A disc weighed by how far its stars climb

A spiral galaxy’s rotation curve measures the total mass inside each radius and nothing about how that mass is divided. The same curve fits a heavy disc with a light halo and a light disc with a heavy one, and the stellar mass-to-light ratio that trades one against the other is not constrained by anything in the kinematics. The usual escape is to choose a ratio — from stellar population models, from the minimum scatter of a scaling relation, or by making the disc as heavy as the curve allows, the “maximum disc” — and every one of those is an assumption about the stars rather than a measurement of the disc.

There is a measurement, and it uses a direction the rotation curve does not. Stars in a disc move up and down through the midplane as well as round the centre. How high they climb depends on how fast they are moving vertically and on how strongly the disc pulls them back — and the disc’s pull, near the midplane, comes almost entirely from the disc itself. The halo is round and diffuse, and over the few hundred parsecs of a disc’s thickness its gravity barely changes. So the vertical motions weigh the disc alone.

What the plane cannot decide #

It is worth seeing the degeneracy once before escaping it, because its exactness is the reason the escape has to come from somewhere else.

The reason the trade is so exact is that a rotation curve measures one function and a decomposition asks for two. The disc’s contribution has a characteristic shape — rising to a peak at 2.2 scale lengths and then declining — and the halo’s is smooth and rising, so any amount of the first can be swapped for a compensating change in the second as long as the halo is flexible enough. Halo models with two free parameters, a core radius and an asymptotic speed, are always flexible enough. The rotation curve fixes their sum and leaves the split to the observer.

Every route out of the degeneracy brings in information from outside the plane of the disc: the colours of the stars, which constrain their mass-to-light ratio through a population model; the gas, whose mass is known from its emission; lensing, which weighs the total mass along a different direction; or the vertical motions, which weigh the disc alone. Of these, the vertical motions are the only dynamical measurement that sees the disc without the halo, and that is why they were given a dedicated survey.

The weighing has no halo in it #

A disc in vertical equilibrium is a sheet of stars supporting itself against its own gravity with its own random motion. The arithmetic is that of a self-gravitating slab, and it is short. A star at the midplane with vertical speed $\sigma_z$ rises to a height set by the potential of the sheet; the sheet’s thickness is set by how high its stars rise; and its surface density $\Sigma$ fixes the potential. Put together, the three give

$$\sigma_z^2 = c\,\pi G\,\Sigma\,h_z,$$

with $h_z$ the scale height of the disc and $c$ a number between 1.5 and 2 that depends on the shape of the vertical density profile — 1.5 for a profile falling exponentially away from the midplane, 2 for the isothermal sheet, whose profile is a squared hyperbolic secant. Measure $\sigma_z$, know $h_z$, and $\Sigma$ follows.

Two features of this matter for everything after it.

The halo is absent. A round halo contributes to the vertical force near the midplane an amount proportional to its local density times the height, and at the radii where discs are measured the halo’s density is a few per cent of the disc’s midplane density. The equation neglects it, correctly to that accuracy, and so the surface density it returns is the disc’s own.

The answer goes as $\sigma_z^2/h_z$. The surface density is quadratic in the dispersion and inverse in the thickness. The disc’s contribution to the rotation speed goes as the square root of its mass, so its share of the speed goes as $\sigma_z/\sqrt{h_z}$. A dispersion wrong by twenty per cent makes the share wrong by twenty per cent; a thickness wrong by a factor of two makes it wrong by forty.

That is why the first figure has straight lines: the share is simply proportional to the dispersion, with a slope set by the thickness. The task is to measure both.

Thirty discs, seen face on #

The dispersion is measured best where it is largest relative to everything else, which is in a galaxy seen nearly face on. A face-on disc shows its vertical motions directly along the line of sight, uncontaminated by rotation, and an integral-field spectrograph — a bundle of fibres spread across the galaxy’s image — gives a spectrum at each position from which the stars’ velocity spread can be read off the width of their absorption lines. The method was applied to about thirty face-on spirals in a programme designed for exactly this purpose, with the velocity spread measured from the stars’ absorption lines at a few disc scale lengths and the rotation curve from the gas.

The result was clear and, to many, surprising: the discs were not maximal. Their share of the circular speed at 2.2 scale lengths came out between about a half and three-quarters, with a typical value near 0.6, which means that even at the radius where a disc’s own rotation peaks, the dark halo supplies most of the mass. A maximal disc would supply about 85 per cent of the speed and therefore about seventy per cent of the mass there; a disc at 60 per cent of the speed supplies about thirty-six per cent of the mass.

That result removed the most comfortable resolution of the disc–halo degeneracy. If discs were maximal, the inner parts of spiral galaxies would be made mostly of stars, the halo would enter only where the light fades, and the inner profile of the halo — the subject of a long argument about cusps and cores — would be poorly constrained and largely irrelevant. Sub-maximal discs mean the halo matters everywhere, and its inner shape can be read from the curve once the disc has been weighed.

