The mass that is not the light

The quantity that carries the argument about galaxies is a ratio of two things that are measured in completely different ways: a mass, obtained from a speed, and a luminosity, obtained from a brightness and a distance. Divide one by the other and the answer is in solar units — how many solar masses there are for each solar luminosity — which is a convenient unit because the Sun is, by definition, one.

For a population of stars that ratio is not free to be anything. That is the constraint the whole subject hangs on.

What a population of stars is allowed to weigh #

The mass–luminosity relation is steep: luminosity goes roughly as the fourth power of mass along the main sequence. A star of half the Sun’s mass is about a twentieth of its luminosity; a star of a tenth is around a thousandth. So the ratio for a stellar population depends almost entirely on how many low-mass stars there are per bright one. That distribution — the initial mass function — is measured directly in star clusters and in the solar neighbourhood, and it is steep enough that most of the number is at low mass while most of the light is at high mass.

Adding up such a population gives a mass-to-light ratio of roughly one to two in the V band for a young, actively star-forming disc, and three to six for an old population with no hot stars left. The uncertainties are real, mostly in the number of the faintest dwarfs and in how much of the mass is in stellar remnants, and a determined argument can push an old population towards eight or so. It cannot push it to fifty.

That is the ceiling. And the dynamical ratio for a galaxy measured out to its last gas is fifteen to thirty; for a cluster of galaxies it is two to four hundred.

The difference between dark and missing #

Two words get used for this and they mean different things.

Dark means “emits too little to detect”. A brown dwarf is dark. A cold cloud of molecular hydrogen is dark. A stellar remnant is dark. All of them are made of the same protons and neutrons as everything else, and their darkness is a statement about temperature or about how little of them there is per unit of mass.

Missing would mean the accounting is wrong — that the light has been underestimated, or the distance, or the inclination.

The rotation curve rules out missing by its shape, as the previous rung argued: an accounting error rescales a curve and does not change a falling curve into a flat one. So the question is which sort of dark thing it is, and this is where the argument stops being about galaxies and starts being about census-taking.

Ruling out the ordinary candidates #

The candidates that are made of ordinary matter were taken seriously and were tested, and the tests are among the most satisfying pieces of observational reasoning in the subject because each one had a real chance of finding the mass.

Faint stars. Deep counts of the faintest red dwarfs in the solar neighbourhood, and later in the halo, simply do not find enough of them. The luminosity function turns over; there is no invisible reservoir of hydrogen-burning stars, and adding one would also add light in the near infrared, where it is not seen.

Compact objects — old white dwarfs, neutron stars, stellar-mass black holes. These are genuinely dark and genuinely made of ordinary matter, and if the halo were full of them a star in the Large Magellanic Cloud would occasionally be magnified as one drifted across the line of sight. That experiment was done.

Cold gas. Molecular hydrogen emits nothing at the temperatures in question, and it was for a while a respectable proposal. It fails on a different ground: a disc of gas that heavy would be gravitationally unstable and would form stars, which would then be visible.

And ordinary matter in total. The sharpest limit is not astronomical at all. The abundances of deuterium, helium and lithium produced in the first minutes of the universe depend on the density of protons and neutrons at that time, and they fix it. The number that comes out is about a sixth of the total mass density that dynamics requires. That single argument closes the whole class at once, which is why the modern candidates are not made of baryons.

The observation behind the number #

The dynamical half of the ratio is a rotation curve, and the previous essay set out what that costs. The photometric half deserves the same treatment, because a luminosity is not observed either.

What is measured is a flux in a filter. To turn that into a luminosity requires a distance — squared, so a ten per cent distance error is a twenty per cent luminosity error — and a correction for the dust in the way, both in the Milky Way and in the target galaxy, which for an inclined spiral is substantial and is estimated from the inclination itself. It requires a correction from the filter’s band to the band the model predicts. And to compare galaxies at different redshifts it requires a further correction for the shifting of the spectrum through the filter.

None of these is large enough to matter for the argument here, and that is the point worth making: the gap being explained is a factor of ten or more, and the systematic uncertainties in a luminosity are tens of per cent. This is a rare and comfortable situation in astronomy — the effect is far larger than the error budget of the hard part of the measurement.

