An atomic ceiling set by the dust

Maps of the 21-centimetre line across nearby spiral galaxies share a feature that is easy to miss because it is an absence. The atomic hydrogen they show is spread over enormous areas, often far beyond the stars, and it is remarkably flat: its surface density rarely rises above about ten solar masses per square parsec, however much gas the galaxy has and however close to the centre the map is made. Where the total gas is denser than that — measured through dust, or through the carbon monoxide that traces molecular gas — the excess is molecular. The atomic gas has a ceiling, and the molecules take over above it.

The ceiling is not a coincidence of the galaxies that happened to be mapped. It is the column of gas a cloud needs in front of its interior to keep ultraviolet starlight from breaking molecules apart, and it can be computed from three rates: how fast molecules form, how fast starlight destroys them, and how quickly the gas shields itself. What comes out is a number that depends almost entirely on how much dust there is per atom, and hardly at all on how bright the starlight is.

Molecules made on dust, broken by starlight #

Two hydrogen atoms meeting in empty space cannot simply combine. The molecule they would form has no way to lose the binding energy, and the collision ends with the atoms flying apart. In the interstellar medium hydrogen molecules form on the surfaces of dust grains instead: an atom sticks to a grain, wanders across its surface until it meets another, and the pair leaves as a molecule with the grain absorbing the excess energy. The formation rate is proportional to the number of grains, and so to the gas’s metallicity — the fraction of its mass in elements heavier than helium, from which the grains are made. At the Sun’s metallicity the rate coefficient is about $3\times10^{-17}$ cubic centimetres per second.

The destruction is done by starlight in the Lyman–Werner band, ultraviolet photons between 11.2 and 13.6 electronvolts. A molecule absorbs one in a specific electronic transition, and about one time in seven the excited molecule decays into the vibrational continuum and flies apart. In the interstellar radiation field near the Sun, an unshielded molecule survives about five hundred years. Molecules made on grains at a rate set by the dust, destroyed in centuries by starlight: at the surface of any cloud, the gas is atomic.

Deeper in, two things protect the molecules. The dust absorbs ultraviolet light of all wavelengths, so the field weakens exponentially with the total column. And the molecules themselves absorb the light in precisely the lines that would destroy molecules further in. Because those are discrete lines, the absorption saturates — the line cores go black and only the wings keep absorbing — exactly as a strong spectral line stops deepening and only widens. A column of molecules a millionth of the column needed for dust to matter is already enough to cut the dissociation rate by orders of magnitude. Balancing formation against shielded destruction at each depth gives the molecular fraction everywhere in the cloud.

The transition is sharp and its depth depends on the metallicity. At solar metallicity the gas is half molecular about four solar masses per square parsec into the cloud; at a tenth of solar, about fifty. The calculation is the standard one for photodissociation regions, and it reproduces the analytic result of Sternberg and collaborators for the total atomic column to within ten per cent.

The ceiling is a shield’s thickness #

A cloud lit from both sides holds its atomic gas in two skins, one on each face, and the thickness of each skin is the column at which the transition happens. A thin cloud — thinner than two skins — is atomic throughout. A thick one has the same two skins and a molecular core. Adding gas to it adds only to the core.

That is the ceiling seen in the galaxy maps, and the solar-metallicity value of about ten solar masses per square parsec matches what the 21-centimetre surveys show. A map of a metal-rich spiral, smoothed over a kiloparsec, is a map of many clouds, each saturated at its own two skins. Where the clouds are crowded the total gas rises and the atomic gas does not.

The 21-centimetre measurement itself is part of the picture. The atomic skins are cold gas at columns where the optically thin formula starts to undercount — a ceiling of ten solar masses per square parsec is a column of about $10^{21}$ atoms per square centimetre. Some of the apparent flatness in the maps could be saturation of the line rather than saturation of the gas. The emission–absorption surveys say that this effect is tens of per cent rather than a factor of several, so the ceiling is real, but it is not quite as flat as the maps suggest.

Dust per atom sets the ceiling #

The ceiling’s dependence on metallicity is the most testable prediction, and it follows from the fact that the dust appears in both halves of the balance.

The slope is the inverse of the metallicity, very nearly exactly. The reason is that the final barrier is dust attenuation: the molecules shield themselves efficiently, but a small residual dissociation rate persists until the dust cuts off the remaining field, and the column at which the dust does that is the inverse of the dust per atom. Lowering the metallicity also slows the formation of molecules on grains, which raises the atomic column further; the two effects together give the inverse slope.

The prediction is borne out where it has been tested. The Small Magellanic Cloud, at about a fifth of solar metallicity, has atomic surface densities several times the Milky Way’s ceiling over large areas. Metal-poor dwarf galaxies hold most of their gas atomic even where its total surface density is high enough that a spiral would have turned it molecular. And within a single spiral galaxy, where the metallicity falls with radius by a factor of a few from centre to outskirts, a gradient laid down by the galaxy’s history, the atomic ceiling rises outward with it.

