A cooling rate read off a pulsation clock

A white dwarf has no fuel. It shines by losing the heat stored in its carbon and oxygen ions, and the rate at which it does so is set by how fast that heat can leak through a thin, opaque envelope. The theory of that leak — Mestel’s, from 1952 — gives the luminosity as a function of age, and the ages of the oldest white dwarfs in the Galaxy’s disc and halo are read from it. Every one of those ages is an inference: a population of white dwarfs is counted by brightness, and the counts are compared with a cooling model. No one has watched a white dwarf cool.

Almost no one. A white dwarf passing through a narrow range of surface temperature — near twelve thousand kelvin for those with hydrogen atmospheres — becomes unstable to pulsation. It rings in gravity modes, oscillations in which buoyancy rather than pressure is the restoring force, with periods of a few minutes, and the brightest of these pulsators are among the most stable clocks in the sky. As the star cools, the periods lengthen. The rate is tiny: a few parts in $10^{15}$ of a second per second. But a clock can be timed for decades, and an effect that grows as the square of the time can be seen if the timing is long enough.

This essay follows that measurement: why the period changes, how fast it should change, how long it takes to see, what the rate measures once it is seen, and why a slightly-too-fast clock would be evidence of something in the star’s core that the cooling theory left out.

Why a cooling star rings slower #

In a gravity mode, a parcel of material displaced vertically is pushed back by buoyancy and oscillates about its equilibrium height at the buoyancy frequency. In the degenerate core of a white dwarf, the buoyancy comes from the thermal part of the pressure — the small fraction that is not degeneracy pressure — because a parcel moved upward without exchanging heat is slightly warmer than its new surroundings, and the difference is set by the temperature. As the core cools, the thermal part shrinks, the buoyancy weakens, and the oscillation slows. The periods of the modes go roughly as the inverse of the buoyancy frequency, so they lengthen as the temperature falls.

To first order the rate is

$$\frac{\dot P}{P} \approx -a\,\frac{\dot T}{T} + b\,\frac{\dot R}{R},$$

with $a$ near a half. The second term — the star’s residual contraction, which shortens the periods — is negligible for a cool white dwarf with a hydrogen atmosphere, whose radius is essentially fixed by degeneracy. What remains is a direct measure of the fractional cooling rate of the core. Under Mestel’s law the luminosity goes as the seven-halves power of the core temperature, and the core’s cooling timescale $T/|\dot T|$ comes out as five-halves of the star’s cooling age. So $\dot P/P$ is about one fifth of the inverse of the cooling age.

The rate along the cooling sequence #

That relation turns the cooling theory into a prediction for every pulsator.

A white dwarf of 0.6 solar masses at twelve thousand kelvin has been cooling for about a third of a billion years, and a 215-second mode in it lengthens by about four parts in $10^{15}$ each second — a second of period every eight million years. Longer-period modes lengthen in proportion. Hotter pulsators are younger and cool faster, and their periods drift faster: the white dwarfs with helium atmospheres that pulsate near twenty-five thousand kelvin should drift ten times more, and at those temperatures the cooling is dominated by neutrinos rather than light, so the drift there is a measurement of the neutrino luminosity of a stellar core.

Across the whole sequence, from the hot stars still contracting towards the white dwarf state to the cool hydrogen pulsators, the predicted rates span four orders of magnitude. The hottest have been measured for decades, because their rates are large; the coolest took forty years.

A drift that grows as the square of the time #

A period that lengthens steadily makes each pulse arrive a little later than a constant-period clock predicts, and the lateness accumulates:

$$O - C = \frac{1}{2}\frac{\dot P}{P}\, t^2 .$$

After ten years the pulses of a 215-second mode drifting at four parts in $10^{15}$ are less than a second late. After forty-five they are nineteen seconds late. A season of observing a bright pulsator times its main mode to about a second, so the drift is invisible for the first decade, marginal in the second and unmistakable after the third. The simulated points show what that looks like: scatter about a flat line for years, and then a curve that the line cannot follow.

