Astronomy & Celestial Mechanics
the sky, computed
Concept

Photoevaporation — where it appears

The removal of a planet's atmosphere by a wind driven by absorbed extreme-ultraviolet radiation from its star. It is limited by the energy arriving rather than by the tail of a thermal distribution, and it is one of the two standard accounts of the gap in the planet radius histogram.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there.

A gap in a histogram that says how planets are built

Between the super-Earths and the sub-Neptunes there is a radius at which planets are markedly rarer. The gap is not a gap in what can be detected, and its slope with orbital period names the process that made it.

exoplanets · Radius valley Pluto and Triton sit on the nitrogen line, and the Earth sits between helium and nitrogen. Escape speed against exospheric temperature, with a criterion line for each molecular species at v_esc = 6 v_th — the speed at which the Jeans loss time is comparable to the age of the solar system. Every line has slope one half, because the thermal speed goes as √T; a body above a line keeps that gas and a body below it does not. The horizontal axis is the temperature at the exobase, which for the Earth is near 1000 K rather than the 255 K of its equilibrium — the difference is extreme-ultraviolet heating, and using the wrong temperature puts the Earth above the hydrogen line, keeping an atmosphere it observably lost. Three readings are worth making. The Earth falls between helium and nitrogen and does exactly that: it loses helium as fast as radioactive decay supplies it, and keeps nitrogen for ever. Titan sits just above nitrogen and just above methane, which is why it has a thick nitrogen atmosphere and is slowly losing its methane. And Pluto and Triton sit on the nitrogen line, within 1 per cent — which is why both have atmospheres that are marginally bound and measurably escaping. Where it fails it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have, because retention is necessary and not sufficient: a body also needs a source, and needs to survive the non-thermal losses this criterion says nothing about. Venus is the sharpest case — it sits above the hydrogen line and has still lost an ocean, because that hydrogen left by charge exchange with the solar wind rather than by moving fast enough.

The gas a planet cannot keep

Escape is a statement about the tail of a distribution rather than about its mean, so the threshold is not a speed but a dimensionless number near thirty — and past that number the loss rate falls by twelve orders of magnitude. Then, for a hot Jupiter, the whole picture fails and the atmosphere leaves as a wind.

exoplanets · Atmospheric escape The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

exoplanets · Atmospheric escape A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous.

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

exoplanets · Planet composition How long a planet's hydrogen lasts, against how much of it there is. The time a young Sun-like star's saturated X-ray and ultraviolet output would take to remove a planet's whole hydrogen envelope, at 100 times the Earth's insolation, against the envelope's share of the planet's mass, for cores of 3, 5, 8 Earth masses. It is computed with energy-limited escape and an interior fit for the envelope's thickness, and it is not monotonic. A heavy envelope takes long to remove because there is a lot of it. A very light one takes long because the planet is small and intercepts little light. In between the time peaks, and it peaks where the envelope has swollen the planet most for its mass: 3 Earth masses at an envelope of 1.9 per cent, 129 Myr, where the envelope is 1.31 times as thick as the core's radius; 5 Earth masses at an envelope of 2.8 per cent, 364 Myr, where the envelope is 1.30 times as thick as the core's radius; 8 Earth masses at an envelope of 4.0 per cent, 946 Myr, where the envelope is 1.30 times as thick as the core's radius. The peak is what makes a valley. A planet above it losing gas moves up the curve, its remaining envelope lasting longer and longer, and settles; a planet below it moves down, lasting less and less, and loses everything. The horizontal lines are the saturated phase, 100 Myr, and the 300 Myr of saturated-equivalent exposure the whole history delivers: a core whose peak lies below the second line cannot keep any envelope at all.

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

exoplanets · Radius valley When the stripping happens, in a model population. For the same 3,000 model planets, the share that have lost their whole envelope by each age, scaled to the share bare at 5.0 Gyr (35 per cent of the population), beside the share of the star's lifetime XUV energy delivered by then. Half of all the stripping in this model is finished by 72 Myr and nine tenths by 457 Myr, while the star has delivered 27 and 76 per cent of its XUV energy. That is the clock photoevaporation keeps: the valley is essentially finished within the first few hundred million years, because the planets near the boundary are the ones that run away, and they run away early. A mechanism powered instead by the slow cooling of the planets' own cores would keep moving planets across the valley for billions of years. The difference is in when, not where, and it is why the ages of the stars hosting planets on either side of the valley are the measurement that can separate the two.

The stripping runs ahead of the starlight that drives it

If young stars carve the radius valley with their X-ray light, the valley should be finished early — half of it before the star has delivered a third of that light, and nine tenths of it within a few hundred million years. If the planets' own cooling cores carve it instead, planets should still be crossing it billions of years later. The two accounts put the valley in the same place, and they are separated by the one thing a histogram cannot show — when.

exoplanets · Radius valley Where photoevaporation puts the valley, round stars of different mass. The radius of the largest core stripped bare, against orbital period, round stars of 0.5 M☉, 0.75 M☉, 1 M☉, 1.25 M☉, in the same energy-limited model, with each star's luminosity taken as its mass to the fourth power and its saturated phase lengthened for smaller stars as the mass to the −1.5, 100 Myr for the Sun. At ten days the valley is at 1.22 Earth radii round the 0.5 M☉ star, 1.38 Earth radii round the 0.75 M☉ star, 1.50 Earth radii round the 1 M☉ star, 1.60 Earth radii round the 1.25 M☉ star, and every one of them tilts as the −0.176 power of the orbital period, because the tilt comes from the exponents of the escape law and the interior fit, which do not depend on the star. A lower-mass star is far fainter, so at a given period its planets receive much less light, and even its longer active phase does not make up the difference: this model puts the valley at smaller radii round smaller stars while keeping the same sign of tilt. The stellar scalings are rough, and they are the least certain part of the calculation; what is robust is that photoevaporation ties the valley to the XUV energy a planet received, which falls with the star's mass at fixed period, and a mechanism tied to something else would tie it differently.

