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A field that would have arrived ten thousand times too strong

A collapsing cloud carries its magnetic field with it, and the arithmetic of that is not negotiable — the field grows as the two-thirds power of the density. Run it from a diffuse cloud to a protostellar core and the answer is four orders of magnitude above anything measured on a young star.

Assumes Molecular clouds and Star formation.

18 min read 10 figures Scales that defeat intuitionMeasuring the unreachable

A molecular cloud is threaded by a magnetic field of about ten microgauss. That is a hundred thousand times weaker than a refrigerator magnet, it exerts a pressure a laboratory would struggle to measure, and it is the single most awkward fact in the theory of how stars form.

The awkwardness is not the field’s strength. It is what happens to that strength when the cloud collapses.

The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 10 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 1 Field strength against density over the eight decades between diffuse interstellar gas and the core of a forming star. The steeper line is what a collapse conserving its magnetic flux must produce; the shallower one is what measurements find. By the density at which a protostar exists the two differ by four orders of magnitude, and the whole of the argument below is about where that factor went.

Two terms, and the number that decides between them

The evolution of a magnetic field in a moving conductor is written in one equation with two terms on the right. The first carries the field along with the fluid; the second lets it diffuse through the fluid and decay. Which of the two matters is settled by their ratio, and the ratio is a pure number: the flow speed times the size of the system, divided by the magnetic diffusivity.

The one number that decides whether a field is a fluid. The magnetic Reynolds number, vL/η, for five plasmas, on a logarithmic axis spanning 22 decades. It is the ratio of the term in the induction equation that carries a field with the flow to the term that lets it slip through, so a large value means a field line is a material line — it moves with the gas, it cannot break, and its flux through any surface carried along with the fluid is conserved. Astronomical values run from 10⁹ to 3·10²⁰; the laboratory value is 100, which is why the behaviour that dominates every plasma in this collection is the one that is hardest to arrange on a bench. The size of the number is doing the work: at 10¹⁶ the diffusion time across a molecular cloud is ten million times the age of the universe, so freezing is not an approximation to be checked but a constraint to be worked around, and the interesting question everywhere below is where and how the coupling is allowed to fail.
Fig. 2 The magnetic Reynolds number for five plasmas. The astronomical values are enormous and the laboratory value is not, which is why the behaviour that governs every object in this collection is the one that is hardest to arrange on a bench.

In a molecular cloud that number is around ten to the sixteenth. The consequence is worth stating without the equation: the field is stuck to the gas. A field line is not a mathematical convenience there but a material object — it moves where the gas moves, it can be stretched and folded, it cannot be cut, and the total flux threading any surface carried along with the fluid stays exactly what it was.

The size of the number is doing all the work, and it is worth dwelling on why it is so large. The diffusivity in the denominator is set by the collision rate of electrons with everything else, and it has units of length times speed; in a partly ionised gas it comes out at something like ten to the third or fourth in centimetre-gram-second units, which is not a small number. What overwhelms it is the length. A cloud is three parsecs across, which is ten to the nineteenth centimetres, and any quantity multiplied by that is enormous compared with a laboratory. This is the recurring reason that astrophysical plasma behaves nothing like the plasma in a discharge tube, and it applies again to the turbulence that transports angular momentum in an accretion disc and to the storage of cosmic rays in the galactic disc.

That is called flux freezing, and it is not an approximation to be checked. At ten to the sixteenth the time for the field to slip through the gas by one cloud radius exceeds the age of the universe by six orders of magnitude. Nothing that happens on any timescale relevant to star formation can undo it.

The arithmetic of a frozen collapse

Take a spherical cloud and let it collapse. Two things are conserved. The mass inside the sphere is fixed, so the density goes as the inverse cube of the radius. The magnetic flux through the sphere’s cross-section is fixed, so the field goes as the inverse square of the radius. Eliminate the radius between them and the field goes as the two-thirds power of the density.

There is no free parameter in that exponent. It is the ratio of a two to a three, and both integers come from counting dimensions.

