An envelope that boils down before the starlight strips it

Most of the planets found around other stars are between the Earth and Neptune in size, and many of them have kept a few per cent of their mass as hydrogen and helium wrapped round a rocky core. The envelope is a small share of the mass and a large share of the radius: a core twice the size of the Earth wearing an envelope of two per cent of its mass looks three times the size of the Earth from outside. Which planets kept their envelopes, and how much of them, is written into the gap in the histogram of planetary radii, and the standard account of the gap is a story about losing them.

The losing is usually told in the star’s ultraviolet. A young star is magnetically active and bright in X-rays and extreme ultraviolet for its first hundred million years; that light is absorbed high in a planet’s envelope, heats the gas there to thousands of kelvin, and drives a wind that strips the planets closest to the star down to bare rock. The rate follows the absorbed light up to a point, the stripping is finished early, and the valley is carved. That account starts with an envelope already in place and a star already shining on it.

It skips the first million years, and the first million years are where a planet’s envelope is least secure. The planet gathered its hydrogen from the gas disc around the young star, and while the disc was there, the envelope’s outer edge sat in the disc’s gas at the disc’s pressure. When the disc disperses — over a few hundred thousand years, a few million years after the star formed — that confining pressure goes. What is left is an envelope that is hot from its own formation, swollen to many times its eventual size, and suddenly facing vacuum.

The same number near thirty, met from the other side #

The quantity that decides how fast any atmosphere leaks is the escape parameter, the ratio of the gravitational energy binding a molecule at the top of the atmosphere to its thermal energy there,

$$\Lambda = \frac{G M \mu m_{\mathrm{H}}}{k T R}.$$

For a gas that is not heated beyond its equilibrium temperature the outflow takes the form of a slow, isothermal wind — the solution Eugene Parker found for the solar wind in 1958 — which accelerates from rest at the base through the speed of sound at a radius $GM/2c_s^2$ and coasts out from there. Its rate is set almost entirely by the density at the sonic point, and that density is the base density reduced by the hydrostatic factor for climbing out to it: $e^{-\Lambda}$, to within a factor of order one. So the mass-loss rate is exponential in $\Lambda$, and so is the time to lose any given envelope.

That is the opening figure’s whole content. Every unit of Λ costs a factor of 2.7 in time. Between a planet whose envelope boils off in a million years and one whose envelope outlasts a billion lies a range of only seven units, from about 18 to about 26 for a five-Earth-mass planet, and the band hardly moves with the planet’s temperature because the temperature is already inside Λ. Envelopes with Λ well below the band are not stable on any timescale anyone would call a planet’s lifetime. Envelopes well above it are effectively sealed against this kind of loss.

The band sits close to the threshold that appeared when Jeans escape was traced from the tail of the Maxwell distribution — the dimensionless number near thirty past which loss rates collapse by orders of magnitude. That is not a coincidence. Jeans escape is what happens to the few molecules at the top of an atmosphere moving fast enough to leave; a Parker wind is what happens when the top of the atmosphere is so weakly bound that the whole gas, not just its tail, can flow. Both are governed by the same exponential, and both shut off at the same order of magnitude of Λ. The young envelope approaches that threshold from the side where it is losing everything.

The wind needs a power supply, and here it differs from the winds of later life. A Parker wind lifts gas out of a gravitational well, and the energy has to come from somewhere. For the solar wind it comes from the corona’s heating; for photoevaporation, from absorbed ultraviolet. For boil-off it comes from two ordinary sources: the star’s bolometric light, which keeps the outer envelope at the equilibrium temperature however much gas flows through it, and the heat stored in the envelope and core from their formation, which is released as the envelope contracts. Neither is in short supply in the first million years. A young planet’s internal luminosity is thousands of times what it will be at a billion years, and the star’s light is the same light that will later set the planet’s climate. The wind is limited by how fast it can climb out, not by how much energy it is offered — the opposite of the energy-limited escape that governs the ultraviolet-driven winds later on.

An envelope that throttles its own wind #

Where does a young planet start? Its envelope at the moment of disc dispersal is hot, because the gas was heated as it was accreted and has not had time to radiate that heat away, and hot gas is extended. A planet of five Earth masses holding a tenth of its mass in hydrogen has, at this stage, a radius of seven or eight times the Earth’s, and at a few hundred kelvin its escape parameter is near ten. By the opening figure, such an envelope should be gone in a few thousand years.

