The support and the seed are the same motions

A molecular cloud at ten kelvin has a sound speed of about 190 metres a second. Its carbon monoxide lines are two to five kilometres a second wide. Something inside it is moving at ten to twenty times the speed at which information travels through it, and whatever that is, it is not heat.

The widths were measured before anybody knew what to call them. What turned them from an anomaly into a subject was a second measurement: the width is not a property of clouds but a function of the size of the region observed, and it grows as roughly the square root of that size over three decades of scale.

$$\sigma(R) \approx 1.0\ \mathrm{km\,s^{-1}}\ \left(\frac{R}{\mathrm{pc}}\right)^{1/2}$$

A power law spanning three decades is not a collection of independent objects that happen to be moving. It is a cascade — energy injected at one scale and passed down to smaller ones — and the exponent is close to what a supersonic cascade gives.

What supersonic means for support #

Turbulent motions support a cloud in the crude sense that they keep it from collapsing, and the way to count them is to replace the sound speed with an effective one that contains both:

$$c_{\rm eff}^2 = c_s^2 + \frac{\sigma^2}{3},$$

the factor of three because only the line-of-sight component is measured and the motions are taken as isotropic. Every threshold in the thermal argument then carries $c_{\rm eff}$ in place of $c_s$.

The consequence is large because the critical mass goes as the cube of the support speed. At Mach 10 the effective speed is about six times the thermal one, and the critical mass is two hundred times larger. A giant molecular cloud that is a thousand thermal Jeans masses is only a few turbulent ones, which is why it does not collapse in a free-fall time and why the galaxy still has gas in it.

The scale where the support runs out #

The crossing in the first figure is the most consequential number in this essay and it is worth being precise about what it is.

It is the scale at which the turbulent velocity dispersion equals the thermal sound speed — the sonic scale. Above it, motions dominate; below it, they have decayed to nothing and thermal pressure is all that is left. Given the linewidth–size relation with an exponent of a half, it lands near a few hundredths of a parsec, which is between five and ten thousand astronomical units.

The size of an observed dense core is a few hundredths of a parsec.

That is either the most important coincidence in star formation or it is the reason cores have the size they do, and the second reading is the one the subject has settled on. A region smaller than the sonic scale has no turbulent support left; it is a piece of gas held up by nothing but ten-kelvin pressure, with a Bonnor–Ebert mass of about one solar mass, and it is the object the thermal calculation was always describing. The turbulent cascade sets the boundary at which the thermal argument becomes true, and the mass scale of stars is the Bonnor–Ebert mass evaluated at the density the cascade delivers, and the bottom of the observed mass function sits an order of magnitude above the opacity limit.

So the two accounts are not competing accounts. One says what a piece of gas does once it is isolated and quiet; the other says how big such a piece is.

Why the exponent is a half #

The power law has a derivation, and it is short enough to give because it explains why this cascade is not the one most people have met.

Kolmogorov’s argument for incompressible turbulence takes the energy flux per unit mass through the cascade as a constant, $\varepsilon \sim v_\ell^3/\ell$, which gives $v_\ell \propto \ell^{1/3}$ and an energy spectrum falling as $k^{-5/3}$. That is the law of a stirred fluid at low Mach number and it is not what a molecular cloud does.

Supersonic turbulence is dominated by shocks, and a field of randomly placed discontinuities has a different spectrum for a reason that has nothing to do with the cascade: the Fourier transform of a step is $1/k$, so the energy spectrum of a gas full of steps falls as $k^{-2}$ regardless of what produced them. Integrating that over wavenumbers larger than $1/\ell$ gives $v_\ell \propto \ell^{1/2}$ — the observed exponent, from the shape of a discontinuity rather than from an energy flux.

The limiting case is Burgers turbulence, a compressible flow with no pressure at all, which is nothing but shocks and gives exactly a half. Real molecular clouds sit between the two, with Mach numbers of ten to twenty and some pressure, and the measured exponents between 0.35 and 0.65 are consistent with that. The exponent is a measurement of how shock-dominated the medium is, and it is closer to the Burgers value than to Kolmogorov’s, which is the observational statement that the motions are genuinely supersonic rather than merely fast.

That also explains something the first figure cannot show. In an incompressible cascade the energy is spread smoothly across the volume; in a shock-dominated one it is concentrated in thin sheets that occupy a small fraction of it. So the same total kinetic energy has a completely different spatial arrangement in the two cases, and the arrangement is what the next section is about.

