A planet small enough to lose its rock

Planets lose atmospheres. Starlight heats the upper layers of a gas a planet cannot keep, the heated gas expands and flows away as a wind, and over billions of years a planet close to its star can lose a hydrogen envelope entirely and be left as bare rock. That process — photoevaporation, and its relatives — has a natural end point: the rock. Rock is bound far more tightly than hydrogen, because its molecules are heavier and a rocky planet’s surface gravity is strong, and a planet stripped to its core keeps the core.

Unless the rock itself is hot enough to boil. In 2012 the Kepler telescope found a star in the constellation Cygnus that dimmed every 15.7 hours by an amount that changed from one orbit to the next — sometimes by more than one per cent, sometimes by nothing at all — with a sharp start and a long, slow recovery. No solid planet makes a transit that changes its depth from orbit to orbit. The explanation that has held up is that the dimming is caused by a cloud of dust trailing behind a small planet on that orbit, and that the dust is rock: condensed from vapour that the planet’s dayside, at about two thousand kelvin, is boiling off its surface. A few more such objects have been found since, in the Kepler data and by the survey satellites that followed it.

This essay asks why such planets must be small, how fast they evaporate, and what their tails say about them. The answer to the first question is the sharpest: the escape of rock vapour is exponential in the planet’s gravity, and the exponential puts a boundary near a tenth of the Earth’s mass that no reasonable uncertainty in the rock’s chemistry can move very far.

Close enough to melt rock #

The planets are on orbits of a few hours to a day round K dwarfs, stars somewhat cooler and smaller than the Sun.

At a period of sixteen hours the planet is about three stellar radii from the star’s surface, and its dayside, assuming no atmosphere to carry heat round, sits above two thousand kelvin. That is hot enough to melt silicate rock and to give it a vapour pressure — small, a fraction of a pascal to a few pascals depending on the composition — of sodium, silicon monoxide, magnesium, iron and oxygen. A dayside as hot as bare rock is the condition for there being any vapour at all. Whether the vapour escapes depends on the planet.

How tightly a planet holds its own vapour #

A gas escapes from a planet in proportion to how much of it is moving fast enough, and the fraction is set by the ratio of the gravitational binding of one molecule to its thermal energy — the escape parameter $\lambda = G M \mu m_{\rm H}/(R k T)$.

For rock vapour at 2,100 kelvin, the Earth’s escape parameter is over a hundred: a molecule needs a hundred times its typical thermal energy to escape, and the fraction that has it is about $e^{-100}$, which is zero for any practical purpose. Mars’s is about twenty. Mercury’s is about thirteen. The Moon’s is about four, and a body that small, that hot, would lose its vapour freely.

The dependence on mass is steep because both the mass and the radius enter. A rocky planet’s radius grows roughly as the quarter power of its mass, so the binding per unit mass grows as about the three-quarter power, and the escape parameter with it. Every factor of ten in mass is a factor of about five in the exponent.

Which planets last a billion years #

The escaping vapour flows away as a thermal wind, accelerating from the surface through a sonic point at a distance set by the escape parameter, like the solar wind. For an isothermal flow the mass-loss rate follows from the vapour’s density at the surface and the escape parameter alone, and dividing the planet’s mass by it gives the time the planet would take to evaporate completely at its present rate.

The lifetime rises extremely steeply with mass: a factor of two in mass changes it by four or five orders of magnitude. That makes the result insensitive to the least certain input. The vapour pressure of molten rock at two thousand kelvin depends on its composition and is uncertain by more than an order of magnitude; a hundredfold change in it moves the mass at which the lifetime reaches a billion years by less than a factor of two, from about five hundredths of the Earth’s mass to about eight. Any planet still evaporating after a billion years must be smaller than about a tenth of the Earth — smaller than Mars, and possibly smaller than Mercury.

That is what makes the disintegrating planets a measurement. Their masses are far too small to detect by the wobble of their stars, and their radii cannot be read from their transits, because the transit is the dust and not the planet. The fact that they are evaporating at all bounds their mass from above.

