A gap in a histogram that says how planets are built

For four years the distribution of exoplanet radii was a smooth hump. Then the radii of the stars were measured properly, every planetary radius moved by the amount its own star had moved, and a gap opened in the middle of the distribution that has been there all along.

It sits at about 1.8 Earth radii, and it is the sharpest structural feature in the exoplanet census.

Why it was invisible #

Every planetary radius is a stellar radius multiplied by a measured ratio. That much is inherent to the transit method: the depth is $(R_p/R_\star)^2$ and nothing else.

Before 2017, the Kepler host stars’ radii came from broadband photometry and a stellar model, and were typically uncertain by 25 per cent or worse. A 25 per cent error in every $R_\star$ is a 25 per cent error in every $R_p$, applied independently to each planet — which is a convolution of the true distribution with a wide kernel. A gap 20 per cent wide, smeared by a kernel 25 per cent wide, disappears.

Two things fixed it. The California–Kepler Survey took high-resolution spectra of 1,305 host stars and re-derived their parameters, cutting the radius uncertainty to about 10 per cent. Gaia then supplied parallaxes, which give a luminosity directly and a radius through the Stefan–Boltzmann law, reaching a few per cent.

The gap appeared immediately, in data that had been public for years.

Nothing about the planets changed. The instrument that resolved the feature was a spectrograph pointed at the stars, and a satellite measuring parallaxes — neither of which observed a planet at all. This is the single clearest example in the field of a measurement being limited by a quantity that appears in it only as a multiplier.

Why it is not a selection effect #

The obvious worry about a deficit in a histogram is that the survey found fewer planets there because it was less able to find them.

That worry is answerable here, and the answer is the reason the feature is believed. Detection efficiency for transiting planets rises smoothly and monotonically with radius: a 2.4 Earth-radius planet is easier to find than a 1.8, which is easier than a 1.3. There is no feature in the completeness at 1.8, and no mechanism that could put one there — the depth goes as the square of the radius with no scale in it.

So the correction that reveals the valley is smooth, and the corrected distribution is not. That is the strongest form the argument can take with this kind of data, and it holds.

What the valley means, in terms of composition #

The mass–radius plane supplies the interpretation, because it says what a planet of each radius has to be.

A rocky planet of Earth-like composition has $R \approx M^{0.27}$, so the whole range from 1 to 1.8 Earth radii spans 1 to 6 Earth masses of rock. Above about 1.6 Earth radii, a purely rocky composition becomes implausible — the masses required are large and the objects are not that dense — and the planets have to carry hydrogen.

And a very small amount of hydrogen goes a very long way. An envelope of one per cent of the planet’s mass, sitting on the outside where the compression is least, adds roughly 30 per cent to the radius. Two per cent adds about half again.

So the two peaks in the histogram are:

• Below the valley: bare rocky cores, with no detectable envelope at all.
• Above it: the same cores, wearing one to a few per cent of hydrogen by mass.

The valley is the region between “no atmosphere” and “enough atmosphere to matter”, and the reason it is empty rather than merely sparse is that an envelope small enough to place a planet there is unstable — it is either retained, in which case it is bigger than that, or lost, in which case there is none.

Two mechanisms, and the measurement that tells them apart #

Something removes the envelopes from the smaller planets. Two candidates have been on the table since the valley was found, and they make different predictions.

Photoevaporation. A young star emits far more X-ray and extreme-ultraviolet radiation than an old one — a hundred to a thousand times more in the first hundred million years. That radiation heats the top of a hydrogen envelope past the escape speed, and the envelope streams away. The energy available falls as the inverse square of distance, so the effect is strong close in and weak far out.

Core-powered mass loss. A newly formed planet’s core is hot from accretion, and it heats the envelope from below over a billion years. The escape again depends on the ratio of thermal energy to gravitational binding, and the equilibrium temperature — set by the star’s bolometric output — controls how far the atmosphere puffs out.

Both strip small planets close to their stars, and both produce a valley. What distinguishes them is how the valley’s position moves with orbital period.

The measured exponent is negative, near $-0.09$ in period, which corresponds to about $+0.11$ in the incident flux. That is what both escape mechanisms predict and what a formation-based explanation does not, so the valley is a carved feature rather than a birthmark. Separating photoevaporation from core-powered loss is harder, since the two agree on the sign and nearly on the magnitude; the current discriminants are the valley’s dependence on stellar mass and its behaviour at long periods, and neither is yet decisive.