It is worth being precise about what the sphere assumes, because a real collapse is not spherical and the exponent is sensitive to that. If the cloud instead flattens into a sheet perpendicular to the field, compressing only along the field direction, the cross-sectional area threaded by the field never changes and the exponent is zero. If it contracts only across the field, forming a filament along it, the exponent is one. Two thirds is the isotropic case, and it is the largest exponent a collapse can produce without amplifying the field by shear. So two thirds is not merely a plausible value; it is the ceiling on what conserved flux can deliver, and the discrepancy below is therefore a lower bound on the flux that has to be lost.

The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 10 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.333 power of the density above it. By the density of a core the two differ by a factor of 47, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 3 The same pair of lines with the observed branch drawn at one third rather than two thirds — a collapse halfway between the spherical case and the flattened one. The gap at protostellar density widens from four orders of magnitude to more than six, which is the arithmetic behind the sentence above: any geometry flatter than a sphere makes the discrepancy larger, not smaller, because it removes field amplification without removing any density. Geometry cannot be the escape route. Whatever is happening has to actually destroy flux rather than merely rearrange it, and the flattest imaginable collapse only sharpens the requirement.

Run it. A diffuse cloud at a hundred particles per cubic centimetre with a ten-microgauss field, collapsed to the ten-to-the-tenth of a protostellar core, gains eight decades of density and therefore five and a third decades of field. Ten microgauss becomes about two gauss, spread over the whole of a forming star. Measured fields on young stellar objects are a few kilogauss over a fraction of the surface, which sounds larger until the flux is worked out: the flux a star of that size carries is roughly four orders of magnitude less than the cloud handed it. The dimensionless statement of the same fact is more useful than the scaling. Divide a cloud’s mass by the magnetic flux through it. Gravity pulls as mass squared over radius squared; the magnetic force pushes as flux squared over radius to the fourth, times radius squared. Both go as the inverse square of the radius, so their ratio is fixed for ever, and a cloud whose mass-to-flux ratio is below the critical value is supported against its own gravity permanently, at any temperature, at any size. The critical value is one over two pi root G — a constant, with nothing fitted in it.

What a telescope actually measures

The field in a cloud is not photographed. It is inferred, and the inference used here is the Zeeman effect on the twenty-one-centimetre line of neutral hydrogen or on the eighteen-centimetre lines of the hydroxyl radical. The splitting these fields produce is minute — parts in ten thousand of the line width — and it cannot be seen in the intensity. It is seen in Stokes V, the circularly polarised component, whose shape is not free: in the weak-field limit it must be a fixed multiple of the derivative of the intensity profile, and the multiple is the line-of-sight field. So the measurement is a fit of a known shape with one free amplitude, which is why fields of a few microgauss are recoverable at all. The same argument in a stellar photosphere is the standard technique for measuring a star’s field, and it is being used here two thousand times further down in sensitivity.

Three cautions come with every such measurement and all three matter below. It gives the line-of-sight component only, so a field lying in the plane of the sky reports zero. It averages along the whole path, so reversals cancel. And it is a detection at two or three standard deviations in most clouds, which means the distribution of measured values carries information that no single value does.

The field a collapse would arrive with, and the one it has. Field strength against hydrogen density, both logarithmic, over the eight decades between diffuse gas and a protostellar core. The steeper line is what perfect flux freezing demands: a sphere collapsing conserves both mass and flux, so B goes as R⁻² while ρ goes as R⁻³, and therefore B goes as the two-thirds power of the density exactly — an exponent with no free parameter in it. The shallower locus is what Zeeman measurements find: a flat branch at about 3 µG up to 300 per cubic centimetre, where the density is rising and the field is not, and a rise as the 0.65 power of the density above it. By the density of a core the two differ by a factor of 1, and a star built at the frozen-flux value would carry a field four orders of magnitude beyond anything measured on one. The flat branch is the important half: it says the gas is moving along field lines without dragging them, which is what gravity does to a cloud that is still magnetically supported, and it locates where the freezing has to break.
Fig. 4 What a non-detection is worth, drawn as one. Set the diffuse field to three microgauss rather than ten — the level at which a Zeeman measurement in a typical cloud returns an upper limit rather than a value — and both lines drop together while the gap between them stays exactly what it was. That is the useful property of the argument: it is about a ratio of a measured field to a predicted one, so a systematic error in the field scale cancels out of it, and the four-order-of-magnitude discrepancy survives being wrong about the absolute calibration by a factor of three. The upper limits, which are most of the data, therefore constrain the flux problem almost as tightly as the detections do.

