Astronomy & Celestial Mechanics
the sky, computed
Exoplanets

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.

Assumes Atmospheric escape, Transits and Exoplanet atmospheres.

16 min read 8 figures Measuring the unreachableLight is the only messengerSmall perturbations, large consequences

A transit depth is a ratio of areas. The planet blocks the fraction of the star’s disc that it covers, so the depth is the square of the ratio of radii — a planet measured by the light it removes — and for a hot Neptune around a Sun-like star that is a little over one per cent.

Observe the same transit in the ultraviolet, in the core of the hydrogen Lyman-α line, and the depth is ten to fifteen per cent. The planet has not changed size. What has changed is what is doing the absorbing, and the arithmetic of how large it must be settles the question of what is happening.

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.
Fig. 1 The regime the observation belongs to. Below a threshold in the ratio of gravitational binding to thermal energy, an atmosphere does not merely leak from its tail — it flows. The whole upper atmosphere accelerates outwards as a transonic wind, carrying material away at a rate set by how much stellar ultraviolet is being absorbed rather than by anything about the tail of a velocity distribution. The same wind strips a mini-Neptune close in and leaves a hot Jupiter intact, because the binding energy differs by more than the heating does.

It is worth registering how unusual it is for a transit measurement to be qualitative. Nearly everything in the transit literature is a matter of parts per million and careful systematics — four contact points and what they fix, a depth of a hundredth, a secondary eclipse of a thousandth. A signal ten times larger than the planet’s own is not a refinement of that programme; it is a different object being observed through the same aperture.

Why the depth is the argument

Take the numbers at face value. A fifteen per cent depth means an absorbing region covering fifteen per cent of the stellar disc, which for a Sun-like star is a radius of about 0.39 stellar radii — some 2.7 × 10⁵ kilometres, or roughly four Jupiter radii.

Now compute the planet’s Roche lobe. For a Neptune-mass planet at 0.05 astronomical units around a solar-mass star, the lobe is a few planetary radii, and the absorbing region is comfortably outside it.

Material outside the Roche lobe is not bound to the planet. So whatever is absorbing is not part of a hydrostatic atmosphere, however extended; it is material the planet has already lost, still close enough to be seen against the star.

The alternative — that it is a bound atmosphere — fails a second, independent test. A hydrostatic atmosphere’s extent is set by its scale height, which is the thermal energy divided by the surface gravity, and reaching four Jupiter radii from a Neptune would require a temperature of order a million kelvin. Nothing observed in the same planet’s optical or infrared transmission spectrum supports that.

Both tests are geometric rather than spectroscopic, which is what makes them robust. Neither requires a model of the outflow, a stellar spectrum, or an abundance. They require the depth, the star’s radius, the planet’s mass and its orbital distance, all of which are measured to a few per cent by ordinary means, and the conclusion follows by arithmetic.

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.
Fig. 2 The same wind calculation with the escape criterion halved, which is the sensitivity test the argument above deserves. The stripping boundary moves outward and the shape of the curves does not, because the criterion enters as a threshold on a quantity that varies by orders of magnitude across the plot: a factor of two in where the line is drawn moves it by a small fraction of a decade in orbital distance. That insensitivity is why the population-level prediction survives despite nobody knowing the criterion well. It is the same property the two geometric tests above have — a conclusion that follows from the size of a number rather than from its precise value is a conclusion that a factor of two cannot overturn.

What the line shape adds

The absorption is not centred on the line.

The core of Lyman-α from a nearby star is destroyed twice over — absorbed by neutral hydrogen in the interstellar medium and contaminated by geocoronal emission — so what is measured is the two wings. And the absorption is asymmetric: it is deep in the blue wing, at Doppler velocities of −40 to −140 kilometres a second, and weak in the red. The blue-shifted asymmetry says the material is moving towards the observer, which for a transiting planet means towards the star — or, more usefully, that the escaped material has been pushed into a comet-like tail. Radiation pressure in the Lyman-α line itself and charge exchange with the stellar wind are the two mechanisms proposed, and distinguishing them is an active question, since they predict different tail geometries and different transit durations.

