Stars that weigh and do not shine

Every measurement of dark matter in a galaxy is a subtraction. The total mass comes from the dynamics — a rotation curve, a velocity dispersion, the bending of light behind the galaxy — and the mass of the stars is taken away from it, and what is left is the dark matter. The mass of the stars is never weighed directly. It is the light, which is measured, multiplied by a mass-to-light ratio, which is computed.

When the rotation curve of a spiral is split into its components, that multiplier is the one number nobody measures, and a whole family of models fits the same curve because it can be anything from nearly nothing to the maximum the inner curve allows. This essay is about where the multiplier comes from and why it is uncertain by a factor that is not small. The short answer is that most of the mass in an old population of stars is in stars that emit almost none of its light, and the number of those stars was set at their birth by a distribution that was assumed, for decades, to be the same everywhere — and is now thought, in the most massive galaxies, not to be.

Light from the few, mass in the many #

A star’s luminosity rises with its mass far faster than the mass does — roughly as the fourth power near the Sun’s mass — so a population’s light is dominated by its most massive living stars and its mass by its least massive ones. In a young population the most massive living stars are blue and brief. In an old one they have died, and the light comes from the stars just at the main-sequence turnoff, a little below a solar mass, and from the giants they become. The dwarfs below half a solar mass — three stars in every four by number — contribute a few per cent of the light between them.

That split makes the mass-to-light ratio a statement about stars that are not seen. The light a population gives is fixed by the stars near the turnoff and above it; the mass it holds depends on how many stars there are below that, down to the hydrogen-burning limit at 0.08 solar masses and past it into brown dwarfs. Two populations with identical light, identical colours and identical spectra to the first approximation can differ in mass by a factor of two or more, because they differ only in stars that were never contributing to the light at all.

The number of those stars is set by the initial mass function — the distribution of masses stars are born with. Edwin Salpeter measured its upper part in 1955 from the stars of the solar neighbourhood: a power law, the number per unit mass falling as the 2.35 power of mass. Continued down to a tenth of a solar mass, that law puts a large share of the mass in the smallest stars. Later counts of nearby dwarfs showed that the law flattens below about half a solar mass, and the modern forms of the function — Kroupa’s broken power law, Chabrier’s lognormal — have fewer small stars than Salpeter’s by a factor that is about a factor of 1.4 in mass-to-light ratio. The bottom-heavy function drawn here does the opposite: it steepens below a solar mass, to a slope of 3, and loads the low-mass end with far more stars than any count in the Milky Way has found.

Where the function was counted, and where it was not #

The mass function has only ever been measured star by star in the Milky Way and its nearest neighbours, and the places it has been counted are a narrow sample of the conditions stars form in.

In the solar neighbourhood the counting is done volume by volume: every star within twenty or so parsecs is catalogued, its mass estimated from its luminosity, and the present-day distribution corrected for the stars that have died. Below about a solar mass nothing has died, so the counts there are the birth distribution directly, and they are complete to the hydrogen-burning limit only within a few parsecs, where the faintest red dwarfs can be found. In young star clusters — the Orion Nebula Cluster, the Pleiades — the counts reach into the brown dwarfs, because young low-mass objects are still warm from contraction and bright enough to see. In old globular clusters the counts can be made with the Hubble Space Telescope down to the bottom of the main sequence, but the clusters have lost many of their smallest stars to the Galaxy’s tides over ten billion years, so what remains is a mass function eroded from below.

Every one of those counts, corrected, is consistent with a Kroupa-like function flattening below half a solar mass. That consistency was the basis for assuming the function is universal, and it was never a test of universality in the conditions that matter: none of the counted regions resembles the dense, fast, metal-rich starbursts in which the stars of a massive elliptical were formed ten billion years ago. The function was measured where it could be measured and assumed everywhere else.

Below the hydrogen-burning limit the brown dwarfs add numbers but little mass. Even in the most generous counts they hold only a few per cent of a population’s mass, which is why the old hope that faint compact objects could be the dark matter did not survive: the microlensing surveys of the 1990s, watching for the brief magnification of background stars by anything compact in the Milky Way’s halo, found far too few events for faint stars, brown dwarfs or stellar remnants to make up the halo. The low-mass stars matter for the mass-to-light ratio because they are a large share of the stars; they do not matter for dark matter because they are a small share of the halo.

The ratio, and how little its colour says #

That figure says what makes the problem hard. The usual way to estimate a stellar mass-to-light ratio is from a colour. Young populations are blue and light for their mass; old ones are red and heavy; metal-rich ones are redder at a given age. A relation between colour and mass-to-light ratio can be calibrated with population synthesis and applied to any galaxy with a measured colour, and it works well — for a fixed initial mass function. Every such relation is computed for one mass function, and the colour of a population is set by the stars near the turnoff and on the giant branch. The dwarfs that would change the ratio change the colour by a few hundredths of a magnitude at most.

