Astronomy & Celestial Mechanics
the sky, computed
Concept

Flux freezing — where it appears

The condition, holding when the magnetic Reynolds number is large, under which a magnetic field line moves with the fluid as if it were a material object. The flux through any surface carried along with the flow is then exactly conserved, which is why a collapsing cloud would arrive with far more field than any star is observed to have.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

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.

stars · Flux freezing 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.

The seed that cannot be remembered

A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters.

cosmology · Primordial fields A radial wind from a rotating star draws a spiral. The interplanetary field out to 5 astronomical units, drawn as the Archimedean spiral it is. Nothing here rotates: the plasma moves radially outward at 400 kilometres a second and the field is frozen into it, so each parcel remembers the longitude it left from and the pattern winds up while the material does not. The pitch angle is arctan(Ωr/v), which is 47° at one astronomical unit — the radius where the star's rotation has carried the footpoint through one radian in the time the wind takes to arrive. The practical consequence is a matter of hours: a flare's particles follow the field rather than the line of sight, so the ones that reach a given planet left a longitude about 61° to the west of it. A magnetically well-connected flare on the western limb delivers a particle storm and a larger one at disc centre does not, and the difference is this geometry rather than anything about the flare.

The flare that arrives from somewhere else

The solar wind blows radially outward and the Sun rotates, so the field frozen into the wind is wound into a spiral making forty-five degrees to the radius at the Earth. Energetic particles follow the field rather than the line of sight, which is why the flares that deliver particle storms are the ones on the western limb rather than the ones facing the planet.

sky · Stellar winds One dimensionless number decides whether a cloud may collapse. Field strength against hydrogen column density, with three loci of constant mass-to-flux ratio. The ratio of mass to magnetic flux is conserved under flux freezing, so it cannot be changed by anything that happens during a collapse — which is what makes it a criterion rather than a description. Its critical value is the pure constant 1/(2π√G), and dividing by that gives the dimensionless λ drawn here: below λ = 1 the field can hold the cloud up for ever, however cold it gets, because gravity and the magnetic force scale the same way with radius; above it no field strength suffices. Each locus is a straight line of slope exactly one, because at fixed λ the required field is exactly proportional to the column. Zeeman measurements of dense cores put them a little above the line and their envelopes a little below it, which is the arrangement a slow leak of flux out of the centre would produce and is the observational case for ambipolar diffusion.

A threshold with no free parameter in it

Divide a cloud's mass by the magnetic flux threading it. Gravity and the magnetic force both fall as the inverse square of the radius, so the ratio cannot change during a collapse — and the critical value that separates a cloud which must collapse from one that never can is one over two pi root G, a pure constant.

stars · Star formation The one number a cloud cannot change by squeezing. The mass-to-flux ratio in units of its critical value, against column density, for clouds threaded by fields of 3, 10, 30, 100 microgauss. λ below one is subcritical, and the field alone holds the cloud up, and no amount of compression changes that, because squeezing raises the magnetic and the gravitational energy at the same rate and leaves their ratio exactly where it was. λ above one is supercritical and the field is irrelevant to whether the cloud collapses. Each line has slope exactly one because λ is proportional to the column density at fixed field, and each crosses the boundary at 3.9·10²⁰ cm⁻² for 3 µG, 1.3·10²¹ cm⁻² for 10 µG, 3.9·10²¹ cm⁻² for 30 µG, 1.3·10²² cm⁻² for 100 µG. The Jeans mass is a threshold a cloud can cross by contracting; this one is a label it is born with, and the only way past it is to let the field leak out.

Support that cannot be squeezed away

A cloud held up by pressure can always be defeated by compressing it, because gravity gains faster than heat does. A cloud held up by a magnetic field cannot — squeezing raises both energies at exactly the same rate, so the ratio a cloud is born with is the one it keeps, and the only way out is to let the field leak.

galaxies · Molecular clouds

Named alongside it

The objects these essays reach for when they reach for this one.

Ambipolar diffusionMass-to-flux ratioZeeman effectCosmic raysJeans massMagnetic critical massMolecular cloudsStar formationAlfven radiusBiermann batteryBlazarCore-collapse

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