Because the drift grows as the square of the time, the time needed to see it goes as the square root of the timing error over the rate. Better timing helps only slowly: three times the precision saves less than half the years. The only way to measure the coolest pulsators’ rates is to have started timing them long ago, which is what happened. The hydrogen-atmosphere white dwarf G117-B15A was found to pulsate in 1976, its 215-second mode has been timed every season since by a team led by S. O. Kepler, and after more than four decades its period derivative is measured at about five parts in $10^{15}$ — consistent with the cooling theory to within its errors. ZZ Ceti itself, the prototype of the class, has a similar rate measured over a similar span. A clock with that stability — losing a second in ten million years — was for some time described as the most stable optical clock known.

What the rate is a measurement of #

A measured cooling rate is an age.

The age of a white dwarf is otherwise always a model output: a luminosity or temperature measured, and a cooling curve read to find how long it took to get there. The drift measures the same thing differently — not the time since the star was formed, but the rate at which it is cooling now. A cooling curve that predicted the right luminosity at the right age, but by the wrong path, would get the rate wrong, and the drift would show it. That makes the pulsators the one direct calibration point for the cooling theory that the ages of the Galaxy’s oldest populations rest on.

At the coolest temperatures the calibration matters most, because the cooling theory has its largest corrections there: the latent heat released as the core freezes, the sinking of neutron-rich neon through the liquid, the transition to a regime in which the heat capacity of the crystal falls. The hydrogen pulsators sit before most of that, at temperatures where the theory is simplest, so their agreement with it is a test of the foundation rather than of the refinements. Massive pulsators that have already begun to crystallise exist, and their drifts, if they could be measured, would test the refinements directly.

A clock that would run fast #

The most consequential use of the measurement is as a bound on energy losses that the theory does not include. If something other than light carried energy out of a white dwarf’s core — a light particle produced in the hot, dense plasma and escaping freely, as neutrinos do from hotter stars — the core would cool faster than the photon luminosity alone allows, and the period would drift faster.

The relation is simple: an extra loss equal to a fraction $f$ of the star’s luminosity shortens its cooling timescale by a factor $1+f$ and raises the drift by the same factor. A drift measured to twenty per cent, agreeing with the prediction, rules out any unseen loss larger than about half the star’s own luminosity. The candidate most often tested this way is the axion, a hypothetical particle that would be produced by electrons scattering in a degenerate plasma and would escape from the core unimpeded. The G117-B15A and ZZ Ceti measurements have been turned into upper limits on the axion’s coupling to electrons, comparable to those from the brightness of red giants at the tip of their branch and from the luminosity function of white dwarfs, and some analyses have found a mild preference for slightly faster cooling than the standard model gives — not enough to claim anything, and in the same direction as hints from other stellar systems.

A clock cannot say what is speeding it up. The bound applies to any extra loss with the right dependence on temperature, and it is only as good as the theory of the modes and the knowledge of the star’s mass and envelope. That is the general shape of stellar particle physics: a star’s energy budget is a calorimeter, and anything that leaks energy shows up as a star that evolves too fast.

Why the clock rings only at twelve thousand kelvin #

The measurement depends on a coincidence of temperatures, and it is worth knowing why the pulsators exist where they do. A white dwarf is a quiet star; most of them do not pulsate at all. The ones that do are found in a narrow strip of surface temperature, from about 12,300 down to 10,800 kelvin for those with hydrogen atmospheres, and every white dwarf of the right mass that passes through the strip seems to pulsate while it is there.

The reason is in the envelope. At those temperatures hydrogen in the outer layers is partly ionised, the opacity is high, and a convection zone develops just beneath the surface. As the star is compressed in an oscillation, the convection zone absorbs heat and releases it a fraction of a cycle later, which is the timing needed to drive the oscillation rather than damp it — the same mechanism, broadly, that drives the larger and more famous pulsations of Cepheids, with partly ionised helium as the valve there. Above the strip the convection zone is too thin to drive anything; below it, the zone is so deep that its thermal response is too slow, and the modes damp out.

For timing, the strip has a useful property. The stars that pulsate near its hot edge ring in a few modes of short period with very stable amplitudes and phases; those near the cool edge ring in many longer modes whose amplitudes wander. The hot-edge pulsators are the clocks. G117-B15A and ZZ Ceti both sit near the hot edge, which is part of why their modes have kept phase for forty years and why a cumulative drift of a few tens of seconds can be followed through them without losing count of the cycles.