Why the starlight hardly matters #

The same calculation predicts something that is at first surprising. The dissociating field near the centre of a spiral galaxy, or in a starburst, is ten or a hundred times stronger than near the Sun. A stronger field should destroy more molecules and push the transition deeper. At fixed gas density it does.

But the density of the cold gas is not fixed. It is set by the two-phase balance between the cold atomic medium and the warm one, and the same ultraviolet photons that break molecules also heat the gas, through electrons knocked off dust grains. A stronger field heats more, raising the pressure at which a cold phase can exist and so the cold gas’s density. The ratio of the field to the density — which is what decides how deep the dissociating photons penetrate — then stays roughly constant. Krumholz, McKee and Tumlinson pointed this out, and it is what makes the ceiling look universal across a disc where the starlight varies a hundredfold. In the model drawn here the cancellation is exact, because the density is taken to be proportional to the field; in real gas it is only approximately so, and the residual dependence is logarithmic.

The cancellation is a small example of a pattern that runs through the interstellar medium: the quantities that look like independent parameters — field, density, pressure — are tied together by balances, and the observable that results depends on fewer things than the equations do. Here the outcome depends on the metallicity and almost nothing else.

Which shield does the work #

The two shields operate on very different columns, and it is worth separating them, because the argument for the metallicity scaling rests on the weaker one.

At the point where the gas turns half molecular, the molecules have done nearly all the shielding: the self-shielding factor is about one in ten thousand, the dust’s factor about one half. The dust matters anyway, twice over. It is where the molecules are made, so a cloud with less dust needs a stronger shield to reach the same balance. And it is what finally cuts off the residual dissociation that slips through the wings of the saturated lines. The atomic column ends where the dust’s exponential becomes significant, so it scales with the dust even though the molecules supply most of the shielding.

That separation also explains the dark gas found around molecular clouds. Carbon monoxide, the molecule normally used to trace molecular gas, is destroyed by ultraviolet light at different wavelengths and does not self-shield as effectively at the columns where hydrogen has already turned molecular. Between the hydrogen transition and the carbon monoxide transition there is a layer of molecular hydrogen with no carbon monoxide in it — invisible to both the 21-centimetre line and the usual molecular tracer. At solar metallicity it is a modest fraction of the molecular mass. At low metallicity, where the dust is scarce and the carbon monoxide transition is pushed much deeper than the hydrogen one, it can be most of it.

A rival explanation in the weight of the disc #

The shielding argument is not the only account of where the molecules begin, and the other one is older. Across the discs of nearby galaxies the ratio of molecular to atomic gas correlates tightly with the pressure at the disc’s midplane — the weight of the gas and stars above it per unit area — and the relation is nearly linear, with the gas half molecular where the pressure is about $4\times10^4$ kelvin per cubic centimetre. On that account the molecular fraction is set mechanically: the weight of the disc squeezes the clouds, a squeezed cloud is denser, and a denser cloud turns molecular sooner.

The two accounts are not as different as they sound, because the shielding calculation also depends on density, and density and pressure travel together. They separate in two places. In the outer discs of spirals the midplane pressure falls steeply with radius, because the stars thin out, while the metallicity falls only gently. The pressure law predicts that the molecular fraction should collapse with the pressure; the shielding law predicts that it should fall more slowly, tracking the dust. And in metal-poor dwarfs, whose midplane pressures can be similar to those of spiral outskirts, the shielding law predicts far less molecular gas than the pressure law does, because the dust is scarce.

In both regimes the measurements are made with the gas that is hardest to see. The carbon monoxide that would show the molecules fades with the dust, so a low measured molecular fraction is partly a low measured tracer. Where the molecular gas has been estimated independently — through dust emission corrected for the atomic gas, or through the star formation rate under the assumption that the molecular depletion time is constant — the metal-poor galaxies hold less molecular gas than the pressure law predicts, in the direction the shielding argument requires. The pressure law remains the better description of where the molecules are within a single metal-rich disc. The shielding law is what explains why that description stops working where the metals run out, and the two are probably the same physics seen through different variables.

The edge where star formation stops #

The atomic ceiling also offers an account of something observed long before the molecular gas could be mapped. The ultraviolet and hydrogen-alpha light of young stars in a spiral galaxy, the standard tracers of its current star formation, fades sharply at a radius well inside the edge of the atomic gas. Beyond that radius there is plenty of gas and very little star formation, and the edge was first interpreted as a gravitational threshold: the gas disc becomes stable against collapse when its surface density falls below a critical value set by its velocity dispersion and the galaxy’s rotation.

