Astronomy & Celestial Mechanics
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Exoplanets

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.

Assumes Escape, Exoplanet atmospheres and Radius valley.

19 min read 10 figures Everything is a conicScales that defeat intuitionLight is the only messenger

The escape speed is one of the first quantities anyone learns about a planet, and it is almost useless for deciding what that planet’s atmosphere contains.

The mean speed of a nitrogen molecule at any temperature the Earth has ever had is about half a kilometre a second, against an escape speed of eleven. The mean speed of a helium atom is one and a half. Neither is anywhere near escaping, and the Earth has kept the nitrogen for four and a half billion years and loses the helium as fast as radioactive decay in the crust supplies it.

The difference is not in the mean. It is in the tail.

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. 1 The right way to ask the question. Escape speed against exospheric temperature, with a criterion line for each molecule: a body above a line keeps that gas, a body below it does not. Every line has slope one half because the thermal speed goes as the square root of the temperature. The Earth falls between helium and nitrogen and does exactly that. Pluto and Triton sit on the nitrogen line, within a few per cent, which is why both have atmospheres that are marginally bound and measurably leaving.

The tail is an exponential

A gas in equilibrium has a Maxwell–Boltzmann distribution of speeds, the same tail that decides which atoms in a stellar core are moving fast enough to fuse, and the fraction of molecules above any given speed falls exponentially once that speed is well past the mean. The controlling quantity is the ratio of a molecule’s gravitational binding energy to its thermal energy,

λ=GMmkTR,\lambda = \frac{GMm}{kTR},

which is also the square of the ratio of escape speed to thermal speed. The escaping flux is proportional to (1+λ)e−λ(1+\lambda)e^{-\lambda}.

Between λ=5\lambda = 5 and λ=40\lambda = 40 that factor falls by seventeen orders of magnitude. There is no threshold anywhere in it. What there is instead is a very steep slope, on which a factor of two in temperature is a factor of 10710^7 in the loss rate.

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 = 4 is gone in a geological instant, a gas at λ = 40 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. 2 The same slope with the marks moved to the places the argument uses. At λ=4\lambda = 4 the escaping fraction is a few per cent and the gas is gone in years; at 15 it is a part in a hundred thousand and the gas is gone in a geological interval; at 40 it is a part in 101610^{16} and nothing leaves in the age of the universe. Between the second and the third is a factor of under three in λ\lambda, which is a factor of under three in temperature at fixed mass — and eleven orders of magnitude in outcome. That compression is the whole reason a criterion drawn as a line works at all.
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 slope, drawn. The escaping fraction of a Maxwellian against the Jeans parameter, on a logarithmic axis spanning seventeen decades. The conventional criterion — vesc=6vthv_{\text{esc}} = 6v_{\text{th}}, which is λ=36\lambda = 36 — is a place on a slope rather than a boundary, chosen because it is roughly where the loss time for a planet-sized body crosses the age of the solar system. Anything an order of magnitude either side of it is either gone within a geological instant or safe for the age of the universe.

The consequence for a planet is that the question “does it keep its atmosphere” has no answer, and the question “does it keep this gas” has a very sharp one. A body retains everything above some molecular mass and loses everything below, with the boundary set by one number.

The temperature that matters is not the one anybody quotes

Here is the trap, and it is worth dwelling on because getting it wrong gives an Earth that keeps hydrogen.

Escape happens at the exobase — the level where the gas becomes collisionless, so that a molecule moving upward at more than the escape speed actually leaves rather than hitting something. That level is high in the atmosphere, and it is heated by extreme-ultraviolet photons rather than by the visible light that sets a planet’s equilibrium temperature.

For the Earth the two numbers are 255 kelvin and something near 1,000, varying by a factor of two with the solar cycle. Put 255 into the criterion and the Earth retains hydrogen and helium, which it observably does not. Put 1,000 in and it falls between helium and nitrogen, which is exactly right: helium escapes as fast as it is produced, and nitrogen never will.

Venus is the opposite case and is worth the entry on its own: a surface at 737 kelvin under an exosphere near 275, because carbon dioxide radiates away the upper atmosphere’s heat almost as fast as it arrives. A planet’s exosphere is not a hot version of its surface, and there is no general relation between them.

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 30 to 600 K, so Jupiter, Earth are outside it and not plotted.
Fig. 4 The same criterion over the exospheric temperatures the outer solar system actually has, from thirty kelvin to six hundred. Nothing about the lines has changed — the slope is still one half and the ordering of the molecules is still by mass — and the bodies that were crowded into the bottom left corner of the hero figure now occupy the plot. Titan at ninety-four kelvin, Pluto at about seventy, Triton at forty: all of them sit near the nitrogen line, and the reason they cluster there is that a body far from it either has a thick atmosphere or none, and the ones worth measuring are the marginal ones.

Where the criterion works, and where it fails

Read the retention diagram carefully and three readings stand out.

The Earth between helium and nitrogen. Exactly the observed behaviour, with no adjustment.

Titan just above nitrogen and just above methane. Which is why it has a thick nitrogen atmosphere at 94 kelvin, and why it is slowly losing its methane — a fact independently required, because the methane in its atmosphere would be destroyed by photochemistry within tens of millions of years and something has to be resupplying it.

