The stripping runs ahead of the starlight that drives it

The radius valley separates bare rocky cores from cores wrapped in a few per cent of hydrogen, and its emptiness comes from a peak: the time needed to strip an envelope is longest for one that doubles the planet’s size, so planets on either side of that peak are driven away from it. Both halves of the argument are about where the valley is and why it is empty. Neither says when it forms, and the timing turns out to be the property that tells the candidate mechanisms apart.

There are two. In photoevaporation the energy that drives the gas off is the star’s X-ray and extreme-ultraviolet light, which is intense while the star is young and rapidly rotating and fades as a magnetised wind brakes its rotation. In core-powered mass loss the energy is the heat left in the planet’s own interior from its formation, released as the core cools over billions of years. Both predict a valley in about the right place with about the right tilt. They keep entirely different clocks.

To see the photoevaporation clock, a whole population has to be followed rather than a single planet: three thousand model planets round Sun-like stars, each with a rocky core drawn from a spread of masses, a hydrogen envelope of a few per cent, and an orbital period between one and a hundred days, each losing gas by energy-limited escape as its star ages.

Stripping that runs ahead of the energy #

The two curves in the figure measure different things on the same scale: the fraction of all the planets that will ever be stripped, and the fraction of all the XUV energy the star will ever deliver. If each unit of energy stripped planets at the same rate, the two curves would lie on top of each other. They do not. The stripping curve runs ahead of the energy curve at every age: half the planets that will end bare are bare by 72 million years, when the star has delivered only 27 per cent of its XUV energy, and nine tenths by 457 million years, when it has delivered 76 per cent.

The reason is the runaway. Planets close to the boundary between keeping and losing an envelope are the ones for which a little energy tips the balance, and once tipped they strip themselves rapidly as the envelope’s shrinking radius and falling mass make each unit of gas easier to remove. Those planets go first, while the star is in its saturated phase. By the time the star’s output has faded, the planets that remain are the ones whose most resilient envelope outlasts everything the star has left to give, and the rest of the XUV light — nearly a quarter of it — falls on planets it can no longer strip. The valley is a record of the first few hundred million years, and the light delivered after that is almost wasted on it.

The shape of the energy curve itself is simple. The young star emits at a saturated level, a fixed fraction of its total luminosity, for its first hundred million years, and after that its XUV output falls as the age to the minus one and a half. The saturated phase delivers a little over a third of the star’s lifetime XUV energy, and the long decline delivers the rest, most of it within the next billion years.

Why a young star’s X-rays saturate and then fade #

The X-rays and extreme ultraviolet come from the hot corona above the star’s surface, heated by magnetic fields that the star’s own rotation and convection generate — the same dynamo whose cycle the Sun’s spots record. A faster-spinning star generates stronger fields and a brighter corona, up to a point. Beyond a certain rotation rate the X-ray output stops rising and holds at about a thousandth of the star’s total light, whatever the spin: the star is saturated. The model’s XUV fraction during saturation is three ten-thousandths, of the same order.

The saturation cannot last, because the magnetised wind that the corona drives carries off angular momentum. As the star slows, it eventually drops below the saturation threshold, and from then on its X-ray output falls with its rotation. Since a slower star loses angular momentum more slowly, the spin-down itself decelerates, and the X-ray output settles into a gradual power-law decline with age. By the Sun’s present age its X-ray luminosity is thousands of times below its saturated value. The energy that stripped the planets was spent almost entirely in the first billion years, and the clock the planets keep is the clock of the star’s rotation.

How few planets cross after the first billion years #

The histograms below make the point numerically, and it is worth putting in absolute terms first. In the model population of three thousand planets, the share of bare cores rises by about one percentage point between one billion years and five. That is some thirty planets crossing the valley in four billion years — one every hundred million years or so, in a population that saw nearly seven hundred cross in its first hundred million.

Each crossing, once begun, is also quick. A planet driven past the peak of its loss-time curve removes its remaining envelope in a fraction of the time it spent approaching the peak, because the loss accelerates relative to what is left. At any moment, then, the valley contains only the few planets caught in the act, and after the first billion years there are almost none to catch. The valley seen round a five-billion-year-old star is not a balance between planets entering and leaving it. Under photoevaporation it is a finished structure, and the stars have moved on.

A valley opening #

The same population drawn as histograms at four ages shows the valley opening.

At ten million years there is no valley, only a single population of planets with envelopes and a handful already bare. By a hundred million years nearly a quarter have been stripped and a second peak of bare cores has appeared, with a gap between. From one billion years to five, the share of bare cores changes by one percentage point. The histogram at five billion years, which is roughly the age at which most observed planets are seen, is essentially the histogram at one billion — and most of the way to it at a hundred million.

The deepest point of the gap moves between neighbouring bins of the histogram, from 1.73 to 1.83 Earth radii. That is the resolution of a histogram of three thousand planets in bins six per cent wide, not a movement of the valley; the planets that keep their envelopes do go on shrinking slowly as they cool, but by much less than a bin.

The intermediate ages fill in the story. At thirty million years the stripping has begun but the bare population is still too sparse to form a peak of its own. At three hundred million years, 31 of the eventual 35 per cent are already bare. Between three hundred million years and three billion, a factor of ten in age, the share rises by four percentage points.

The clock depends on how long the star stayed young #

The one parameter of the star’s history that the clock is sensitive to is the length of its saturated phase, and that is not a fixed number. Stars born spinning slowly leave saturation within a few tens of millions of years; stars born spinning fast stay saturated for several hundred. Which the Sun was is not known directly.

A shorter saturated phase delivers less energy overall, so fewer planets end bare, 26 per cent rather than 35, and the ones that are stripped are stripped sooner: half by 50 million years, nine tenths by 380. The stripping still runs ahead of the energy, by a smaller margin.

