A threshold with no free parameter in it

The Jeans criterion is the standard answer to whether a cloud collapses, and it has a property that is easy to miss: it is a scale rather than a threshold. Cool a cloud and the Jeans mass falls; compress it and the Jeans mass falls; wait long enough and almost any cloud becomes unstable.

The magnetic criterion is different in kind. A cloud below it will not collapse, ever, at any temperature, at any density, however long anyone waits — and the value that separates the two cases contains no free parameter at all.

Why the ratio is conserved #

Take a cloud, and take a surface through it perpendicular to the field. The mass enclosed by a bundle of field lines is fixed, because the gas cannot leave the bundle without crossing field lines, which flux freezing forbids. The magnetic flux through that bundle is fixed, by the definition of freezing.

So the ratio of the two is a constant of the motion, and it stays constant through any amount of collapse, expansion, distortion or rotation. It is a label the cloud carries.

That is a strong statement and it is worth checking against the alternative. The Jeans mass is not conserved: it is a combination of temperature and density, both of which change during a collapse. The virial ratio is not conserved. The free-fall time is not conserved. Almost nothing in the collapse problem is, and the mass-to-flux ratio is.

One conservation is worth comparing directly, because it produces a similar-looking argument with a different outcome. Specific angular momentum is also conserved, and it also produces a barrier — a centrifugal radius inside which a collapsing core cannot go. But the centrifugal barrier is at a definite radius, so a cloud with too much spin collapses partway and stops; the magnetic criterion is scale-free, so a subcritical cloud does not collapse at all. Two conserved quantities, two barriers, and only one of them applies at every scale at once.

The scale-free character is the unusual feature and is worth restating. There is no radius in the magnetic criterion, no density, no temperature. It compares a mass with a flux and returns a verdict that is true of the cloud rather than of its current state, which makes it more like a conserved charge than like a stability condition.

Why the critical value is a pure number #

The competition is between two energies or, equivalently, two forces.

Gravity pulls with a force going as the square of the mass divided by the square of the radius. The magnetic force pushes with a force going as the square of the flux divided by the square of the radius — the field is flux over area, its energy density is the square of the field, and the volume brings the radius back.

Both go as the inverse square of the radius. So their ratio is independent of radius, and shrinking the cloud changes nothing about which wins.

Setting them equal gives a critical ratio of mass to flux, and every dimensioned quantity cancels except the gravitational constant. The result is one over two pi root G, times a coefficient of order one that depends on the geometry assumed.

That coefficient is the one place a choice enters, and it is worth being honest about it: for a uniform sphere threaded by a uniform field the number is 0.13 over root G; for a flattened sheet it is 0.16. The difference is twenty per cent, which is small compared with the measurement errors and is not zero.

The comparison with the Jeans criterion’s own coefficient is instructive. That one is a similar factor of order unity, arising from the same kind of geometric choice, and nobody worries about it — because the Jeans mass depends on the temperature to the three halves and the density to the minus a half, so the geometric factor is swamped by everything else. Here nothing else varies, so the geometric factor is the whole uncertainty in the threshold, and it is the reason the criterion is usually quoted as a dimensionless ratio normalised to the critical value rather than as an absolute number.

There is a neater way to write the whole thing that makes the pure-number character obvious. Express the mass as a column density times an area, and the flux as a field times the same area; the area cancels, and the criterion becomes a comparison between a column density and a field strength with the gravitational constant as the only conversion. That is the form the figure above is drawn in, and it is why the critical locus is a straight line of slope exactly one.

What subcritical means #

A cloud below the critical ratio is supported for ever against collapse along the field. It is not supported against motion along the field lines: gas can slide down them freely, and does, which is what flattens such clouds into sheets and filaments.

So a subcritical cloud is not static. It contracts in one dimension, becomes flattened, becomes denser, and remains unable to collapse in the other two. That is the origin of the flat branch in the observed field–density relation: density rising at fixed field is exactly what one-dimensional contraction produces.

The consequence for star formation is that a subcritical cloud must first become supercritical, and there are only two ways. It can accumulate more mass along the same field lines, which raises the mass at fixed flux. Or it can lose flux, which requires the freezing to fail.

Both happen. Accumulation along field lines is what turbulent flows do; flux loss is ambipolar diffusion, slow and steady, at a rate set by the cosmic rays that keep a shielded core ionised.

The two routes leave different signatures and that is what makes the distinction testable. Accumulation raises the mass without changing the flux, so it moves a region along a line of constant field on the diagram at the head of this essay — rightward, at fixed height. Flux loss lowers the field at fixed mass, moving a region downward. A population of cores whose ratios were raised by accumulation should therefore lie at higher column at similar field, and one raised by diffusion at lower field at similar column.

