A field that waits for the core to freeze

Hot stars, white dwarfs and young neutron stars carry almost the same magnetic flux, and that is what a field frozen into a star would keep as the star shrank. It is the strongest argument that white-dwarf fields are fossils, carried down from the parent star’s envelope or from the cloud before it. A fossil has one property that can be tested. It must be there from the start. A white dwarf that formed with a fossil field should show it at any age, since nothing about cooling creates or destroys flux, and its strength should fade only by the slow decay set by the white dwarf’s conductivity.

Surveys that can test this have recently become possible. They count every white dwarf within twenty or forty parsecs, so the sample is complete rather than biased towards the bright and young. Spectropolarimetry then checks each star for a field, and a cooling model dates each one. They found that strongly magnetic white dwarfs of ordinary mass, near 0.6 solar masses, are rare among those younger than about a billion years and markedly more common among older ones. Ordinary fossils would not behave like that. This essay works through the leading explanation: that the field switches on when the white dwarf’s core begins to freeze.

When the centre freezes #

A white dwarf is a star held up by electrons that cannot be squeezed, with nothing left to burn. It cools by radiating the heat stored in its carbon and oxygen ions. Those ions are not a gas in the ordinary sense. At densities of millions of grams per cubic centimetre they are packed close enough that the electrostatic repulsion between neighbours competes with their thermal motion. The ratio of the two is the coupling parameter,

$$\Gamma = \frac{(Ze)^2}{a\,kT} = 2.275\times10^5\,\frac{Z^2}{T}\left(\frac{\rho}{A}\right)^{1/3},$$

where $a$ is the spacing between ions, $Z$ their charge and $A$ their mass number. When $\Gamma$ reaches about 175 the ions lock into a lattice. The interior freezes, from the centre outwards, because the centre is densest.

In the figure, $\Gamma$ climbs as the white dwarf cools and crosses 175 at different ages for different masses. The cause is density. A heavier white dwarf is smaller, because more mass compresses degenerate matter further. Its centre is denser by an order of magnitude between 0.6 and 1.0 solar masses, so the ions are closer and freeze at a higher temperature. A higher temperature is reached sooner on the way down. In this toy model, which leaves out the envelope’s real opacities and the heat freezing itself releases, a 1.2-solar-mass white dwarf begins to freeze a hundred million years after it forms and a 0.6-solar-mass one after 2.6 billion. Detailed models move those numbers but not the ordering, and the ordering is what matters here.

The freezing has been seen directly. The latent heat it releases slows the cooling, so white dwarfs linger at the stage where it happens, and the Gaia satellite’s distances showed them piling up in a band across the Hertzsprung–Russell diagram, in the place where cooling models put the onset of crystallisation. Freezing is not a hypothesis. The question is whether it makes fields.

Freezing stirs the liquid above #

Freezing a single substance releases heat and changes nothing else. A white dwarf’s core is not a single substance. It is a mixture of carbon and oxygen, and the solid that forms is richer in oxygen than the liquid it comes from. The new crystals are denser and settle onto the solid core. The liquid just above is left richer in carbon, and so lighter than the liquid further out. Light fluid beneath heavy fluid is unstable, and it overturns. As long as freezing continues, it drives a steady compositional convection in the liquid above the solid core.

Convection in an electrically conducting fluid is the ingredient a dynamo needs, as in the Earth’s core, where the freezing of the inner core similarly drives the outer core’s convection with the light elements it rejects. White-dwarf matter is an excellent conductor, because its degenerate electrons move nearly freely. The proposal, made in 2017, is that the convection set off by freezing builds a field the way the Earth’s liquid core does. Such a field would not exist before freezing begins.

Why nothing stirs the core before it freezes #

The Earth is the obvious precedent, and the comparison shows why the white-dwarf case is sharper. The Earth’s solid inner core grows as the planet cools, rejecting light elements into the liquid outer core and driving its convection, the same mechanism proposed here. But the Earth had a magnetic field for billions of years before its inner core began to freeze, probably within the last billion years. Before then its outer core was stirred by heat alone: the core lost heat to the mantle faster than conduction along its own temperature gradient could carry it, so it convected.

