Support that cannot be squeezed away

The Jeans argument has a property that is so convenient it is easy to mistake for a law of nature: the threshold it sets can always be crossed. Squeeze a cloud and its density rises, and the critical mass falls as the inverse square root of the density, so a cloud that was stable becomes unstable and a cloud that was already unstable becomes more so. Compression is a one-way door.

The reason is a mismatch of exponents. Gravitational energy goes as $1/R$; thermal energy at fixed temperature goes as the logarithm of the volume, which is to say hardly at all. Gravity wins every contraction, and it wins by more each time.

A magnetic field is not like that, and the difference is not one of degree.

Why compression does not help #

The result turns on flux freezing, which is worth stating carefully because it is an idealisation and the rest of this essay is about how it fails.

In a perfectly conducting fluid the magnetic flux through any surface moving with the fluid does not change. Interstellar gas is not perfectly conducting, but it is ionised to a part in $10^{7}$, and the induction equation is the competition between carrying a field and letting it slip and the ions are collisionally coupled to the neutrals, so over short times the field goes where the gas goes. Contract a cloud and the field lines threading it are dragged inward with the matter, and the flux through it stays what it was.

Now count energies. The gravitational binding energy of a cloud of mass $M$ and radius $R$ goes as $GM^2/R$. The magnetic energy is the field energy density times the volume, and at fixed flux $\Phi = \pi R^2 B$ the field strength goes as $R^{-2}$, so the energy goes as $B^2R^3 \propto \Phi^2/R$.

Both go as $1/R$. Their ratio does not contain the radius at all, so it is the same before and after any compression whatever, and it depends only on

$$\lambda = \frac{M/\Phi}{(M/\Phi)_{\rm crit}},\qquad (M/\Phi)_{\rm crit} \simeq \frac{0.13}{\sqrt{G}} .$$

A cloud with $\lambda < 1$ — subcritical — cannot be made to collapse by squeezing it, or by cooling it, or by waiting. A cloud with $\lambda > 1$ — supercritical — will collapse regardless of the field, which merely slows it down. There is no continuum of difficulty between the two cases; there is a boundary, and which side of it a cloud sits on is fixed at the moment the cloud acquires its mass and its flux.

That is a qualitatively different kind of criterion from a Jeans mass, and it changes what the question about star formation is. It stops being when does a cloud collapse and becomes how does a cloud get rid of its flux.

The conclusion was reached in 1956 and it was reached as an objection. Mestel and Spitzer were not proposing a theory of star formation; they were pointing out that the interstellar field, at the strength then estimated, made star formation impossible. A cloud of the observed mass threaded by the observed field is subcritical by a comfortable margin, and by the argument above nothing in the cloud’s own evolution can change that. The paper’s conclusion was that either the field estimates were wrong or some process not yet identified removed flux from the gas.

Both halves turned out to matter, and the second became a subject. That is a useful shape to notice: an argument published as a difficulty becomes the framework in which the difficulty is answered, and the answer — ambipolar diffusion — was worked out over the following two decades by people taking the objection seriously rather than looking for a way round it.

The field is tied to a trace population #

The escape is that flux freezing is not exact, and the way it fails is specific.

The field is tied to the charged particles. The neutrals are not charged and feel no magnetic force at all; they feel gravity, and they feel the ions only through collisions. So a subcritical cloud is a two-fluid system in which the neutral gas is being pulled inward by gravity, the ions and the field are resisting, and the neutrals drift slowly past the ions toward the centre.

That drift is ambipolar diffusion, and its rate is set by how often a neutral meets an ion — which is set by how many ions there are.

The arithmetic that follows is the most surprising thing in this essay. The drift time is

$$t_{\rm AD} = \frac{3\gamma\rho_i}{4\pi G\rho},$$

with $\gamma$ the ion–neutral drag coefficient. Both densities are proportional to the total number density, so the density cancels exactly. The time a cloud takes to shed its field depends on its ionisation fraction and on nothing else about it — not its mass, not its size, not how far it has already collapsed.

Twenty, and why the number is a coincidence worth distrusting #

The ratio that comes out is the single most quoted number in this subject, and it deserves examination because it is an agreement between two quantities with no reason to agree.

Cosmic rays penetrate a molecular cloud and ionise its gas at a rate per particle that is roughly independent of density, because the cloud is transparent to them over the relevant column. Recombination removes ions at a rate proportional to the product of the ion and electron densities, which is the square of the ionisation fraction times the density squared. Setting the two equal gives an ionisation fraction proportional to $n^{-1/2}$.

That exponent is the whole coincidence. The free-fall time also goes as $n^{-1/2}$, for an entirely unrelated reason — it comes from $\sqrt{1/G\rho}$ and has nothing to do with chemistry. So the two times track each other exactly, and the factor between them, about twenty, is the same in a diffuse cloud and in a dense core.

