A smaller star puts the valley lower

The radius valley was found in the planets of Sun-like stars, and its emptiness and its tilt follow from how a young star’s X-ray and ultraviolet light strips hydrogen from planets of different core masses. The stripping is finished early, within the star’s first few hundred million years, and that clock is one test of the mechanism. The mass of the star is another, and it does not need ages at all.

Stars of lower mass differ from the Sun in two ways that matter here, and they push in opposite directions. They are far fainter, so a planet at a given orbital period receives much less light. And they stay magnetically active for much longer, so that light is delivered over a longer saturated phase. Photoevaporation says exactly how those trade against each other, and the result is a prediction for where the valley sits round stars of every mass.

A tenth of the light at the same period #

The first effect is large, and it comes from two relations between a star’s mass and its planets. A main-sequence star’s luminosity rises steeply with its mass — roughly as the fourth power across the range of Sun-like and smaller stars, a relation calibrated on the few stars whose masses are measured directly — and by Kepler’s third law a planet with a given orbital period orbits closer to a less massive star, as the cube root of the mass. The light a planet receives goes as the luminosity divided by the square of its distance, so at fixed period it scales as the star’s mass to the power $4 - 2/3$, or $10/3$.

A factor of two in stellar mass is therefore a factor of ten in insolation. A planet on a ten-day orbit round the Sun receives about 120 times the Earth’s insolation; round a star of half the Sun’s mass, about twelve; round a star of three tenths, about two. The valley’s radius rises with insolation as its 0.13 power, so on this effect alone a star of half the Sun’s mass would put the valley at a radius smaller by a quarter.

A longer youth partly makes up for it #

The second effect runs the other way. Smaller stars have deeper convective envelopes and spin down more slowly, because the magnetised wind that brakes them carries angular momentum away less effectively relative to their store of it. They stay in the saturated, X-ray-bright phase for longer: in the model, as the mass to the minus one and a half, so a star of half the Sun’s mass is saturated for 283 million years rather than 100, and a star of three tenths for about six hundred.

A longer active phase means a larger total dose of XUV energy for the same insolation. That raises the valley, and the size of that effect can be read off directly by holding the insolation fixed and changing only how long the star stays saturated.

This is the valley for a star saturated for the Sun’s assumed hundred million years. The next two figures change only that.

Tripling the saturated phase triples the total XUV energy, and the boundary core rises from 5.1 to 8.0 Earth masses — a heavier core, stripped by a longer dose — and from 1.51 to 1.68 Earth radii. The tilt does not change, because it comes from the exponents of the escape law and the interior fit, which do not know how long the star was active.

Halving the saturated phase lowers the boundary core to 3.9 Earth masses and 1.41 Earth radii. Across the factor of six from 50 to 300 million years, the valley’s radius changes by a factor of 1.19. That is a real effect, but it is a gentle one: the dose enters the boundary through the same weak power as the insolation does.

Which effect wins #

At fixed orbital period the two effects arrive together, and the first figure is their sum. Going from the Sun to a star of half its mass cuts the insolation by ten and nearly triples the saturated phase; the net dose falls by a factor of about three and a half, and the valley at ten days falls from 1.50 Earth radii to 1.22.

Across a factor of five in stellar mass the valley’s radius at ten days changes by a factor of 1.6, and the four points are described well by a single power: the valley scales as the star’s mass to about the 0.3 power — 0.29 from the Sun down to three tenths of its mass, 0.30 down to half, and 0.30 up to one and a half. The insolation effect wins by a comfortable margin, and a smaller star puts the valley lower.

The whole family of lines shares one tilt. That is not a coincidence of the chosen stars: the tilt comes only from how the stripped core mass depends on insolation, and the star’s mass shifts every line up or down without rotating it.

