Astronomy & Celestial Mechanics
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Flux freezing — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · stars
  2. Field strength against radius, from the Sun to a magnetar. Surface magnetic field against radius, both on logarithmic scales, radius in solar radii and field in gauss, for five kinds of star: the Sun's dipole, Ap and Bp stars, magnetic white dwarfs, radio pulsars, magnetars. The diagonal lines are constant magnetic flux, πR²B = 10²³, 10²⁵, 10²⁷ gauss square centimetres, along which a star shrinking with its field frozen in would slide. The Ap and Bp stars, the magnetic white dwarfs and the magnetars all fall close to the upper lines: a field of a few kilogauss on a star of three solar radii, squeezed to the size of a white dwarf, becomes a hundred million gauss, and squeezed to twelve kilometres becomes a hundred trillion. The Sun's dipole and the ordinary pulsars carry a hundred to a thousand times less flux.

    The same flux in three sizes

    A few per cent of hot main-sequence stars carry kilogauss fields that do not change over decades. So do a similar share of white dwarfs, at a million times the strength, and of young neutron stars, at a hundred billion times it. Multiply each field by its star's area and the three classes carry nearly the same magnetic flux — what a field frozen into a star would keep as the star shrank by a factor of a hundred thousand. A collapse makes a field enormous, but it cannot make it important.

    part 2 · stars
  3. The share of white dwarfs whose centres have begun to freeze. The fraction of a population of carbon–oxygen white dwarfs, with masses spread as a Gaussian of mean 0.62 and width 0.12 M☉, whose centres have begun to crystallise, against cooling age, overall and in three mass ranges, in the toy model of the other figures. If a field appears when freezing begins, this is the shape its incidence must take: it switches on first among the heaviest white dwarfs, half of which have begun freezing by 0.5 Gyr, and last among the typical ones near 0.6 M☉, by 2.5 Gyr. A fossil field, present from the white dwarf's birth, would show no such delay and no such ordering by mass.

    A field that waits for the core to freeze

    A field inherited from the parent star would be present from a white dwarf's birth. Yet magnetic white dwarfs of ordinary mass are rare among the young and common among the old, as if their fields were switched on later. The proposed switch is the freezing of the carbon–oxygen core, which stirs the liquid above it. If that is right, the delay is set by when each mass begins to freeze, and the heaviest should go first.

    part 3 · stars

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