A planet read in the ashes of its star

Everything known about the composition of planets round other stars, apart from their atmospheres, comes from two numbers: a mass and a radius. A planet’s density, set against the densities that iron, rock, water and gas would give, says roughly what fraction of it is metal and what fraction is rock, and with assumptions about the star’s own composition it says how big the core is — the method that found planets with more iron than their star could have given them. It cannot say what the rock is made of. Magnesium and silicon, the elements of the mantle, and calcium and aluminium, the elements of a crust, have nearly the same density in rock and leave no separate signature in a radius.

There is one place where that composition can be read element by element: the atmospheres of white dwarfs. A white dwarf is the exposed core of a star like the Sun, a mass of carbon and oxygen about the size of the Earth held up by the pressure of electrons that cannot be squeezed into the same state, with a surface gravity a hundred thousand times the Earth’s. In that gravity anything heavier than helium sinks out of the visible layer, quickly, and the atmosphere is left as pure hydrogen or pure helium. Yet between a quarter and a half of white dwarfs show heavy elements in their spectra — magnesium, silicon, iron, calcium, sometimes a dozen others. The only way to keep them there is to keep adding them, and the material being added is the debris of rocky bodies from the white dwarf’s own planetary system, torn apart as they pass too close.

The spectrum of such a white dwarf is a chemical assay of an asteroid or a planet. This essay is about what it takes to read it, and about the one thing that makes the reading uncertain: every element sinks at its own rate, so the composition in the atmosphere is the composition of the body only at certain moments.

Why a white dwarf eats rock #

A white dwarf’s planets were there before it was a white dwarf. As the star swelled into a giant and then shed half or more of its mass, the planets that survived — those beyond a few astronomical units — moved outward in proportion to the mass lost, and the orbits of anything round it were reshaped by the change in the central mass. The new architecture is no longer stable in the same way. Over the white dwarf’s cooling life of billions of years, planets perturb asteroids and comets onto orbits that plunge inward, and any rocky body that comes within about a solar radius of the white dwarf is torn apart by its tide.

The fragments form a disc of dust and gas, seen in infrared emission round many polluted white dwarfs, and the disc drains onto the star. Material is delivered at rates from about a hundred million to ten billion grams a second — modest, the mass of a large asteroid over a million years — and it arrives as the bulk of whatever body was disrupted, crust, mantle and core mixed. Then it sinks.

How long a metal stays in sight #

The rate at which an element sinks out of sight depends on two things: how fast it diffuses downward through the white dwarf’s outer layer, and how deep the layer it is mixed through. White dwarfs with hydrogen atmospheres hotter than about twelve thousand kelvin have thin, radiative outer layers; metals arriving there are mixed only through a sliver of the envelope and sink out of it in days. Cooler hydrogen atmospheres, and helium atmospheres at every temperature, have convection zones that stir a much larger mass, and the metals are diluted through it and leave it slowly.

The range is enormous: from a day to a million years, nine orders of magnitude, depending on which kind of white dwarf is looked at and how hot it is. The two extremes are the two kinds of evidence. A hot hydrogen white dwarf with metals in its spectrum is accreting now, this week; if accretion stopped, the metals would be gone in a fortnight. A helium white dwarf with metals could have stopped accreting a million years ago and still show them.

Different elements sink at different rates. Heavier ions, and those that carry less charge for their mass in the hot plasma, diffuse faster; iron leaves faster than magnesium, by about half as much again, and calcium and silicon sit between. The differences are modest, factors of one to two, but they compound with time in exactly the place where time is long.

Three stages of one meal #

Consider a single accretion episode: a disc draining steadily for a while, delivering every element in the proportions of the body that made it, then running out. The mass of each element in the visible layer is set by a balance between what arrives and what sinks, and the balance passes through three stages.

Early in the episode, before either element has had time to sink, everything that arrives stays, and the atmosphere’s composition is exactly the body’s. This is the rising stage. Once accretion has continued for several sinking times, each element reaches a level where it sinks as fast as it arrives: its mass in the layer is its accretion rate times its own sinking time. In this steady state the atmosphere’s ratio of iron to magnesium is the body’s ratio multiplied by the ratio of their sinking times, so iron, sinking faster, is under-represented by a known factor. And when accretion stops, both elements decay exponentially, each at its own rate, and the faster one disappears first; the ratio falls further the longer the decline goes on.

