A molecule that forms too slowly to wait for

A galaxy’s hydrogen sits in two forms, as single atoms and as molecules, and the boundary between them is set by a balance. Ultraviolet starlight breaks molecules apart; dust grains, on whose surfaces two hydrogen atoms can meet and stick, make them. Deep enough inside a cloud the molecules already formed absorb the dissociating light and shield what lies behind them, and the gas turns molecular. That shielding column is why the atomic hydrogen in a disc stops piling up at about ten solar masses per square parsec, and why the ceiling rises as the metals — and the dust that goes with them — become scarce.

Every step of that argument is about where molecules would be if the gas had waited long enough. The rates balance; the profile through the cloud is computed with the molecular fraction at every depth set to the value at which formation equals destruction. Nobody asks how long the formation takes, because in the Milky Way’s gas it takes a few million years and that is short enough not to matter.

In gas with fewer metals it is not short. A grain surface is the only efficient place for hydrogen molecules to form, the rate is proportional to the grain surface per hydrogen atom, and that is proportional to the metallicity. A tenth of the Sun’s metals means a tenth of the dust and ten times the time. The question that follows is whether a cloud waits.

Two clocks, and only one of them reads the metals #

The figure sets two clocks side by side. One is the time to turn the cloud’s hydrogen into molecules, $1/(2 R Z n)$, where $R \approx 3 \times 10^{-17}$ $\mathrm{cm^3\,s^{-1}}$ is the rate coefficient measured for the Milky Way’s dust, $Z$ is the metallicity relative to the Sun and $n$ the density. At a hundred atoms per cubic centimetre — a typical cold cloud — it is about five million years for gas like the Sun’s. The other is the free-fall time, the time a cloud with no support would take to collapse under its own gravity, $\sqrt{3\pi/32G\rho}$, which at the same density is a little over four million years. It depends on the density and nothing else.

At solar metallicity the two are comparable, which is a coincidence that has been hiding the question. Molecules in Milky Way clouds form on about the same timescale on which the clouds evolve, and since clouds are not in free fall — they are supported for a few free-fall times by turbulence and magnetic fields — the molecules have time. The equilibrium arguments work in the Milky Way because the clock that counts molecule formation is not much slower than the clock the cloud lives by.

Move to a tenth of the Sun’s metals, as in the Small Magellanic Cloud or the outer discs of spirals, and the molecule clock runs ten times slower. A hundredth, typical of the most metal-poor dwarf galaxies nearby and of galaxies in the first billion years, makes it a hundred times slower: half a billion years to make molecules in a cloud that would collapse in four million. The free-fall clock has not moved. So the cloud that the shielding argument says should be molecular is, for most of its life, still atomic, waiting for a reaction that its dust can only run slowly.

A rate measured once, along a few sightlines #

The rate coefficient that sets the molecule clock has an unusually narrow foundation for a number used so widely. It was measured in the 1970s by the Copernicus satellite, which recorded the ultraviolet absorption lines of molecular hydrogen against bright stars in the Sun’s neighbourhood. Along each sightline the molecular column and the atomic column could be read directly, and the ultraviolet field that destroys the molecules estimated from the stars nearby. Setting formation equal to destruction gave the formation rate, and Edward Jura’s 1975 analysis of those sightlines produced the value still in use, $3 \times 10^{-17}$ $\mathrm{cm^3\,s^{-1}}$, uncertain by a factor of about two.

That measurement already assumes equilibrium — it infers a formation rate from a balance — and it was made in gas of the Sun’s composition. Everywhere else the rate is scaled in proportion to the dust, on the reasoning that the reaction happens on grain surfaces and the grain surface per atom follows the dust-to-gas ratio. The scaling is natural and has never been checked directly at low metallicity, because the ultraviolet spectroscopy that measured it locally has been possible in only a few sightlines through the Magellanic Clouds. It is the one place where the whole argument could be wrong in its premise: if grains in metal-poor gas were more efficient per unit surface, or if some other route made molecules without dust, the clock would run faster than the figures show.

The most metal-poor gas has a second route, the one that made the first molecules in the universe before any dust existed: a free electron attaches to a hydrogen atom, and the negative ion then combines with a second atom. It is slow, and it needs free electrons, so it matters only where there is no dust at all and some ionisation — gas at a few thousandths of solar metallicity and below. Above that, dust wins.