A ratio that is also a clock #

There is a second reason the stellar mass-to-light ratio is bounded, and it is one that makes the quantity more interesting than a bookkeeping constant: it changes with time, in a direction that is known.

A population of stars formed in one burst begins bright, because the massive stars dominate the light. Those stars are also the ones that die first, and by an enormous margin: lifetime falls roughly as the inverse cube of mass, so a twenty-solar-mass star is gone in ten million years while a solar-mass star lasts ten billion. The light of the population therefore fades rapidly while its mass barely changes, and the ratio climbs. So a mass-to-light ratio of one to two says “this population is forming stars now” and a ratio of five says “this population stopped long ago”. Which means the same measurement that fails to account for a galaxy’s dynamical mass succeeds beautifully at dating its stars — and the two uses pull in opposite directions, because the age uncertainty is exactly what makes the stellar ratio uncertain.

The colour that stands in for the model #

Summing a stellar population requires a spectrum, an age and a composition, and a survey holds a hundred million galaxies with none of those. What it holds is two or three brightnesses in different filters, which is a colour — and a colour turns out to be enough.

The reason is the clock of the previous section, seen from the photometric side. A population’s colour and its mass-to-light ratio are driven by the same thing: which stars are still alive. Losing the massive stars removes the blue light and the luminosity together, so a population reddens and grows heavier per unit light in step, and the two quantities move along a common track.

Empirically the relation is close to linear in the logarithm — $\log(M/L) \approx a + b\,(B-V)$, with $b$ near 1.7 in the optical and considerably shallower in the near infrared — and it holds across a wide range of star-formation histories rather than only for single bursts. That is the useful part: a galaxy is not one population, and the relation survives being applied to a mixture.

One accident makes it better than it deserves to be. Dust reddens a galaxy and dims it, and it does so in almost the same proportion as ageing does, so a dusty young population sits close to where an old dust-free one sits and the estimated mass is nearly right for the wrong reason. The vector of the error runs roughly along the relation rather than across it.

What does not cancel is the initial mass function, because adding faint dwarfs to a population changes its mass and changes its colour hardly at all. The relation’s zero point $a$ therefore carries the same factor-of-two assumption as everything else in this essay, applied uniformly to every galaxy that uses it — a systematic that shifts an entire survey’s stellar masses together rather than scattering them.

The candidate that came back #

Compact objects were closed off in the 1990s by microlensing, and the case was reopened in 2015 by an entirely unrelated observation.

The gravitational-wave detectors’ first binary merger involved two black holes of about thirty solar masses each, which is heavier than the stellar-evolution channels of the time comfortably produced, and the rate implied a substantial population. That revived a hypothesis older than the microlensing surveys: black holes formed not from stars but from density fluctuations in the very early universe, before any nucleosynthesis — so not made of baryons, and therefore not subject to the deuterium argument at all.

Such objects are the one dark-matter candidate that requires no new particle, which is why the idea keeps returning. The constraints against it now come from several directions that share nothing. Microlensing surveys still exclude the mass range from about $10^{-7}$ to 10 solar masses as the dominant component. Accretion onto such holes in the early universe would have injected energy into the gas and altered the cosmic microwave background’s small-scale structure, which is not seen. And a halo full of them would disrupt wide binary stars and the star clusters in dwarf galaxies over a Hubble time, which have survived.

Between them these leave only narrow windows open, none of them at the mass the detectors found. The episode is worth keeping for what it shows about the shape of the argument: the candidate list is closed not by one decisive experiment but by an accumulation of unrelated exclusions, and a proposal that evades any single one of them usually falls to another that was designed for something else entirely.

Where the picture stops #

The stellar mass-to-light ratio is a model, and it is the weakest link. Everything above compares a measured dynamical ratio against a predicted stellar one, and that prediction comes from summing a population with an assumed initial mass function. Change the assumed function at the low-mass end and the stellar ratio moves by a factor of two. That is not enough to close a factor of ten, but it is enough to make the decomposition of a rotation curve into disc and halo genuinely uncertain — the disc–halo degeneracy again, arriving from the photometric side this time.

