The gas runs out before the galaxy does

A galaxy’s gas is its fuel, and the rate at which it burns it is measurable. Dividing one by the other gives a time, and the time is uncomfortably short.

The law #

Maarten Schmidt proposed in 1959 that the rate of star formation per unit volume should go as a power of the gas density, and Robert Kennicutt established the surface-density version empirically in 1998 across a wide range of galaxies:

$$\Sigma_{\text{SFR}} = A\,\Sigma_{\text{gas}}^{N}, \qquad N \approx 1.4.$$

The exponent is the interesting part, and it has an argument behind it.

If star formation proceeds at some fraction of the free-fall rate, then the rate per unit volume is $\rho/t_{\text{ff}}$, and since $t_{\text{ff}} \propto \rho^{-1/2}$ that is $\rho^{3/2}$. Projecting a disc of roughly constant thickness turns a volume density into a surface density without changing the exponent, so $\Sigma_{\text{SFR}} \propto \Sigma_{\text{gas}}^{1.5}$.

Measured: $1.4 \pm 0.15$. This is one of the few places in the subject where a dimensional argument with no free parameters lands on the observed exponent.

The time that falls out of it #

The depletion time is the gas divided by the rate consuming it:

$$t_{\text{dep}} = \frac{\Sigma_{\text{gas}}}{\Sigma_{\text{SFR}}} \propto \Sigma_{\text{gas}}^{1-N} = \Sigma_{\text{gas}}^{-0.4}.$$

Since $N > 1$, the exponent is negative: a gas-richer disc has a shorter depletion time. A fuller tank empties sooner, which is the opposite of the intuition the phrase invites and is a direct consequence of the law being super-linear.

For normal spirals the number comes out at one to three billion years. The discs themselves are ten to thirteen billion years old and are still forming stars at rates comparable to their long-term average.

Something must therefore be supplying gas. The candidates are accretion of fresh material from the intergalactic medium, recycling of gas returned by dying stars, and minor mergers bringing gas-rich dwarfs. All three operate; between them they roughly balance consumption, which is why the star-formation rates of most discs have declined only slowly rather than collapsing.

The numbers for one galaxy #

The Milky Way makes the arithmetic concrete, and its numbers are among the best known in the field.

It contains roughly $10^{10}$ solar masses of gas — about seven billion in atomic hydrogen spread through a wide, flaring disc, and one to two billion in molecular form concentrated inside the solar circle.

Its star-formation rate is one to two solar masses per year. That number is small enough to be surprising: a galaxy of sixty billion solar masses of stars, forming one new star’s worth of material a year, has assembled itself at an average rate several times higher than its present one.

Dividing gives a depletion time of five to ten billion years for the total gas, and about one to two billion for the molecular component that is actually forming stars. The Galaxy is therefore a middle-aged system running down slowly, and it is one of the quieter spirals of its size.

For comparison, the most extreme starburst galaxies form stars at hundreds of solar masses per year and have depletion times under a hundred million years. Those systems cannot be in a steady state on any reading — they are consuming a supply that a merger has just delivered to their centres, and they will be quiescent within a hundredth of a Hubble time.

Where the star formation actually happens #

The law above is written in terms of total gas, and refining that changes its shape in an informative way.

Stars form in molecular clouds, not in atomic hydrogen. When the same measurement is made against the molecular surface density alone, the relation becomes close to linear — the exponent drops to about one — and the depletion time becomes roughly constant at two billion years everywhere.

That is a substantially different statement, and it splits the original law into two questions. First: what fraction of a galaxy’s gas is molecular? That fraction depends on pressure and metallicity, rises steeply towards a galaxy’s centre, and is what supplies most of the super-linearity in the total-gas version. Second: how fast do molecular clouds turn into stars? That appears to be a nearly universal rate, and a slow one — a few per cent of a cloud’s mass per free-fall time.

The inefficiency is the thing to be surprised by. A molecular cloud left to itself would collapse in a few million years and convert most of its mass into stars. What is observed is a few per cent, which means something is holding the collapse back or dispersing the cloud: turbulence, magnetic fields, and the radiation and winds of the stars already formed.

The observation behind the number #

Neither axis of the plane is measured directly, and both deserve stating.

The gas is two observations. Atomic hydrogen comes from the twenty-one centimetre line, whose intensity is proportional to the column density because the line is optically thin. Molecular hydrogen emits nothing at these temperatures — it is a symmetric molecule with no dipole moment — so it is traced by carbon monoxide instead, and converted using a factor calibrated locally and known to vary with metallicity. That conversion factor is the largest systematic in the whole subject.

The star-formation rate is inferred from the light of the most massive stars, which live only a few million years and therefore trace the present rate rather than the history. Several indicators are used: ultraviolet continuum, hydrogen recombination lines, and far-infrared emission from dust heated by young stars. Each requires the rest of the population to be inferred from the massive stars via an assumed initial mass function, so every star-formation rate in the literature is a mass function multiplied by a luminosity.