A ten-day orbit round a red dwarf #

The numbers round the smallest star in that figure describe a very different place. A ten-day orbit round a star of three tenths of a solar mass is about six hundredths of an astronomical unit from it, closer than Mercury is to the Sun by a factor of six. Yet the star is so faint that a planet there receives only about twice the light the Earth does, and it spends its first six hundred million years bathed in the X-rays of a star that has not yet slowed down. The valley there sits at 1.06 Earth radii: in this model a planet barely larger than the Earth, on that orbit, has been stripped, and one of two Earth radii has kept its envelope.

That orbit is not far inside the range where liquid water could exist on a planet’s surface, which round a star this faint lies at orbital periods of a few weeks. The red-dwarf planets most studied for their habitability are exactly the ones this calculation is about, and the long, active youth that decides where their valley sits also decides how much atmosphere of any kind they keep.

A smaller star also makes a smaller planet easier to see. The fraction of the star’s light a transiting planet blocks is the square of the ratio of their radii, so an Earth-sized planet crossing a star of three tenths of the Sun’s radius dims it by about a tenth of a per cent — eleven times the dip it would make crossing the Sun. The valley round the smallest stars is at the smallest radii, but those radii are measured against the smallest stars.

The same prediction, turned round #

The dependence has a second form that is easy to confuse with the first. At fixed orbital period, smaller stars put the valley lower. At fixed insolation — comparing planets that receive the same light, which round a smaller star means planets on shorter orbits — the insolation effect is gone, and only the longer active phase is left.

So photoevaporation predicts the opposite sign at fixed insolation: round a smaller star, for the same light, the valley should sit slightly higher, by the factor the longer saturated phase buys. For a star of half the Sun’s mass, saturated nearly three times as long, the boundary figures put that at a radius larger by about a tenth — close to the step from 1.51 to 1.68 Earth radii that tripling the saturated phase produced. A valley that falls with stellar mass at fixed period and rises with it at fixed insolation is a specific signature of a mechanism driven by a young star’s activity.

Core-powered mass loss, in which the energy comes from the planet’s own cooling, depends on the star only through the planet’s temperature — its bolometric insolation — and not on how long the star was active. At fixed period it too predicts a valley that is lower round smaller stars, because the planets are cooler there, with a dependence on stellar mass of broadly similar size. At fixed insolation it predicts little or no dependence on the star, since two planets at the same temperature cool the same way whatever they orbit. The discriminating comparison is therefore not the one at fixed period, where the two agree in sign and roughly in size, but the one at fixed insolation, where one predicts a shift of about a tenth and the other next to none.

What has been measured #

Surveys of the planets found by the Kepler mission, with stellar masses refined by later spectroscopy and parallaxes, have found the valley moving to smaller radii round lower-mass stars, in the direction and roughly by the amount both mechanisms predict at fixed period. When the host stars of the first precise sample were divided into bins of mass, the valley in the highest-mass bin sat at a larger radius than in the lowest, and later samples including the planets of the Kepler mission’s successor, which surveys smaller and nearer stars, have extended that trend towards the red dwarfs. The dependence at fixed period is therefore measured, at least in sign, and its size is consistent with a power of a few tenths. The test at fixed insolation needs planets round stars of different masses receiving the same light, and the differences sought are a few per cent in radius — comparable to the uncertainty in the stellar radii from which every planetary radius is derived.

The commonest stars in the Galaxy are red dwarfs of a few tenths of a solar mass, far more numerous than stars like the Sun, and they are the stars for which the model’s assumptions are weakest. Their luminosity is not the fourth power of their mass, their active phases are long and poorly calibrated, and their planets are found in large numbers only by surveys that have been running for a few years. Studies of the planets of these stars have suggested that the valley there tilts the other way with orbital period — rising further out rather than falling — which is the signature of planets formed with and without gas rather than stripped. If that holds, the valley round the smallest stars may have a different origin from the valley round the Sun.

