A colour that runs backwards at the end of cooling

A white dwarf has nothing left to burn — its weight is held up by the pressure of electrons that cannot share a state, not by heat — and its brightness falls steadily as it radiates away the heat it was born with. That makes it a clock: its luminosity says how long it has been cooling, and the faintest white dwarfs in the Galaxy date the oldest stars that made them. The clock is read through the white dwarf’s temperature, and temperature is read through colour. A hot surface peaks in the blue and a cooler one in the red, and for almost every star colour is a thermometer that reads the same way from the hottest to the coolest.

For the coolest white dwarfs it stops reading that way. As a hydrogen-atmosphere white dwarf cools below about four thousand kelvin, its infrared colours stop reddening and turn blue, and they keep turning bluer as it cools further and gets fainter. On a diagram of brightness against infrared colour, the cooling track does not run off into the red, as a blackbody’s would; it hooks back.

Molecules that absorb only in a crowd #

The cause is molecular hydrogen, and an unusual way for it to absorb light. An isolated hydrogen molecule is symmetric: its two identical atoms share their electrons equally, it has no electric dipole moment, and it cannot absorb infrared radiation by changing its vibration or rotation the way a water molecule can. Cold hydrogen gas is almost perfectly transparent in the infrared.

When two molecules collide, their electron clouds are distorted by the encounter, and for the duration of the collision the pair has a small, transient dipole. During that instant the pair can absorb a photon, and the absorption appears in broad bands at the wavelengths of the molecule’s vibrational transitions: the fundamental near 2.4 microns, overtones near 1.2 and 0.8 microns, with a rotational band far out beyond ten microns. The process is called collision-induced absorption. The bands are broad because a collision lasts only a fraction of a picosecond, and a transition that exists for so short a time has no sharp frequency.

Because it needs two molecules at once, collision-induced absorption scales with the square of the density. In most astrophysical gases that makes it negligible. In the atmosphere of a cool white dwarf it does not, for a reason that is specific to these stars. A white dwarf’s surface gravity is a hundred thousand times the Sun’s, so its atmosphere is compressed into a layer a few hundred metres thick with a pressure at the visible surface of order the pressure at the bottom of the Earth’s oceans. And as the star cools, the opacity of its atmosphere falls — the free-free and bound-free processes that make a hot atmosphere opaque die away — so the visible surface moves deeper, into gas that is denser still. By four thousand kelvin the photosphere sits at densities of a tenth of a gram per cubic centimetre, liquid-like for a gas, and the molecules there are colliding constantly.

The switch-on is steep for two compounding reasons. The absorption grows as the square of the photospheric density, and the photospheric density grows steeply as the star cools and its other opacities fade. In the model drawn here that combination is a stand-in, an optical depth growing as the eighth power of falling temperature, but full model atmospheres show the same behaviour: the effect is invisible above about five thousand kelvin and dominant below four thousand.

Why the visible surface sinks #

The density at which a star’s light escapes is not a fixed property of the star. A star’s surface is a depth: the level at which the gas above is just transparent enough, about two-thirds of an optical depth, for light to leave. Where the opacity is high, that level is reached high up, in thin gas; where it is low, the light comes from deep down. What sets the mean opacity is mostly the windows between absorbers, not the absorbers themselves.

In a hot white dwarf, the opacity comes from hydrogen’s bound-free and free-free absorption and from the negative hydrogen ion, all of which need free electrons, and free electrons need either high temperature or easily ionised metals. As the star cools, hydrogen recombines, the electrons disappear, and the atmosphere becomes transparent at a rate that outruns the thinning. The photosphere falls through the atmosphere into ever denser layers, the molecules form there because density favours them, and the collisions that make the molecules absorb become more frequent in proportion.

So the collision-induced opacity arises precisely because the other opacities have gone. A white dwarf cooling below five thousand kelvin exposes deeper and denser gas, and in that gas a new absorber appears that did not exist higher up. The atmosphere is regulating its own transparency: the more transparent it becomes in the continuum, the deeper the light comes from, and the stronger the molecular absorption that re-opaques it in the infrared.

Where the absorbed light goes #

A white dwarf’s luminosity is fixed by its cooling. The atmosphere does not change how much energy leaves the star; it changes the wavelengths at which the energy leaves. Light absorbed in the infrared bands heats the gas a little, and the gas radiates the energy back out wherever the atmosphere is most transparent. The flux is redistributed, not removed.