A smaller star puts the valley lower

Round stars of half the Sun's mass, the gap between bare rocky cores and sub-Neptunes should sit at a smaller radius than round the Sun, because at the same orbital period their planets receive a tenth of the light. Their stars also stay young and active far longer, which pushes the other way. Photoevaporation weighs those two against each other in a definite proportion, and the answer is a valley that scales as the star's mass to about the power three tenths.

exoplanets · Radius valley The XUV energy that reached one astronomical unit, for three young Suns. The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth's distance from the Sun, against age, for three histories that differ only in how long the young Sun stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator, 300 Myr for a fast rotator. After saturation each declines to a common track by 1 Gyr, as rotation histories are observed to converge. By 4.5 Gyr the totals are 2.06·10¹⁵ J/m² for the slow rotator, 3.66·10¹⁵ J/m² for the medium rotator, 6.47·10¹⁵ J/m² for the fast rotator: the fast rotator delivered 3.1 times as much as the slow rotator, nearly all of it in the first few hundred million years. The Sun's own rotation at that age is not measured; it is inferred from the spread of rotation periods in young clusters, and all three histories are consistent with a Sun that ends up rotating as it does now.

The young Sun's spin decides what the Earth kept

The calculation that strips sub-Neptunes applies just as well to a planet with a trace of hydrogen. An Earth that captured a few hundredths of a per cent of its mass from the gas it formed in — several times the hydrogen now in its oceans — would have lost all of it, or kept most of it, depending on something nobody has measured — how fast the Sun was spinning in its first few hundred million years.

exoplanets · Radius valley The heaviest atom a hydrogen wind can carry, against the strength of the wind. The crossover mass — the heaviest atom an escaping hydrogen flow drags out with it — against the hydrogen flux at the base of the flow, on a logarithmic axis, for the Earth's gravity and gas at 1000 K. Atoms lighter than the crossover leave with the hydrogen; heavier ones stay. The crossover rises in proportion to the flux: lifting neon takes 9.7·10¹² hydrogen atoms per square centimetre per second, argon 1.8·10¹³, krypton 4.3·10¹³ and xenon 6.6·10¹³. The Earth loses about 3·10⁸ today, which lifts nothing heavier than hydrogen. Driven by a young Sun at its saturated X-ray and ultraviolet output, an energy-limited flow would reach 2.2·10¹³ on Mars, a crossover of 114; 2.3·10¹³ on Venus, a crossover of 50; 10¹³ on the Earth, a crossover of 21 — the smaller planet's weaker gravity both raises the flux per unit area and lowers the force holding the heavy atoms back.

The heaviest atom a hydrogen wind can lift

A planet losing hydrogen fast enough to blow off as a wind drags other gases with it, but only those lighter than a crossover mass set by how strong the wind is. Below it every gas leaves in proportion to its distance from the limit and its isotopes are sorted by the same amount per mass unit; above it nothing goes. The Earth's xenon is sorted and its krypton is not, which no wind of neutral atoms can arrange.

exoplanets · Radius valley An envelope that boils away in a million years, or keeps for longer than the universe. The time a 5-Earth-mass planet takes to lose an envelope of 5.0% of its mass through an isothermal wind driven by its own heat and the star's light, against the escape parameter Λ at the envelope's outer edge — the ratio of the gravitational binding of a molecule there to its thermal energy — for equilibrium temperatures of 500, 800, 1200 K, on a logarithmic time axis. The rate is exponential in Λ, so the time grows by a factor of e for every unit: each step of ten in Λ is more than four decades of time. The envelope goes in under a million years below Λ ≈ 19, 18, 18 and lasts longer than a billion years above Λ ≈ 26, 26, 26. The window between a million and a billion years is a band only about seven units of Λ wide, so a planet is either boiling or effectively sealed.

An envelope that boils down before the starlight strips it

The usual story of how small planets lose their hydrogen starts with the star's ultraviolet light and runs for a hundred million years. It skips the first million. When the gas disc that fed a young planet disappears, an envelope that was swollen and held in by the disc's pressure is suddenly exposed, and it pours off as a wind powered by ordinary heat — until it has shrunk to the size at which the same number near thirty that governs every escaping atmosphere shuts the flow off.

exoplanets · Atmospheric escape

Named alongside it

The objects these essays reach for when they reach for this one.

Energy-limited escapeRadius valleyXUV fluxSub-neptuneHydrodynamic escapeSuper-EarthCore-powered mass lossSaturationScale heightAtmospheric escapeEnvelope fractionOccurrence rate

All concepts