Astronomy & Celestial Mechanics
the sky, computed
Concept

XUV flux — where it appears

The X-ray and extreme-ultraviolet radiation a planet receives from its star, absorbed high in its atmosphere and heating the gas enough to drive it away. It is brightest while the star is young and rotating fast, and it falls steeply once the star's spin slows.

Named by 9 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 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 A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it.

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

exoplanets · Habitable zone A loss rate that stops following the starlight. The rate at which a hot Jupiter of 1.38 Jupiter radii and 0.69 Jupiter masses loses its atmosphere, in grams per second, against the ionising flux it receives, in erg per square centimetre per second, both on logarithmic axes. The dashed line is the energy-limited rate with a heating efficiency of 0.3: linear in the flux, 1.04·10¹⁰ g s⁻¹ at 10³. The solid curve follows it at low flux and turns over near 10⁴, where the base of the wind becomes nearly fully ionised and each recombination radiates its energy as Lyman-α instead of heating the flow; above that the rate grows only as the square root of the flux. At 10⁶ erg cm⁻² s⁻¹, the flux such a planet receives from a young, active star, the energy-limited formula overestimates the loss by a factor of 11. A single efficiency cannot describe both ends of the curve: fitted at one flux, it is wrong at the other.

An escape that stops following the starlight

The standard estimate of how fast a planet loses its atmosphere assumes the loss grows in proportion to the ultraviolet light it absorbs. It does, up to a point. Beyond about ten thousand erg per square centimetre per second the escaping gas is fully ionised, its heat leaves as Lyman-α light instead of lifting anything, and the loss grows only as the square root of the flux — so the one efficiency every calculation needs is really a function of where the planet sits.

exoplanets · Atmospheric escape 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 worn down by a young star's ultraviolet. The mass of a primordial hydrogen envelope, as a fraction of the planet's mass, against age, for an Earth-mass planet in the middle of the eventual habitable zone of a 0.1 M☉ star, starting with 0.1%, 0.3%, 1%, 3% of its mass in hydrogen. The star contracts toward the main sequence and stays saturated in X-rays for 1.5 Gyr; the envelope inflates the planet, so it intercepts more of that light and loses mass faster than a bare rock would. The shaded span is the 75 Myr during which the planet receives more light than the runaway-greenhouse limit, when any exposed ocean would be steam. 0.1% is gone at 463 Myr; 0.3% is gone at 1.5 Gyr; 1% outlasts 5 Gyr; 3% outlasts 5 Gyr. An envelope that outlasts the shaded span has shielded the ocean through the whole of the runaway; one that also outlasts the planet's present age has left it a small gas-rich planet rather than a rocky one.

An envelope spent so an ocean could stay

A planet in the eventual habitable zone of a small red dwarf spends its first tens of millions of years too close to a star that is still contracting and still bright, with its ocean boiled into steam and the steam being split and lost. A wisp of hydrogen left over from the nebula — a few hundredths of a per cent of the planet's mass — would change that, because the star's ultraviolet would spend itself removing the hydrogen first. The envelope that shields the ocean and the envelope that never leaves bracket a range a factor of twenty wide, and the amounts a rocky planet is expected to capture fall inside it.

exoplanets · Habitable zone

Named alongside it

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

Energy-limited escapePhotoevaporationSub-neptuneRadius valleySuper-EarthAtmospheric escapeCore-powered mass lossPrimordial atmosphereSaturationEnvelope fractionHabitable zoneM dwarf

All concepts