Near-infrared light is a better mass tracer than optical light, and it is still not a mass. The near-infrared is dominated by old, low-mass stars, so it tracks the stellar mass with less sensitivity to a recent burst of star formation. That is why the tightest form of the Tully–Fisher relation is measured there. It reduces the scatter; it does not remove the assumption.

And the running ratio in the first figure is a ratio of enclosed quantities. It rises partly because the numerator keeps growing and partly because the denominator has stopped. A local mass-to-light ratio — mass per unit light in a shell rather than inside a sphere — rises far more steeply still, and is the more physical quantity but the less measurable one, because differentiating a noisy enclosed mass is a bad idea.

Two systems where the ratio is extreme #

The argument sharpens at the two ends of the galaxy population, and both ends were surprises.

Dwarf spheroidals. The faint satellites of the Milky Way — objects with a few hundred thousand solar luminosities, containing perhaps a thousandth of the stars of a normal galaxy — have velocity dispersions of ten kilometres per second where their stellar mass predicts one or two. Their mass-to-light ratios run from tens to over a thousand. They are the most dark-matter-dominated objects known, and they are the cleanest test of the halo profile, because there is almost no baryonic complication to model.

Clusters of galaxies. Zwicky’s original 1933 measurement of Coma gave a ratio of several hundred, which he reported and which was largely set aside for forty years. It is now the least controversial mass measurement in the subject, because three independent methods agree on it and only one of them depends on anything being in equilibrium.

Between those extremes sits the ordinary spiral, whose ratio is the mildest of the three and which is nevertheless where the case was made, because a rotation curve gives a profile and a dispersion gives a number.

The Milky Way’s own accounting #

The Galaxy allows a measurement that no external galaxy does: the mass density in the plane, here, locally, from the vertical motions of the stars around the Sun.

The logic is Oort’s and it is entirely local. Stars oscillate vertically through the disc; the amplitude of that oscillation depends on the vertical gravitational force, which depends on the mass density in the plane. Measure the vertical velocities and the vertical distribution of a well-defined set of stars and the density follows, without any reference to a rotation curve, a distance ladder or an inclination.

The answer is about 0.1 solar masses per cubic parsec in the disc, of which the counted stars and gas supply nearly all. Locally, the dark component contributes only about a hundredth of a solar mass per cubic parsec — a few per cent of the total — because the halo is round and the disc is thin, so most of the halo’s mass at the Sun’s radius is far above and below the plane.

That number is the reason the effect is invisible in the solar neighbourhood and unmistakable at thirty kiloparsecs, and it is also the number every direct-detection experiment on Earth is designed around: the flux of halo particles through a laboratory is set by it. It is a striking place for the argument to end up — a quantity read off the sky becoming the normalisation of an experiment in a mine.

The generalisation #

The pattern is worth naming, because it recurs whenever two measurements of the same object have different sensitivities.

Every quantity in astronomy is inferred from light, with one exception: mass is inferred from motion. So whenever the two disagree, the disagreement is informative in a way that a disagreement between two photometric measurements would not be — they are not two estimates of the same thing but two different physical channels, and only one of them cares whether a thing shines.

The reflex velocity of a star finds a planet by exactly this logic and nobody calls the planet dark. A spectroscopic binary’s unseen companion is found the same way. The galactic case is the same measurement carried to a scale at which the unseen component is not a companion to the visible one but the structure the visible one lives in — and the reason it took forty years to accept is that this is a large thing to conclude from a curve.

Where the ladder goes next #

The next rung is the halo’s shape rather than its mass: what profile the dynamics actually constrain, how a cusp differs from a core in a rotation curve, and why the dwarf galaxies are where that argument is fought.

Later rungs on this anchor: the baryonic Tully–Fisher relation and the coupling it implies; the acceleration scale at which the discrepancy sets in, which is a genuine empirical regularity whatever explains it; weak lensing as a mass profile beyond where any gas is left; the bullet cluster and what a collision separates; the Milky Way’s own halo mass, from satellites and streams; and the laboratory searches, which are the only rung on this ladder that could end it.