A white dwarf’s core cannot do that. Its heat is carried by the degenerate electrons, which conduct so well that the whole core sits at very nearly one temperature, the isothermal core that makes the cooling a clean clock. The temperature drops across the thin non-degenerate envelope at the surface, not inside. With no temperature gradient in the core there is nothing for heat-driven convection to work against, so before freezing starts the core is still. A dynamo needs moving conductor, and the core gets its first motion from the composition left by freezing.

So the switch is sharper in a white dwarf than in a planet. The Earth’s inner core changed the source of an existing dynamo. A white dwarf’s crystallising core would start one in a liquid that had never moved. That is what makes the age of a magnetic white dwarf a test: before its mass’s freezing onset, the dynamo has not started.

A field that cannot leak #

A field made inside a white dwarf is useful only if it can be seen at the surface, and here flux freezing works against the dynamo as firmly as it once worked for the fossil. In a conductor a field moves relative to the matter only by diffusing, at a rate set by the magnetic diffusivity $\eta = c^2/4\pi\sigma$. The time to cross a distance $\ell$ is about $\ell^2/\eta$.

The conductivity of a white dwarf’s interior is between about $10^{21}$ and $10^{22}$ per second, and the figure shows what that implies. In the age of the Universe a field diffuses between 600 and 1,800 kilometres. That is less than a fifth of the radius of a typical white dwarf, and much less than the distance from the convecting layer above a young solid core to the surface. The field is frozen to the matter as firmly as the fossil was. It can appear at the surface only if the convection itself carries it there, and the convection reaches only as far as the composition gradient that drives it.

That makes the timing more interesting than “at the onset”. A dynamo running in a thin shell just above a small solid core is deep, under thousands of kilometres of non-convecting material, and its field stays hidden. As freezing proceeds the solid core grows and the convecting shell moves outward with it. Its outer boundary is where the falling composition stops driving convection. One analysis has argued that the field breaks out only once the shell comes within diffusion reach of the surface layers. That adds a delay after the onset, from hundreds of millions to a billion or more years, depending on the mass and on how far the convection penetrates.

Energy is not the limit #

The dynamo has plenty of energy available. The latent heat of freezing is about 0.77 of $kT$ per ion, and over a 0.6-solar-mass core that is $2\times10^{46}$ erg, released over billions of years. The phase separation releases gravitational energy of the same order as the oxygen settles. The magnetic energy of a dipole with a polar field of $B$, outside the star alone, is $B^2R^3/12$. For the strongest white-dwarf fields measured, near a gigagauss, that is a quarter of one per cent of the latent heat. For the megagauss fields most magnetic white dwarfs carry it is a hundred-millionth.

So the limit on the field is not energy but efficiency: how much of the convection’s kinetic energy a dynamo turns into field. That depends on the convective speed, the rotation and the geometry. The convection driven by freezing is extremely slow, since the composition changes gradually over billions of years. Scaling laws calibrated on planetary dynamos give fields from about a tenth of a megagauss to a few megagauss from such slow convection, the larger values for rapidly rotating white dwarfs. The strongest white-dwarf fields, hundreds of megagauss, seem beyond what freezing alone can do, and must come from elsewhere: from fossils, or from the merger of two white dwarfs, whose violent mixing can amplify a field quickly.

The order in which the fields should appear #

If every white dwarf’s field appeared at the onset of freezing, the share of a population carrying a field would rise with cooling age in a definite way. The hero figure computes that share in the toy model, for a population with masses spread around 0.62 solar masses as the local white dwarfs are, and separately for three mass ranges.

The prediction has two parts, and the second is the one a fossil cannot imitate. First, the incidence rises with age, from nothing among young white dwarfs of typical mass to a plateau once nearly all have frozen. Second, the rise happens at different ages for different masses: half the heavy white dwarfs have begun freezing by half a billion years, half the typical ones only by two and a half. A fossil field, present from birth, gives a flat incidence with age in every mass range. A merger origin would make magnetic white dwarfs heavier than average and, at least at first, young and fast-rotating, the opposite of a delay.

The surveys see the first part: magnetism is rare in the young white dwarfs of ordinary mass and common in the older ones, with the rise setting in at ages broadly like those in the figure. They also see that magnetic white dwarfs are, on average, heavier than non-magnetic ones. The heavy magnetic white dwarfs, though, are found at all ages, including very young ones. That fits a mixture of origins: fields that appear with freezing in ordinary white dwarfs, and fields born with merger products among the heavy ones. A dynamo delay in the heavy group would be hidden under the mergers’ fields.