The escape calculation, in one line #

The physics of both mechanisms reduces to a comparison the site has met before: is the thermal energy per particle comparable to the gravitational binding energy per particle?

Escape speed is $v_{\text{esc}} = \sqrt{2GM/R}$, and a gas at temperature $T$ has particles moving at around $\sqrt{3kT/\mu}$. Hydrogen is the lightest gas and therefore the fastest at any temperature, which is why it is always the first thing a planet loses and why the Earth has none left. For a hydrogen envelope on a 2 Earth-radius planet the two are uncomfortably close, and the escape is not a slow Jeans leak but a hydrodynamic outflow — the whole upper atmosphere expanding and flowing away, dragging heavier species with it. That regime has been observed directly: hydrogen absorption during transits of some hot Neptunes shows the planet trailing a comet-like tail, and the inferred mass-loss rates are enough to strip a per-cent envelope in a few hundred million years.

What a per cent of hydrogen is worth #

The arithmetic behind “one per cent of the mass is thirty per cent of the radius” is worth doing, because it is the reason the valley exists at a radius rather than at a mass.

A hydrogen envelope in hydrostatic equilibrium on a rocky core has a scale height $H = kT/(\mu g)$, and for hydrogen at 1,000 K on a 2 Earth-radius, 5 Earth-mass planet that is of order a thousand kilometres — a substantial fraction of the core radius itself. So the envelope is not a thin skin; it is a puffy shell whose thickness is set by temperature and gravity rather than by how much gas is present. Adding mass to it raises the density throughout but changes the radius only slowly.

The consequence is a lever: the radius responds to the presence or absence of an envelope far more than to its amount. That makes radius an excellent detector of whether a planet has an atmosphere at all, and a poor one for how much. It also means a distribution in radius separates the two populations cleanly while a distribution in mass would not — and it is exactly why the valley shows up in the quantity a transit measures rather than in the quantity a velocity curve measures.

The same lever runs backwards for the density inference: a bulk density is dominated by the core and nearly blind to the envelope, while a radius is dominated by the envelope. Two measurements sensitive to different components of the same object is the most useful arrangement there is.

What was actually measured #

The CKS radius distribution, 2017. Benjamin Fulton and colleagues published the completeness-corrected radius distribution for 2,025 planets around 1,305 spectroscopically characterised stars, and it showed a bimodal distribution with a deficit at 1.5–2.0 Earth radii. The feature was immediately recognised as the prediction several groups had made from photoevaporation models years earlier — which is unusual in this field, where predictions more often follow the data.

The slope, 2018. Vincent Van Eylen and colleagues used asteroseismic stellar parameters for a smaller, much better characterised sample, and measured the valley’s location as a function of period: $R \propto P^{-0.09}$, with the negative sign established at high confidence. That is the number that settled the direction of the argument.

Ultra-short-period planets. Planets with periods under a day are almost all smaller than 1.8 Earth radii, with essentially no exceptions. They are the extreme case of the same process: at those distances no hydrogen envelope survives at all, so the population is bare cores by construction.

And the direct observation of the leak. Ultraviolet transit observations of GJ 436 b show an absorption depth in the hydrogen Lyman-α line of over 50 per cent — against a 0.7 per cent optical transit — lasting for hours before and after the planet itself passes. The planet is enveloped in a cloud of escaping hydrogen far larger than its own disc. The mechanism in the paragraphs above is not inferred; it has been watched.

Where the picture stops #

The valley’s depth is not zero. Planets do exist between the peaks, and whether the residual population is real or is the tail of the measurement errors is still argued about. A feature that is a deficit rather than an absence is harder to interpret than one that is empty.

A corrected histogram inherits its correction’s assumptions. Everything here rests on the completeness calculation, which is measured by injection and recovery and is itself a model of the pipeline. A correction that was smooth by construction cannot manufacture a valley, which is the argument made above — but a correction that was systematically wrong in normalisation would move the peaks’ relative heights, and the ratio of super-Earths to sub-Neptunes is one of the numbers formation models are asked to reproduce.

The sample is close-in planets around Sun-like stars. The valley is measured for periods under 100 days around FGK stars. Around M dwarfs the picture differs — the valley appears shifted and may have a different origin — and beyond 100 days the completeness is too poor to say anything.