The plane-of-sky component is reachable by an entirely different route — the alignment of dust grains, which reports a direction and no strength at all — and the two techniques are so nearly disjoint that they are usually reported separately rather than combined. A cloud with a Zeeman detection and a polarisation map has been measured twice by methods sharing no physics, which is the closest thing to a check available.

The flat branch, which is the actual finding

Assemble a few hundred such measurements against density and the result is not a single power law. Below about three hundred particles per cubic centimetre the field does not rise with density at all: it sits at ten microgauss or so across two decades of density. Above that it rises, with a best-fitting exponent near two thirds — the analysis that gives 0.65 is a statistical one over upper limits and detections together, and the honest statement is that it is consistent with the frozen-flux value and inconsistent with zero.

The flat branch is where the argument lives, and it is easy to skim past. Gas is being compressed by a factor of a hundred and the field is not responding. Under flux freezing that is impossible in a spherical collapse — but it is exactly what happens if the gas moves along the field lines rather than across them. A cloud that is still magnetically supported can only contract in one direction, sliding down the field like beads on a wire, and sliding beads compress no wires.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 5 The other criterion a cloud has to satisfy, drawn for comparison. The Jeans argument asks whether thermal pressure can hold a mass up and gives a threshold that depends on temperature and density. The magnetic criterion asks a different question and gives a threshold that depends on neither, which is why a cloud can be subcritical and Jeans-unstable at once, and simply sit there.

So the two branches are two geometries rather than two physical regimes: a flattened, field-supported contraction that changes the density and not the field, and a three-dimensional collapse that changes both. The break between them is where the cloud stops being supported.

That reading makes a prediction, and it is one the maps bear out. If the flat branch is gas sliding along field lines, the resulting structures should be flattened perpendicular to the field, and the field direction should be systematically perpendicular to the long axis of the densest material. Polarisation maps of nearby clouds show exactly that geometry in the quiescent regions and lose it in the actively collapsing ones, where the field is dragged into an hourglass pinched at the centre. The hourglass is the visible signature of a field that has stopped being able to resist, and it appears at about the density where the two branches meet.

Cosmic rays set the clock

Something has to let the field go, and the mechanism has a name that undersells it. Ambipolar diffusion is the slow drift of neutral gas through the ions, which alone are tied to the field.

The neutral molecules feel gravity and not the field. The ions feel the field and not, appreciably, gravity. What couples them is collisions, and in a dense core the fraction of the gas that is ionised is around one part in ten million. So the neutrals fall inward past a nearly stationary ionic scaffold, dragging the field only through the friction of those rare encounters, and the field is left behind at a rate set by how rare the encounters are.

The leak that lets a cloud go. Two timescales against the fractional ionisation of a dense core. The rising line is the ambipolar diffusion time — how long the neutral gas takes to drift through the ions that are tied to the field — and it is proportional to the ionisation fraction, because a rarer ion is collided with less often and holds the neutrals less firmly. The flat line is the free-fall time at 10⁴ molecules per cubic centimetre, which knows nothing about the field. They cross at an ionisation of 1.8e-6. Cosmic rays keep the ionisation of a shielded core near 1e-7, which puts the drift time 0 times the free-fall time — slow enough that the core is supported and fast enough that it is not supported for ever. That is the resolution of the flux problem and it is a quantitative one: the field leaks out on a schedule set by how many ions the cosmic rays make, so the star formation rate of a magnetised cloud is set by a particle flux arriving from outside the galaxy.
Fig. 6 The drift time against the fractional ionisation, with the free-fall time drawn flat for comparison. At the ionisation a shielded core actually has, the drift is about ten times slower than free fall — slow enough that the core is supported, fast enough that it is not supported for ever.