The velocity is also the reason the measurement is possible at all. A slow-moving cloud of neutral hydrogen would absorb only in the line core, which is exactly where the interstellar medium has already removed the stellar flux; there would be nothing left to absorb. It is the acceleration to a hundred kilometres a second that Doppler-shifts the absorption into the wings, where photons still arrive. The signal exists because the material is moving fast, not merely because it is there — which is a selection effect built into the technique and a reason non-detections are hard to interpret.

17 orders of magnitude between λ = 2 and λ = 45. The fraction of a Maxwellian gas that is moving fast enough to escape, against the Jeans parameter λ = GMm/kTR — the ratio of a molecule's gravitational binding energy to its thermal energy, and the only quantity this problem has. The curve is (1 + λ)e^−λ, and the reason it is drawn on a logarithmic axis spanning 17 decades is the whole argument: escape is not a threshold that a molecule is or is not over. The mean speed of a gas at any temperature a planet has is far below its escape speed — the tail is what leaves, and the tail is an exponential. So a gas at λ = 3 is gone in a geological instant, a gas at λ = 25 is there for the age of the universe, and there is no sharp line between: the conventional criterion of λ ≈ 36 is a place on a slope, chosen because it is where the loss time crosses the age of the solar system for a body of planetary size. A factor of two in the temperature is a factor of 10⁴ in the loss rate, which is why the exospheric temperature — the one the extreme ultraviolet sets, not the one the sunlight sets — is the only temperature that matters.
Fig. 3 The tail. Escaped material shares the planet’s orbital motion but is no longer bound, so it trails behind on a slightly different orbit and spreads. A tail long enough to cover a substantial fraction of the star produces a transit that begins before the planet’s optical ingress and ends long after its egress — which is exactly what the ultraviolet light curves of the best-studied systems do.

The rate, and how it is estimated

The quantity actually wanted is the mass-loss rate, and the observation gives an absorbing column rather than a rate.

The standard estimate is energy-limited. A fraction of the stellar extreme-ultraviolet flux intercepted by the planet is absorbed, ionising hydrogen and heating the gas; some efficiency of that energy goes into lifting material out of the potential well; and the rate follows from dividing the available power by the binding energy per unit mass.

The result scales as the extreme-ultraviolet flux times the planet’s radius cubed over its mass, which is to say inversely as its mean density. Low-density planets close to active stars lose atmosphere fast, and the scaling is steep enough that the population divides rather than spreading.

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.
Fig. 4 The classical version of the same question, and the boundary it draws. Whether a body retains a given gas depends on its escape speed and its equilibrium temperature, and the line separating retention from loss accounts for the atmospheres of nearly every body in the solar system — the gas a planet cannot keep. What the hydrodynamic regime adds is that a planet can lose gas it would comfortably retain against thermal escape alone, because the escape is driven rather than statistical.

Two aspects of that scaling deserve emphasis. It is a rate at one moment, and what matters for a planet’s history is the integral over time — and stellar ultraviolet output falls by orders of magnitude as a star spins down, so nearly all the loss happens in the first few hundred million years. A planet observed losing mass now is losing far less than it did, and one observed losing nothing may have lost an ocean.

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 1 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 3.8·10¹⁵ J m⁻² at one astronomical unit and is some 26 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 0.0068 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.
Fig. 5 The same integral stopped at one billion years rather than five, which is the drawn version of that paragraph. Almost nothing changes. The saturated phase of the star’s ultraviolet output lasts a hundred million years and the decay afterwards is steep enough that the fluence has essentially converged by a gigayear — so four fifths of the elapsed time contributes a small fraction of the damage. A planet’s envelope is decided in its first few hundred million years and then observed for billions, which is why the radius gap is a fossil rather than a process, and why measuring how it depends on system age is the discriminating test the next section describes.

And the rate depends on the mean density rather than on the mass or the radius separately, which is why the population sorts itself so cleanly: two planets of the same mass and different envelope fractions have very different densities, and the one with the envelope loses it.