That redistribution is why the colours turn blue rather than merely dimming in the infrared. With the infrared bands blocked, the energy that would have left there leaves in the optical and near the one-micron window, and the spectrum of a 3300 K white dwarf is concentrated at shorter wavelengths than a 3300 K blackbody’s. In the extreme case, at the lowest temperatures, a white dwarf can put more of its light into the optical than a blackbody of the same temperature would.

The consequence for colours is asymmetric. A colour that compares two infrared bands, such as J−K, is dominated by the strongest absorption, which falls on K, so it turns blue early and steeply. A colour that compares two optical bands, such as V−I, is barely affected until the lowest temperatures, because the absorption in the optical is weak; it keeps reddening, only more slowly than a blackbody would.

That split is itself a diagnostic. A star whose optical colour says it is cool and whose infrared colour says it is hot is a star with an atmosphere absorbing in the infrared, and the combination picks out ultracool white dwarfs from surveys in which neither colour alone would. The first such star identified, in 1999, had an infrared flux so depleted that its near-infrared colours were bluer than those of a star three times hotter.

In the optical the hook is gentle #

The same cooling track drawn in an optical colour looks almost normal, and that is the point of drawing it.

That is why the problem hid for so long. The large white dwarf samples were built from optical photometry and proper motions, and in the optical the coolest hydrogen white dwarfs sit where a slightly extended blackbody sequence would put them. Only the infrared shows the hook, and infrared photometry deep enough to reach stars of absolute magnitude seventeen at tens of parsecs came decades after the optical surveys. When it arrived, the coolest optical white dwarfs split into two groups: those with helium atmospheres, whose infrared continued the blackbody trend, and those with hydrogen, whose infrared had collapsed.

Why the oldest were missed #

White dwarfs are found by combining faintness with colour and motion: a star that is faint for its colour, and moves fast enough across the sky to be nearby, is a white dwarf. The search criteria were written around blackbody-like cooling, in which old white dwarfs are faint and red. Cooling tracks that hook back to the blue put the oldest ones in the colour range of much younger, hotter white dwarfs, and in the colour range of other kinds of faint blue object, where they were not being sought.

The oversight had a sharp consequence in the late 1990s. Microlensing surveys toward the Magellanic Clouds had found more lensing events than the Galaxy’s known stars could produce, suggesting that part of the dark halo was made of compact objects of about half a solar mass — the mass of a white dwarf. If a fifth of the mass that is not the light were ancient white dwarfs, they had to be somewhere, and they had to be cool. The recognition that ancient hydrogen white dwarfs would be blue in the infrared, and only moderately red in the optical, redirected the search: blue, faint, high-velocity objects were looked for, and some were found, and a claim was published in 2001 that they made up a few per cent of the halo’s mass.

The claim did not survive. The velocities of most of the candidates fitted the thick disc better than the halo; the white dwarfs’ own cooling ages were too short for halo objects in several cases; and later microlensing surveys found fewer events than the first, pushing the implied halo fraction toward zero. What survived was the method. The oldest white dwarfs of the disc and halo are now identified by precisely the infrared-blue signature, and the faint end of the white dwarf luminosity function, which is the Galaxy’s most direct age measurement, depends on counting them correctly.

A second track for mixed atmospheres #

Not every cool white dwarf has pure hydrogen at its surface. Some have helium atmospheres, in which no molecules form and the spectrum stays close to a blackbody all the way down. And some have mixtures, in which the absorption is dominated by collisions between hydrogen molecules and helium atoms rather than between two hydrogen molecules. Those collisions absorb more strongly and at lower hydrogen abundance, because helium keeps the atmosphere transparent and therefore denser at the photosphere.

That matters for the clock. A white dwarf of a given luminosity has a temperature that depends on what its atmosphere is made of, and the temperature inferred from colours depends on which absorption model is assumed. A class of white dwarfs found in the 2000s with extreme infrared deficits, named for the property, was first fitted with pure hydrogen atmospheres and assigned very low temperatures and very old ages — ages that exceeded the Galaxy’s. Refitted with mixed hydrogen–helium atmospheres, and with improved absorption coefficients for the helium collisions, they came out warmer, less massive, and of ordinary age. The colours had not changed. The inference from colour to temperature to age had, because it runs through a model of how molecules absorb in a crowd.

