The metals a galaxy keeps measure what it threw away

The solar neighbourhood’s stars say that a galaxy is not a closed box: gas keeps arriving, and small galaxies lose enriched gas they made. Those were two corrections to one model, inferred from one galaxy and a handful of binned systems. The survey that turned them into a law measured the oxygen abundance of the gas in fifty thousand star-forming galaxies and found that it rises with stellar mass along a single curve — steeply below about a billion solar masses, flattening above ten billion — with a scatter of only a tenth of a dex.

A relation that tight between a galaxy’s mass and the composition of its gas needs a reason that does not depend on each galaxy’s individual history, and there is one. A galaxy that is continuously fed with gas, continuously forming stars and continuously driving some gas out does not accumulate metals indefinitely. It settles into an equilibrium in which the metals it makes are balanced by the metals it loses and the pristine gas it takes in, and the equilibrium abundance depends on the ratios of those flows rather than on the galaxy’s past.

That turns the relation from a correlation into a measurement. Every point on it is a statement about how much gas a galaxy of that mass throws away for each star it forms.

The regulator #

Take a galaxy with a reservoir of gas. Pristine gas flows in; stars form from the reservoir at a rate set by its depletion time; a fraction of each generation’s mass — about forty per cent for a standard mix of stars — is returned promptly by massive stars, carrying new metals with it; and a wind removes gas at some multiple λ of the rate at which mass is locked into long-lived stars. The wind carries the reservoir’s own metallicity out.

If the reservoir is in a steady state, its metallicity satisfies a balance: metals produced equal metals locked into stars plus metals blown out plus the dilution needed to keep the reservoir’s abundance steady while fresh gas pours in. The result is

$$Z_{\rm eq} = \frac{y}{1 + \lambda + \tau_{\rm dep}\,{\rm sSFR}/(1-R)}$$

where $y$ is the yield, $R$ the returned fraction, $\tau_{\rm dep}$ the gas depletion time and sSFR the specific star-formation rate — the rate at which the galaxy’s stellar mass is growing, as a fraction of itself.

Each of the three terms in the denominator has a physical meaning. The one is the metals locked into stars. The λ is the metals thrown away. The last term is dilution: a galaxy growing quickly must be taking in fresh gas quickly, and fresh gas is metal-poor. For galaxies today that dilution term is small, about 0.18 on the representation used here, so at the present epoch the relation is almost entirely a statement about λ.

The model is an idealisation and it is worth saying which way it errs. It assumes the inflow is pristine; gas falling back after an earlier wind is not, and raises the equilibrium. It assumes the wind carries away gas of the same abundance as the reservoir; winds driven by supernovae are thought to be more enriched than the gas they leave, which makes each unit of λ more effective. And it assumes equilibrium, which a galaxy reaches on roughly its gas depletion time — a few billion years today, a few hundred million at early times. None of those changes the shape of the argument; each changes how λ should be read.

Why the slope is a wind #

Only one quantity in the denominator varies strongly with a galaxy’s mass, and it is the one that is not a property of stars. A wind is gas lifted out of a potential well, and the depth of the well goes as the square of the circular speed. How the mass loading depends on speed is a question about what is doing the lifting.

If supernovae drive the wind with their energy, a fixed energy per unit of star formation can lift a mass that goes as the inverse square of the escape speed: $\lambda \propto v^{-2}$. If they drive it with their momentum — because most of the energy is radiated away in dense gas before it can do work — the mass goes as the inverse first power: $\lambda \propto v^{-1}$. For disc galaxies, stellar mass rises roughly as the cube of the circular speed, the Tully–Fisher relation read backwards, so the two scalings are λ ∝ M^(−2/3) and λ ∝ M^(−1/3).

Far below the turnover, where λ dominates the denominator, the equilibrium abundance goes as 1/λ, and the relation’s slope is two-thirds or one-third. Near the turnover the one and the dilution term take over and the curve saturates at the yield. The turnover mass is where λ is about one — where a galaxy throws away about as much gas as it keeps in stars — and it is the single fitted number in the model curves.

The measured slope at a few times 10⁷ solar masses, from electron-temperature abundances, is 0.57, beside 0.59 for the energy-driven wind and 0.24 for the momentum-driven one at that mass. On this calibration the data prefer winds driven by supernova energy.