The largest uncertainty in any published rate is the extreme-ultraviolet flux of the host star, which cannot be measured directly for anything but the nearest stars: the interstellar medium absorbs the band almost completely. It is inferred from proxies — X-ray flux, Ca II emission, rotation period — with a scatter of a factor of several, and it propagates straight through. The rotation-period proxy is the most useful of the three, because a star’s activity is driven by a dynamo whose strength falls as the star spins down, and the spin-down of a star is a clock with a calibration of its own.

What the population says

The observations are one system at a time and expensive. The statistical evidence is elsewhere, and it is much stronger. The gap’s position depends on orbital distance and on stellar mass, and both dependences were predicted before they were measured. Recovering it at all required stellar radii good to a few per cent for thousands of stars, since a planet’s radius is only ever known as a fraction of its host’s — a transit depth is a ratio and nothing more, and the gap was invisible in the same data until the stellar radii improved. That is the strongest evidence the escape process shapes the population, and it is considerably better evidence than any individual Lyman-α detection. It is also worth being clear about what the census can and cannot say. Every survey draws a different sky, and a transit survey is biased towards short periods and large planets, so the observed radius distribution is not the intrinsic one until a completeness correction has been applied. The gap survives the correction, which is the reason it is believed.

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. This drawing covers 100 to 3000 K, so Pluto is outside it and not plotted.
Fig. 6 The classical retention diagram redrawn over the temperature range the hot exoplanets actually occupy rather than the solar system’s. Every criterion line has slope one half, because the thermal speed goes as the square root of temperature, and the effect of pushing the axis to three thousand kelvin is to sweep the whole solar system into the bottom left corner and leave the interesting region empty of anything measured. That emptiness is the honest statement of what this figure can do for the exoplanet case: the thermal criterion is the right physics for a planet losing gas molecule by molecule and the wrong physics entirely for one whose atmosphere is flowing, and the population the radius gap is about is in the second regime.

There is a competing explanation and it has not been eliminated. A planet’s own formation heat can drive the same envelope loss from below — core-powered mass loss — with no stellar ultraviolet involved. The two mechanisms predict very similar gaps, and separating them requires either the dependence on stellar age, or direct detections of escape in the act, which is what the Lyman-α work supplies.

What the two mechanisms would look like if separated

It is worth setting out the discriminating predictions, because this is a case where the observations exist and the question is still open.

Photoevaporation is driven by the star and therefore depends on the star’s history. Its effects should be almost entirely finished within a few hundred million years, so the radius gap should be fully formed in young systems and unchanged afterwards. It should also depend on the host star’s activity level, so planets around active stars should be stripped further out.

Core-powered mass loss is driven by the planet’s own residual formation heat leaking through its envelope, and it operates on the planet’s cooling timescale — a gigayear or more. So the gap should deepen with system age, and should depend on the planet’s own properties rather than on the star’s ultraviolet output.

Both predict a gap in roughly the same place, which is why the population statistics have been unable to choose. What separates them is timing, and measuring the gap’s depth as a function of system age requires ages for thousands of field stars — which is why the question has become, in practice, a question about asteroidal seismology and gyrochronology rather than about planets.

An individual detection of escape settles nothing statistically and settles something else: it establishes that the ultraviolet-driven mechanism is real and operating at a measurable rate, which converts one of the two candidates from a hypothesis into an observed process.

The wider argument

That overlap is the single most consequential result of the escape literature, and it is not a comfortable one. A planet in the habitable zone of an M dwarf receives hundreds of times the Earth’s extreme-ultraviolet flux during its star’s long active phase, and integrated over a billion years the energy available to drive escape exceeds the binding energy of an Earth-mass atmosphere by a wide margin. Whether such a planet keeps anything depends on how much it started with, on its magnetic field, and on resupply from its interior — all of which are unknown. The escape rate scaling makes the pessimistic reading the default and does not make it certain: a planet that began with a thick envelope, lost it, and retained a secondary atmosphere outgassed from its interior is a perfectly ordinary outcome of the same physics, and it is the one the solar system’s own terrestrial planets are examples of.