A cloud supported by its field therefore takes about twenty free-fall times to shed it, wherever it is in its own collapse. That is a statement no single calculation predicts; it is the product of a nuclear-physics rate, an atomic recombination coefficient, an ion–neutral cross-section and Newton’s constant, and it comes out a small number.

The number is also the reason the theory was attractive for thirty years. Molecular clouds live for something like twenty to thirty million years and their free-fall times are one or two, so a factor of twenty is exactly what is needed to explain why only a few per cent of a cloud’s gas becomes stars before the cloud is destroyed. A theory that produced a factor of three or a factor of a thousand would have been discarded on sight.

That is a reason for suspicion as well as for confidence. A mechanism whose one free number lands on the observation is doing what it was selected to do, and the case for it has to rest on something else.

What is actually measured #

Three quantities in the argument are observable and each is measured badly, which is why the subject was argued about for three decades.

The field strength comes from the Zeeman effect, and it is the hardest of the three. The relevant lines are the 21 cm line of atomic hydrogen, which is also how the cold phase of the interstellar medium is weighed and the 18 cm lines of hydroxyl and, in dense gas, the 1.3 cm lines of cyanide. The splitting is a few hertz per microgauss against line widths of tens of kilohertz, so what is measured is not a split line but a tiny difference between the two circular polarisations, and the measurement yields the line-of-sight component only. A non-detection is therefore compatible with a strong field in the plane of the sky, and a detection has to be corrected for geometry by a statistical argument about orientation.

The column density is measured through a tracer, with all the difficulties that implies — a carbon monoxide brightness converted by a factor calibrated three ways, or a dust continuum converted by an assumed temperature and emissivity. Factors of two are routine.

The ionisation fraction is never measured directly. It is inferred from the abundance ratios of molecular ions — the deuterated forms of $\mathrm{HCO}^+$ and $\mathrm{N_2H}^+$ against their normal counterparts — through a chemical network with dozens of reactions whose rate coefficients are themselves uncertain.

What the measurements say, after all that, is that the mass-to-flux ratio in the envelopes of molecular clouds is around one to three times critical, and in the dense cores inside them it is higher. The gradient is in the right direction for the theory — the cores have shed flux relative to their surroundings — and the significance of it is marginal, because every one of the three quantities carries a factor-of-two systematic.

A field strength read off the scatter of a direction #

The Zeeman measurement is not the only route, and the other one is worth describing because it gets a field strength out of an observation that contains no field strength at all.

Interstellar dust grains align with their short axes along the local field, so the thermal emission from a cloud of dust is polarised perpendicular to the field, and a map of polarisation angles is a map of the field’s direction projected on the sky. A direction is all it gives — nothing in the polarisation fraction is a field strength.

What supplies the strength is the scatter. If the field were perfectly uniform every polarisation vector would point the same way. Turbulent motions bend the lines, and how far they bend them depends on the competition between the kinetic energy of the motions and the tension in the field: a strong field is stiff and the angles stay ordered, a weak one is pushed about and the angles scatter. Equating the two energies gives

$$B \simeq \sqrt{4\pi\rho}\,\frac{\sigma_v}{\delta\theta},$$

which is the Chandrasekhar–Fermi estimate — a field strength from a gas density, a line width, and the standard deviation of a set of angles.

Its virtues and its vices are both large. It works where Zeeman measurements do not, over whole clouds at once, and it gives the plane-of-sky component that Zeeman is blind to, so the two are complementary rather than competing. And it is a factor-of-two estimate at best: it assumes isotropic turbulence, a single dominant field direction along the line of sight, and that the observed angle dispersion is not diluted by averaging several independent regions within the beam — which it generally is, so the method systematically overestimates the field unless corrected.

Two measurements of the same quantity with unrelated systematics, neither of which can be improved by observing longer, is the situation this subject has been in since the 1990s.

The exponent that tests the assumption #

There is one test of flux freezing that does not require believing any single measurement, and it comes from the shape of the relation rather than its normalisation.

If a cloud contracts spherically with its flux frozen, the field goes as $ho^{2/3}$: mass conserved in a shrinking sphere gives $ho \propto R^{-3}$, flux conserved gives $B \propto R^{-2}$. If it contracts along the field lines only — which is what a subcritical cloud does, since the field opposes motion across the lines and not along them — the field does not change at all and $B$ is independent of $ho$.

So the measured $B$–$ho$ relation is a statement about the geometry of the collapse, and it has now been compiled over six decades in density.

What it shows is two regimes. Below about $300$ particles per cubic centimetre the maximum field is roughly constant with density, which is the flattening case: the gas is subcritical, it settles along the lines, and the flux does not change. Above that the maximum field rises as roughly $ho^{0.65}$, which is close to the spherical value of two-thirds and not close to zero.

