The envelope that doubles a planet lasts longest

The distribution of small planets’ radii has a gap near 1.8 Earth radii, between bare rocky cores and cores wrapped in a few per cent of hydrogen, and its tilt with orbital period says that the gap was carved by the star rather than built in. What the tilt does not explain is why the gap is so empty. A star that removed gas steadily from every planet would leave a smear of planets with every envelope from full to none, and the histogram would sag where the stripping had been strongest rather than open a valley.

The emptiness needs a mechanism of its own, and the mechanism is a peak in a curve. For a planet of a given core, the time its star would take to remove the whole of its hydrogen envelope is not simply longer for a bigger envelope. It rises, reaches a maximum, and falls — and the maximum sits where the envelope has swollen the planet to about twice its core’s size. A curve with a maximum turns gradual loss into a runaway on one side and into a settling on the other, and a runaway leaves nothing in between.

Two reasons a small envelope is fragile #

The curve is built from two ingredients, and each is simple.

The first is how fast gas escapes. A young star’s X-ray and extreme-ultraviolet light is absorbed high in a planet’s atmosphere and heats it, and the heated gas expands and flows away — the hydrodynamic escape that removes the lightest gases first. In the regime where the heating is the bottleneck, the mass lost per second is the energy intercepted, a fraction of it put to work, divided by the energy needed to lift a kilogram out of the planet’s well:

$$\dot M = \frac{\eta\,\pi R^3\,F_{\rm XUV}}{G M}.$$

The planet’s radius enters as its cube — once from the area catching the light, twice from the weaker gravity at a larger radius — and that cube is the lever.

The second ingredient is how large an envelope makes the planet. A hydrogen envelope on a rocky core is not a thin skin. Its scale height is set by its temperature and the core’s gravity, not by how much gas there is, so even a tiny envelope stands hundreds of kilometres tall and a few per cent of the planet’s mass can double its radius. A few per cent of a planet is not a little gas: the 2.8 per cent that swells a 5 Earth-mass core most for its mass is about 160,000 times the mass of the Earth’s whole atmosphere.

Put the two together. The time to strip an envelope is its mass divided by the rate of loss, and the rate of loss grows as the cube of the radius. At a very small envelope the planet is barely larger than its core, the loss rate is roughly fixed, and the time grows in proportion to the gas there is to remove. At a large envelope the radius is dominated by the envelope, the cube of the radius grows faster than the mass of gas, and the time falls. In between is the maximum.

Where the peak is, and why it is always in the same place #

Written out, the loss time is proportional to the envelope’s mass fraction $f$ divided by the cube of the planet’s radius, and the radius is the core’s plus an envelope thickness $\Delta R$ that grows as the fraction to a power $\beta$ of about 0.59. The peak is where a small change in $f$ changes $f$ and $R^3$ in the same proportion, which gives a condition that does not involve the core’s mass, the insolation or the star at all:

$$R_{\rm core} + \Delta R = 3\beta\,\Delta R \;\;\Longrightarrow\;\; \Delta R = \frac{R_{\rm core}}{3\beta - 1} \approx 1.3\,R_{\rm core}.$$

The longest-lived envelope is always the one about 1.3 core radii thick — a planet a little over twice its core’s radius. Everything else in the problem moves the peak’s height, and nothing moves its position in radius. The figure computes the peak directly rather than using the approximation, and for every core it finds the envelope thickness within a hundredth of 1.30 core radii.

What changes from core to core is the mass fraction that makes that thickness. A heavier core compresses its envelope harder, so it needs more gas to swell by the same factor: 1.9 per cent for 3 Earth masses, 4.0 per cent for 8.

The peak’s height changes much more, and its change can be taken apart. The loss time is the envelope’s mass over the loss rate, $f M^2 G/(\eta\pi R^3 F)$, so going from a 3 Earth-mass core to an 8 multiplies it by $(8/3)^2$, or 7.1, from the core’s mass; by 2.1 more from the larger fraction of gas at the peak; and divides it by 2.0 for the larger radius the peak sits at. The product is 7.3, and the figure’s peaks, 129 and 946 Myr, are in the ratio 7.3. A core less than three times heavier keeps its most resilient envelope more than seven times longer, and that disproportion is what makes the valley a boundary in core mass.