The physics of each stage is simple and the corrections in each are known. The difficulty is that a single spectrum does not say which stage it was taken in.

A white dwarf that can be read at once #

For the hottest hydrogen white dwarfs the difficulty vanishes, because the sinking time is so short compared with any plausible accretion episode that the atmosphere is in steady state essentially all the time it shows metals.

The rising stage lasts days; the decline lasts weeks; the steady state lasts as long as the disc does, which is tens of thousands of years or more. The chance of catching such a star in anything but steady state is the ratio of days to millennia — negligible. So the correction from atmospheric to accreted composition is simply the ratio of sinking times, which is computed from the physics of diffusion rather than guessed. The cost is that hot hydrogen white dwarfs have so little mass in their visible layer, and so short a memory, that only a few elements are strong enough to measure; their spectra in the ultraviolet give magnesium, silicon, iron, carbon and a few more, but rarely the full set.

A white dwarf that needs a guess about time #

Helium white dwarfs are the opposite. Their deep convection zones hold enough metal to show many elements at once — some show fifteen or more, enough for a nearly complete chemical analysis — but their sinking times are comparable to the accretion episodes themselves.

A helium white dwarf showing metals might be early in an episode, with its atmosphere holding the body’s own proportions; it might be in steady state, with iron under-represented by the sinking-time ratio; or it might be millions of years past the end of accretion, with the faster-sinking elements depleted by large factors. The spectrum alone does not say. Infrared emission from a surviving disc argues that accretion is still going on, which rules out the decline. Its absence rules out nothing, since many accreting white dwarfs show no infrared excess at all.

A mantle or a core, depending on when #

What the uncertainty about stage does to an inferred composition is easy to see by running the calculation backwards. Suppose a helium white dwarf is observed with an iron-to-magnesium ratio exactly equal to the bulk Earth’s — about two to one by mass. What was the body it swallowed?

Caught during the rising stage, the body was Earth-like: a differentiated planet or asteroid swallowed whole, core and mantle together. In steady state it held one and a half times Earth’s iron, a planet richer in metal than the Earth, like the dense planets whose radii demand big cores. Caught well into the decline, the body must have held many times the Earth’s iron — a fragment of a metal core with only a little mantle attached, the kind of object that the collisions in a planetary system would strip from a differentiated body. One spectrum, three different objects.

The figure also carries the one inference that does not depend on stage. Iron sinks faster than magnesium in every stage, so the atmosphere can only ever under-represent iron relative to the body. The observed ratio is therefore a floor on the body’s iron content, and a white dwarf showing less iron than bulk Earth genuinely swallowed something iron-poor: a piece of mantle or crust. The asymmetry is useful. Calcium-rich, iron-poor compositions, which have been found, can only be crust; iron-rich ones may be core, or may be ordinary rock seen at the wrong moment.

What the assays have found #

Read with this in mind, the polluted white dwarfs give a consistent picture. Most of the bodies they have accreted are rocky and dry, with compositions close to the bulk Earth’s or to the chondritic meteorites: magnesium, silicon, iron and oxygen in roughly the proportions of the inner solar system. The rock in other planetary systems is made of the same few things as the rock in this one, which was not known before these measurements and could not have been learned from masses and radii. Some bodies are clearly fragments of differentiated objects — iron-rich and core-like, or calcium- and aluminium-rich and crust-like — which means that planetary bodies elsewhere melted and separated into core and mantle as the Earth did. A few show more oxygen than their metals can bind as oxides, which means water: a body that was tens of per cent ice, like the asteroids of the outer belt or the material beyond the line where ice counts as rock.

The masses involved are worth stating, because they make the whole thing concrete. A helium white dwarf’s convection zone at a given time holds up to about $10^{22}$ grams of metal, the mass of an asteroid a couple of hundred kilometres across. That is a lower limit on the parent body, since the zone holds only what has not yet sunk, and summed over a sinking time it implies bodies comparable to the largest asteroids in the solar system. Nothing in this sample requires a whole planet; the material is planetesimals and their fragments, which is what the dynamical picture of a white dwarf’s surviving system predicts should be thrown inward.