The shield that does not shield. Above: measured ion escape rates for three planets against their surface magnetic field. Venus and Mars have no dynamo at all and Earth has one, and the three rates lie within a factor of 9 — with the magnetised planet losing the most. The intuition that a magnetosphere protects an atmosphere is not a small correction away from being right; the measurement does not support it. Below: why. A dipole's field lines are not all closed. Those emerging within a polar cap reconnect with the wind's own field and lead straight to space, and the cap's area is set by how far the magnetosphere reaches — a boundary at ten planetary radii still leaves 5.1 per cent of the surface open. So a magnetosphere is both a shield and a funnel: it deflects the wind from most of the planet and collects ions from the whole ionosphere into the polar wind, which is exactly what an instrument above the poles measures leaving. Whether the net is protection depends on quantities nobody can compute from the field strength alone, and the three points above are the state of the evidence.
Fig. 7 The magnetic field in that list, measured rather than assumed. Ion escape rates have been counted by spacecraft at Venus, Earth and Mars — two of them with no dynamo at all and one with a strong one — and the three rates lie within a factor of nine of each other, with the magnetised planet losing the most. A magnetosphere is not a lid. It is a structure with open field lines over the poles, and it funnels as much as it deflects, so the intuition that a field protects an atmosphere is not what the only direct measurements of the question say. That removes one of the three unknowns above from the optimistic side of the ledger, and it does so with in-situ ion counts rather than with a model.

The same process, measured from inside

Everything above is inferred from a transit depth at a distance of tens of parsecs. The same process is happening in the solar system, where it can be measured with instruments flown into it, and the comparison is worth making because it shows how much of the exoplanet inference is model and how much is observation.

Mars is the clearest case. Its escape has been measured directly: a spacecraft in orbit counts the ions leaving, sorted by species, as a function of solar wind conditions and of solar activity. The measured rate is of order a kilogram a second — small enough to be undetectable by any remote technique, and large enough to have removed a substantial atmosphere over four billion years given that the early Sun’s ultraviolet output was orders of magnitude higher.

Venus supplies the complementary measurement. It has a hydrogen corona detectable in Lyman-α from Earth orbit, and the ratio of deuterium to hydrogen in its atmosphere is about a hundred times the terrestrial value — which is the fingerprint of escape, because the lighter isotope leaves preferentially and the heavier one is left behind. That ratio is a measurement of an integral rather than a rate: it says how much has been lost over the planet’s history, without requiring anybody to have been watching.

The Earth is the awkward member of the set. It loses hydrogen too, at a rate limited by how much water vapour reaches the stratosphere rather than by anything about the upper atmosphere, and the cold trap at the tropopause is what keeps that number small. That is a mechanism with no analogue in a hot Neptune, and it is a reminder that escape can be limited at the bottom as well as at the top.

The transferable lesson is about which measurements are available at which distance. Close up, the rate is measured directly and the integral is inferred from isotopes; far away, neither is measured and both are modelled from a transit depth and a stellar flux estimate. The solar system supplies the calibration and cannot supply the regime, since no planet in it is anywhere near the hydrodynamic case the exoplanet observations are about.

There is a second lesson in the isotope argument, and it is the more useful one for the exoplanet case. A ratio of two isotopes is an integral over a history and requires no time series at all — a single spectrum of a present-day atmosphere records how much has escaped since it formed. That is exactly the kind of measurement that would settle the exoplanet question, and it is beginning to be possible: deuterium-to-hydrogen ratios can in principle be measured in a transmission spectrum, and for the nearest and most favourable systems the precision required is within reach of the current generation of instruments. A rate observed once tells almost nothing about a history; an isotope ratio tells almost everything, and the reason the exoplanet literature is full of rates is that rates are what a transit depth gives.

Where the picture stops

The star gets in the way. Lyman-α is emitted by the star’s chromosphere, which is spatially inhomogeneous and time-variable, so a transit depth measured in the line core is a difference between two epochs of a variable source. That is the same difficulty that turns up whenever a planetary signal has to be separated from an active star, and it has the same resolution: repeat the measurement at a different phase of the star’s own cycle, as the metre per second that is not the star has to be. Several claimed detections have not reproduced, and the ones that have are those with multiple transits observed.

Neutral hydrogen is a tracer, not the outflow. What is observed is neutral hydrogen at high velocity; what escapes is mostly ionised. The conversion from one to the other requires modelling the ionisation state of the outflow, and the neutral fraction can be small and variable — so the rate inferred from a Lyman-α depth is model-dependent by a factor that has been argued about for two decades.

