A star that distils itself as it freezes

A white dwarf is a clock with no fuel in it. It shines only by losing the heat it was born with, and since that heat leaks out through its thin envelope at a rate that depends on how much is left, its brightness is a record of how long it has been cooling. That record is how the ages of the Galactic disc, of old star clusters and of the halo are read from their faintest white dwarfs.

The clock does not run evenly. When the interior becomes cold enough to freeze, the latent heat of crystallisation holds the star up: the freezing releases energy that must be radiated before the star can cool further, and because it is released at a luminosity of a ten-thousandth of the Sun’s, a little energy takes a long time to get rid of. The settling of the heavy isotope neon-22 adds a second delay of the same kind.

This essay is about a third. A carbon–oxygen white dwarf is a mixture, and a mixture does not freeze at one composition. As its interior crystallises it separates, and the separation releases gravitational energy that delays the clock by about as much again.

Why the crystals take the oxygen #

The interior of a white dwarf is a plasma of carbon and oxygen nuclei in a sea of degenerate electrons. The electrons provide the pressure that holds the star up; the nuclei, too massive to be degenerate, behave as a classical fluid whose structure is set by their Coulomb repulsion. As the star cools, the ratio of that repulsion to the nuclei’s thermal energy grows, and when it reaches about 175 for a single species the nuclei lock into a lattice.

The ratio scales as the square of a nucleus’s charge. Oxygen, with eight protons against carbon’s six, is more strongly coupled at a given temperature, so a pure oxygen plasma freezes at a higher temperature than a pure carbon one — by a factor of about one and a half at the same density. A mixture freezes somewhere between, and the solid that first forms out of the liquid is enriched in the species that freezes more readily. The two compositions — the liquid and the solid in equilibrium with it — are linked by the mixture’s phase diagram, which for carbon and oxygen is a spindle: at every composition, the solid takes more oxygen than the liquid.

The phase diagram cannot be measured in a laboratory. The densities involved — a tonne per cubic centimetre — and the temperatures, millions of degrees, are out of reach of any experiment, so it is computed, from Monte Carlo and molecular-dynamics simulations of the plasma. The calculations agree on the spindle’s shape and disagree in detail on its width, which matters, because the width is how much oxygen moves.

Distillation from the inside out #

A white dwarf freezes from the centre outward, because its centre is densest and the coupling grows with density. So the first solid forms at the centre, oxygen-enriched, and the liquid immediately above it is left carbon-enriched.

In a degenerate plasma, carbon-rich liquid is slightly less dense than oxygen-rich liquid at the same pressure. The electrons set the pressure and both nuclei bring two nucleons per electron, so the difference is small, coming from the ions’ own contribution to the pressure — but it is a difference, and it is enough. The carbon-enriched liquid just above the crystallisation front is lighter than the liquid over it, and it rises, mixing with the layers above until it reaches liquid of its own composition. The front moves outward, the next shell freezes with extra oxygen, the next residue rises.

The process is a fractional crystallisation, the same thing that happens when sea water freezes into ice that is fresher than the water, or when a cooling magma deposits crystals of one mineral and leaves the melt enriched in another. In a white dwarf the separation runs over billions of years and a fraction of a solar mass, and the residue moves not by flowing out of a vessel but by rising buoyantly through the star.

Where the energy comes from #

Moving heavier material down and lighter material up releases gravitational energy, just as sinking stones release energy into a pond. The amount depends on how much oxygen moves, how far, and how large the density difference is — all of which are small, but they act on a fraction of a solar mass in a gravitational field a hundred thousand times the Earth’s.

Published calculations put the energy comparable to the latent heat of crystallisation for a standard carbon–oxygen white dwarf. The figures here take it as 0.8 of the thermal energy per ion at the freezing point, against 0.85 for the latent heat — a value chosen to be of the right size, and varied below.

That energy is released where the crystallisation front is, and so at the luminosity at which the white dwarf is freezing: about a ten-thousandth of the Sun’s for a typical white dwarf. A star that faint radiates slowly, and energy delivered to it there extends its life by an amount proportional to the energy divided by the luminosity.

The delays are not corrections. Together they nearly double the time a typical white dwarf takes to finish freezing, and the separation term on its own is more than a billion years. A white-dwarf age that left it out would be too young by that amount for every star that has finished crystallising — which, for the old populations whose ages matter most, is every star at the faint end.