An evaporation that accelerates to its end #

The planet does not evaporate at a constant rate. As it loses mass its gravity weakens and its hold on what is left loosens.

The escape parameter falls as the mass falls, the escaping fraction rises exponentially, and the evaporation runs away: a planet loses its first tenth of its mass slowly, over most of its life, and the rest in a small fraction of the time. A planet whose present rate would take a billion years to remove it is in fact gone in a few hundred million, because most of that billion years is the slow early phase it has already passed through or is in.

That shapes what is seen. Most evaporating planets caught by a survey should be in the long, slow phase, with masses near the upper end of what can evaporate and modest mass-loss rates; the rapid end is brief and rarely observed. A population of such planets is a sample of the slow phase, and their number tells how often small rocky planets end up on orbits of hours.

A mass-loss rate read from a shadow #

The tail gives the rate directly. The dust it contains blocks a fraction of the star’s light equal to its total cross-section divided by the star’s area. Dust grains condense from the cooling vapour, are pushed away from the star by its light, and leave the region in front of the star in a few hours; the tail has to be replenished at the rate it clears.

For micron-sized grains, which is the size the colour of the transits and the forward-scattering bump suggest, a tail half a per cent deep requires the planet to lose about two Earth masses of dust every billion years — a few hundred thousand tonnes a second. At that rate a planet of a tenth of the Earth’s mass would be gone in fifty million years. Since the escape argument forbids an evaporating planet from being much larger than that, the tail is evidence that the planet is in the last few per cent of its existence on astronomical timescales, and its own mass is shrinking appreciably over the lifetime of the human species — by a few parts in a million a century, not enough to see.

The transit depth varies from orbit to orbit because the outflow does. A wind of vapour carrying dust is sensitive to the exact conditions at its base — the temperature of the dayside, the cloud of dust already shading it, the rate at which grains condense — and it fluctuates. Some orbits show nothing; the tail was thin that time. Others show more than a per cent.

A planet seen only through what it loses #

It is worth pausing on how invisible the planet itself is. A rocky body of a twentieth of the Earth’s mass has a radius a little under half the Earth’s, about three thousand kilometres, comparable to Mercury’s. In front of a star two-thirds the Sun’s size it would block about four thousandths of one per cent of the light — a transit of forty parts in a million, below what Kepler could detect for a star that faint in a single orbit and barely detectable by folding many. The dust tail blocks a hundred times more. The planet would not have been found at all if it were not destroying itself.

That reverses the usual relation between a planet and what surrounds it. For most transiting planets the solid body is the signal and any atmosphere or escaping gas a small correction, measured with difficulty in the narrow wavelengths where the gas absorbs. Here the escaping material is the signal and the body is an inference: its period from the regularity of the dimming, its mass bounded from above by the fact of evaporation, its size bounded from above by the absence of any transit at the orbits when the tail is thin. On the orbits when the tail fails entirely and the light curve shows nothing, the observers are looking at the planet’s own transit and failing to see it, which limits its radius to a few thousand kilometres.

So the disintegrating planets are a class defined by an absence as much as by a presence: rocky bodies too small to see, made visible by the dust their stars boil off them, on orbits where nothing larger could stay rocky and nothing smaller could stay whole.

A transit with a slow end #

The shape of the dimming is the second signature.

A solid planet makes a symmetric dip: it enters and leaves the disc at the same speed. A trailing tail enters behind the planet and leaves behind it, so the dimming starts as the planet’s own small shadow crosses the limb and deepens as more of the tail follows, then fades gradually as the tail’s long, thinning end crosses the disc after the planet has gone. The deepest point lags the planet’s mid-transit. The observed light curves have exactly that asymmetry, and a small brightening just before the dip — light scattered forward by the tail’s dust towards the observer as it approaches the line of sight — confirms that the obscuring material is grains of about a micron and not a gas.

