A clock with no fuel in it

Almost everything in this collection that tells the time does so by moving. A period repeats, an orbit precesses, a pulsar spins down. A white dwarf does none of those things. It sits still and gets colder, and the rate at which it gets colder is so completely determined by physics that has nothing to do with astronomy — a heat capacity, an opacity, a surface area — that its brightness is a clock.

The clock has one very unusual property. It contains no nuclear physics at all. Every other age in astronomy runs through a reaction rate somewhere: the main-sequence turnoff of a cluster is a statement about how fast hydrogen burns, and a radiometric date is a statement about a decay constant. A cooling white dwarf is burning nothing. What it is doing is radiating away the thermal energy its ions had on the day the star’s envelope was removed, and that is a problem in heat conduction.

Why the interior is a reservoir and the envelope is a valve #

A white dwarf is held up by degeneracy pressure, which is a pressure that does not care about temperature. That single fact separates the star into two parts that behave completely differently, and the separation is what makes the cooling law simple.

The interior is degenerate, so its pressure is fixed by its density alone. Cooling it changes nothing structural: the star does not shrink as it cools, the way a normal star would. The electrons in a degenerate gas are also excellent conductors — they have long mean free paths and no low-energy states to scatter into — so the interior is very nearly isothermal, a single temperature from the centre out to within a per cent or so of the radius.

The outermost sliver is not degenerate. It is an ordinary gas, thin and opaque, and it is the only thing between the hot interior and space. The whole rate of cooling is set there, by how fast heat can be pushed through a few hundred kilometres of partially ionised helium and hydrogen. Everything the star is made of, and everything it once was, enters the answer only through the total heat available; the rate is a property of the skin.

The opacity of that skin is the Kramers law, the free–free and bound–free absorption that goes as density over temperature to the seven-halves, and it is the same averaged opacity that appears in every radiative stellar envelope. That the cooling exponent comes out as a simple rational number is entirely because the opacity law is a power law; a real envelope with hydrogen ionising in it departs from that, which is one of the reasons published cooling tracks are computed rather than quoted from the formula.

There is a further consequence of the isothermal interior that is easy to pass over. Because the electrons conduct so well, essentially all of the star’s heat content is available: there is no deep reservoir insulated from the surface, waiting. The entire body cools together, and the luminosity at any moment is a statement about the whole object rather than about a layer of it. Very little else in astronomy has that property.

Working the radiative transfer through the envelope with a Kramers opacity gives the luminosity as the seven-halves power of the interior temperature, and equating that to the rate at which the ions lose their thermal energy — three-halves $kT$ each, with the electrons contributing almost nothing because they are degenerate — gives Mestel’s law:

$$\frac{L}{L_\odot} \approx 1.7\times10^{-3}\left(\frac{M}{M_\odot}\right)^{5/7}\left(\frac{A}{12}\right)^{-5/7}\left(\frac{t}{\text{Gyr}}\right)^{-7/5}.$$

The exponent, and the population it produces #

Minus seven-fifths is a large exponent, and its practical meaning is that a white dwarf is faint for nearly all of its existence — an exponent with the same flavour as the one that makes the biggest stars die first, and the opposite consequence, since here the steepness works to fill the faint end rather than to empty the bright one. A factor of ten in luminosity costs a factor of only 5.2 in age. Put the other way: the star spends the first per cent of its cooling life brighter than the Sun and the remaining ninety-nine per cent below a thousandth of it.

That has a consequence for any population. If white dwarfs have been made at a roughly steady rate, then the number found in any interval of brightness is simply the time a star spends crossing that interval — a census turns into a stopwatch — and the count rises steadily toward the faint end because the crossing takes longer there.

The drop at the faint end is therefore not about white dwarfs. It is about how long there have been white dwarfs to cool, which is a statement about the Galaxy. The observed cut-off sits near bolometric magnitude 15.3, a luminosity around $10^{-4.4}$ solar, and reading it back up the cooling track gives a number between eight and ten billion years.