Why heavier white dwarfs are delayed more #

The figure’s slope has a simple cause. In the cooling law that governs white dwarfs, a heavier white dwarf reaches a given luminosity later than a lighter one, because it has more heat to lose through an envelope of similar opacity. Every delay is an energy divided by a luminosity, so a heavier white dwarf, which also has more ions and freezes with more stored energy, radiates it more slowly.

That near-constancy is worth noting. A fractional error that does not depend on mass shifts every age in a population by the same factor, which changes the age of a disc or a cluster but not the shape of its luminosity function. It can therefore hide inside a calibration: a cooling model tuned to give the right age for one cluster would absorb most of the missing separation energy into whatever parameter it was tuned with, and give approximately the right ages for other populations of similar composition, for the wrong reason.

The pile-up the Gaia mission saw #

Delays leave a mark on a population. If white dwarfs are born at a steady rate and slide down the cooling sequence, the number found at each luminosity is proportional to the time each spends there. Anything that slows the slide over a range of luminosities piles white dwarfs up in that range.

The Gaia satellite’s parallaxes made that measurement possible. With distances for hundreds of thousands of white dwarfs, their colour–magnitude diagram shows a band of excess density running across the massive white dwarfs, at exactly the luminosity and colour where crystallisation is expected to begin for each mass — the first direct evidence that white dwarfs crystallise at all, published in 2019. The height of the excess could not be explained by the latent heat alone: the analysis required an additional energy source at the same luminosities, of the size phase separation provides.

The same diagram contained a surprise. Among the most massive white dwarfs, a subset lingers on the crystallisation band for far longer than any of the three energies here can explain — billions of years longer. The leading explanation is a further distillation, of neon-22 rather than oxygen, which can drive a much larger energy release in white dwarfs born with more neon than usual. That is a separation of a minor species rather than a major one, and its size depends on the three-component phase diagram of carbon, oxygen and neon — computed rather than measured, like everything else here.

Where the oxygen came from #

The oxygen a white dwarf distils was made in the star that became it. During helium burning, three helium nuclei fuse into carbon through a resonance in carbon-12 that had to exist for carbon to be made at all, and some of that carbon captures a fourth helium nucleus to become oxygen. How much becomes oxygen depends on the rate of that capture, and on how long the core burns helium; it leaves a core with roughly two-thirds oxygen at its centre, falling outward, where the burning was shorter or cooler.

The comparison shows that the separation energy is not a property of the phase diagram alone. A white dwarf whose progenitor left oxygen concentrated at the centre has already done part of the sorting before it froze — the heavier material is already low — and gains less by doing the rest. One left with a flat profile gains more. So the separation energy, and with it the white dwarf’s age, depends on the history of the star that made it, through a nuclear reaction rate measured only at energies well above the stellar ones and extrapolated down.

How much the answer depends on the phase diagram #

The separation energy is the least certain of the three, because it rests on the width of a phase diagram nobody can measure.

The oxygen profile matters as much as the phase diagram. A white dwarf born with oxygen concentrated at its centre — as helium burning in the progenitor tends to leave it — has less to gain by moving oxygen further inward than one born with a flat profile, and releases less energy. The profile depends on the rate of a single nuclear reaction, carbon-12 capturing an alpha particle to become oxygen-16, whose rate at stellar energies is uncertain by tens of per cent and which is among the most studied and least measured reactions in nuclear astrophysics.

So a white-dwarf age depends, through the separation energy, on a phase diagram computed by simulation and a nuclear reaction rate extrapolated from laboratory energies far above the stellar ones. Neither is a property of the white dwarf being dated. Both are being constrained by the white dwarfs themselves, through the pile-up, which is a reversal of the usual direction: the stars measure the physics rather than the other way round.

What would catch a missing billion years #

The way to test a white-dwarf age is to compare it with an age measured another way for the same population, and open star clusters provide the comparison. A cluster’s main-sequence turnoff — the point where its stars are leaving the main sequence — gives an age from the theory of hydrogen burning, independent of anything about white dwarfs. The faintest white dwarfs in the same cluster give a second age from cooling theory. Where the two agree, the cooling physics is calibrated; where they disagree, something is missing.

The comparison is not clean. A white dwarf’s total age is its cooling age plus the lifetime of the star that made it, and how massive that star was must be inferred from the white dwarf’s own mass through a relation that is itself calibrated on clusters. And the clusters old enough for their faintest white dwarfs to have crystallised are few and distant, with faint white dwarfs at the limit of what large telescopes can measure. The agreement between turnoff and cooling ages in the clusters where both have been measured is at the level of ten per cent — enough to show that the cooling clock is basically right, and not quite enough to detect a missing seventeen per cent with confidence on its own. That is why the Gaia pile-up, which measures the energy release directly in the shape of the population, was the first observation to require the separation rather than merely allow it.