The drift time is proportional to the ionisation fraction, so the whole schedule of star formation in a magnetised cloud reduces to one question: what keeps a cold, dark, dust-shielded core ionised at all? Ultraviolet light does not reach it. The answer is cosmic rays, which penetrate grams per square centimetre of material and maintain an ionisation of a part in ten million more or less independently of depth.

The leak that lets a cloud go. Two timescales against the fractional ionisation of a dense core. The rising line is the ambipolar diffusion time — how long the neutral gas takes to drift through the ions that are tied to the field — and it is proportional to the ionisation fraction, because a rarer ion is collided with less often and holds the neutrals less firmly. The flat line is the free-fall time at 10⁴ molecules per cubic centimetre, which knows nothing about the field. They cross at an ionisation of 1.8e-6. Cosmic rays keep the ionisation of a shielded core near 1e-8, which puts the drift time 0 times the free-fall time — slow enough that the core is supported and fast enough that it is not supported for ever. That is the resolution of the flux problem and it is a quantitative one: the field leaks out on a schedule set by how many ions the cosmic rays make, so the star formation rate of a magnetised cloud is set by a particle flux arriving from outside the galaxy.
Fig. 7 The same two timescales at a core ten times more thoroughly shielded, where the ionisation has fallen to a part in a hundred million. The drift time falls with it — proportionally, since the coupling is by collision — and the core that was supported for ten free-fall times is now supported for one. Deeper shielding does not merely slow the ionisation; it releases the collapse. So the cosmic-ray flux is not a background condition of star formation but a control on its rate, and a cloud in a galaxy with a different cosmic-ray flux forms stars on a different schedule for no reason internal to the cloud at all.

That is a strange chain of causation and it is worth stating plainly. The rate at which a molecular cloud turns itself into stars is set by the flux of relativistic particles arriving from elsewhere in the galaxy — particles whose own confinement depends on the same magnetic field that the ionisation is holding in place.

Where the problem came from

The flux problem was posed before there was any measurement to pose it against. In 1956 Leon Mestel and Lyman Spitzer worked out the consequence of flux freezing for a collapsing cloud and concluded that stars could not form at all — that the interstellar field, if carried inward, gives every prospective star a magnetic energy exceeding its gravitational binding energy by orders of magnitude. Their paper proposed the resolution in the same breath: the ions and the neutrals must be able to slip past one another, and they estimated the rate.

What is striking in retrospect is how much of that was arrived at with almost no data. The interstellar field strength was known to within an order of magnitude from the polarisation of starlight, discovered eight years earlier by chance during a search for something else. There were no Zeeman detections in clouds; the first came two decades later. The cosmic-ray ionisation rate that sets the drift time was not measured until the molecular era. The argument was made from a conservation law, a dimensional estimate, and the refusal to accept a conclusion that contradicted the existence of stars — and it has survived seventy years of measurement essentially unaltered in structure, with the numbers filled in.

Where the argument is weakest

Three things are unsatisfactory and the honest version says so.

The first is that turbulence does much of what ambipolar diffusion was invented to do. A supersonically turbulent cloud reconnects and tangles its field constantly, and the effective diffusion in such a medium is faster than the laminar estimate by an amount that is still argued about. Clouds are observed to be supersonically turbulent, so the laminar drift time is an upper bound rather than a prediction.

The leak that lets a cloud go. Two timescales against the fractional ionisation of a dense core. The rising line is the ambipolar diffusion time — how long the neutral gas takes to drift through the ions that are tied to the field — and it is proportional to the ionisation fraction, because a rarer ion is collided with less often and holds the neutrals less firmly. The flat line is the free-fall time at 10⁴ molecules per cubic centimetre, which knows nothing about the field. They cross at an ionisation of 1.8e-6. Cosmic rays keep the ionisation of a shielded core near 1e-6, which puts the drift time 1 times the free-fall time — slow enough that the core is supported and fast enough that it is not supported for ever. That is the resolution of the flux problem and it is a quantitative one: the field leaks out on a schedule set by how many ions the cosmic rays make, so the star formation rate of a magnetised cloud is set by a particle flux arriving from outside the galaxy.
Fig. 8 The objection drawn as what it does to the schedule. Turbulent reconnection does not change the ionisation, but its effect on the drift is the same as raising it: the field slips faster than the laminar rate, and reading the figure at a part in a million rather than a part in ten million is a fair proxy for that. At this point on the curve the drift time is comparable to the free-fall time, and the whole magnetic pacing of star formation disappears — the cloud collapses on the gravitational timescale and the field is simply carried away with whatever cannot keep up. That is the alternative theory, and the figure shows how little has to change for it to hold.