So a colour measures the age and the metals and nothing about the IMF, and every mass-to-light ratio from colours carries the assumed mass function as a hidden multiplier. The rotation-curve decompositions that separate stars from dark matter in spirals, the stellar-mass functions of galaxy surveys, and the stellar masses in the Tully–Fisher relation all use it. A change from Kroupa to Salpeter moves every one of them by the same factor of 1.4; a change to the bottom-heavy function by 2.7.

Weighing the stars of an elliptical #

The mass function can be tested only where the total mass is known independently and the dark matter can be argued away. The centres of massive elliptical galaxies are the best such place: the stars there are densely packed, the dark matter is thought to be a minor share of the mass inside the half-light radius, and the total mass can be measured two ways — from the velocity dispersion of the stars and the shape of the light, by solving for orbits that reproduce both, and from gravitational lensing, when a background galaxy happens to be imaged into an Einstein ring by the elliptical.

Both methods were applied to hundreds of ellipticals in the decade after 2010, and both found the same trend: the stellar mass-to-light ratio needed to account for the dynamics rises with the galaxy’s velocity dispersion, and in the most massive ellipticals it exceeds what a Kroupa population of the right age and metals would give by a factor of two or more.

The figure’s last sentence has an alternative, and it is the reason the dynamical result was not accepted on its own. The dynamics measure the total mass inside a radius; the split into stars and dark matter is part of the model. If the dark matter in the centre of an elliptical is more concentrated than assumed — which it would be if the galaxy’s formation dragged the halo inward — the extra mass the fits assign to the stars could be dark matter instead. A heavier mass function and a contracted halo produce nearly the same dynamics. This is the disc–halo degeneracy of spirals moved into the centre of an elliptical, and it needed a measurement that sees the stars themselves rather than their gravity.

The dwarfs in the spectrum #

The measurement exists, and it is hard for the reason the first figure gives: the stars it seeks give almost none of the light.

Cool dwarfs and cool giants of the same temperature have different spectra, because a dwarf’s atmosphere is denser. Some absorption features are strong in dwarfs and weak in giants: the sodium doublet at 819 nanometres, whose strength depends on the atmospheric pressure, and the Wing–Ford band of iron hydride near 991 nanometres, a molecule that forms only in dense, cool atmospheres. In an old population, where the light at those wavelengths is dominated by giants, the dwarfs add a signal of a few per cent to each feature. The difference between a Kroupa and a bottom-heavy population is a change of that few per cent by a factor of several — a spectral feature measured to a fraction of a per cent of the continuum, in the integrated light of a galaxy.

That was first done in 2010, by van Dokkum and Conroy, on the cores of eight massive ellipticals in the Virgo and Coma clusters, and they found both features stronger than any Milky Way-like population could produce — strong enough for a mass function at least as bottom-heavy as Salpeter’s and possibly steeper. The dynamics and the spectra agreed, from entirely independent directions: one weighed the stars through their gravity, the other counted the small ones through their atmospheres. Neither alone would have been convincing. Together they are the strongest evidence that the initial mass function is not universal.

They also leave room for doubt, and the doubt is instructive. Both spectral features depend on the abundance of their element as well as on the dwarfs: the sodium line is stronger in a sodium-rich population, and massive ellipticals are rich in the lighter elements relative to iron because they formed fast, before the iron from white-dwarf explosions arrived. Untangling abundance from gravity needs many features fitted together with a population-synthesis model detailed enough to carry both, and the answer depends on the model’s own dwarf spectra — which are calibrated on the few nearby cool dwarfs whose spectra can be measured at all.

What a spiral can rule out #

The ellipticals leave a question about the spirals, and the rotation curve of a spiral answers half of it.

The logic is one-sided. A rotation curve limits the stars from above, because stars alone cannot make a galaxy rotate faster than it does. It cannot limit them from below, because a halo can always make up the rest. So a spiral can exclude a mass function heavier than some bound and cannot confirm a light one. The bound depends on where the Kroupa disc sits — at 35 per cent of the maximum-disc mass instead of 45, the bottom-heavy disc would reach 96 per cent of the observed curve and scrape underneath it — and that normalisation is the thing the vertical velocity dispersions of face-on discs were measured to supply. Those measurements, in the DiskMass survey of about thirty spirals, put typical discs well below maximal, with stellar mass-to-light ratios consistent with a Kroupa-like function and not with anything much heavier.