Held up by electrons, sorted by nuclei #

There is a division of labour in a white dwarf that makes the separation possible, and it is worth stating because it is not obvious. The star is held up by its electrons: degenerate electrons, packed as tightly as the exclusion principle allows, provide nearly all the pressure, and their pressure depends only on their density. The nuclei provide nearly all the mass and almost none of the pressure. Their arrangement — liquid or solid, mixed or sorted — barely affects the star’s structure.

That is why a white dwarf can sort its nuclei without changing its size. The distillation rearranges the mass among the nuclei while the electrons, which hold the star up, are indifferent to it; carbon and oxygen each bring the same number of electrons per nucleon, so the electron density, and the pressure, and the radius, are nearly unchanged. Only the small difference in the ions’ own contribution to the pressure makes carbon-rich liquid lighter, and that small difference, acting over the whole star in its enormous gravity, is the source of a billion years of light.

A history of a small effect #

The possibility that a crystallising carbon–oxygen white dwarf separates was raised around 1980, once it was clear that white dwarfs crystallise and that the plasma is a mixture. The early estimates disagreed about whether the phase diagram was a spindle, with no composition at which solid and liquid agree, or had an azeotrope, a composition at which they do, which would have halted the separation partway. The difference mattered: an azeotrope would have produced a white dwarf with a layer of fixed composition and a much smaller energy release.

The computations of the 1990s and 2010s settled on a spindle for the carbon–oxygen mixture, with a width that the different methods now agree on to within a factor of order unity. The effect was included in white-dwarf cooling models from the 1990s onward, and the cooling ages of the oldest white dwarfs in the disc came out older, by of order a billion years, in the models that included it. It was not observed directly until the Gaia diagram, nearly forty years after it was first proposed.

Where the model stops #

One number for the energy. The figures express the separation energy as a single value per ion, released at the crystallisation luminosity, and the delay as that energy times two-fifths of the cooling age. A full cooling calculation integrates the energy release shell by shell as the front moves out, with the star’s structure and luminosity evolving under it; the delays drawn are of the right size and the right dependence on mass, not the precise values a full model gives.

A simplified freezing point. All white dwarfs here begin to crystallise at the same luminosity in the delay figures, while heavier white dwarfs actually begin earlier and at higher luminosity; the luminosity-function figure shifts the freezing luminosity with mass to show the pile-up where it belongs.

A single enrichment. The distillation profile uses one parameter for the phase diagram’s width and assumes the carbon-rich residue mixes upward exactly until it is neutrally buoyant. The real mixing is a form of convection driven by composition differences, whose efficiency is uncertain, and an incomplete mixing would leave a different profile and a different energy.

No envelope physics. The rate at which energy leaks out is set by the envelope’s composition and opacity, and by whether the outer convection zone has reached the degenerate core. Those determine where on the cooling track the delays fall, and they are handled in the simplest way here.

The surprising connection #

The separation of carbon and oxygen in a freezing white dwarf is the process that separated the Earth’s continents from its mantle and its mantle from its core. In all three, the heavy material sank because a cooling mixture crystallised or melted one component more readily than another, and the density difference between the two drove them apart under gravity: iron sinking to the centre, light silicates floating up to make the crust. The gravitational energy of the Earth’s core formation is a large part of the heat the Earth still radiates. A white dwarf is doing the same thing at a hundred thousand times the gravity, with nuclei instead of minerals, and the heat it releases is, like the Earth’s, something it radiates for billions of years after the separation began.

Still open: what the oldest white dwarfs freeze into #

The faintest white dwarfs in the disc have finished crystallising and are now cooling as solids, their colours turning back towards the blue as their atmospheres change. Their ages, which date the disc, depend on everything that delayed them on the way down — the latent heat, the neon settling, and the carbon–oxygen separation — and on whether the neon distillation that explains the massive white dwarfs’ excess also acted, to a lesser degree, on ordinary ones. Whether the three-component phase diagram of carbon, oxygen and neon, now being computed, confirms a distillation in ordinary white dwarfs large enough to shift the disc’s age by another billion years, or confines it to the rare neon-rich ones, is the question on which the oldest dates in the Galaxy’s disc now turn.