Separating the three origins #

The fossil, the merger and the freezing dynamo are not mutually exclusive, and the evidence suggests all three operate. What the ages offer is a way to sort them. A white dwarf with a strong field that is younger than its mass’s freezing onset cannot have a freezing dynamo; it is a fossil or a merger. A field in an old white dwarf of typical mass with no sign of fast rotation fits freezing best. Rotation helps further, because a merger spins the remnant up, and a fossil from a slowly rotating Ap star does not.

The field strength gives another handle. Fossils from Ap stars, by the flux argument, are expected from kilogauss to hundreds of megagauss; dynamo fields from freezing are expected to be weaker. If freezing makes a population of weaker fields that appears at the right age, its distribution of field strengths should be shifted low compared with the young magnetic white dwarfs. That is a comparison complete surveys are now large enough to make.

A population that should break the flux coincidence #

The argument that white-dwarf fields are fossils rested on a number: multiplied by the star’s area, the fields of magnetic hot stars, white dwarfs and neutron stars give nearly the same flux. A field made by freezing has no reason to respect that number. Its strength is set by the convection’s power and the dynamo’s efficiency, not by any flux carried down from the parent star.

For a 0.6-solar-mass white dwarf with a radius near 8,700 kilometres, a one-megagauss field over its disc carries a flux of about $2\times10^{24}$ gauss square centimetres. The fossil fields of the strongly magnetic Ap stars, a few kilogauss over a star twice the Sun’s radius, carry nearly a hundred times as much, and so does a white dwarf of a hundred megagauss. The common flux of the three classes sits there. If a substantial share of the older magnetic white dwarfs have dynamo fields near a megagauss, they should form a group sitting below the common flux of the three classes. It would be a cluster of weaker fields in older stars, inconsistent with collapse of a fossil by the same factor.

The size of the gap is the point. A factor of a hundred in flux cannot be produced by the scatter of fossil fields between parent stars, which spans about that range already but is centred on the common value; a separate group of weak fields would show as a second peak at the bottom of the distribution, not as a wider first one. Fields this weak are also the hardest to detect in a cool white dwarf, so the second peak should become visible only as the surveys’ sensitivity improves towards a hundred kilogauss. That is also where most of the weak, old fields would be found.

That makes field strength and age together a sharper test than either alone. Measured in complete samples, the old magnetic white dwarfs of typical mass should be weaker on average than the young magnetic ones. The young ones can only be fossils or mergers, and they should be the ones that carry the fossil flux.

What the toy leaves out #

The white dwarf in the figures is a toy, stated in full so that every number can be checked. Its central density comes from a scaled polytrope, its envelope from a single radiative relation, its cooling from Mestel’s law, which ignores the latent heat that freezing itself releases and the extra delay from phase separation. Detailed models can differ substantially from the toy’s onset ages, and from each other, according to the core’s oxygen fraction, the envelope’s hydrogen layer, and the trace of neon-22 that sinks through the liquid. The ordering by mass survives every version, because it follows from density alone: a heavier white dwarf is denser and freezes hotter.

The surveys have their own limits. Fields below about a hundred kilogauss are hard to detect in a cool white dwarf with weak spectral lines. The coolest white dwarfs, with almost featureless spectra, can hide fields that a warmer one of the same strength would show. A rise of incidence with age could be partly a rise of detectability, if older white dwarfs were systematically easier to measure, and they are not: the coolest are the hardest. That makes the observed rise more striking, not less.

Still open: how long the field takes to reach the surface #

The freezing dynamo explains why fields appear late in ordinary white dwarfs, and the energy budget allows it. What is not settled is the delay between the onset of freezing and the field’s emergence. That delay is set by how a convecting shell above a growing solid core rises towards the surface. It depends on the composition profile left by the white dwarf’s parent star, on the trace elements that also separate, and on whether the convection can carry flux through the layers above it faster than diffusion could.

If the delay is short, the incidence of fields by age should track the figure’s freezing curves closely, mass range by mass range. If it is long, the curves should be shifted to later ages, by a different amount for each mass, and the lightest white dwarfs might not show their fields until well after they have begun to run their colours backwards in the last stage of cooling. Measuring the incidence separately in narrow ranges of mass, in samples large enough to contain old white dwarfs of every mass, would measure the delay directly. It would also reveal the inside of a star from the timing of a field at its surface.