What the figures leave out #

The figures use a parametric stand-in for the absorption: three bands of fixed shape, an optical depth growing as a power of falling temperature, and a blackbody underneath. Real white dwarf atmospheres are computed by solving the radiative transfer in a stratified atmosphere with the collision-induced opacities measured in the laboratory and calculated from quantum chemistry, and the temperatures at which the colours turn in those models differ from these by a few hundred kelvin. The direction of every effect drawn here is theirs; the exact temperatures are not.

The figures also assume a pure hydrogen atmosphere and a fixed radius. The first assumption matters most: as the preceding section describes, helium changes the absorption qualitatively. The second matters less; the radius of a white dwarf depends on its mass and scarcely at all on its temperature, and a more massive white dwarf simply sits lower on the magnitude axis along the same shape of track.

And they cannot show the observation behind every temperature quoted: the distance. A white dwarf’s absolute magnitude, which places it on the track, needs a parallax, and before the space astrometry missions only a few hundred white dwarfs had one. With parallaxes for hundreds of thousands, the ultracool population now appears as a distinct sequence on the brightness–colour diagram, turning blue at the faint end as the figures predict, and its numbers can finally be counted without depending on a colour cut that was written for blackbodies.

The clock and its thermometer #

It is worth tracing exactly how an error in the absorption would reach an age, because the chain is short and every link amplifies. A white dwarf’s cooling age depends steeply on its luminosity — in the simplest theory the age goes as the luminosity to the power of minus five-sevenths — and its luminosity is its radius squared times its temperature to the fourth power. The radius comes from its mass, the mass from its surface gravity or its position on the brightness–colour diagram, and the temperature from fitting its colours with model spectra. A two per cent error in temperature is an eight per cent error in luminosity and a six per cent error in age. At ten billion years that is six hundred million years.

For a star whose colours are those of a blackbody, the temperature can be fitted to a per cent. For a star whose infrared has collapsed, the fit depends on the absorption model, and different models of the same star have disagreed by ten per cent in temperature. That is the scale at which the faint end of the luminosity function is uncertain, and why the last magnitude of it — the stars that date the start of the disc — is argued over in terms of molecular physics.

A thermometer that depends on a crowd #

The general point is that a colour is a thermometer only through a model of where the light escapes. For most stars the model is nearly a blackbody, perturbed by lines, and colour and temperature move together over the whole range. For cool white dwarfs the model has an ingredient that depends on density rather than on temperature — the rate at which molecules collide — and that ingredient switches on exactly where the old white dwarfs live.

The same physics appears wherever a molecular gas is dense and cool. The coolest brown dwarfs, whose atmospheres are cooler still, show collision-induced absorption by hydrogen as a suppression of their K-band flux that is used to estimate their surface gravity: a denser atmosphere absorbs more. The giant planets’ infrared spectra are shaped by it, and on Titan it is the main source of opacity in the thermal infrared, warming the lower atmosphere. It has been proposed as a greenhouse term for the early Earth, when a hydrogen-rich atmosphere could have kept the surface warm under a fainter Sun. And in the transmission spectra of exoplanets, collision-induced continua set the baseline against which molecular bands are measured.

In every one of those cases a gas that is transparent molecule by molecule becomes opaque in company, and a spectrum that would be read as a temperature becomes, in part, a reading of pressure. The white dwarf is the most extreme instance, because its gravity compresses its atmosphere further than any planet’s, and because its temperature is the quantity that everything downstream — its age, and through it the age of the Galaxy — depends on.

Still open: how old the oldest disc white dwarfs are #

The faint end of the white dwarf luminosity function is where the disc’s age is read, and the ultracool white dwarfs are its last few magnitudes. Counting them requires that each be recognised, separated from the massive white dwarfs that pile up where their cores freeze, placed at the right temperature, and assigned the right cooling age, and every step passes through the atmosphere’s composition and the absorption coefficients for its collisions. The coefficients for hydrogen colliding with helium at the densities of these atmospheres, where three-body collisions and the non-ideal behaviour of a dense gas begin to matter, are still being revised, and each revision moves temperatures by a few hundred kelvin at the faint end — where a few hundred kelvin is a billion years. Whether the oldest disc white dwarfs are ten billion years old or eleven is at present a question about molecular physics as much as about the Galaxy.