The calibration that decides the physics #

That conclusion depends on a step the figure makes explicit: how an abundance is read from a spectrum.

The most direct method uses a faint emission line of doubly ionised oxygen at 436.3 nm whose strength relative to the bright lines depends steeply on the electron temperature; with the temperature known, the bright lines give the abundance with few assumptions. The faint line is too weak to measure in most individual galaxies, so the direct relation drawn here was built by stacking thousands of spectra in bins of mass. The alternative is to calibrate ratios of the bright lines against photoionisation models or against the direct method in a smaller sample, and apply those “strong-line” calibrations to every galaxy.

The calibrations disagree. Applied to the same spectra, different strong-line methods give abundances that differ by up to 0.7 dex at a given mass, and they differ in shape as well as in level.

Read through the strong-line calibration, the slope near 10⁹ solar masses is 0.40 — between the two wind scalings and closer to momentum. Read through the direct method, it is closer to energy. The question “how are galactic winds driven?” therefore has two answers from one data set, and which one is right depends on atomic physics in the ionised gas: on whether the temperature measured from the faint line is representative of the whole nebula, which it is not if the gas has temperature fluctuations, and on whether the photoionisation models behind the strong-line calibrations have the right structure. Neither possibility is exotic. Temperature fluctuations bias direct abundances low by an amount that is still argued about, and it is the largest single uncertainty in the absolute oxygen abundance of any galaxy but the Milky Way.

What the wind would have to be #

A loading of four is a large number and it can be checked, with difficulty. Outflows from star-forming galaxies are seen directly: absorption lines of cool gas blueshifted by hundreds of kilometres a second against the galaxy’s own starlight, and X-ray emission from hot gas above the discs of nearby starbursts. Converting what is seen into a mass flux needs a geometry, a column density and an ionisation correction, and each is uncertain by factors of a few. The measured loadings for the cool phase range from well below one to several, broadly larger in smaller galaxies, and the hot phase — which carries most of the metals — is almost unobservable outside the nearest starbursts.

The ejected metals should still be somewhere. A census of where the metals ever produced by galaxies near a Milky Way mass now reside found only about a quarter of them in the stars, gas and dust of the galaxies themselves, and a comparable or larger amount in the diffuse, ionised gas of the surrounding halo, detected by ultraviolet absorption against background quasars — with a substantial remainder still unaccounted for. The regulator’s λ is the flow into that halo, and the halo is where the accounting has not closed.

The same galaxies, earlier #

The relation is not fixed in time. At redshift two, galaxies of a given mass have oxygen abundances about a third of a dex lower than galaxies of the same mass today — and the regulator explains that without changing anything about stars or winds.

The dilution term is the ratio of how fast a galaxy is growing to how fast it can process its gas. Galaxies at redshift two were growing, in proportion, twenty times faster than today’s, and forming stars from gas-rich discs whose depletion times were shorter — but not twenty times shorter. Their reservoirs were therefore dominated by recently arrived, unprocessed gas. That is not a statement about a different kind of galaxy; it is the same equilibrium at a different ratio of inflow to consumption.

The distinction is testable because the two explanations predict different shapes. A change in the winds would change the slope below the turnover and leave the massive end alone. A change in dilution lowers everything, and lowers the massive end most in relative terms, because that is where the wind term is smallest and dilution is the only thing left.

Measurements of the relation at redshift two and beyond, from near-infrared spectroscopy of the same rest-frame optical lines, find an offset of roughly that size and a turnover that moves to higher mass. The newest measurements at redshifts above three, from infrared spectra that reach the faint temperature-sensitive line directly in some galaxies, suggest further decline but rest on small samples, and every one of them inherits the calibration problem above, now compounded by the fact that high-redshift nebulae have harder radiation fields and higher pressures than those the calibrations were built on.

The bottom of the relation is a laboratory #

The low-mass end of the relation matters well beyond galaxies, because it is where the gas is nearest to what came out of the Big Bang. A dwarf galaxy of a few million solar masses has an oxygen abundance a thirtieth of the Sun’s or less; it has thrown away most of what it made, and what it kept is diluted by pristine gas. Its helium is therefore almost all primordial.