So the mass function appears to vary with the kind of galaxy: Milky Way-like in spirals, where the discs are sub-maximal, and bottom-heavy in the cores of the most massive ellipticals, where the dynamics and the spectra agree. That is exactly the variation a theory of star formation would have to explain, and no theory yet does it convincingly. The candidates involve the conditions in which the ellipticals’ stars formed — gas that was denser, more turbulent, more pressured and possibly warmer from cosmic rays than in any star-forming region in the Milky Way today — which would change the mass at which a collapsing clump of gas fragments into stars.

What changes if the multiplier moves #

It is worth tracing a factor of two in the stellar mass-to-light ratio through the quantities that depend on it, because they are most of what is known about galaxies’ dark matter.

In spirals, the stellar mass sets how much dark matter is needed inside the optical disc. A Salpeter mass function makes most spiral discs close to maximal and the inner dark matter nearly negligible; a Kroupa one leaves the dark matter a large share even inside the disc. The debate about whether dark matter halos have cusps or cores is fought exactly there, and the answer moves with the multiplier: a heavier stellar disc leaves less room for a cusp.

In ellipticals, the stellar mass is most of the central mass whatever the mass function, and the dark-matter fraction inside the half-light radius can go from a third to nearly zero between a Kroupa and a bottom-heavy population. The ratio of a galaxy’s stellar mass to its halo mass — the efficiency with which it turned its share of baryons into stars — is built on stellar masses computed with one mass function applied to every galaxy, and would be tilted if the function changes with mass.

And across cosmic time, the stellar mass density of the universe, the rate of star formation that produced it, and the comparison between them all carry the same multiplier. A census of stellar mass is a census of light times an assumed distribution of stars that cannot be seen, and the distribution is now known to be at least sometimes wrong.

The same multiplier in a star-formation rate #

The initial mass function enters the history of galaxies twice, and the second entry is in the opposite direction from the first.

A star-formation rate is measured from the light of the most massive young stars — their ultraviolet, the hydrogen recombination lines their radiation powers, the infrared from the dust it heats — and each indicator counts only the stars above some mass. Converting that count into a total rate of star formation in solar masses per year requires the mass function below the counted stars, exactly as the mass-to-light ratio does. A bottom-heavy function raises both the stellar mass of an old population and the star-formation rate inferred for it when it was young, by similar factors.

That creates a consistency test that does not depend on dynamics. The stellar mass in a galaxy today should equal the integral of its past star-formation rate, minus what the stars have returned to the gas. Summed over the whole universe, the integrated cosmic star-formation history and the present stellar mass density have been compared many times, and they agree to within the uncertainties of each — with a universal Kroupa or Chabrier function. A function that became bottom-heavy in the galaxies that dominate the stellar mass would move both sides, but not by identical factors, because the ultraviolet indicators and the old-population light weight the low-mass stars differently. The test is currently too coarse to settle anything, and it is one of the few that could be applied across cosmic time rather than only to the nearest ellipticals.

What the figures simplify #

The populations are single bursts with crude giants. Main-sequence luminosities follow the broken power law used throughout these figures, and the giant branch is represented by a single factor on the light; a real population-synthesis model follows every phase of stellar evolution with its own spectra. The absolute ratios drawn — 2.35 for an old Kroupa population — are close to published bolometric values but not computed with the same care; the ratios between mass functions, which are the argument, are much less sensitive to it.

The dynamical trend is representative. The published relations between the mass-to-light excess and velocity dispersion differ in slope and normalisation, and they depend on assumptions about the dark matter’s profile that the curve drawn does not carry.

The spiral is a single model galaxy, with its disc normalisation taken from the average of a survey rather than measured for that galaxy, and the conclusion that it excludes the bottom-heavy function holds only for discs at least that close to maximal.

And remnants are counted with fixed masses. A population’s white dwarfs, neutron stars and black holes hold a significant share of its mass at old ages, and a top-heavy mass function — the opposite departure, argued for in some starbursts — would raise the ratio through remnants rather than dwarfs. The figures here compare only departures at the low-mass end.

Still open: whether a mass function can be read from one galaxy’s inside #

The cores of massive ellipticals are where the evidence for a varying mass function is strongest, and the evidence is integrated light — a single spectrum summing billions of stars. The spatially resolved spectroscopy of the last few years has begun to show that the dwarf-sensitive features weaken from the centre of an elliptical outward, which would mean the mass function varies within a galaxy as well as between galaxies: bottom-heavy in the dense core that formed first and fastest, Milky Way-like in the outskirts assembled later from accreted smaller systems. If that holds, the mass function is set by the local conditions of star formation rather than by the galaxy as a whole, and it can be measured as a gradient. It also means that the running mass-to-light ratio that rises outward through every galaxy — the signature from which dark matter was inferred in the first place — has a stellar component that falls outward underneath it, and the dark matter in the inner parts of ellipticals has been measured against a multiplier that was changing in the other direction.