The mass dependence runs the wrong way, for a good reason #

The tracks in the hero figure are ordered with the heaviest at the top, which surprises people who expect a bigger star to be brighter. A heavier white dwarf is not bigger; it is smaller, because the degenerate mass–radius relation has a negative exponent. So it has more ions to cool — more mass — and less surface through which to lose them. At any given age it is therefore hotter and brighter than a lighter one, and it takes longer to reach the cut-off.

The practical difficulty is that the mass distribution of white dwarfs is not flat. It peaks sharply near 0.6 solar masses, because the mass a star does not keep is most of it, and a wide range of progenitor masses is funnelled into a narrow range of remnants. That is helpful — it means the cut-off is dominated by one mass — and it is also the largest systematic, because the tail matters and the tail is poorly measured.

Where Mestel’s law stops being enough #

The interior of a white dwarf is a plasma of carbon and oxygen nuclei in a degenerate electron sea, and a plasma with strong enough electrostatic coupling freezes. The relevant parameter is the ratio of the Coulomb energy between neighbouring ions to their thermal energy, and when it reaches about 175 the ions arrange themselves into a lattice.

Two things happen at that point and they pull in opposite directions.

Freezing releases latent heat — of order $kT$ per ion — which the star has to radiate before it can cool any further. That is a delay, and it is not small: it is a substantial fraction of all the thermal energy the star still had.

Below the Debye temperature of the new lattice, the ionic heat capacity stops being the constant $3k/2$ of a gas and falls as the cube of the temperature, the same way a cold solid’s does in a laboratory. There is then far less heat left to lose, and the star runs down to the bottom much faster than Mestel’s law says.

There is a third correction that is the least certain of the three. Carbon and oxygen do not freeze together: as the lattice forms it is enriched in oxygen, the heavier species, which sinks. That is a release of gravitational energy on top of the latent heat, and how much depends on the interior’s composition profile, which depends in turn on a nuclear reaction rate — the carbon-alpha capture cross section — that is one of the worst-measured numbers in stellar physics. So the clock with no nuclear physics in it has, at the third decimal place, some nuclear physics in it after all.

Which skin the star is wearing #

The whole cooling rate is set in the envelope, so the composition of the envelope is not a detail. It is the single largest thing separating one white dwarf’s clock from another’s of the same mass.

About four-fifths of white dwarfs have atmospheres of essentially pure hydrogen and are classified DA; most of the rest have essentially pure helium and are DB. Neither is a remnant of what the star was made of — the envelope is stratified by gravitational settling within days of the star forming, and whichever element is lightest floats to the top and hides everything beneath it. A hydrogen layer a ten-thousandth of the star’s mass is enough to make it a DA, so which class a star belongs to is decided by a trace.

That trace decides the cooling rate, because hydrogen and helium have very different opacities at the temperatures the envelope passes through. Helium is the more transparent below about twelve thousand kelvin — it has no bound electrons left to speak of at the relevant ionisation, and no molecules to form — so a helium-atmosphere star loses heat faster and reaches any given faintness sooner. The difference in inferred age between assuming one and the other reaches a gigayear at the faint end, which is a tenth of the number the cut-off is being used to measure.

Worse, a star can change class. The ratio of DA to DB is not constant along the cooling sequence: it has structure, with a well-known gap between about thirty and forty-five thousand kelvin in which almost no helium-atmosphere stars are found, and a rise in the helium fraction at low temperatures. The accepted reading is that convection in the deeper layers periodically dredges helium up and dilutes the thin hydrogen skin, converting a DA into a DB. A clock whose rate depends on its surface, and whose surface changes during the run, is a considerably more delicate instrument than the opening derivation makes it sound.

There is a related event in every cooling track that has nothing to do with composition and everything to do with the envelope’s structure. As the star cools, its surface convection zone deepens, and eventually it reaches down to the degenerate interior. From that moment the heat no longer has to diffuse radiatively through the whole envelope — convection carries it to within a short distance of the surface — and the star’s cooling briefly accelerates. The convective coupling appears in a computed track as a shallow bend, near the luminosity at which crystallisation is also under way in the interior, and disentangling the two is one of the reasons published tracks are trusted more than the closed-form law.