What a rate of one solar mass a year is made of #

The rate is quoted as a mass per year, and unpacking what that mass consists of exposes the assumption the whole quantity rests on.

Take the initial mass function measured in the solar neighbourhood: many low-mass stars, few high-mass ones, with the number per unit mass falling as roughly $m^{-2.35}$ above a solar mass. Forming one solar mass of stars per year then means forming, each year, something like two or three stars of a few tenths of a solar mass, a fraction of a solar-type star, and — once in several hundred years — a star above eight solar masses.

It is that last, rarest category that is observed. The ultraviolet, the recombination lines and the infrared all come from stars above about five solar masses, which are a few per cent of the mass formed and essentially all of the light.

So every star-formation rate in the literature is a measurement of the rare bright stars multiplied by the number of faint ones assumed to accompany them — a factor of twenty or so, taken from an initial mass function measured in one galaxy and assumed universal.

What the law does not do #

It is a correlation between two surface densities averaged over a galaxy or over kiloparsec-sized regions. At the scale of individual clouds it breaks down entirely: a map of a nearby galaxy at fifty-parsec resolution shows gas peaks and star-formation peaks that are anti-correlated, because a region either has a cloud or has already turned it into stars and blown the remains away. The law is a statistical statement about ensembles, and applying it below the scale at which the ensemble exists gives nonsense.

There is a threshold at low density. Below about ten solar masses per square parsec, star formation shuts down far more steeply than the power law predicts. The outer discs of spirals are gas-rich and nearly sterile, and the threshold is thought to be where the gas can no longer become molecular or where the disc becomes gravitationally stable against collapse.

And there is a starburst branch. Merging systems lie above the relation drawn for normal discs — the same gas surface density produces ten times the star-formation rate. Whether that is a genuinely different mode, or the same law with the molecular fraction and the cloud properties changed, is unsettled.

Where this leaves the census #

Put the pieces of this field together and a picture emerges of what determines whether a galaxy is blue or red.

A disc has a supply, a consumption rate, and a depletion time of a couple of billion years. As long as the supply continues, the disc keeps forming stars at a rate that declines slowly as the accretion rate declines. Cut the supply, and the disc reddens on the depletion timescale — which is a few hundred million years for the most active galaxies and a couple of billion for the rest.

That is the timescale the green valley measures, arrived at from a completely different direction: one from counting galaxies at intermediate colours, one from dividing a mass by a rate. The two agreeing is a check on both.

It also explains why quenching mechanisms work by removing or heating gas rather than by stopping star formation directly. Nothing needs to interfere with the physics inside a molecular cloud; it is enough to stop the resupply, and the depletion time does the rest.

An efficiency that is low everywhere #

The most durable result in this field is a negative one, and it is worth stating on its own.

Take a molecular cloud of $10^5$ solar masses. Its free-fall time is a few million years. If it converted its mass into stars in that time, the Milky Way’s molecular gas — a billion solar masses or more — would produce hundreds of solar masses of stars per year.

It produces one or two. The efficiency per free-fall time is therefore about one per cent, and the same number turns up in clouds of very different sizes, in other galaxies, and in the starburst systems once their shorter free-fall times are accounted for.

Explaining that number is an open problem with several partial answers. Turbulence supports clouds against collapse and cascades energy down to small scales. Magnetic fields resist compression across field lines. And the stars that do form immediately begin dispersing their surroundings — through radiation pressure, ionising photons, stellar winds, and eventually supernovae — which is a self-limiting arrangement of exactly the kind that produces a low, stable efficiency.

The last of those is the same class of mechanism as the feedback that shapes the luminosity function, acting three orders of magnitude further down in scale. A galaxy regulates itself because its clouds do.

A last consequence, and it is the one that connects this field to the whole history of the universe.

If a galaxy’s star formation is set by its gas supply rather than by anything internal, then the star-formation history of the universe is a history of supply. That is what is observed: the cosmic star-formation rate density peaked some ten billion years ago, at about ten times its present value, and has fallen ever since. Galaxies were not better at forming stars then; there was more gas arriving, into haloes that were denser and closer together. The decline since is the slow exhaustion of a reservoir no longer being refilled at the rate it once was — the same arithmetic as this essay’s, applied to everything at once.

The generalisation #

The reasoning here is a reservoir argument, and its general form is worth extracting: a rate divided into a supply gives a time, and comparing that time against the system’s age says whether the system is closed.

If the depletion time is long compared with the age, the system can be treated as closed and its present state reflects its initial conditions. If it is short, the system is a flow, and its present state reflects the balance between supply and consumption rather than anything about its history.