Where molecules keep up #

The formation clock and the free-fall clock both depend on density, but differently. Formation runs as $1/n$, because the reaction needs an atom to find a grain; free fall runs as $1/\sqrt{n}$. A denser cloud therefore converts faster relative to its collapse, and for any metallicity there is a density above which the molecules keep up.

The dividing line rises as the inverse square of the metallicity, because the formation time scales as $1/(Zn)$ and its ratio to the free-fall time as $1/(Z\sqrt{n})$. At solar metallicity a cloud of a hundred atoms per cubic centimetre sits near the line. At a tenth of solar the line has risen to fifteen thousand, the density of a cloud’s dense core rather than the cloud itself. At a hundredth it has risen to over a million, beyond the densities at which a cloud is still a cloud and not a protostar.

So the region where molecules lag is not a corner of parameter space. It is where all of a metal-poor cloud’s ordinary life is spent. Only the last stage of a collapse, when the density runs up by orders of magnitude in a fraction of a free-fall time, crosses into the region where molecules form fast — and by then the collapse is nearly over.

A collapse that outruns its molecules #

The plane suggests the right calculation. Take a cloud thick enough to be shielded, so that once a molecule forms it survives, and let it collapse in free fall from a hundred atoms per cubic centimetre. Follow the molecular fraction as the density rises.

The curves are all late. A free-falling cloud spends most of its collapse at low density, slowly accelerating, and then runs away in the last tenth of the free-fall time, when the density rises by a factor of a hundred. The molecules form where the density is highest, so they form mostly at the end. At solar metallicity even that last stage is enough, and the cloud arrives at ten thousand atoms per cubic centimetre almost completely molecular. At a tenth of solar, only a third of it has converted. At a hundredth, the cloud that has collapsed to the density of a star-forming core is still ninety-six per cent atomic.

Every one of these clouds is thick enough for the shielding argument. In equilibrium, every one would be molecular. The difference between the figure and the equilibrium picture is the whole effect of time, and it grows as the inverse of the metallicity.

Where it stops matters more than where it started #

Because the molecules form mostly at the end, the answer is insensitive to where the collapse began and sensitive to where it stops. A cloud starting at thirty atoms per cubic centimetre and one starting at three hundred arrive at ten thousand with nearly the same molecular fraction — the early, slow stage contributes little either way. What matters is how dense the gas becomes before it stops being a cloud and becomes stars.

Each curve is an S, a decade wide in metallicity, and the three are spaced by a factor of about three for each factor of ten in final density. The spacing follows from the collapse: the integral of the density over time, which is what the molecule formation counts, is dominated by the final stage and scales as the square root of the final density. A cloud whose star formation begins at ten thousand atoms per cubic centimetre is half molecular at a fifth of solar metallicity; one that collapses to a hundred thousand before forming stars is half molecular at six per cent of solar.

That puts the transition — the metallicity below which gas forms stars while still mostly atomic — between a tenth and a twentieth of the Sun’s, for the densities at which star-forming cores are observed. The more detailed calculations that follow the gas’s chemistry and dynamics together put it lower, near a hundredth, because real clouds are not in free fall but spend several free-fall times supported, giving the molecules longer. Both kinds of estimate agree on the direction and on the order of magnitude: in the most metal-poor galaxies, star formation happens in gas that is mostly atomic.

The gas cools without them #

Molecules matter for star formation because a cloud has to radiate its heat to collapse, and it is tempting to read the lagging molecules as a cloud that cannot cool and so cannot collapse. That is not how cold interstellar gas cools.

At the temperatures of a cold cloud, tens to a hundred kelvin, hydrogen molecules are nearly useless as coolants: their lowest excited state is five hundred kelvin above the ground and is hardly populated. The cooling is done by heavier species — singly ionised carbon in the outer, atomic layers, through a fine-structure line at 158 microns, and carbon monoxide and atomic carbon deeper in. Ionised carbon is present wherever there is carbon and starlight to ionise it, needs no molecules, and cools the gas in a few thousand years at solar metallicity and in half a million years at a hundredth — still a small fraction of the free-fall time.

This is the thermal balance that splits the interstellar medium into a cold gas and a warm one, and it works in atomic gas. The cold neutral medium that makes up the dense phase in every galaxy is cold because of ionised carbon, not because of molecules. A metal-poor cloud that has not formed its molecules is still cold, still dense, and still able to collapse. The correlation between molecules and star formation in spirals, which the shielding argument explains as two consequences of one condition, comes apart here: the shielding keeps the gas cold whether or not molecules have formed in it, and the molecules simply arrive too late to be counted.