A long saturated phase strips nearly half the population, and it takes longer to do it: half by 126 million years, nine tenths by 955. Here the lead of the stripping over the energy is at its largest — half the planets are bare when only 17 per cent of the energy has arrived — because a long, intense start pushes the runaway planets over the edge while most of the star’s output is still to come.

Across a factor of six in the saturated phase, the age by which nine tenths of the stripping is done moves from 380 million years to just under a billion. That is the whole range photoevaporation allows: within the first billion years in every case, and within the first few hundred million for any star that was not an unusually fast rotator.

The share bare is a record of the young star #

The clocks carry a second, less expected result. The final share of bare planets depends strongly on the saturated phase — 26, 35 and 49 per cent — while the valley’s position does not.

At a hundred million years this population is identical to the one round the gentler star, 23 per cent bare, because both stars have been saturated throughout. The difference opens afterwards, while the fast rotator stays saturated for two hundred million years longer, and it ends at 49 per cent bare against 35. The deepest point of the gap, 1.83 Earth radii at five billion years, is the same bin for both.

So the proportions on either side of the valley say something the valley’s position cannot: how much XUV energy the planets received in total. If photoevaporation is responsible, the ratio of bare cores to sub-Neptunes round a group of stars is a fossil record of how fast those stars were spinning when they were young, and it should differ between populations of stars that were born in different environments.

The other clock #

Core-powered mass loss uses the same geometry of peaks and runaways but a different source of energy. A newly formed planet’s rocky core is hot, and the heat escaping through the base of the envelope keeps the envelope warm and puffy and drives gas off its top. The core cools on a timescale set by how fast the envelope lets heat out, which for these planets is of order a billion years, and planets continue to be stripped for as long as their cores stay hot enough.

That puts its clock an order of magnitude later. Under photoevaporation the proportions on either side of the valley are fixed by a billion years and hardly change after. Under core-powered loss they are still changing at several billion. The prediction is sharp, and it is independent of every uncertain number in either model: not the position of the valley, not its slope, but whether planets round stars of one billion years and planets round stars of five billion years look the same.

Planets caught in the act #

If most of the stripping happens in the first hundred million years, the planets round the youngest stars should look different: larger, with envelopes still being lost, and in some cases in the middle of crossing. A handful of transiting planets round stars of that age are known, and they are strikingly large. Planets round stars only ten to twenty-five million years old have been found with radii of four to nine Earth radii, several times the size of typical sub-Neptunes round older stars, in orbits where older stars’ planets are small.

Photoevaporation reads them as planets whose envelopes are still hot and inflated and whose stripping is under way. The test is their masses. A planet of five Earth radii and ten Earth masses has a density a fifth of the Earth’s and a loosely held envelope that the model says will be largely gone within a few hundred million years; the same radius with forty Earth masses would be a planet that keeps it. For planets in chains of near-resonant orbits the masses can come from the transits running early and late, and for at least one such young system they have, placing some of its planets at very low densities. Where an envelope is escaping fast enough, the escaping gas itself can be seen, as an extended cloud absorbing the star’s light during transit at wavelengths where hydrogen or helium absorbs.

These are individual planets rather than a population, and the young stars they orbit are active and spotted in ways that make every measurement harder. They are nonetheless the one place where the clock can be watched running rather than inferred from its end state.

Why the ages are hard #

The test needs the ages of the stars that host transiting planets, and the ages of ordinary field stars are among the hardest quantities in astronomy. A star in a cluster can be dated by the point at which its companions leave the main sequence, but almost all planet hosts are single stars in the field, spread over billions of years in age and occupying nearly the same place on the main sequence throughout.

Two indirect clocks are used. One is the star’s position relative to models of how stars brighten and swell slowly on the main sequence, which is a sequence of mass rather than of age, which gives ages to a few tens of per cent for stars a little hotter than the Sun and very poorly for cooler ones. The other is rotation: stars spin down as they age, at a rate set by the wind, so a rotation period and a colour give an age — the same braking that ends the saturated phase in the first place. Each has its own calibration problems, and a population divided into age bins inherits all of them, in the way that any clock read off a whole population does.

Analyses of the planets found by the Kepler mission, with stellar properties refined by Gaia, have reported that the ratio of stripped to unstripped planets is still changing between one billion years and several, which is what core-powered mass loss predicts and photoevaporation does not. Surveys of planets in young clusters, whose ages are known far better, have found sub-Neptunes more common there than round older field stars — evidence that envelopes are still being lost after several hundred million years, though not, by itself, evidence of which process is removing them. Neither reading is yet decisive, and both are limited by the ages.

What the population leaves out #

One kind of star. Every planet here orbits a Sun-like star with one saturated phase. A real sample mixes stars of different masses, whose activity lasts different lengths of time and whose light falls differently on their planets, and that mixture shifts both the clock and the valley.

One origin for the envelopes. The starting envelopes are drawn from a single smooth spread. If some planets formed with no hydrogen at all, they would sit below the valley from the start and add to the bare population at every age.

And only one mechanism. A real planet is heated both by its star and by its own core, and the two processes act on the same envelope at the same time. The model population keeps them apart so their clocks can be compared; the planets do not, and a measured clock may turn out to be a mixture, early stripping by the star followed by slower stripping from within.

Still open: whether the valley moves with the star’s mass #

Round smaller stars the light falling on a planet at a given period is far fainter, and the star stays active far longer. Photoevaporation weighs those two against each other in a definite way, and so predicts where the valley should sit round stars of different mass — a second test, independent of age, which the population of planets round the Galaxy’s commonest stars is only now large enough to make.