And what changes with the light is how long the peak envelope survives.

Between the two figures the insolation fell by a factor of 3.3 and every peak lasts 3.6 to 3.7 times as long. The loss rate is proportional to the flux, which accounts for the 3.3; the rest is the envelope, which is slightly cooler and less inflated further from the star, so it needs a little more gas to reach the same thickness and catches a little less light while it does.

A factor of ten in insolation moves the peak’s height by about a factor of ten and its position in envelope fraction by a few tenths of a per cent. The geometry of the argument is the same at every distance from the star; only the clock it is compared with changes.

How a peak makes a runaway #

Now follow a single planet. It begins with some envelope and loses gas, so it moves leftward along its curve.

If it starts to the right of the peak, with more than the most resilient envelope, losing gas moves it up the curve: each loss leaves a planet whose remaining envelope lasts longer than before. The loss slows down as it goes. The planet drifts towards the peak and, once the star’s active phase has ended, stops somewhere near it, with an envelope of a few per cent and a radius a little over twice its core’s. That is the population above the valley.

If it starts to the left of the peak, or is driven past it, losing gas moves it down the curve: each loss leaves a planet whose remaining envelope lasts less long than before. The planet shrinks and its loss rate falls, but its envelope falls faster, because on this side of the peak the radius has stopped responding much to the gas that remains; the loss accelerates relative to what is left, and it does not stop until the envelope is gone. That is the population below the valley.

Nothing is stable in between. A planet with half a per cent of hydrogen, sitting at a radius between the two populations, is on the steep downhill side of its curve and is removing its envelope faster, relative to what it holds, than any planet near it. The valley is empty because it is the part of the radius distribution that every planet leaves quickly, in one direction or the other, and it is empty at a radius rather than at a mass because both of the stable endpoints — a bare core, and a core roughly doubled — are defined by the core’s size.

A peak only exists because escape is steep in radius #

The cube in the loss rate did all of this, and it is worth asking what a gentler law would do. Suppose the loss rate grew as the planet’s radius to some power $n$ rather than exactly the third. The same argument puts the peak at

$$\Delta R = \frac{R_{\rm core}}{n\beta - 1},$$

which is finite only if $n\beta$ exceeds one — only if $n$ is larger than $1/0.59$, or 1.7. With $n = 3$ the peak is at 1.3 core radii. With $n = 2$ it would be at 5.6 core radii, a planet more than six times its core’s size, far outside the sub-Neptunes. And with $n$ below 1.7 there is no peak at all: the time to strip an envelope would grow with every addition of gas, the loss would always slow down as it proceeded, and a population of planets would be thinned evenly towards small envelopes rather than split.

This matters because the energy-limited law is not the only regime of escape. When the incident flux is high enough that the upper atmosphere is mostly ionised, the flow is limited by how fast protons and electrons recombine and radiate the heat away rather than by how much heat arrives, and in that regime the loss rate grows more slowly with radius — roughly as its three-halves power. A planet losing its envelope in that regime would not run away. The emptiness of the valley is therefore evidence about more than where it is: it says the planets near it lost their gas in a regime steep enough in radius to have a peak — or, if it was not the star that stripped them, by a process with a runaway of its own.

Where the boundary falls #

The same curve decides which cores end on which side. A core whose most resilient envelope cannot survive the star’s total XUV exposure is stripped whatever it started with; a core whose peak outlasts that exposure keeps something. The boundary between them is where the peak’s height equals the exposure.

Closer to the star the exposure is larger, so heavier cores are stripped, and the boundary sits at a larger radius. That is the tilt of the valley, derived rather than fitted, and its exponent comes from the exponents of the escape law and the interior fit alone, with no parameter adjusted to reach it. It is steeper than the tilt that has been measured — about $P^{-0.09}$, with an uncertainty approaching a factor of two — and that difference is a known one: the simplest energy-limited models predict a steeper valley than the data show, and a real population mixes host stars, core compositions and ages that no single line represents. The sign, which is the part that decides between a valley carved by starlight and one built in at formation, is the same. The boundary’s position, a core of about five Earth masses at ten days, depends on the efficiency and on the star’s XUV output, which are uncertain by factors of a few and move the whole line up or down without tilting it.