The composition inference relies on models. “One per cent hydrogen by mass” is a statement from an interior model with an assumed core composition and an assumed thermal state. A water-rich core would produce planets of the same radius without any hydrogen at all, and whether the sub-Neptunes are rocky-with-envelopes or water worlds is an open question that the radius distribution alone cannot settle.

Radius is not composition, and the valley is stated in radius. Everything above translates a radius into an envelope fraction through an interior model, and the same density can be produced by more than one mixture. A population of water-rich planets with no hydrogen at all would occupy nearly the same part of the radius distribution as rocky cores with envelopes, and the histogram cannot tell them apart. Only masses — and, eventually, atmospheric spectra — can.

And the two escape mechanisms are not distinguished. Both predict a valley in about the right place with about the right slope. They differ in the timescale — a hundred million years against a billion — and in the dependence on stellar mass, and the data are not yet good enough to separate them.

The stellar half of the story #

Both mechanisms make the valley a record of the star as much as of the planet, and that is worth separating out because it connects this figure to a part of the collection that looks unrelated.

A young star’s X-ray and extreme-ultraviolet output is not a fixed fraction of its bolometric luminosity. It is powered by magnetic activity, which is powered by rotation, and a star spins down over time as its magnetised wind carries away angular momentum. So the XUV flux is high and roughly constant for the first hundred million years and then falls steeply — and the total XUV fluence a planet receives is dominated by that early period, which is over before most of the planets in the census were a per cent of their present age.

The dependence on stellar mass is the part still being used to separate the two mechanisms. A lower-mass star is dimmer in the bolometric sense but stays active for far longer, so a planet in the same equilibrium temperature receives a very different integrated dose depending on the host, and the relation between a star’s mass and its whole life sets which. Photoevaporation and core-powered loss weight those two differently, so the valley’s position as a function of stellar mass should differ between them. Measuring it needs a large sample of well-characterised M-dwarf hosts, which is what the current transit surveys are accumulating.

Where the valley sits, and what that alone says #

There is one more piece of information in the feature and it is the least used: not the valley’s slope, but its absolute position.

The valley marks the radius of a bare core stripped of everything it could lose, so its location is the radius of the largest core that a given process can strip. That radius depends on what the core is made of. A core of Earth-like rock and iron of a given mass is smaller than a core of the same mass containing a substantial water fraction, because ice is less dense than silicate — so a population of water-rich cores would place the valley at a noticeably larger radius than a population of rocky ones, for the same stripping physics.

The measured position, near 1.8 Earth radii, sits where rocky cores predict. That is an argument against the sub-Neptunes being water worlds, made without any mass measurement at all, and it is independent of the slope argument the previous sections rested on.

It is not conclusive, because the stripping physics and the composition enter the prediction together and the models carry their own uncertainties. It is nevertheless the one place where the histogram constrains what the planets are made of rather than what happened to them.

The generalisation #

A gap in a distribution, where a smooth process would predict continuity, is one of the most productive kinds of observation there is, because a gap requires a mechanism.

The Kirkwood gaps in the asteroid belt are empty because a resonance clears them. The Hertzsprung gap in the colour–magnitude diagram is nearly empty because stars cross it quickly, so the deficit is a statement about a timescale rather than about a preference. The absence of stable elements at atomic masses 5 and 8 shapes the whole of stellar nucleosynthesis.

In each case the same reasoning runs: a smooth underlying population, plus a process that removes or accelerates through a particular range, produces a deficit — and the shape of the deficit constrains the process more tightly than the population on either side of it does. What is unusual about the radius valley is how recently it appeared, and that it appeared in data nobody re-observed.

Where this goes next #

The valley is a statement about what happens to a planet after it forms. The other great structural feature of the census — giant planets in orbits where they cannot have formed — is a statement about where planets end up relative to where they began, and it is read from a different distribution entirely.

Later rungs on this anchor: photoevaporation models and the energy-limited approximation. Core-powered mass loss. The valley around M dwarfs. Water worlds as an alternative reading. Ultra-short-period planets as stripped cores. Lyman-α transits and escaping atmospheres. The XUV history of young stars. The valley’s dependence on stellar mass. Metallicity and envelope retention. And whether the solar system’s own terrestrial planets are the same population, which is a question about the Sun’s first hundred million years.