The shielding argument gives a second reason for the same edge. In the outer disc the total gas surface density falls below the atomic ceiling, so no cloud is thick enough to shield a molecular core, and the gas stays atomic. Below the ceiling there is no molecular phase to form stars from, whatever the stability of the disc. Both thresholds fall near a gas surface density of a few to ten solar masses per square parsec in a typical spiral, which is why the edge was attributable to either. Deep ultraviolet imaging has since found low-level star formation continuing well beyond the edge, at rates per unit area a hundred times lower, which neither threshold forbids absolutely: a disc can still be locally unstable in places, and a single cloud can still shield itself where the average gas is thin. The edge is where the typical cloud stops being able to, not where every cloud does.

Inside a cloud the story continues with thresholds of a different kind. A molecular cloud must also fail to hold itself up against its own gravity, and its magnetic field must be too weak for the mass it threads. The atomic ceiling is the first of the thresholds in that sequence. Like the others, it reduces to a balance with one dominant parameter. Its parameter is the dust per atom.

Stars follow the molecules, and perhaps not because of them #

The reason anyone cares where the hydrogen turns molecular is star formation. Across the discs of nearby spirals the rate of star formation per unit area correlates tightly with the molecular gas surface density and hardly at all with the atomic: regions above the atomic ceiling form stars in proportion to how much gas they hold above it, and regions below it form very few. The molecular gas in a spiral is consumed at a nearly constant rate, with a depletion time of about two billion years. This is the local version of the observation that a galaxy’s gas runs out before the galaxy does, and it puts the atomic-to-molecular transition at the start of the chain that turns gas into stars.

The correlation has a tempting reading — that hydrogen molecules are needed to form stars — and a more interesting one. Molecules matter for star formation because cold gas must radiate its heat to collapse, and at tens of kelvin the cooling is done by carbon monoxide and by carbon atoms, not by hydrogen molecules at all. What cold, star-forming gas needs is shielding from the ultraviolet field, which would otherwise heat it. The same shielding that lets hydrogen turn molecular lets the gas cool. On this reading the molecules and the star formation are two consequences of one condition, a column deep enough to keep the starlight out, and the molecules are a marker rather than a cause.

The distinction matters in metal-poor gas. There the hydrogen transition is pushed to large columns, while the timescale for molecules to form on the scarce grains — hundreds of millions of years at a tenth of solar metallicity — can exceed the lifetime of a cloud. Calculations that follow the gas’s thermal state find that such gas can cool and form stars while still mostly atomic. If they are right, the first generations of stars in young galaxies formed in atomic gas, and the tight link between molecules and star formation in present-day spirals is a property of metal-rich gas rather than a law. The few metal-poor dwarfs where star formation can be measured against both tracers form stars more efficiently per unit of detected molecular gas than spirals do, which fits either reading.

What the slab leaves out #

The calculation behind every figure here is a one-dimensional slab in steady state, lit by a beam from one side, with the gas density uniform through it. Real clouds are clumpy, and ultraviolet light penetrates between clumps further than through a uniform slab of the same mean column, so a clumpy cloud turns molecular later and holds more atomic gas than the slab suggests. Real fields arrive from every direction, which changes the effective depth by a factor of order two. The density is taken from the two-phase balance at a fixed multiple of its minimum, a choice that sets the solar ceiling at about ten solar masses per square parsec and that a different choice would move by tens of per cent.

The steady-state assumption is the most consequential. Formation on grains takes a time of order $1/(Rn)$, about fifty million years at solar metallicity and the densities drawn, and several hundred million at a tenth of solar. A cloud that is assembled, shocked or dispersed faster than that never reaches the balance the slab assumes. In metal-rich spirals clouds live long enough; in dwarfs they may not.

Still open: whether the ceiling holds where the metals are scarcest #

The ceiling’s inverse scaling with metallicity has been checked from solar down to about a fifth of solar, in the Magellanic Clouds and a handful of nearby dwarfs. Below that, the measurements become hard in every way at once. The galaxies are small and faint, their molecular gas emits almost no carbon monoxide, and their dust — the gauge used to measure total gas where the molecules cannot be seen — is itself uncertain because the dust-to-gas ratio may fall faster than the metallicity once grains stop growing efficiently in the interstellar medium.

The extremely metal-poor dwarfs, at a fiftieth of solar and below, are the nearest analogues of the first galaxies, and the question there is whether the atomic ceiling keeps rising as the inverse of the metallicity — so that their gas is overwhelmingly atomic — or whether the dust-to-gas ratio drops so steeply that the hydrogen never turns molecular at all within any cloud that exists. The two possibilities predict different relations between gas and star formation in the earliest galaxies, and neither has yet been measured in a galaxy metal-poor enough to separate them.