But the measurement depends on two numbers that a face-on spectrum cannot provide, and both have been questioned.

A thickness borrowed from other galaxies #

The disc’s thickness is invisible face on. It has to be assumed, and the assumption is borrowed from galaxies seen edge on, whose thickness can be measured directly from their images. Edge-on spirals of similar type have a ratio of scale length to scale height of about 7.3, with a scatter of about two among galaxies; applied to a face-on galaxy of known scale length, that gives a thickness with a corresponding uncertainty.

The thickness, then, is a real uncertainty but not a decisive one. Its scatter is statistical — it applies to the population rather than to any one galaxy — and the square root halves its effect. A face-on galaxy that happened to be unusually thin would be measured too light by about a fifth; one unusually thick, too heavy by a similar amount. Across a sample of thirty those errors average down. They cannot by themselves turn a population of 60-per-cent discs into 85-per-cent ones, which would need every disc to be about twice as thin as its edge-on counterparts.

There is a subtler version of the same problem: the scale height that belongs in the equation is that of the population whose dispersion is measured. A disc is not one population. Its old stars form a thick, kinematically hot layer; its young stars a thin, cold one; and the edge-on scale heights are measured in light dominated by the old stars at red wavelengths. If the dispersion belongs to a different population from the thickness, the equation is mixing two things.

A dispersion that belongs to the wrong stars #

That turned out to be the larger problem. The dispersion is measured from absorption lines in the integrated light of the disc — all the stars along the line of sight, weighted by their contribution to the light at the wavelength of the lines. In the blue-green region where the strongest stellar absorption lines lie, young and intermediate-age stars contribute a large share of the light, and young stars are kinematically cold: they were born recently from gas moving on nearly circular orbits in a thin layer, and have not yet been heated by encounters with spiral arms and molecular clouds.

So the measured dispersion is a blend. It mixes the hot old disc, which dominates the mass and sets the thickness, with a cold young layer that contributes light but little mass. The blend’s dispersion is lower than the old disc’s, and putting it into an equation whose thickness belongs to the old disc gives a surface density that is too low by the square of the ratio.

Why young stars are cold is itself a dynamical result. Stars are born from gas, and the gas in a disc moves on nearly circular orbits with a vertical velocity spread of ten kilometres a second or less — kept that low by dissipation despite everything that stirs it. A star born from that gas inherits its small dispersion. Over billions of years it is scattered by spiral arms, which pump up its in-plane motions, and by massive molecular clouds, which redirect some of that motion out of the plane. The result, measured star by star in the solar neighbourhood, is an age–velocity relation: the dispersion of a population grows roughly as its age to the third to half power, from about ten kilometres a second for stars a few hundred million years old to forty or more for the oldest thin-disc stars.

So a disc is not one kinematic population but a sequence, and the light at any wavelength samples that sequence with weights set by how bright stars of each age are. Young stars are few in mass and luminous; old stars are numerous and faint. The light-weighted dispersion therefore sits well below the mass-weighted one, and the gap is largest where star formation is most active.

The numbers in the figure are illustrative rather than measured for any particular galaxy, but their size matches what has been argued. Measurements of a face-on spiral using planetary nebulae — the remnants of old, low-mass stars, which trace the old disc’s kinematics and contribute no continuum light of their own — found an old-disc dispersion substantially larger than the integrated-light value, enough to raise the inferred surface density by a factor of about two. With that correction, the disc moved from sub-maximal toward maximal.

The disc is measured right and weighed wrong. The dispersion of the light is a correct measurement of the stars that dominate the light. The error lies in assuming that the stars dominating the light are the stars holding the disc up, and that assumption fails in exactly the galaxies — star-forming spirals — where the measurement is easiest to make.

The same galaxy, weighed with the old stars #

Put the old disc’s dispersion into the first figure instead of the blended one and the answer moves across the line.

The figure shows something the earlier ones did not: the thickness matters more once the dispersion is large. At a dispersion of 24 km/s the three thickness lines sit between 38 and 54 per cent, all comfortably sub-maximal; at 40 km/s they sit between 63 and 91 per cent, straddling the maximal line. The uncertainty in the thickness, which was a minor correction to the original measurement, becomes one of the two numbers that decide the corrected one.

That is a general feature of a measurement whose answer goes as a product. The fractional error in the result is the same whatever the size of the result, but whether that error matters depends on where the result sits relative to the threshold being tested. A sub-maximal disc measured with a twenty per cent uncertainty is sub-maximal; a near-maximal disc measured with the same uncertainty may be either. The correction that moved the dispersion upward therefore did two things at once: it made discs heavier, and it made the question of whether they are maximal much harder to answer for any single galaxy.