It is not, because losing mass changes Λ. An envelope that loses part of its mass shrinks — less gas is supported above the core — and a smaller radius means a larger Λ. The wind reduces the very quantity that sustains it.

The tracks show the consequence. An envelope of thirty per cent loses most of itself in the first hundred thousand years after the disc goes, when its radius is largest; by a million years it has shrunk to under five per cent of the planet’s mass, and from then on it loses almost nothing. An envelope of fifteen per cent arrives at the same place by a shorter path. An envelope of six per cent loses a little. An envelope of two per cent, compact and well bound from the start, keeps nearly all of it.

The striking feature is convergence. Two planets that left the disc with envelopes of fifteen and thirty per cent — a factor of two in their hydrogen, the kind of difference that might be set by how long the disc lasted or how fast the core grew — end with envelopes of the same size. The boil-off does not remove a fixed amount, or a fixed fraction. It removes whatever is necessary to bring the planet to one particular condition, and that condition is a value of Λ.

Plotted against Λ, the convergence is exact. Every planet that boiled climbs to the same escape parameter, and the reason is the one the opening figure gives: the wind shuts off where the time to lose what is left first exceeds the time that has passed. At a hundred thousand years that happens near Λ = 20, at a million years a few units higher, and so on — the stopping point moves up slowly with age, by about two and a half units for every factor of ten in time, because each decade of time is a factor of e to the 2.3 in the allowed loss rate. The planets ride that slowly rising threshold as their envelopes cool and contract, and after a hundred million years all three that boiled sit at Λ ≈ 31.

What the disc decided, and what it did not #

The convergence means that much of what a planet’s formation set is erased in the first million years.

The figure has two regimes, joined at a corner. Below the corner, a planet’s envelope is compact enough that its Λ starts above the shutting value, and nothing happens: the final envelope equals the initial one. Above the corner the curve flattens, and for a five-Earth-mass core it is flat outright — any envelope above about five per cent is boiled down to about four and a half. A planet that formed in a long-lived disc and accreted forty per cent of its mass in hydrogen, enough to be a small Saturn, ends up indistinguishable from one that accreted ten.

The corner depends on the core’s mass, and steeply. A three-Earth-mass core boils down below one per cent, close to bare; a ten-Earth-mass core, whose gravity gives it a Λ three times larger at the same radius, keeps nearly thirty per cent and barely boils at all. Boil-off is selective in mass in the same direction as photoevaporation, and for the same reason: Λ is proportional to mass.

There is a corollary for the planets that sit just below the corner. An envelope of two or three per cent on a five-Earth-mass core never boils, so it reaches the age of photoevaporation intact; an envelope of ten per cent on the same core boils down to four and a half, and also reaches it. The planets that formed with the least gas are therefore not the ones that end with the least: the corner caps the rich, and leaves the poor alone. In a population drawn from the disc with a broad spread of envelope fractions, boil-off narrows the spread from above, and the distribution that enters the next hundred million years is piled up against the corner.

This bears directly on the comparison that photoevaporation models make with the radius valley. Those models need an initial envelope for every planet, and the usual choice is a distribution of envelope fractions motivated by how much gas a core of a given mass could accrete from the disc. The figure says that choice matters less than it looks. Above the corner the envelope a photoevaporation model starts with is not the envelope the planet accreted but the one boil-off left, and it is set by the core’s mass and the planet’s temperature rather than by the disc.

A hotter orbit boils further #

The equilibrium temperature enters Λ in the denominator, and a planet closer to its star starts its post-disc life with a lower Λ at the same radius.

For cool planets, a few hundred kelvin, boil-off touches only the smallest cores. As the temperature rises, the envelope’s outer edge needs to be smaller to reach the shutting value, so more of the envelope must go: at fifteen hundred kelvin, a three-Earth-mass planet is boiled to bare rock and a five-Earth-mass one to half a per cent. Only the ten-Earth-mass core keeps a substantial envelope at that temperature, and it keeps a third of what it had.

This is the trend in temperature that photoevaporation also produces, and for the same reason the two are hard to separate in a single snapshot of today’s planets. But it is produced by different light. Photoevaporation runs on the extreme ultraviolet, a tiny fraction of a young star’s output that varies by a factor of ten between stars of the same mass according to how fast they spin. Boil-off runs on the bolometric light, which sets the equilibrium temperature and is fixed by the star’s luminosity and the planet’s distance alone. A boil-off trend has no scatter from stellar activity; a photoevaporation trend does, because the young Sun’s spin — and every young star’s — decided how much ultraviolet it emitted while it was strong, and stars of the same mass span a factor of ten.