The same motions make the structure #

The half of the argument that makes turbulence different from any other support is that it is not smooth.

Supersonic motion means shocks. Where two flows meet, the gas piles up in a sheet whose density contrast is of order the square of the Mach number — at Mach 10, a factor of a hundred — and the sheets intersect in filaments, and the filaments intersect in knots. A supersonically turbulent medium is not a uniform gas with a large pressure; it is a froth of thin dense structures separated by rarefied voids, and the density distribution is roughly log-normal with a width that grows with the Mach number.

That has two consequences pulling in opposite directions.

Globally it is support. The kinetic energy is real and it resists collapse, and the effective critical mass is genuinely raised.

Locally it is the opposite. The dense sheets and filaments have densities a hundred times the mean, so their local Jeans masses are ten times smaller, and some of them exceed their own thresholds immediately. Star formation in a turbulent cloud happens in the shocks, promptly, in the small fraction of the gas that the motions have compressed.

The net effect is a cloud that converts a few per cent of its mass per free-fall time rather than most of it — which is the one per cent efficiency the whole subject is trying to explain — and does so in filaments rather than uniformly, which is what every map of a nearby cloud shows.

Three relations, of which two are the same one #

The linewidth–size law arrived in 1981 alongside two others, and the relationship between them is a lesson in what a correlation between measured quantities is worth.

Larson’s three relations were: the velocity dispersion grows as a power of the size; the density falls inversely with the size, so the column density is roughly constant; and the clouds are close to virial equilibrium. They were presented as three empirical findings.

They are not three. Take the virial relation, $\sigma^2 \approx GM/5R$, and substitute a constant column density, $M \propto R^2$. Out comes $\sigma \propto R^{1/2}$ — the first relation, exactly, with no turbulence in the derivation at all. Any two of the three imply the third.

That matters in both directions. It means the linewidth–size relation is not independent evidence of a cascade: a population of virialised clouds of similar column density produces it whatever their internal motions are doing. And it means the constant column density is the substantive and under-examined member of the trio, since it is the one with no dynamical argument behind it and it is at least partly a selection effect — a survey in a particular tracer sees clouds above a threshold column and below an opacity limit, which is a narrow range by construction.

A set of correlations among quantities that are not independent is one correlation and two restatements, and deciding which of the three is doing the work requires a sample selected some other way. The relations have held up, with scatter, in surveys that tried; the exponent has not tightened, which is the honest summary of forty years.

The problem the support has #

Turbulent support has a defect that thermal and magnetic support do not, and it is fatal unless something fixes it.

Supersonic turbulence dissipates. Shocks convert kinetic energy into heat, the heat is radiated away by the same molecular lines that keep the cloud isothermal, and the energy is gone. The dissipation time is about one crossing time at the driving scale, which for a giant molecular cloud is a few million years — comparable with its own free-fall time.

So a cloud stirred once and left alone loses its support within a free-fall time and collapses. Turbulence cannot be a static support in the way a magnetic field can; it has to be driven, continuously, by something.

The candidates are argued about and none is obviously sufficient. Outflows from the stars already forming inject momentum locally and are probably enough to maintain motions inside a cluster-forming clump, and are certainly not enough at the scale of a whole cloud. Supernovae inject enormous energy and do so mostly outside molecular clouds, in the warm medium. The passage of a spiral arm compresses and stirs the gas on a galactic timescale. And the accretion of fresh gas onto a cloud from its surroundings delivers kinetic energy at the rate the cloud is growing.

The honest position is that the driving is not understood, and that the difference between a driven cloud and a decaying one is not observable in a snapshot — both show supersonic linewidths, both look bound by the virial test, and the test measures the kinetic energy present without saying how long it has left.

The number that says a cloud is bound #

There is a single statistic that most of this is argued through and it deserves a definition, because it is quoted constantly and is weaker than it looks.

The virial parameter is the ratio of a cloud’s kinetic energy to the gravitational energy it would need to be bound,

$$\alpha = \frac{5\sigma^2 R}{GM},$$

which is one for a uniform sphere in exact virial balance, well under one for something strongly bound and collapsing, and over two for something unbound and flying apart. Every quantity in it is measured: the dispersion from a line width, the radius from a map, the mass from a tracer.

Observed clouds cluster near one, with the smaller clumps inside them scattering to larger values. That is the result that made the supported picture attractive, and its weakness is in the word “near”. A parameter of one means the two energies are comparable, which is what any self-gravitating object that has had time to respond to its own gravity will show — a collapsing cloud passes through $\alpha \approx 1$ on its way in, and a cloud in free fall has $\alpha$ of order two by the time it has fallen an appreciable distance.