The practical upshot is that the atmosphere has to be classified before the star can be used as a clock, and classification means a spectrum. For the objects at the faint end that is a spectrum of a magnitude-sixteen star with almost no features in it, taken to decide which of two cooling tracks to read an age from. The measurement that sets the clock’s rate is therefore harder than the measurement the clock is being used to make, and on the faintest objects it is often not available at all — so the age is quoted with the composition marginalised over rather than known.

What was actually measured #

Two observations underlie everything above, and neither is the age.

The first is a luminosity function, which means a complete count of white dwarfs per unit volume in each bin of brightness — the same kind of object as the galaxy counts with a knee in them, and subject to the same difficulty, which is that a count is only as good as the volume it is divided by. Completeness is the whole difficulty: the objects at the cut-off have absolute magnitudes around 16, so a survey sees them only within a few tens of parsecs, and the volume is small enough that the faintest bin holds a handful of objects. The early determinations rested on samples of a few dozen stars, and the cut-off was a statement about the last two or three of them.

The second is a distance for each of those objects, because a luminosity function needs absolute magnitudes and a space density needs a volume. Before trigonometric parallaxes were available at these magnitudes this was done photometrically, and the systematic uncertainty in the answer was dominated by it. Gaia’s parallaxes changed that: the local white dwarf sample within a hundred parsecs is now essentially complete and individually distanced, and the cut-off is defined by hundreds of stars instead of by a handful.

What emerged when the sample became large was a feature the earlier data could not have shown — a pile-up of stars at the luminosity where crystallisation happens, exactly where the delay above says objects should accumulate. The latent heat is not an inference from theory any more; it is visible as an excess of stars sitting at one brightness because they have to radiate something before they can move on.

Two clocks, and the argument between them #

The white dwarf cut-off is one of three independent ways of putting a lower bound on the age of the Galaxy, and the value of having three is that they fail differently. The third is cosmological: the age implied by the expansion history, which must exceed the age of anything in it. That constraint was once the sharpest thing in cosmology — there was a period in which the oldest globular clusters came out older than the universe — and it is now comfortable, with the disc’s white dwarfs at eight to ten gigayears, the halo’s globular clusters at twelve to thirteen, and the universe at 13.8. It is worth noticing how differently that tension would read today, when the two ways of measuring the expansion rate disagree by five sigma and an age is one of the few quantities on which they are not being asked to arbitrate.

The white dwarfs’ own contribution to that reconciliation was to say something about the disc rather than the halo, which no cluster age could do: the thin disc is billions of years younger than the oldest stars in the Galaxy, and the shape of its white dwarf luminosity function encodes not merely a single age but the star formation history that produced it.

Where the ladder goes #

A cooling track is a mapping from brightness to age, and every use of it downstream is an application of that mapping to a different population. Applied to the white dwarfs in an open cluster, it gives an age that can be checked directly against the cluster’s turnoff, which is how the method was calibrated — and the check is a strong one, because a cluster’s turnoff age comes from where the bend in its main sequence sits and shares no systematic whatever with an envelope opacity. Applied to the halo, it gives an age for the oldest stars in the Galaxy and a limit on how many faint remnants the halo can contain — one of the exclusions behind the argument that the missing mass is not faint stars.

And applied in reverse, it is a probe of matter under conditions no laboratory reaches. The crystallisation pile-up is a direct observation of a phase transition in a plasma at $10^{9}$ kilograms per cubic metre; the shape of the faint end constrains the heat capacity of that lattice; and any exotic cooling channel — an axion emission, say, or a neutrino magnetic moment — would show itself as a deficit of stars where the cooling ran too fast. The clock is now sensitive enough that its disagreements are used as particle physics.

There is one more thing the cut-off is quietly sensitive to, and it is the reason the method needed Gaia rather than merely more telescope time. The number of white dwarfs at any luminosity is proportional to the rate at which they were made, so the luminosity function is the star formation history of the disc convolved with a cooling law and with the mass distribution of the stars that formed. A single cut-off compresses all of that into one number. A well-measured function does not: its shape between magnitudes ten and fifteen carries the history, and the bumps in it are epochs. That is a considerable amount of Galactic archaeology to extract from objects whose only remaining activity is getting colder.