The second is the measurement itself. If clouds were uniformly subcritical the Zeeman detections would cluster near the critical line; if uniformly supercritical they would sit well above it. What is seen is cores just above and envelopes just below, which is what a slow leak of flux out of the centre produces — but it is also what a selection of lines of sight through a tangled field produces, and separating those requires the statistics of the whole sample rather than any one object. The third is that the observed exponent is a fit to a scatter plot spanning eight decades, with a large majority of the points being upper limits. Two thirds and 0.65 are not distinguishable in that data. What the data do establish, and establish firmly, is the flat branch and the four-order-of-magnitude discrepancy at the top — and those are the two facts the theory has to account for.

The same shape as the other problem

A cloud that collapses at frozen flux arrives with far too much field. A cloud that collapses at conserved angular momentum arrives spinning far too fast, stopping at a centrifugal barrier near the orbit of Neptune. The two statements are the same statement about two different conserved quantities, and they have the same resolution: the conservation law must fail, and the interesting physics is the rate at which it does.

There is even a common mechanism. The magnetic field is what carries the excess angular momentum outward — a core still tied to its envelope by field lines is braked by them, and the same lever that slows a star’s rotation over its lifetime works on a collapsing core in its first hundred thousand years. So the field is simultaneously the quantity that has to be shed and the agent that sheds something else. It is not a coincidence: both problems are about a quantity that couples the collapsing material to material it is trying to leave behind. There is a second endpoint that puts a number on the losses from the other direction. The magnetic white dwarfs, of which a few per cent of the population qualifies, carry fields of a million to a billion gauss; run those back to the main sequence at conserved flux and they correspond to a few thousand gauss on a star of ordinary size, which is what the magnetic A stars have, and those are also a few per cent of their population. Two independent counts agreeing on a fraction is a stronger argument than either field strength alone, and it says that once a star exists its flux is conserved rather well. The losses all happen before it does.

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.
Fig. 9 For contrast, what happens where flux is not conserved. A galactic dynamo regenerates a field rather than carrying one, and its exponential budget erases the memory of whatever it started from — the opposite situation to the one this essay is about, and the reason the two subjects need different instruments.

What survives the objections

Strip away what is uncertain and a firm residue remains, and it is worth separating from the rest.

Flux freezing holds in interstellar gas to an accuracy of many orders of magnitude, and is not in doubt. The mass-to-flux ratio is therefore a conserved, dimensionless label with a critical value that is a pure constant, and no fitting enters it. Measured fields in dense cores are four orders of magnitude below what a frozen-flux collapse from cloud densities would deliver, so flux is lost, and the loss is large. The observed field-density relation has a flat branch, which means the loss is not uniform: something happens at a few hundred particles per cubic centimetre.

What is genuinely open is the rate, and therefore whether star formation in a galaxy is paced by magnetic diffusion or by turbulent dissipation.

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 10⁹ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 13 billion years the budget is 13 e-foldings, which is a factor of 4·10⁵. That is the finding: the three tracks reach the same ceiling within 18.4 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.
Fig. 10 The contrast case at a slower turnover, and the reason this essay’s problem has an answer at all. A dynamo running one e-folding per billion years for the age of the universe delivers thirteen e-foldings — a factor of four hundred thousand — and the three seed fields eight orders of magnitude apart still have not converged. The star-formation problem is the opposite situation and is better off for it: flux is conserved to many orders of magnitude, so the field a core arrives with is a calculation from the field its cloud had, and the discrepancy between that calculation and the measurement is a number rather than an unknown. A quantity that is exactly conserved is a quantity whose loss can be counted.

