A pulsating star that freezes while it rings

A white dwarf’s pulsations are a clock that measures its cooling. Its gravity modes ring at periods of minutes, set by the buoyancy in its interior, and as the star cools the buoyancy weakens and the periods lengthen by a few parts in $10^{15}$ each second. After forty years of timing, that lengthening has been measured in the brightest pulsators of ordinary mass, and it agrees with the theory of how a star with no fuel radiates away the heat stored in its ions.

The theory’s most distinctive prediction is about something that happens later, at lower temperatures, when the ions stop moving freely and lock into a crystal lattice. Freezing releases latent heat, and that heat has to be radiated before the star can continue to cool, so the cooling clock stalls while the interior freezes. The stall is seen in the colour–magnitude diagram of white dwarfs measured from space, as a pile-up of stars at the luminosities where massive ones are freezing. But it is seen there statistically, as a population effect, inferred from how many stars sit where. No one has watched a single star freeze.

A pulsator would do it — if it froze while it rang. For an ordinary white dwarf the two events are separated by billions of years: it pulsates near twelve thousand kelvin and freezes near six thousand. For a heavy one they overlap. This essay works out where, how much of the star is frozen while its clock is running, how much the freezing slows the clock, and how long the slowdown takes to read.

Where each white dwarf would freeze #

A white dwarf’s interior is a plasma of carbon and oxygen nuclei in a sea of degenerate electrons. The nuclei repel each other, and whether they move freely or lock into a lattice depends on how their electrostatic energy compares with their thermal energy. The ratio is the coupling parameter Γ, the Coulomb energy between neighbouring ions divided by $kT$; in a one-component plasma, the ions crystallise when Γ reaches about 175. The Coulomb energy rises with density as the cube root, because neighbours are closer, so the temperature at which a layer freezes rises as the cube root of its density.

A white dwarf is densest at its centre, and how dense depends strongly on its mass, because a heavier white dwarf is smaller. A star of 0.6 solar masses has a central density of about three million grams per cubic centimetre, and its centre freezes at about 3.8 million kelvin. A star of 1.1 solar masses, half the radius and sixteen times the central density, freezes at the centre near ten million kelvin. Each star’s interior is close to isothermal — its degenerate electrons conduct heat so well that the core shares one temperature — so as the core cools, the freezing front starts at the centre and moves outward through progressively less dense layers.

The core temperature at a given surface temperature depends on the mass too. A heavier white dwarf is smaller, so at the same surface temperature it radiates less, and the thin insulating envelope then requires a lower core temperature to supply it. At a surface temperature of 11,700 kelvin, in the middle of the band where hydrogen-atmosphere white dwarfs pulsate, a 0.6-solar-mass core is at about ten million kelvin, far above its freezing point. A 1.1-solar-mass core is at about five and a half million — below the freezing temperature of the inner two-thirds of its mass.

Only the massive pulsators freeze while they ring #

Following each star down its cooling track gives its frozen fraction as a function of its surface temperature. A star of 0.6 solar masses begins to freeze at about six thousand kelvin, long after it has left the instability strip; its pulsations and its crystallisation never overlap. A star of 0.9 solar masses begins near eleven thousand kelvin, inside the strip’s cool edge. A star of 1.1 solar masses begins near seventeen thousand kelvin, before it has even reached the strip, and is between three-fifths and three-quarters solid during the whole time it pulsates.

Such stars are known. BPM 37093, a white dwarf of about 1.1 solar masses at 11,700 kelvin, was found to pulsate in the 1990s and was immediately recognised as a test case; detailed models of its interior have put its crystallised fraction anywhere from about a third to nine-tenths, depending on the composition assumed — carbon and oxygen, or the oxygen and neon that the heaviest white dwarfs are expected to have — and on the details of the phase diagram. The value here, two-thirds, sits in the middle of that range. Since then the space-based photometry that measures pulsations in thousands of stars has added several more massive pulsators, and the colour–magnitude diagram has shown where the population of massive white dwarfs piles up as it freezes.

The frozen core has a second effect on the pulsations, separate from its effect on the cooling. A gravity mode is an oscillation of fluid driven by buoyancy, and a solid has no buoyancy of that kind: the modes are excluded from the crystallised core and live only in the liquid shell above it. As the front advances, the cavity in which the modes ring shrinks, and the spacing between modes of successive radial order changes. That is a static signature — a snapshot of how far the front has advanced — and it is what the models of BPM 37093 were fitted to. The drift is the dynamic one.