17 orders of magnitude between λ = 2 and λ = 45. The fraction of a Maxwellian gas that is moving fast enough to escape, against the Jeans parameter λ = GMm/kTR — the ratio of a molecule's gravitational binding energy to its thermal energy, and the only quantity this problem has. The curve is (1 + λ)e^−λ, and the reason it is drawn on a logarithmic axis spanning 17 decades is the whole argument: escape is not a threshold that a molecule is or is not over. The mean speed of a gas at any temperature a planet has is far below its escape speed — the tail is what leaves, and the tail is an exponential. So a gas at λ = 3 is gone in a geological instant, a gas at λ = 25 is there for the age of the universe, and there is no sharp line between: the conventional criterion of λ ≈ 36 is a place on a slope, chosen because it is where the loss time crosses the age of the solar system for a body of planetary size. A factor of two in the temperature is a factor of 10⁴ in the loss rate, which is why the exospheric temperature — the one the extreme ultraviolet sets, not the one the sunlight sets — is the only temperature that matters.
Fig. 8 The Jeans fraction against the binding-to-thermal ratio, spanning seventeen orders of magnitude between λ=2\lambda = 2 and λ=45\lambda = 45, which is why the distinction in the paragraph above is not a detail. The curve is (1+λ)e−λ(1+\lambda)e^{-\lambda} and it is drawn logarithmically because nothing else would fit: a factor of two in temperature is a factor of ten thousand in escaping fraction at the steep end. So a quantity inferred from a tracer whose abundance depends on the local temperature and ionisation inherits that steepness, and a neutral fraction wrong by a factor of three is a rate wrong by much more than three. The exponential is the reason every number in this literature carries an order-of-magnitude uncertainty rather than a percentage one.

And helium has changed the subject. A metastable helium line in the near infrared, at 1.083 microns, probes the same escaping material from the ground, without the interstellar absorption that ruins Lyman-α and without needing a space telescope. It is now the workhorse, it has produced detections and non-detections in systems where Lyman-α gave neither, and the two lines do not always agree — which is informative rather than embarrassing, since they trace different levels of the same outflow.

The helium line’s own selection effect is worth noting for the same reason the Lyman-α one was. The metastable level it arises from is populated by recombination after ionisation by extreme-ultraviolet photons in a narrow band, and depopulated by photons in the near ultraviolet. So the line is strong around stars with a particular spectral shape — active K dwarfs — and weak around others, whatever the outflow is doing. A non-detection is a statement about the star at least as much as about the planet, and comparing rates across a sample means comparing quantities that were not measured on the same footing.

Where this ladder goes next

Later rungs on this anchor: the metastable helium line and what a ground-based escape measurement costs and buys; the energy-limited approximation and where it fails; charge exchange with the stellar wind, and what a tail’s shape says about that wind; the core-powered alternative and the observations that would separate it; and the long-term question of atmospheric retention around low-mass stars, which is the point at which this anchor stops being about planets and starts being about whether the commonest habitable-zone worlds in the galaxy have any air.

Where this goes next

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. A radius that depends on the colour it is measured in. Transit depth against wavelength for a planet of 1.38 Jupiter radii at 1400 K, whose atmosphere has a scale height of 538 km — computed from H = kT/µg, not assumed. One scale height of extra opacity adds 153 parts per million to a transit of 1.40 per cent, so the whole spectral signal is 862 ppm at its strongest: one part in 16 of the transit that carries it. The features are at real band centres — sodium at 0.589 µm, water at 1.4 µm, carbon dioxide at 4.3 µm — with the rise at the blue end the Rayleigh slope of scattering off the smallest particles. A radius that depends on the colour it is measured in Measure a transit in one colour and then another, and the planet is a different size. The difference is a few atmospheric scale heights, which is a few hundred parts per million of an already tiny signal. 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.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Charge exchangeEnergy-limited escapeExosphereHydrodynamic escapeLyman-alpha transitMass loss ratePhotoevaporationRadius valleyRoche lobeScale heightStellar activityTransit depth