That is the argument’s best piece of evidence and it is an exponent rather than a number. It says that at low density the gas is being supported and rearranged by the field, and at high density it is dragging the field in with it, and the transition is where clouds become supercritical. Neither half depends on the absolute calibration of any Zeeman measurement, because a slope survives a systematic that a normalisation does not.

What replaced it, and what survived #

The magnetic picture was the standard account of star formation into the 1990s, and it lost its position without being shown to be wrong.

What displaced it was the recognition that molecular clouds are supersonically turbulent and that the turbulence does much of what the field was invoked for — supporting the cloud as a whole while producing dense regions inside it. The turbulent picture predicts cloud lifetimes of a few free-fall times rather than twenty, and the observational case has moved in that direction.

But the field did not go away, and three things from this argument survive intact.

The invariance is a theorem, not a model. Whatever else is happening, a subcritical cloud cannot be made to collapse by compression, and any account of star formation has to explain how clouds become supercritical.

The ambipolar time is still the flux-loss time, and it is what sets how much field a collapsing core carries inward. That matters enormously downstream: a core that kept all its field would produce a protostar with a field of $10^{7}$ gauss, which is six orders of magnitude above what young stars have, and a young star’s rotation is set by the field it keeps. The magnetic flux problem — where the other six orders of magnitude go — is unsolved, and ambipolar diffusion is one of the two candidates.

And the field removes angular momentum, by magnetic braking along the lines connecting a contracting core to the medium around it. That is a separate problem from support and it is one the turbulent picture does not address at all.

Where the picture stops #

The critical ratio depends on geometry and the geometry is assumed. The number $0.13/\sqrt{G}$ is for a uniform sphere flattened along the field; a uniform disc gives $1/2\pi\sqrt{G}$, which is twenty per cent larger, and a centrally concentrated cloud gives something else. Since the measurements sit within a factor of two of the boundary, a twenty per cent uncertainty in where the boundary is is not negligible.

Turbulence and fields are not separable. Supersonic motions in a magnetised medium generate field structure, and a field changes what the turbulence does — it makes the cascade anisotropic and it carries waves that the hydrodynamic picture has no analogue for. Treating one as support and the other as a correction is a convenience of exposition rather than a division the physics respects.

And ambipolar diffusion is not the only way to lose flux. Turbulent reconnection, in which field lines of opposite sense are brought together by the motions and annihilate, removes flux at a rate that does not depend on the ionisation fraction at all — so the twenty free-fall times above is an upper limit on how long a cloud has to wait rather than a prediction of it.

The habit worth extracting #

There is a generalisation here about supports, and it is worth stating because it recurs in several places here.

A support that scales more weakly than gravity under compression can always be overcome, and the system’s fate is decided by whether it is currently above a threshold. A support that scales the same way cannot be overcome at all, and the system’s fate is decided by a ratio it cannot change. The two kinds of support produce completely different subjects: one is about states and the other about histories.

The same distinction appears with degeneracy pressure in a white dwarf, which scales as $R^{-4}$ against gravity’s $R^{-2}$ in the non-relativistic case and defeats it at every radius, and as $R^{-1}$ exactly in the relativistic case, at which point a ratio decides everything and the Chandrasekhar mass exists. Wherever two energies share an exponent there is a critical number rather than a critical size, and the number is a property of the object rather than of its circumstances.

Rotation is the third case and it goes the other way: angular momentum conservation makes rotational energy go as $R^{-2}$, which beats gravity’s $R^{-1}$, so rotation is a support that strengthens under compression and cannot be beaten by squeezing either — which is why a collapsing cloud makes a disc rather than a point, and why the angular momentum has to be exported before anything can accrete.

Still open: where the six orders of magnitude of flux go #

A dense core threaded by a field of a hundred microgauss, contracted to the size of a young star with its flux intact, would arrive with a surface field of order $10^{7}$ gauss. Young stars have fields of a few kilogauss. Somewhere between the core and the star, a factor of $10^{4}$ in field strength is lost, and there is no agreed account of where.

Ambipolar diffusion in the core removes some of it and not that much. Magnetic reconnection in the disc removes more, at a rate nobody can compute from first principles because it depends on the resistivity of a partially ionised, dusty, turbulent gas. Ohmic dissipation takes over at the very highest densities, where the ionisation fraction falls low enough for the field to slip freely.

Each of the three dominates a different density range and the ranges are known only roughly, so the flux a star is born with is a prediction nobody makes. Since the field is also what launches the outflows that end a star’s accretion and what sets its early rotation, the gap is not a detail at the end of the story — it is in the middle of it.