A drift slowed by the heat of freezing #

The cooling of the core is a matter of book-keeping. The star radiates its luminosity from the surface, and the energy comes from the ions’ heat: about three times $k$ per ion per kelvin, in the liquid near freezing. When a layer freezes it releases a latent heat of about 0.77 $kT$ per ion, which must be radiated too, and while the front is moving the star loses that extra energy without its core temperature falling. The rate at which the core cools is therefore slower than it would be without freezing, by a factor that depends on how much mass freezes for each degree of cooling:

$$D = 1 + \frac{0.77}{3}\left|\frac{df}{d\ln T_c}\right|,$$

where $f$ is the frozen fraction of the mass. The period drift is proportional to the fractional rate of cooling, so it is divided by the same factor.

How large $D$ becomes depends on where the front is. When it is near the centre, a small amount of mass freezes for each degree, because the central region is small. When it is in the star’s dense middle, a drop in temperature freezes a large slice of mass at once, and $D$ is largest. For the 1.1-solar-mass star in the instability strip, the front is in that middle, and the drift is slowed to about four-fifths of what it would be without latent heat. For a 0.9-solar-mass star at the cool edge of the strip, just as freezing begins in its centre, the slowdown briefly reaches a quarter.

The mass at which the clock starts measuring the freezing #

Fixing the surface temperature in the middle of the strip and varying the mass shows how sharply the effect switches on. Below about 0.94 solar masses nothing is frozen at 11,700 kelvin, and a pulsator’s drift measures its cooling alone — that is the regime of the pulsators timed so far. Above it, the frozen fraction rises steeply and the drift’s shortfall rises with it, peaking at about a quarter for stars just over a solar mass, where the front is crossing the region holding most of the star’s mass.

The heaviest pulsators have nearly finished freezing, and their drift is slowed less: at 1.2 solar masses nine-tenths of the star is solid and the front is in the thin outer layers where little mass remains to freeze. So the measurement is best made at the middle of the massive range, around 1.0 to 1.1 solar masses, which happens to be where the colour–magnitude diagram shows the pile-up of freezing stars.

Two drifts that part over decades #

The drift itself is slow, because a massive white dwarf at the strip is old and cooling slowly — its cooling time is over a billion years, more than three times that of an ordinary pulsator at the same temperature, because it is smaller, fainter and holds more heat. A 600-second mode lengthens by about four parts in $10^{15}$ each second, close to the ordinary pulsators’ rate, and accumulates a lateness of five or six seconds over forty years of timing, growing as the square of the time.

The latent heat’s contribution is the difference between two such curves, and it is about a fifth of either. After forty years the freezing star’s pulses are 1.2 seconds less late than a non-freezing star’s would be. That is a small difference to extract, and it is a difference from a prediction: the expected drift without freezing has to be computed from the star’s mass, its temperature, its envelope and its mode structure, and the measured drift compared with it. The latent heat appears as a shortfall.

How long the freezing takes to show #

Because the effect is a fifth of a drift that itself takes decades to see, it takes longer still. With one second of timing error per season, the shortfall reaches three times the error after about sixty-five years. Better timing helps only as the square root: a precision of 0.3 seconds, achievable for a bright star with a large telescope or with a long continuous run from space, brings it to thirty-five years. That is the same arithmetic that made the ordinary pulsators take forty years: a curvature that grows as the square of the time rewards an early start far more than a better instrument.

There is a further difficulty that the timing error does not capture. The shortfall is measured against a computed drift, and the computed drift depends on the star’s mass to within a few per cent, its core composition, the thickness of its hydrogen layer, and which mode is being timed — trapped modes in the outer layers drift differently from modes sampling the core. A twenty-per-cent effect has to be separated from a prediction uncertain by an amount of the same order. The cleanest version compares several modes in the same star, since they share the cooling but differ in how much of the frozen and liquid regions they sample.

What the freezing is made of #

The latent heat is only part of what a freezing white dwarf releases. Carbon and oxygen do not freeze in the same proportion they had as a liquid: the crystal that forms is richer in oxygen, and the liquid left above it richer in carbon. The oxygen-rich solid is denser and sits at the bottom; the carbon-rich liquid is lighter and mixes upward. The redistribution moves mass inward in the star’s gravity, releasing gravitational energy — a star that distils itself as it freezes — and the heavy neon-22 present at about a per cent can separate and sink as well, releasing more. Together these can add as much energy as the latent heat itself, and on the population level they are what makes the pile-up in the colour–magnitude diagram as high as it is.