Why the composition can be trusted at all #

A natural worry is that the white dwarf’s own material contaminates the measurement. It does not, for the same reason the method works: the white dwarf’s own heavy elements sank out of its atmosphere long ago, within its first few million years as a white dwarf, and its interior is carbon and oxygen buried under the hydrogen or helium layer. The cooler white dwarfs, which have had billions of years to settle, freeze and separate their interiors, but nothing from that process reaches the surface. Every metal in a polluted atmosphere was delivered from outside.

A second worry is radiative levitation: in the hottest white dwarfs, above about twenty-five thousand kelvin, the outward push of radiation on heavy ions can hold some of them up without accretion. That is why the cleanest measurements come from white dwarfs cooler than that, and why the hottest polluted ones need a separate correction for each element. A third is the disc itself: if its dust is sorted by size as it spirals in, the way drag sorts any disc, the material reaching the star at a given time might not be the bulk body. Gas and dust in the discs seen so far drain together rather than separately, which suggests the sorting is weak, but it is measured on few systems.

A clock that ages the meal #

There is one more way the stage problem can be attacked, and it uses the white dwarf’s own cooling. A white dwarf’s temperature is a clock that runs without fuel, and the fraction of white dwarfs that show pollution at each temperature reflects how long they keep accreting as they age. For hot hydrogen white dwarfs that fraction is the fraction accreting now. For helium white dwarfs it is the fraction that accreted within the last sinking time. Comparing the two, at matched cooling ages, measures how long a typical accretion episode lasts — and the answer, from the samples so far, is long: comparable to or longer than the helium sinking times, at least a hundred thousand years and probably a million.

The alternative is worth seeing, because it is what a short episode would do.

A meal that ends before the atmosphere can settle never shows the steady-state ratio at all. The atmosphere holds close to the body’s own proportions while the disc drains and then spends most of the time it is visible in decline, with the iron leaving first. If episodes were usually this short, a helium white dwarf with metals would most often be seen in that decline, and the assays of helium white dwarfs would systematically find less iron than the bodies held — a bias towards mantle-like compositions, the opposite of the one that worries the core fragments. That the helium and hydrogen samples give similar average compositions is evidence against short episodes.

If episodes are typically long, most polluted helium white dwarfs are in or near steady state, and the decline stage, with its large corrections, applies to a minority. That is a statistical answer to a question about individual stars, and it is the best available. It means that the population of compositions inferred from helium white dwarfs is probably right on average, while any single extreme composition read from one of them — a core fragment, say — remains open to the alternative that an ordinary body was caught late.

Where the sinking times come from, and how far to trust them #

The figures follow two elements through one idealised episode: constant accretion, a sharp end, sinking times that are fixed numbers. Real discs probably drain unevenly, and an episode can begin while the previous one’s metals are still sinking. The sinking times are themselves calculated, from models of diffusion and convection in white dwarf envelopes whose depths depend on treatments of convective overshoot that have shifted by factors of several in the last decade; the values plotted are approximate to within such a factor. And the conversion from atmospheric abundance to the mass of the parent body needs the mass of the mixing zone, which comes from the same models. None of this changes the structure of the argument — three stages, one of them always available on hot hydrogen white dwarfs, all three possible on helium ones — but every number on the vertical axes carries it.

Still open: whether the planets that made the rock were the planets that survive #

The white dwarfs say what the rocky material in other planetary systems is made of. They do not say whether that material came from planets or only from asteroids — whether the bodies being swallowed are the leftovers of planet formation or the remains of planets broken up in the white dwarf’s long cooling. The masses inferred favour asteroids. The differentiated compositions say the bodies melted, which needs either the heat of short-lived radioactive elements early on, as in the solar system’s own asteroids, or the size of a planet. The next measurement is the comparison of the composition of a white dwarf’s accreted rock with the composition of the star it used to be, read from a wide stellar companion that formed from the same cloud. If the rock matches what the star’s iron, magnesium and silicon predict, the white dwarfs are sampling ordinary planetesimals; if it is systematically richer in iron, as some of the dense planets round living stars are, then the processes that strip mantles were at work in those systems too, and the white dwarfs are reading what is left of planets.