measured point is known to ten per cent, and its dark halo’s central density is not known at all. The precision of the two quantities differs by more than an order of magnitude, and only one of them is usually quoted with an error bar.

What breaks it, and what it costs #

Four routes out, none of them free.

The vertical structure of the disc. The stars’ velocity dispersion perpendicular to the plane, together with the disc’s thickness, gives the surface density of the disc directly, through a one-dimensional hydrostatic argument that has no halo in it. This is the cleanest measurement in principle: it weighs the disc alone. In practice it requires resolved stellar kinematics perpendicular to the plane, which is available for the Milky Way and for a handful of nearby edge-on systems and nowhere else. Gas-dominated dwarfs. In a low-surface-brightness dwarf the stars contribute a small fraction of the baryons and the gas contributes most, and gas has a mass-to-light ratio that is essentially known — a 21-centimetre flux is a hydrogen mass, with a small correction for helium. There is then almost nothing to argue about, and the halo is measured. The price is that such galaxies have the noisiest rotation curves and the most severe beam-smearing problems, so the cleanest systems dynamically are the hardest observationally.

The baryonic Tully–Fisher relation. Plot the total baryonic mass of a galaxy against the fourth power of its flat rotation speed and the scatter is remarkably small — smaller, it is claimed, than the scatter the mass-to-light uncertainty ought to produce. Choosing the mass-to-light ratios that minimise the scatter is therefore a way of estimating them, and it gives values in the middle of the allowed range rather than at the maximum-disc end. Strong or weak lensing. For an elliptical galaxy acting as a lens, the total projected mass inside the Einstein radius is measured with no dynamics at all, and subtracting the stellar mass expected from the light constrains the IMF. The same technique run statistically at large radius extends it past where any rotation curve reaches. Results from this route have suggested that massive ellipticals have bottom-heavier mass functions than the Milky Way’s, which if true breaks the assumption of a universal IMF underneath every decomposition ever published.

The core–cusp problem, which is the same degeneracy wearing a hat #

Cosmological simulations of collisionless dark matter produce halo density profiles that rise steeply towards the centre — a cusp, with density going roughly as $1/r$. Many observed rotation curves of dwarf galaxies are better fitted by a flat central core.

That has been treated as a crisis for the dark-matter model, and part of it may be. But the inner rotation curve is exactly where the disc’s contribution is being subtracted, and how much is subtracted depends on the mass-to-light ratio.

Adding to that, the measurement itself is hardest there: a rotation curve’s innermost points are the most affected by beam smearing, by non-circular motions from a bar, and by the assumed centre and inclination.

None of that means the discrepancy is not real. It means the observational claim and the mass-to-light assumption cannot be separated by the curve alone, which is the same statement this whole rung has been making.

The place the argument is cleanest is therefore the place with the least starlight to subtract: the gas-rich dwarfs again, where the stellar contribution is a correction rather than the main term. There the cores are found, and there the mass-to-light freedom cannot hide them. Where the degeneracy is weakest, the discrepancy survives — which is a considerably stronger form of the claim than a survey of all galaxies would give, and it is the version worth arguing about.

The curve is not the circular speed #

Everything above treats the rotation curve as an exact statement about the gravitational field and argues only about how to divide it. There is a prior question, and it bites hardest in the same place: what is measured is a set of line-of-sight velocities, and turning those into a circular speed requires the gas to be on circular orbits.

Two effects break that, and both are worst in the inner regions.

A bar. A substantial fraction of disc galaxies have one, and gas inside a bar does not move on circles: it follows elongated orbits aligned with the bar, with strong streaming along them. A velocity field measured across such a region gives a curve that is too high or too low depending on the angle between the bar and the line of nodes — by tens of kilometres a second, which in the inner few kiloparsecs is a large fraction of the signal.

Pressure support. Gas has random motions as well as ordered ones, and a component supported partly by that pressure orbits slightly slower than the circular speed. The correction — asymmetric drift — is small for cold atomic gas in the outer disc and not small for the warmer, more turbulent gas in the centre.

The consequence is uncomfortable given the previous section. The core-versus-cusp question is decided by the innermost few points of a rotation curve; those points are where the disc subtraction is most uncertain, where beam smearing is worst, and where the assumption of circular motion is least safe. Three independent difficulties concentrate in the same place, and each of them is capable of producing the difference being argued about.

One more construction shows what the same degeneracy looks like measured from inside a galaxy rather than from outside one.

Amount is not shape #

The second unstated assumption is geometric. A rotation curve constrains the mass interior to each radius in the plane, and a spherical halo and a flattened one containing the same interior mass produce nearly the same curve.

So the decomposition returns a halo’s radial profile and says almost nothing about its shape — which is a pity, because shape is the observable that distinguishes theories. A halo of collisionless dark matter should be triaxial; one of self-interacting particles should be rounder; and a modification of gravity produces no halo to have a shape at all.

Measuring it requires something out of the plane. The flaring of the atomic hydrogen layer with radius depends on the vertical restoring force, which depends on the halo’s flattening. A warped outer disc precesses at a rate set by the same quantity. And a tidal stream from a disrupting satellite traces an orbit through the three-dimensional potential, so its track on the sky and its velocities along it constrain the shape directly — which is why the Sagittarius stream has had more attention paid to it than any other structure in the Milky Way’s halo.

Each of those is a harder measurement than a rotation curve and each answers a question the rotation curve cannot ask, and the results so far disagree with one another by more than their errors, which is roughly where the amount-versus-division argument stood thirty years ago.

It is worth being clear that none of this is an argument against the halo’s existence, which the total mass establishes on its own. It is an argument about how much of what is published as a halo profile is a measurement, and the answer is that the outer part is and the inner part is a subtraction with three unresolved difficulties sitting in it.

The distinction is worth keeping because the two claims are argued in the same papers and defended with the same figures, and only one of them is what a rotation curve is competent to establish.

There is a corollary about how such results should be read. A paper reporting a halo profile has fitted a model with a stellar mass-to-light ratio in it, and the profile it reports is conditional on that value; two papers reporting different profiles for the same galaxy are usually reporting different assumptions rather than different data. The useful thing to look for is not the fitted halo but the range of halos the data admit, which good analyses publish as a contour rather than a curve — and which is broad in exactly the direction this essay has been describing.

The degeneracy the essay is about is a statement about a family of fits, so it is worth drawing that family with the scalings spread further apart and reading the same construction for a second galaxy.

Where this ladder goes next #

This rung has established that a rotation curve determines a total and not a division, that the free parameter is the stellar mass-to-light ratio, and what has to come from outside to fix it.

The rung above is the response of the halo to the baryons: adiabatic contraction as gas cools inwards, and the feedback that may undo it. Both change the inner profile, and both mean the halo a decomposition recovers is not the halo a dark-matter-only simulation predicts even if the dark matter is exactly as simulated.

Beside it lies the acceleration scale — the empirical observation that the discrepancy between the observed and baryonic curves sets in below a particular acceleration rather than at a particular radius or density. That regularity is real whatever explains it, and any successful account of galaxy formation has to produce it.

And below it, the habit: a fit with more parameters than the data constrains returns a number for each of them anyway. The output of a decomposition is not a measurement of a halo; it is a measurement of a total, plus an assumption, and quoting the halo without the assumption is quoting half a result.