For a pulsator’s drift they add in the same way: more energy to radiate per degree of cooling, a slower drift. A measured shortfall therefore measures the sum of latent heat and distillation energy, and dividing the two requires knowing the phase diagram of the carbon–oxygen mixture — how much richer in oxygen the solid is than the liquid at each temperature — which is calculated rather than measured. The pulsator’s drift does not separate them; the population’s pile-up, the mode spacings and the drift together might.

A freezing core may also make a field #

The distillation has a consequence beyond energy. The carbon-rich liquid released at the front is lighter than the liquid above it, so it rises, and a rising, electrically conducting fluid in a rotating star is the ingredient of a dynamo. It has been proposed that the fields that appear late in white dwarfs — magnetic white dwarfs are disproportionately cool and old — are generated by exactly this compositional convection once the core starts to freeze. If that is right, the massive pulsators are freezing in the regime where the dynamo should switch on, and some of them should be developing fields while they pulsate.

A field of the strength the dynamo might produce, a few hundred thousand gauss to a few megagauss, would be detectable in the pulsation spectrum, because a magnetic field splits and shifts the modes as rotation does, and in the spectrum of the star’s surface, where it splits the hydrogen lines. Whether the field reaches the surface quickly or takes billions of years to diffuse out through the layers above the convection is not settled, so the absence of a field in a freezing pulsator would not refute the dynamo. Its presence, in a star with a drift measured to be slowed by latent heat, would tie together three consequences of one event: the energy of freezing, the separation of the elements, and the field the separation drives.

Where the population already shows it #

The freezing of massive white dwarfs is not in doubt, because it has been seen in the population. The space-astrometry catalogue measured distances to hundreds of thousands of white dwarfs, and in their colour–magnitude diagram the massive ones pile up along a narrow band where their cooling has stalled, at the luminosities predicted for crystallisation — and with more stars in the pile-up than latent heat alone can hold, which is the evidence that distillation adds energy too. The colours of the coolest white dwarfs and their luminosity function bear on the same theory from the faint end.

The population fixes the total delay, integrated over the freezing. The pulsator drift fixes the rate of energy release at one moment in one star of known mass and temperature. The two are complementary in the way a census and an individual measurement always are: the census is robust and averages over unknowns, the measurement is fragile and specific, and an agreement between them would confirm that the energy budget is understood at each stage rather than only on average. The seismology needed to pin a pulsator’s mass and composition finely enough for the comparison is the same reading of an interior from a comb of frequencies that measured the Sun’s.

What the model leaves out #

The white dwarfs here are cold Chandrasekhar models, which neglect the thermal contribution to the pressure and the effect of the composition on the electron count; real models at these temperatures are a per cent or two larger. The crystallisation criterion is the one-component value Γ = 175 applied to a half-and-half carbon–oxygen mixture with an averaged charge, rather than the real two-component phase diagram, and the heaviest white dwarfs may have oxygen–neon cores, which freeze at higher temperatures. The relation between core and surface temperature is a single power law with its constant set so that a 0.6-solar-mass star begins to freeze at 6,000 kelvin, a value taken from detailed models; it is not computed from an envelope model. The heat capacity is fixed at $3k$ per ion, and the distillation energy and neon settling are omitted from the drift. The drift is the simple fractional rate of cooling, without the mode-by-mode coefficients that depend on where each mode is concentrated.

Still open: whether the measurement is a measurement of the phase diagram #

The pulsators that have been timed long enough to measure a drift are of ordinary mass and have liquid cores; their drifts test the foundation of the cooling theory. The massive pulsators, a few of them known and more being found, have been observed for a decade or two at most, intermittently, and their drifts have not been measured. They are the only stars in which the energy released by freezing could be seen as a rate rather than inferred from a census.

What the rate would test is the part of the theory least tied to experiment: how much energy carbon and oxygen release as they freeze and separate at densities of millions of grams per cubic centimetre, which no laboratory can reach and which is computed from simulations of the plasma. The population pile-up already says that the energy is large. Whether it comes mostly from latent heat or mostly from distillation — whether the white dwarfs’ clock stops because their ions lock in place or because their oxygen sinks — decides how the ages of the oldest stars in the Galaxy are read, and a clock that keeps running while its interior freezes is the one instrument that could say. It will say it to the astronomers who start timing the massive pulsators now, sometime in the second half of the century.