Galaxies are emphatically in the second regime, and so are several other things in this collection. The Sun’s own fuel is in the first — it has consumed half its core hydrogen in 4.6 billion years, so a star is a closed system. A hot Jupiter’s atmosphere is in the second, and is stripped on a timescale much shorter than the star’s life. The interesting cases are always the ones where the ratio is near one, because then the answer depends on the history rather than on the limit.

Where the supply comes from #

The essay has established that a disc must be fed and has not said by what. The candidates are two, they operate in different regimes, and the observational evidence for either is thinner than the argument requires.

Hot-mode accretion applies to massive haloes. Gas falling in from the intergalactic medium is shocked at the virial radius, heated to the halo’s virial temperature, and then has to cool before it can reach the disc. The cooling time is long, so the supply is slow and steady, and it is what a hot X-ray-emitting halo around a massive galaxy is a picture of.

Cold-mode accretion applies to lower masses and to earlier times. Gas arriving along a filament is dense enough to cool faster than it can be shocked, so it never heats up at all — it flows in along the filament as a cold stream, penetrating to the disc directly.

The distinction predicts a transition at a halo mass around $10^{12}$ solar masses, above which the hot mode dominates and below which the cold one does, and the transition is invoked as one reason star formation shuts down in the most massive galaxies.

What is observed is much less than that. Around the Milky Way there are high-velocity clouds of neutral hydrogen falling in, and their inferred accretion rate is a few tenths of a solar mass a year — a fraction of what is needed. Around other galaxies, absorption against background quasars detects cool gas in the halo in quantity, but converting an absorption line into an inflow rate requires knowing the geometry and the velocity, and neither is measured.

The cold streams themselves have never been imaged. They are predicted to be faint in emission and to cover a small fraction of the sky around a galaxy, and searches for them have produced candidates rather than detections.

The supply that the depletion time requires is a theoretical necessity with weak direct evidence, which is an uncomfortable position for a quantity every model of galaxy evolution depends on.

The gas the stars give back #

There is a term missing from the depletion arithmetic, and including it lengthens the answer substantially.

Not all the mass that goes into stars stays there. A star returns material to the interstellar medium through winds and, if it is massive enough, in a supernova — and the fraction returned is large. For a standard mass function about a third of the mass formed comes back within a hundred million years, and something approaching half within a few billion.

That changes the accounting. If a disc converts a solar mass of gas into stars and gets a third of it back within the lifetime of the calculation, the effective consumption rate is only two thirds of the star-formation rate, and the depletion time is correspondingly longer.

The correction is not small and it is not enough. Multiplying a two-billion-year depletion time by about one and a half gives three, which is still a fraction of the disc’s age — so recycling extends the deadline rather than removing it, and external accretion is still required.

What recycling does change is the chemistry. Gas returned by stars is enriched, so a disc that runs partly on recycled material has a metallicity that rises with time in a way a closed system would not, and the observed relation between a galaxy’s mass and its metallicity is partly a statement about how much of its gas has been through a star.

The return is also not instantaneous, and the delay matters. Massive stars return their material within a few million years and enrich it in the elements made in core collapse; low-mass stars return theirs over billions of years and enrich it in different elements entirely. So the composition of the recycled supply changes as a population ages, and the ratio of two element groups in a galaxy’s gas is a clock reading how long ago its stars formed.

The depletion times the law implies are worth reading across the full range the population covers rather than at the three points used above.

That spread is the reason the depletion time is a better summary than the star-formation rate. A galaxy’s rate says how fast it is converting gas now; the depletion time says how long it can keep doing so, and the second is what decides whether a galaxy is a going concern or an object running down. The quantity is the ratio of two measurements and it is dimensionally a time, so it can be compared directly with the age of the system, with the interval since the last merger, and with the timescale on which gas is resupplied.

Read that way the population sorts into three regimes rather than lying on a continuum. Galaxies with depletion times far longer than a Hubble time are not converting their gas in any meaningful sense and are limited by something other than supply. Galaxies near a few gigayears are in the regime the law was fitted on. And galaxies below a gigayear are burning through a reservoir that must have arrived recently and will be gone shortly, which is why every one of them is either interacting or has been.

The law itself makes no distinction between the three, which is both its strength and the reason it explains nothing. A power law fitted across two decades of surface density will describe all three regimes and will attribute the difference between them entirely to how much gas each has — which is a description, and the mechanism that sets how much gas each has is somewhere else entirely.

Where the ladder goes next #

The next rung is the molecular version of the law, the conversion factor it rests on, and the near-constant efficiency per free-fall time that emerges once the atomic gas is set aside.

Later rungs on this anchor: the star-formation threshold and gravitational stability of discs; the initial mass function as the assumption underneath every rate; feedback from massive stars as the regulator of the efficiency; the starburst branch and whether it is a separate mode; the cosmic star-formation history and its peak ten billion years ago; gas accretion from the intergalactic medium and the evidence for it; and the metallicity of a galaxy as the integrated record of everything above.