A depletion time shortened by the missing molecules #

The difference shows up in the number most often used to describe star formation, the depletion time — the mass of molecular gas divided by the rate at which stars form. In the discs of nearby spirals it is about two billion years and remarkably constant, which has been read as evidence that stars form from molecular gas at a fixed efficiency.

If stars form from all the cold, shielded gas at the spiral rate, but the census counts only what has turned molecular, the measured depletion time is the true one multiplied by the molecular fraction. At solar metallicity nothing changes. At a tenth of solar the census sees under half the gas that is forming stars, and the depletion time it reports is under a billion years. At a hundredth it is under a hundred million. A metal-poor dwarf would appear to convert its molecules into stars twenty times faster than a spiral does, and a galaxy in the early universe faster still.

Metal-poor dwarfs do show short molecular depletion times, when their molecules can be measured at all. The usual explanation is a different one: that the carbon monoxide line used to trace the molecules fails at low metallicity, because the molecule needs more shielding than hydrogen does, so a CO census misses molecular hydrogen that is there. Both effects push in the same direction, and both grow as the metals fall. One says the molecules are there and not seen; the other says they are not there yet. Separating them needs a tracer of the molecular hydrogen itself rather than of carbon monoxide — and the direct tracers, the ultraviolet absorption lines of the molecule against background stars, are available only along a handful of sightlines in the nearest metal-poor galaxies.

What the collapse leaves out #

The calculation is the simplest version of the argument, and three of its simplifications change the numbers without changing the direction.

The cloud starts entirely atomic. Real clouds form from gas that has been through earlier clouds, and some of their hydrogen may already be molecular when they begin to collapse. That shifts every curve towards molecular, by an amount that depends on the history of the gas and is not known for any metal-poor galaxy.

The cloud is uniform. Real clouds are turbulent, and the densest filaments and clumps within them reach high densities long before the cloud as a whole does. Molecules form fastest in the densest gas, so a turbulent cloud forms its molecules earlier than a uniform one of the same mean density — another shift towards molecular, which simulations that include turbulence find to be a factor of a few in the effective formation rate.

And the cloud is in free fall. Real clouds are supported for several free-fall times, which gives the molecules several times longer. That is the largest of the three corrections and the one that moves the transition from a tenth of solar toward a hundredth.

None of these removes the effect, because the dependence on metallicity is in the rate coefficient itself, and each correction multiplies the time available by a factor that does not depend on the metals. A cloud given ten free-fall times rather than one is, in effect, a cloud with ten times the metallicity in these figures. The transition moves; it does not disappear.

Still open: where the first stars’ gas turned cold #

The argument changes what the molecular gas of a galaxy means. In metal-rich gas, molecules are a reliable census of the gas about to form stars, because they form on the same timescale as the clouds evolve. In metal-poor gas they are a lagging indicator — a record of how long the gas has been dense and shielded, rather than of how much gas is dense and shielded. Below about a tenth of the Sun’s metals, and certainly below a hundredth, the cold gas that forms stars is mostly atomic.

That has a test. The atomic hydrogen of metal-poor dwarfs, measured at twenty-one centimetres, should show a cold component in the regions where stars form, with column densities that the shielding argument says should have turned molecular. The 21-centimetre absorption measurements that separate cold atoms from warm are possible only against background radio sources, and the metal-poor galaxies near enough to resolve are small, so the sources behind them are few. Where the measurements have been made, they find cold atomic gas at columns above the solar ceiling. Whether that gas is forming stars directly, without passing through molecules first, is the question that decides how the first galaxies — with metallicities of a thousandth of the Sun’s, and molecule-formation times longer than their own ages — formed any stars at all.

The answer feeds forward into two other measurements. A galaxy’s metals record how much it has expelled, and the efficiency with which its gas formed stars decides how many metals it made per unit of gas; if metal-poor gas forms stars from atoms at the spiral rate, the early metal enrichment of galaxies was faster than a molecule-limited picture allows. And the threshold surface density below which star formation is observed to fall off in the outer discs of spirals — gas at a fifth of solar metallicity, at large radii — has been explained by a cloud that cannot become a star for want of shielding. The time argument says that even where shielding is reached, the molecules that would mark it may lag, and an outer disc that looks molecule-poor may be forming stars more efficiently than its molecules suggest.