What a shift of a tenth costs to measure #

Every planetary radius in a transit survey is a ratio multiplied by a stellar radius: the depth of the transit gives the planet’s size relative to its star, and the star’s size has to come from somewhere else. When the Kepler mission’s planets were first catalogued, the stellar radii came largely from photometric colours and were uncertain by a quarter or more, and a feature a few tenths of an Earth radius wide was smeared out of the radius distribution entirely. The valley became visible only when spectroscopy of more than a thousand host stars brought their radii to about a tenth, and it sharpened when parallaxes from the Gaia satellite — the oldest geometric method, extended to a billion stars — brought them to a few per cent.

The effects argued here are the same size as those uncertainties. A shift of a tenth in the valley’s radius between two groups of stars is visible only if the radii within each group are known to a few per cent, and a stellar radius from spectroscopy depends on a model of the star’s atmosphere whose systematic errors differ between a red dwarf and a star like the Sun. A comparison across stellar mass is precisely the comparison in which those errors do not cancel. The samples differ in what they can find as well, since a transit survey’s sensitivity to a small planet depends on the star as much as on the planet, and a census is shaped by what each method can see before it is shaped by what is there.

What the model assumes about the stars #

A single mass–luminosity law. The fourth power is a fair average for stars from about half to one and a half solar masses. Below that, luminosity falls less steeply with mass, so the insolation effect is weaker than the model says, and the valley round the smallest stars should be less displaced.

A single activity law. The lengthening of the saturated phase as the mass to the minus one and a half is a rough fit to the rotation of stars in young clusters. The real dependence is steeper for the lowest-mass stars, some of which stay active for billions of years.

XUV in proportion to bolometric light. During saturation the model takes the XUV output as a fixed fraction of each star’s total light. That fraction is roughly constant in X-rays, but the extreme ultraviolet, which does most of the heating, is poorly measured for any star but the Sun.

Hotter stars are different in kind. Above about 1.2 solar masses a star’s outer convective zone becomes thin, its dynamo weakens and its rotation barely slows — the change in behaviour that also separates the stars that realign their planets’ orbits from those that do not. Such stars are X-ray-faint for their luminosity, and the scaling of activity with mass that the model uses does not describe them; the 1.5 solar-mass line is an extrapolation of the law rather than a portrait of such a star.

And the same planets. Every star is given the same population of cores and envelopes. If planets round smaller stars form with systematically smaller cores, as some models of planet formation suggest, the valley would move for that reason too, with no help from the star’s light: smaller stars have less massive discs and snow lines much closer in, and both change what cores can grow.

One mechanism, three fingerprints #

The valley’s tilt with period says it was carved rather than built. Its clock says the carving was early. Its dependence on the star’s mass says how much of the carving tool was the star’s total light and how much was its youth. Photoevaporation makes definite predictions for all three from the same few exponents, and each is measured with a different kind of difficulty — periods are easy, ages are hard, and stellar masses are in between.

What makes the three together more than the sum of their parts is that, in the model, they are not independent. The exponent that sets the tilt with period is the one that sets the response to insolation, and the response to insolation, combined with how luminosity and activity scale with mass, sets the dependence on the star. A population of planets sorted by orbital period, by the age of their stars and by their stars’ masses has to show a tilt near the −0.18 power of period, a valley finished within the first billion years, and a displacement near the 0.3 power of stellar mass — all from one set of numbers. A valley carved by the planets’ own cooling would pass the first test, fail the second, and match the third only at fixed period; a valley built in at formation would fail the first. Each measurement alone is suggestive. The combination, made on one sample, is what can decide.

Still open: what the young Sun did to the planets that are not sub-Neptunes #

The same arithmetic applies to planets with far less hydrogen than a sub-Neptune. An Earth-mass planet that captured even a hundredth of a per cent of its mass as hydrogen from the nebula it formed in would have faced the same young Sun, and whether it kept any depends on how active that young Sun was — a question the terrestrial planets of the Solar System can put to their own star.