A virial parameter of one is evidence that gravity matters and is not evidence that it is balanced. Distinguishing balance from passage requires knowing the sign of the radial motion, and a line width has no sign in it — an expanding cloud and a contracting one of the same speed produce identical profiles.

The other reading, in which nothing is supported #

There is a rival account of the same data and it is worth stating properly, because the observations do not currently choose between them.

On the second reading, molecular clouds are not supported at all. They are young — assembled recently from colliding flows in the diffuse medium — they are collapsing now, and the supersonic motions are not support but the collapse itself seen in projection. A cloud in global gravitational contraction has line-of-sight velocities that grow with the size of the region, for the same reason a free-fall velocity grows with the distance fallen, and the linewidth–size relation comes out with roughly the observed exponent.

The two pictures make the same measurement look identical and different predictions about time. A supported cloud lives for many free-fall times and converts gas slowly throughout. A collapsing one lives for one or two, converts gas at an accelerating rate, and is destroyed by the first massive stars it makes.

The observational discriminant is the relation between a cloud’s age and the fraction of it already turned into stars, which is the efficiency a galaxy’s whole gas supply is set by, and ages are the hardest quantity in the subject to measure. What evidence there is — the small scatter of stellar ages within a cluster, the rapidity with which clouds are cleared — has moved the field toward the shorter timescale, without settling it.

What the linewidth actually measures #

It is worth separating the observable from the interpretation, because the chain has three links and each can fail.

What is recorded is a line profile — intensity against frequency in a rotational transition of carbon monoxide or one of its isotopologues — integrated over a beam. Its width is converted to a velocity dispersion by the Doppler relation, which is exact.

The first difficulty is that the line is optically thick. A thick line’s width is not the velocity dispersion of the gas; it is broadened because the line saturates at its centre and the wings continue to grow, so a thick line is wider than a thin one drawn from the same gas. Rarer isotopologues are used to avoid it, at the cost of being faint, and the conversion to a mass carries a factor calibrated three ways.

The second is that the dispersion measured is a projection. What is wanted is the three-dimensional motion; what is available is one component, and the factor of three in the effective sound speed assumes isotropy that the shocks and the field both violate.

The third is that “the size of the region” is not well defined. A cloud does not have an edge, so its size is an isophote at some threshold, and the threshold differs between surveys — which is a large part of why the measured exponent ranges from about 0.35 to 0.65 and why, as the third figure shows, that range matters so much.

Where the picture stops #

The cascade is not isotropic and the field is why. A magnetised medium carries waves that travel along the field lines and not across them, so the turbulence is anisotropic at every scale below the driving one, with eddies elongated along the field. The isotropy assumed in the effective sound speed is wrong in a way whose size depends on how strong the field that cannot be squeezed away is — so the two accounts are not independent and the corrections do not simply add.

Support by kinetic energy is not the same as support by pressure. A pressure pushes outward everywhere; a set of bulk motions pushes some places outward and others inward. Treating $\sigma^2/3$ as if it were a pressure term is a virial-theorem statement about a whole cloud, and it has no local meaning at all — which is exactly why the same motions can support the cloud and compress its interior, and why no single number describes both.

And nothing here predicts the exponent. The half power is close to what dimensional arguments give for a shock-dominated cascade and it is also close to what a cloud in free fall gives, so it does not discriminate between the two readings of the whole subject. A measured power law with two sufficient explanations is weak evidence for either.

Still open: whether a snapshot can distinguish two histories #

Both accounts of a molecular cloud fit every static measurement there is. The same linewidths, the same virial parameters near unity, the same filamentary structure, the same low efficiency. What separates them is how long the state has lasted, and duration is not in a snapshot.

The routes to an age are all indirect. The spread of pre-main-sequence ages in an embedded cluster gives a lower bound on how long star formation has been going on there, and pre-main-sequence ages carry model uncertainties of a factor of two. Chemical clocks — the abundance ratios of species that take a known time to reach equilibrium — give ages for the gas rather than for the stars, with their own network uncertainties. Comparing the number of clouds with and without young stars gives a statistical lifetime, and depends on knowing the selection.

None of the three is decisive, and the decisive measurement would be a cloud’s age with an error under a factor of two. Nobody has a candidate method, and the question — how long a molecular cloud lives — remains the one whose answer would settle the most.