For comparison, the histogram in which the valley was found:

The sub-Neptune peak at 2.4 Earth radii and the super-Earth peak at 1.3 differ by a factor of about 1.8. A core roughly doubled by its envelope is what the peak of the loss time predicts, and the exact ratio in the data is a mixture of core masses, ages and envelope fractions that no single curve reproduces.

Two cores that bracket the population #

The peak’s position is insensitive to the core, but the envelope’s leverage is not, and two extreme cores show how differently they respond to the first traces of gas.

A light core is inflated far more by the same fraction of gas, because its gravity is weaker and its envelope stands taller. With one per cent of hydrogen the 2 Earth-mass core has more than doubled, while the 10 Earth-mass core has grown by half. The consequence is that light cores reach their peak with less gas and are stripped more easily, and heavy cores need much more gas to reach theirs and are much harder to strip — which is why the boundary climbs so steeply in core mass and so gently in radius.

What would show the runaway directly #

A runaway makes a prediction a histogram of radii cannot test on its own: at a given orbital period, the planets below the valley should have systematically lighter cores than the planets above it, because the boundary between them is a boundary in core mass. That needs masses. For planets in or near orbital resonance they come from the transits running early and late as the planets tug on each other, which weighs a sub-Neptune without a spectrograph, and for the rest from the star’s reflex velocity. Neither is easy for a planet of a few Earth masses, and a mass alone does not settle what a planet is made of, because the same density is reached by different mixtures of rock, water and gas. The test is statistical: the masses of planets on either side of the valley, at the same period, sorted by which side they are on.

What the argument assumes #

Energy-limited escape with a fixed efficiency. The real efficiency depends on how the absorbed energy is shared between heating, ionisation and radiation, and it falls for the most massive and most irradiated planets. The shape of the curve survives that so long as the escape stays steep in radius; its height does not.

One interior fit. The envelope thickness comes from a published fit to planetary interior models, valid for cores of one to twenty Earth masses and envelopes of a tenth of a per cent to twenty. The peak’s position at 1.3 core radii is a consequence of its 0.59 exponent.

The envelope’s own gravity is neglected. For envelopes of a few per cent that is a few per cent error, and the analytic arguments this follows make the same approximation.

No other loss process. A newly formed planet’s core is hot and can drive its own envelope off over a billion years with no help from the star, and a curve of the same kind governs that process too. The two mechanisms produce valleys in similar places with similar tilts, and the gas a planet cannot keep is removed by whichever acts first. Separating them needs a measurement this curve does not provide.

And the composition. The core is taken to be rock and iron in the Earth’s proportions. A water-rich core is larger for its mass, and the radius alone cannot tell a rocky core with hydrogen from a watery one without.

A peak, not a slope #

The measured tilt of the valley says who carved it. The peak in the loss time says why the carving left a gap rather than a gradient. Neither needs the other, and together they reduce the most distinctive feature of the exoplanet census to two exponents — how a planet’s size responds to hydrogen, and how its loss rate responds to its size — and one clock, the young star’s, which runs down as a magnetised wind brakes the star’s rotation and takes the X-ray activity with it.

That clock is the natural next question. A peak that decides the outcome by comparison with the star’s active phase makes a prediction about when the valley forms, and the two candidate mechanisms keep very different clocks.

Still open: whether the valley is finished by a hundred million years or a billion #

If the star’s light does the stripping, almost all of it happens while the star is young and magnetically active, and a population of planets should look essentially the same at five hundred million years as at five billion. If the planets’ own cooling cores do it, planets should still be crossing the valley billions of years later. The two are separated not by where the valley is but by how the population on either side of it changes with the age of the stars it orbits — a measurement that needs the ages of planet hosts, which are among the hardest quantities in stellar astronomy to obtain.