The same separation shows up around stars of different masses. A red dwarf is fainter, so a planet at the same equilibrium temperature orbits closer to it, but the equilibrium temperature is what boil-off sees, and a five-Earth-mass planet at 800 kelvin boils down to the same envelope round a red dwarf as round a Sun-like star. Photoevaporation sees something else: red dwarfs stay active for billions of years rather than a hundred million and deliver more ultraviolet relative to their bolometric output. So the valley’s position around smaller stars is a test in which the two diggers predict different shifts, and the observed shift — a valley that moves with stellar mass roughly as photoevaporation predicts — says that the ultraviolet did real work after boil-off had done its share.

An opacity nobody has measured #

Every model of this kind has a quantity it cannot pin down, and for boil-off it is the density at the base of the wind. The base sits roughly at the envelope’s photosphere, where the gas becomes transparent to its own thermal radiation, and the pressure there is set by the opacity of the gas: a gas that absorbs strongly becomes opaque at low pressure, one that absorbs weakly only at higher pressure. The opacity of a young planet’s outer envelope depends on whether it carries dust grains from the disc, which can raise it a hundredfold, and on the molecular chemistry of the hydrogen and the metals mixed into it. Nobody has measured it for a planet a million years old.

The answer is that it matters, but much less than its uncertainty. A thousandfold change in the opacity changes the base density a thousandfold, and the wind’s rate with it — but the rate is exponential in Λ, so a factor of a thousand in rate is about seven units of Λ, and seven units of Λ is a modest change of radius. What survives moves by a factor of about three for the five-Earth-mass core. For the ten-Earth-mass core, which hardly boiled anyway, it does not move at all.

The same logic was met in the measurement of what a hydrogen wind can carry with it: a process controlled by an exponential reports its threshold robustly and its rate poorly. That is the general character of the result and the reason it is worth trusting despite its simplifications. The process is controlled by an exponential, and an exponential converts large uncertainties in its prefactor into small shifts in its argument. The envelope’s opacity, the details of the wind’s sonic point, the exact radius–mass relation of a hot young envelope — each is uncertain by factors of several, and each moves the shutting value of Λ by a unit or two. None of them moves the conclusion that planets with low Λ lose most of their envelopes in the first million years and stop near thirty.

What the picture cannot show #

The figures follow one planet at a time at a fixed equilibrium temperature, and the real situation is less tidy in three ways.

The radius of a young, hot envelope is taken from fits to models of older planets, extended below a hundred million years with the same dependence on age. Very young envelopes may be larger still, and their thermal state depends on how fast they were accreted and how much heat the core released as the disc vanished. A more extended start makes boil-off more violent and the convergence more complete; a more compact one makes it gentler.

The disc does not disappear instantaneously. If it thins over a few hundred thousand years, the envelope is unconfined gradually, and part of the boil-off happens while the disc’s pressure is falling. The end point is the same, because it is set by Λ, but the time at which the planet reaches it is not.

And the boiled planets go on to face the star’s ultraviolet, which strips them further over the next hundred million years. The figures stop before that. What they establish is the starting envelope photoevaporation acts on — and that starting envelope is already close to the value at which photoevaporation’s own regime does its work, which is why the two processes can produce the same valley.

Still open: whether the valley was dug twice #

The radius valley is read as the record of which planets were stripped. If boil-off has already brought every envelope above the corner down to a value set by core mass and temperature, the valley has two diggers working in sequence: the bolometric light in the first million years, and the ultraviolet over the next hundred million. They push in the same direction and leave the gap in nearly the same place, so the present-day valley alone cannot say how much of the digging each did.

The way to tell them apart is by age, as with the race between starlight and a planet’s cooling core. A planet a few million years old, in a cluster where the disc has only just gone, has been through boil-off and has hardly begun to be photoevaporated. If young sub-Neptunes are already sitting near Λ ≈ 25 with envelopes of a few per cent, boil-off did most of the shaping and the ultraviolet finished it; if they are found still swollen, with envelopes of ten or twenty per cent, boil-off was weak and the familiar story stands. The handful of transiting planets now known around stars of ten to thirty million years are larger than their older counterparts, but they are already older than the boil-off’s first million years. The measurement that would decide this needs planets younger still, transiting stars that have only just lost their discs — and their radii, which are what a transit measures, are exactly what boil-off changes fastest.