The data are not yet good enough to separate those tracks, which is a reasonable summary of the observational state of this whole subject: the predictions are clean, the measurements are two-sigma detections of a part in ten thousand in polarisation, and the sample is a few dozen objects.

What is measured #

The measurement is a Zeeman splitting in a radio line — twenty-one centimetres for atomic hydrogen, eighteen centimetres for hydroxyl, or a millimetre-wave line of cyanide for the densest material — giving the line-of-sight field. Against it goes a column density from dust emission or from a molecular tracer.

Both quantities are hard and the combination has a particular pathology worth naming. The Zeeman measurement gives the line-of-sight component, and the column density is also measured along the line of sight, so the ratio is at least consistent in geometry. What it is not is a volume average, and a cloud whose field is inclined reports a ratio too high by the cosine of the inclination.

Statistically that is correctable: averaging over many clouds with random orientations gives a mean projection factor, and the standard correction is a factor of two or three in the ratio. The correction is the same kind of statistical de-projection that turns an ensemble of measured inclinations into a distribution of true masses. Applied to a single object it is not correctable at all.

What the corrected measurements say is that dense cores are supercritical by a factor of two or three, and the envelopes around them are close to critical or slightly below. That arrangement — supercritical inside, subcritical outside — is the signature the ambipolar picture predicts, because flux leaves the dense centre faster than the diffuse envelope. It is also what a random selection of sight lines through a tangled field would produce, and separating the two requires the statistics of a large sample.

There is one further complication that the projection correction does not address. A Zeeman non-detection is not the absence of a field; it is an upper limit, and the majority of published measurements are upper limits rather than detections. Treating a set of upper limits statistically requires assuming something about the distribution being sampled, and different assumptions have produced published mass-to-flux ratios differing by a factor of two on the same data.

The honest way to report it, and the way it is now usually reported, is as a distribution with the limits handled explicitly rather than as a mean. What survives that treatment is the ordering — cores above their envelopes — and the conclusion that cores are supercritical, though not by how much.

Where the criterion is challenged #

The main challenge to the criterion is not that it is wrong but that it may be irrelevant, and the argument is worth stating fairly.

If molecular clouds are dominated by supersonic turbulence — and they are observed to be — then their structure is a transient, produced by colliding flows and dispersed within a crossing time. On that picture a cloud does not sit quietly becoming supercritical over ten million years; it is assembled in a million, forms stars, and is destroyed.

Turbulence also does the accumulation directly. Gas driven along field lines by a converging flow piles up mass at fixed flux and can carry a region across the critical value in a crossing time, which is far faster than ambipolar diffusion. The two pictures make different predictions about lifetimes and about the fraction of clouds that are subcritical, and the observations sit between them. Cloud lifetimes inferred from the ages of associated stellar populations are a few million years, favouring the turbulent picture; the mass-to-flux measurements put envelopes near critical, favouring the magnetic one.

The current synthesis is that both operate: turbulence assembles and disperses, the field sets how much of the assembled gas can proceed — and the flux it must lose on the way sets the pace of what follows, and ambipolar diffusion matters in the densest, most shielded material where the turbulence has dissipated. That is a synthesis in the sense of containing both, rather than in the sense of being derived.

One thing both pictures agree on is what the field does higher up, and it is worth drawing at a field strength typical of an active region rather than of the quiet Sun.

The height of that crossing is the single number that decides how a magnetic structure behaves, and it moves by kilometres for a factor in the field because the gas pressure falls exponentially with height. Over the quiet Sun the crossing is in the low chromosphere, so photospheric motions shuffle the footpoints of field lines freely and the corona above is braided by convection. Over an active region it is at or below the surface, so the field is rigid all the way down and the convection is what gets pushed around instead — which is the same statement as a spot being dark.

That is why the criterion generalises upward as well as inward. A cloud’s mass-to-flux ratio, a spot’s darkness and a corona’s structure are three readings of one ratio taken at three densities, and in each case the question is which of two pressures is larger. Nothing about the specific physics of ionisation, opacity or self-gravity enters the comparison itself; those decide only where the crossing sits. It is the reason a criterion written for one context keeps turning up in another with the same critical number and a different name — plasma beta in the corona, the mass-to-flux ratio in a cloud, the magnetopause standoff at a planet. Each is the same question asked of a different pair of pressures, and each has a threshold at unity because a ratio of two pressures has nowhere else to put one. That is why the criterion arrives with no fitted constant, and why a measurement of it is a measurement rather than a calibration — the thing being tested is whether nature sits above or below a number that was never chosen.