That is exactly how the primordial helium abundance is measured: helium lines in the ionised gas of the most metal-poor dwarf galaxies, plotted against their oxygen, extrapolated to zero oxygen. The extrapolation is short only because winds have kept these galaxies poor. A universe in which small galaxies held on to their metals would have no star-forming gas close enough to primordial for the measurement to be made, and the helium abundance that tests the number of neutrino species would be inferred through a long and model-dependent correction. The mass loading of four at 10⁸ solar masses is, in that sense, a service to cosmology.

The relation continues in the stars of galaxies too small to have any gas left. The dwarf spheroidal satellites of the Milky Way and Andromeda, down to a few thousand solar masses of stars, have mean stellar iron abundances that rise with stellar mass as roughly the three-tenths power over six orders of magnitude, with small scatter. Those galaxies stopped forming stars billions of years ago, so their relation is a fossil of the same regulation, recorded when they were last active — and it extends the evidence for mass-dependent winds far below any galaxy whose gas can be observed.

The top of the relation, where the regulation stops #

At the massive end the regulator saturates for a different reason from the one the model gives. The model says a massive galaxy stops losing metals because its potential is too deep for supernovae to empty. That is true, and it is not what ends star formation in the most massive galaxies. Above a few times 10¹⁰ solar masses most galaxies are not forming stars at all: the population splits into two colours with almost nothing between, and the red galaxies have had their gas heated or removed by something that is not supernovae — most likely the energy from accretion onto the black hole at the centre, whose mass tracks the galaxy’s own.

Those galaxies are not on the relation drawn here, which is a relation for star-forming galaxies only. Their stars carry the abundance of the gas they formed from before star formation stopped, and the stellar abundances of massive quiescent galaxies are high and nearly flat with mass. The saturation in the figure is the yield showing through once winds fail; the absence of galaxies above it is a second kind of regulation entirely, and a chemical record cannot easily tell the two apart.

A scatter that is also a signal #

The relation’s scatter of a tenth of a dex is small, and it is not random. At fixed stellar mass, galaxies forming stars faster than average tend to be slightly more metal-poor. The regulator predicts exactly that: a higher specific star-formation rate means a larger dilution term, because a galaxy forming stars faster than its neighbours is one that has recently received more gas. The correlation is strongest in the low-mass galaxies where inflow is most variable, and it is part of the evidence that the relation is an equilibrium being perturbed rather than a fixed property of mass.

That the scatter correlates with star formation is also a warning. A sample selected on star formation — as every high-redshift spectroscopic sample effectively is, because only galaxies with strong emission lines are measured — picks out the metal-poor side of the relation at every mass, and part of any measured evolution is that selection.

What the picture leaves out #

The regulator is a one-zone model. A real galaxy has an abundance gradient, so the metallicity measured in a fibre or a slit depends on how much of the galaxy the aperture covers, and a survey at fixed angular size samples different fractions of galaxies at different distances. The yield itself is uncertain by a factor of about two, because it depends on the upper end of the initial mass function and on which massive stars collapse to black holes without exploding; the models here avoid that by normalising to the observed saturation, which makes the yield a fitted number rather than a predicted one. And the representations of specific star-formation rate and depletion time with redshift are smooth descriptions of measured trends, not physics.

What the model does capture is why a relation that looks like it should depend on a galaxy’s entire history does not. The gas runs out before the galaxy does — reservoirs are replaced on a depletion time much shorter than a galaxy’s age — so the abundance of the gas forgets its past and reports its present flows.

An equilibrium measures fluxes, and the history cancels #

A quantity in equilibrium measures fluxes, not totals. The metallicity of a galaxy’s gas is not a record of how many stars it has made; it is a ratio between the rate at which metals are made and the rates at which they are removed or diluted, and the history cancels. The same is true of a planet’s surface temperature under its carbonate–silicate thermostat, which reports a balance between outgassing and weathering rather than how much carbon the planet has ever had, and of the count of haloes against the count of galaxies, whose ratio at each mass reports where feedback is most effective — the same winds, seen through a different census.

Still open: whether a disc’s abundance gradient is its history #

A one-zone equilibrium cannot describe the fact that the inner parts of a disc galaxy are richer in metals than its outskirts. The gradient is not an equilibrium in the same sense: the inner disc has had its gas longer and turned it over more times, and the outer disc is still being built. How steep that makes the gradient, which ingredients of disc growth produce it, and whether the gradient carried by old stars preserves the history or has been blurred by the stars themselves moving, is the question a single number per galaxy cannot reach.