One curve, two halos #

The two decompositions are not small variants of each other. With a light disc, the halo rises steeply from the centre and supplies most of the mass everywhere — the configuration simulations of cold dark matter tend to produce, once the baryons have dragged some of the dark matter inward. With a heavy disc, the halo is nearly irrelevant inside two or three scale lengths and must have a large core — a central region of nearly constant density — to leave room for the stars. Which of the two describes spiral galaxies is a statement about how dark matter and baryons share the inner few kiloparsecs, and it rests, for face-on discs, on whose dispersion was measured.

The Milky Way is the one disc whose vertical structure can be measured star by star, and its weighing comes out near the heavy end. The vertical motions of stars in the solar neighbourhood, combined with their measured density falling away from the midplane, give a local disc surface density that, scaled to the whole disc, puts the Galaxy’s disc close to maximal — supplying most of the circular speed near the Sun’s radius. One galaxy settles nothing about thirty, but it is the one galaxy where the population problem is avoided, because the old stars can be selected individually by their colours and ages.

The same equation in other directions #

The vertical equation is a special case of a broader principle: random motion holds a stellar system up, and the speed of the motion measures the gravity it opposes. The rotation curve applies that principle in the plane, to ordered motion; the vertical dispersion applies it out of the plane, to disordered motion. That they give independent mass estimates is because they sample the gravity in different directions, and the disc and halo contribute to the two directions in different proportions — the flattened disc dominating the vertical pull near the midplane, the round halo contributing to the radial pull everywhere.

The same split appears in the measurement of elliptical galaxies, where there is no disc and no rotation curve, and where the random motions in three dimensions must be disentangled from a projection that hides two of them. There, orbits nobody resolved inflate the measured dispersion — the same kind of error as here, a dispersion that belongs to a different set of orbits from the one the equation assumes. In both cases the measurement is of a moment of the velocity distribution, the equation needs a different moment, and the conversion between the two depends on a population model.

What the figures assume #

The disc is a single exponential, with the same scale length in mass as in light and a constant scale height at all radii. Real discs have breaks in their profiles, their scale heights increase outward in many galaxies, and their mass-to-light ratio changes with radius as the stellar population changes. A flaring disc measured at one radius and extrapolated to 2.2 scale lengths would be weighed incorrectly in a direction that depends on the flaring.

The halo’s vertical pull is neglected. At the radii used it contributes a few per cent of the vertical force; in low-surface-brightness discs, where the halo dominates even locally, the neglect fails and the equation over-weighs the disc.

The two-population model is a blend of two Gaussians. A real disc has a continuous distribution of ages and dispersions, and the light-weighted dispersion depends on the wavelength of the lines used, the star-formation history and the dust. The bias drawn here is of the right size for the published comparison, not a calculation for any one galaxy.

The measured curve in the last figure is constructed from the heavier decomposition — a disc plus a cored halo — so that the heavier disc is maximal by design. The lighter decomposition is computed from the same curve.

Why it matters which disc #

The difference between a disc at 60 per cent of the speed and one at 85 per cent is a factor of two in the disc’s mass and, through it, in the stellar mass-to-light ratio. That ratio is the same number stellar population models predict from the light, and the prediction depends on the mass function at the low-mass end, where most of the mass and very little of the light reside. A maximal disc needs a mass-to-light ratio near the top of what population models allow; a sub-maximal one, near the middle.

So the vertical weighing is also a test of the stellar mass function — the same question the relation between baryonic mass and rotation speed addresses from the other side. If the old-disc dispersions measured with planetary nebulae generalise, spiral discs are close to maximal and their mass-to-light ratios are high, which would mean either more low-mass stars than the solar neighbourhood’s mass function implies or more stellar remnants. If the integrated-light results stand, discs are light, their mass-to-light ratios moderate, and dark matter dominates inside even the brightest parts of spiral galaxies.

Still open: the old disc’s dispersion, galaxy by galaxy #

The decisive measurement is the dispersion of the old disc alone, in many face-on galaxies, with tracers that belong to it. Planetary nebulae do; so do the absorption lines of old stars measured in the near-infrared, where young stars contribute less light. Both are being applied, and the early results suggest the integrated-light dispersions were biased low, by amounts that move discs a long way toward maximal.

If that holds, the question returns to the halo. A maximal disc leaves the inner halo with a large core in nearly every spiral, which is what the dwarf galaxies’ rotation curves also suggest for a different reason, and which the simplest cold dark matter haloes do not produce. If it does not hold, the inner halo in spirals is steep, the discs are light, and the dwarfs’ cores need their own explanation. In either case the answer will come from the direction the rotation curve never looked — straight up out of the disc — and from knowing which stars were moving.