There is a way of putting the whole difficulty that makes clear why it has taken seventy years. The quantity in dispute is not observable at either end of the process. The field a cloud starts with is measured at two or three standard deviations in a handful of objects; the field a young star ends with is measured on its surface, over a fraction of its area, at an epoch when accretion is still going on. Between them is a collapse by six orders of magnitude in radius which nobody has watched and which takes longer than the history of the observations. Every statement about flux loss is therefore an inference from two badly measured endpoints and a conservation law, and the conservation law is the only part that is exact.

That is worth saying because it explains why the flat branch matters so much more than its statistical weight suggests. It is the one part of the story that is directly observed rather than inferred: a set of measurements at a range of densities, showing a field that does not respond to compression over two decades. Whatever the eventual account of flux loss turns out to be, it has to produce that branch, and most simple accounts do not — a slow, steady leak operating at all densities would give a single power law with an exponent below two thirds, not a flat stretch followed by a rise. That is a live question, and it is a quantitative one rather than a conceptual one, which is a considerable improvement on where the subject was when the problem was first stated.

The distinction has consequences well outside one cloud. If the pace is magnetic, the efficiency with which a galaxy converts gas into stars should depend on its cosmic-ray flux and therefore on its own star formation rate, which is a feedback loop; if the pace is turbulent, it should depend on how the gas is stirred, which is a different loop with different consequences for how quickly a galaxy exhausts its gas. Both loops are slow and both give a low efficiency, which is why the observed efficiency — a few per cent per free-fall time — does not by itself decide between them. The measurement that would settle it is the one this essay has leaned on throughout and which remains desperately hard: enough Zeeman detections, in enough cores, to turn a scatter plot into a distribution. Every field strength quoted here came from a circular polarisation of a few parts in ten thousand in a radio line, and the whole structure of the argument rests on that one signal being real. The next rung of this ladder takes the same conserved quantity and asks not how much flux is lost but whether a given cloud is allowed to collapse at all, which turns out to be a threshold with no free parameter in it.

Where this goes next

A collapse that stops 29 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 2.4·10¹⁶ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 4.27·10¹² metres — 29 astronomical units, or 6135 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 1.7·10⁴ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse. A cloud that cannot become a star A molecular cloud core turning once every twenty million years sounds like a body at rest. Conserve its angular momentum through a collapse by a factor of ten thousand and it stops at the orbit of Neptune, spinning, having failed to make anything at all. The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting. The cloud that cannot hold itself up A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star. A 3-gauss field moves the line by 0.33% of its width and is measured anyway. Above: the Fe I 6173 Å line, Landé factor 2.5, at a Doppler width of 0.041 Å. The solid curve is the unmagnetised profile; the dashed curve is the same line in a longitudinal field of 3 gauss, which splits it by 1.3·10⁻⁴ Å — 0.33 per cent of its own width — and is drawn on top of it because the two are not distinguishable. The double-lobed curve underneath them is the Stokes V profile of that same field, magnified 100 times: the two σ components are circularly polarised with opposite handedness, so what is lost in the sum survives in the difference, and the difference of two profiles a hair apart is the derivative of one of them. Below: what that buys. The V amplitude is linear in the field, because it is a first derivative; every signature of the same field in the intensity is quadratic, because a symmetric splitting can only broaden. At a polarimetric precision of 10⁻⁴ the first reaches 0.1 gauss; at a line-width accuracy of 0.001 the second reaches 38, a factor of 420 worse. A quantity far too small to resolve is measured because it is the only thing in the signal that carries a sign — and the same argument run the other way says what polarimetry cannot do: a field of mixed polarity inside the resolution element cancels in V and does not cancel in I, so the 3000-gauss field of a sunspot, which does resolve, is measured the other way round. A field strength read off a line that will not split A few hundred gauss splits a spectral line by a ten-thousandth of its own Doppler width, which no spectrograph will ever resolve. The two components are circularly polarised with opposite handedness, so the split survives in the difference of two polarisations — where it is linear in the field rather than quadratic.

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Ambipolar diffusionCosmic raysFlux freezingInduction equationIonisationMagnetic fluxMagnetic reynolds numberMass-to-flux ratioMolecular cloudsStar formationZeeman effect