What happens after the threshold is crossed #

The criterion decides whether collapse begins. What happens next is a different problem, and the field is still the awkward participant.

A supercritical core collapsing at frozen flux drags the field inward, and the geometry it produces is an hourglass: field lines pinched at the centre and fanning out above and below. That shape has been mapped in a dozen cores by polarimetry, and its degree of pinching is a measurement of how much the collapse has dragged the field. The collapse also has to shed the flux it is dragging, or the resulting star would carry four orders of magnitude too much. So ambipolar diffusion continues to operate through the collapse, faster now because the density is higher and the ionisation lower, and most of the flux is lost during the protostellar phase rather than before it.

And the field does something useful on the way, which is to remove angular momentum. A core connected to its envelope by field lines is braked by them, and that magnetic braking is the leading answer to the other conservation problem — a core’s spin, which would otherwise stop the collapse at a centrifugal barrier. The same quantity that has to be shed is the agent that sheds something else.

The consequence is a genuine tension in the models. Braking that is efficient enough to solve the angular momentum problem is efficient enough to prevent a rotationally supported disc forming at all, which contradicts the fact that discs exist. Resolving that — usually by invoking misalignment between the field and the rotation axis, or non-ideal effects — is one of the standing problems in star formation theory.

Why a threshold is worth more than a scale #

The criterion’s real value is structural, and it is the reason it survives all the caveats.

A cloud’s mass-to-flux ratio is set when the cloud is assembled and cannot change thereafter without a specific, slow, quantifiable process. So it partitions the material permanently: the gas that will make stars and the gas that will not are distinguished before anything happens, by a number that can in principle be measured.

Nothing else in the collapse problem does that. The Jeans mass tells whether a cloud is unstable now, and the answer changes as soon as the cloud moves. The virial ratio tells whether it is bound, and the same applies. The criterion also explains an otherwise puzzling fact about star formation efficiency. Only a few per cent of a molecular cloud’s mass becomes stars, and the rest disperses. A picture in which most of the cloud is subcritical and a small fraction is not accounts for that immediately, and it does so without requiring feedback to remove the rest. Whether that is the actual explanation is another matter — feedback plainly does operate — but it is a mechanism of the right size, available for free.

The same criterion at other scales #

The argument nowhere refers to the size of the object, so it should apply wherever a self-gravitating mass is threaded by a frozen field, and it does.

At the largest scale it applies to a whole galaxy’s gas disc, where the ratio of mass to flux is enormous — supercritical by orders of magnitude — which is why galaxies collapse and form stars at all rather than being held open by their fields — though the same field is a substantial part of what holds the gas layer thick. That is a check on the criterion in the trivial direction, and it is worth making because a criterion that condemned everything would be useless.

At the smallest scale it applies to the fragments a collapsing core breaks into. A fragment inherits its parent’s field, and if it takes a small fraction of the mass with a comparable fraction of the flux it inherits the parent’s ratio; if the geometry is such that it takes more flux than mass, it is subcritical and cannot collapse further. That is a magnetic contribution to setting a minimum fragment mass, alongside the thermal one.

And it applies to the first stars, where the answer is different. Primordial gas has essentially no field — whatever the seed was, it is far below anything dynamically relevant — so the earliest star formation is unmagnetised and the criterion does not bind. That means the first generation formed under a rule the later ones do not obey, which is one of several reasons its mass distribution is expected to have been different.

What is settled and what is not #

Settled: the ratio is conserved under flux freezing; the critical value is a pure constant; a subcritical cloud cannot collapse regardless of its temperature or density. These are consequences of two conservation laws and an inverse-square, and none of them depends on a model of anything.

Measured: cores are supercritical, envelopes are near-critical, and the field–density relation has a flat branch. The measurements carry a factor-of-two systematic from projection and a factor-of-two scatter from everything else. Unsettled: whether the criterion is what regulates star formation, or whether turbulence assembles supercritical regions so quickly that the subcritical state is never occupied for long. That question is about timescales rather than about the criterion, and it will be settled by measuring cloud lifetimes and ionisation fractions rather than by measuring more fields. Both are being measured: lifetimes from the ages of the stellar populations that clouds are next to, and ionisation from the abundances of molecular ions whose chemistry depends on it.

What should be carried away is the shape of the argument rather than its current status. Two conserved quantities, divided; a critical value that is a combination of nothing but the gravitational constant; and a partition of material that is fixed before the interesting physics starts. The flux problem this ladder began with asked where four orders of magnitude of field went. This rung asks a sharper question of the same conserved quantity — not how much is lost, but which side of a line the cloud was on before any of it was.