Where the envelope story ends #

The disintegrating planets are the last chapter of a story told across the rest of this subject. A planet that formed with a hydrogen envelope near its star loses it to starlight over its first few hundred million years, or boils it off from its own heat even sooner, and the helium escaping from such envelopes is now seen directly in absorption. What remains is a rocky core. For most cores that is the end: the escape parameter of rock on a planet of a few Earth masses is in the hundreds, and a stripped super-Earth on a short orbit, however hot, keeps its rock indefinitely. The radius valley — the gap between stripped cores and planets that kept their envelopes — is made of such survivors.

Only at the very bottom of the mass range does the story continue. A rocky body a twentieth of the Earth’s mass on an orbit of hours is in the regime where hydrogen escape would have been instantaneous and where the rock itself is marginally bound, and there evaporation proceeds through the rock. That such bodies exist at all on such orbits is a puzzle in its own right. They are inside the region where planets are thought not to form, and they must have migrated there — perhaps as the cores of larger planets that lost their envelopes and much of their mantles on the way, perhaps as fragments of larger bodies. Whatever their history, they are seen only in their last few per cent.

They also sit close to a second limit. A body on an orbit of fifteen hours round a K dwarf is not far outside the distance at which tides tear a body apart for rock of ordinary strength; a body that strayed slightly closer, or that had been weakened by losing its outer layers, would be disrupted by tides before it could evaporate. Evaporation and tidal disruption are two ways of returning a small planet’s material to dust, and which acts first depends on its distance and its strength.

The surprising connection: rock in the atmosphere of a white dwarf #

The same process, seen from the other end, appears around stars that have died. White dwarfs should have atmospheres of pure hydrogen or helium, since anything heavier sinks out of sight in days to years; a large fraction of them show silicon, magnesium, iron and calcium in their spectra, in the proportions of rock. That rock is the ashes of planetary bodies that strayed close enough to the white dwarf to be torn apart by its tides and accreted, and several white dwarfs show transits by irregular, varying clouds of debris on orbits of hours — disintegrating bodies, broken by tides rather than boiled by starlight, but producing the same kind of dusty, variable, asymmetric transit.

The two cases bracket what can happen to a small rocky body near a star. Round a living star it can boil if it is small and hot; round a dead one it can break if it is weak and close. Both leave dust, both are seen as transits that refuse to repeat, and both turn the composition of a rocky body into something a telescope can measure — in the white dwarf’s spectrum in one case, in the tail’s absorption features in the other.

What the estimate leaves out #

The wind is isothermal and the vapour has a single molecular mass, where the real vapour is a mixture of species with different volatilities and the flow cools as it expands, condensing dust that changes its heating. The vapour pressure is a free parameter spanning two decades; detailed models of the outflow, including the dust’s own shading of the dayside and the momentum transferred by starlight, give mass-loss rates that differ from this estimate by an order of magnitude in either direction, and move the threshold mass by tens of per cent. The radii follow a single rocky mass–radius relation, and the dayside temperature is fixed rather than responding to the dust.

The tail model is a single exponential of grains of one size, crossing the star on the planet’s chord, where the real tail has a size distribution, a curved shape set by radiation pressure and the orbit, and a forward-scattering phase function that adds light as well as removing it.

Still open: what the tails are made of #

The tails are now targets for the James Webb Space Telescope, which can measure the infrared absorption features of the dust as it passes in front of the star. Pyroxene, olivine, iron, corundum — each condensate has its own spectral signature, and which ones appear says which elements the planet’s surface was made of and at what temperature they condensed. A tail rich in iron would say the planet’s mantle has been stripped and the core is evaporating; one of magnesium silicates would say the mantle is still there.

If the composition can be read, the disintegrating planets become the only rocky exoplanets whose interiors are sampled directly — not inferred from a mean density, as the iron-rich planets are, but seen in the material being lost. And the sequence of evaporation predicts that the composition should change as a planet shrinks, from crust to mantle to core, so that a population at different stages of disintegration would be a cross-section of small rocky worlds, peeled one layer at a time.