Keeping time for forty years #

Timing a white dwarf over decades is in principle simple — record the time of maximum light each season, and compare with the prediction of a constant period — and in practice it is an exercise in removing everything except the star. The arrival times have to be referred to the barycentre of the solar system, because the Earth’s orbit moves the observatory by eight light-minutes back and forth each year, a delay several hundred thousand times larger than the effect being sought. The clocks at the observatories used over four decades have changed from quartz to radio-disciplined to satellite-disciplined standards, and each change has to be tied to the next. The photometers have changed from photomultiplier tubes to charge-coupled devices to frame-transfer cameras, each with its own timing conventions.

The result of that bookkeeping is a record of the phase of one mode, season by season, covering about six million cycles of the star’s oscillation without a single cycle lost — a clock that has never once been allowed to stop being counted; the closest astronomical analogue is the timing of radio pulsars, whose spin periods are steadier but whose drifts are dominated by braking rather than cooling. That the white dwarf’s phase can be followed at all depends on the drift being slow: if the period changed by a few parts in $10^{12}$ a second instead of $10^{15}$, the drift would reach a whole cycle within a decade, and the count would be lost unless every season was observed.

The drift itself is now known to about fifteen per cent for the best-timed stars, limited by the length of the record. Each further decade of timing improves it by more than a factor of two, because the curvature grows as the square of the baseline while the timing errors stay fixed. The most useful thing an observer can do for this measurement, as with several others in astronomy, is simply to keep going.

What has to be taken out first #

The observed period derivative is not purely the cooling. A star moving across the sky at a transverse speed $v_t$ and distance $d$ is receding faster and faster from the observer as its line of sight swings round, and that makes every period appear to lengthen by a fractional rate $v_t^2/(cd)$ — the same apparent acceleration that pulsar timing has to remove. For G117-B15A it is a significant fraction of the measured rate, and the proper motion and the parallax, now known precisely from space astrometry, are needed to subtract it.

A planet in orbit round the pulsator would add a periodic delay, the light-travel time across the star’s reflex orbit, and a long-period companion would add a slowly curving term that could imitate or hide the cooling drift. Searches for such delays in pulsating white dwarfs have so far found no convincing planets, which is itself useful: it means the drifts measured are probably what they look like.

And the mode has to be identified. Each white dwarf rings in several modes, with different spherical-harmonic degrees and radial orders, and the coefficient $a$ that converts cooling into period change depends on where in the star the mode is concentrated. Modes trapped in the outer layers respond to the envelope as well as the core, and their drifts can differ in sign. The interpretation rests on models of the star’s structure fitted to its full spectrum of periods — the same seismology that reads the interiors of the Sun and of red giants, applied to a star a hundred times smaller.

What the predictions leave out #

The predicted rates use Mestel’s law with a fixed mass, radius and ionic composition, and a single value of the coefficient $a$. Detailed evolutionary models with realistic envelopes and opacities give cooling ages different by tens of per cent at a given temperature, and rates of period change that depend on each mode’s trapping. The hot pulsators’ rates are dominated by neutrino cooling and residual contraction, neither included here; the value drawn for the pre-white dwarf is an order-of-magnitude estimate. The measured ratio and its error in the energy-loss figure are illustrative, chosen to show how a measurement turns into a bound, and are not the published value for any particular star.

The simulated timing points assume one measurement a season with a one-second error and no gaps; real records have years missing, changes of instrument and of reference clock, and occasional jumps in a mode’s phase that have to be understood before the drift can be fitted.

Still open: watching a star freeze #

The hydrogen pulsators so far measured are of ordinary mass, with liquid cores. The theory of white dwarf cooling makes its most distinctive prediction for heavier stars, whose cores begin to crystallise while they are still hot enough to pulsate: the release of latent heat should slow the cooling, and with it the drift, by an amount that depends on the composition of the crystal and on whether the heavier elements sink as it forms. A massive pulsator caught partway through crystallisation would show a drift slower than its luminosity and temperature predict, and the shortfall would measure the heat released in the freezing directly.

Such stars are known; their periods are being timed; and the drift they should show, at a few parts in $10^{15}$, takes the same two or three decades to see. The measurement that would confirm, star by star, that white dwarfs stop their clocks while their interiors freeze — the effect inferred from the pile-up of white dwarfs in the colour–magnitude diagram — is therefore under way and will be finished in the 2040s, by observers timing pulses that began their drift long before any of them was born.