Pluto and Triton exactly on the nitrogen line. Within a few per cent, which is not a coincidence but a selection: a body far above the line would have a thick atmosphere and a body far below would have none, and the interesting ones are the marginal ones. Both were expected to be losing nitrogen rapidly, and the spacecraft measurement at Pluto found a rate far lower than predicted — because the upper atmosphere turned out to be much colder than the models assumed, and λ\lambda is in an exponential.

Where the criterion fails, it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have.

The small bodies are worth looking at on their own, because the diagram is crowded exactly where they sit and the failures are all in one corner of it.

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 40 to 1400 K, so Jupiter, Earth, Venus are outside it and not plotted.
Fig. 5 The same criterion, with the vertical axis cut to the bodies whose escape speeds are under six kilometres a second. Mercury, the Moon and the Galilean satellites move from a crowd near the bottom of the full diagram to a spread across most of the panel, and every one of them plots above at least one line for a gas it does not have.

Zoomed in, the pattern is unambiguous: the criterion never fails by predicting that a body will lose something it has kept. It fails only by predicting that a body could keep something it does not have, and it fails that way for almost every small body in the solar system. That asymmetry is the diagnostic. A criterion that failed in both directions would be a bad criterion; one that fails in a single direction is a criterion that is right about the physics it describes and silent about a second requirement.

Retention is necessary and not sufficient. A body also needs a source. The Moon could hold carbon dioxide at its exospheric temperature and has none, because nothing supplies any. Ganymede could hold nitrogen and has none, for the same reason.

And there is a second escape route the criterion knows nothing about. Venus sits above the hydrogen line and has still lost an ocean’s worth of water, because that hydrogen did not leave by moving fast — it left by charge exchange with the solar wind, and by being picked up by the wind’s magnetic field — a loss that leaves an isotopic signature rather than a signature in a transit depth. Non-thermal escape is not a correction to Jeans escape; it is a different process with a different scaling, and for an unmagnetised planet in a strong wind it dominates completely.

Where the picture breaks completely

Move a Jupiter to a four-day orbit and the argument above does not merely need adjusting. It stops applying.

The extreme-ultraviolet flux at 0.05 astronomical units is four hundred times what the Earth receives. The upper atmosphere absorbs it and cannot radiate it away — hydrogen has no efficient cooling lines at those temperatures — so the layer heats to about ten thousand kelvin, expands, and flows outward as a wind. There is no exobase in the ordinary sense, no Maxwellian tail doing the escaping, and no exponential in λ\lambda anywhere.

The loss is instead limited by the energy arriving:

The same wind strips a mini-Neptune inside 0.09 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 0.3 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 0.017 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 0.09 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. 6 The wind at twice the heating efficiency, which is the top of the range the simulations allow. Doubling η\eta doubles the mass-loss rate and moves the stripping boundary outward — from six hundredths of an astronomical unit to nine. That is the sensitivity the section on unsettled questions is about, drawn: the efficiency is the term nobody has measured, it enters the rate linearly, and a factor of two in it is a factor of one and a half in the orbital distance inside which a mini-Neptune loses everything. The population boundary is therefore known no better than η\eta is.

M˙=ηπR3FXUVGM,\dot M = \frac{\eta\pi R^3 F_{\text{XUV}}}{GM},

with η\eta an efficiency of order a tenth. The exponential has vanished and what is left is R3/MR^3/M, which is one over the density.

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. 7 The regime change, and what it does to different planets. Fraction of the envelope lost in five billion years against orbital distance, integrating the extreme-ultraviolet flux over the star’s own history — saturated for the first hundred million years at a level ninety times today’s, then declining. A hot Jupiter loses under one per cent of itself even at 0.047 astronomical units. A mini-Neptune loses its entire envelope anywhere inside about six hundredths of an astronomical unit, and what remains is a bare rocky core.

What was actually observed

The prediction that a hot Jupiter should be losing an atmosphere was made before anyone looked, and looking is not straightforward: the mass-loss rate for HD 209458 b is around 101010^{10} grams a second, which sounds enormous and is a ten-thousandth of a per cent of the planet per billion years.

What is observable is not the rate but the extent. An escaping wind fills a volume far larger than the planet, and although it is far too tenuous to block visible light, it is optically thick in the Lyman-α line of hydrogen. So the same planet transits its star and blocks:

  • 1.5 per cent of the light in the optical, which is the planet’s disc;
  • about 15 per cent in Lyman-α, which is a cloud of hydrogen ten times the planet’s radius.

That ratio is the measurement. A transit ten times deeper in one line than in the continuum is not a planet; it is an exosphere overflowing its Roche lobe, and the fact that some of the absorption appears at velocities of a hundred kilometres a second — far above the planet’s escape speed — says that the gas is not bound.

The Roche lobe, which decides how far the wind gets

There is a second geometry in the hot-Jupiter case that has no analogue on the Earth, and it makes the loss substantially easier than the energy argument alone suggests.

A planet close to its star sits deep in the star’s tidal field, and the region it can hold on to is bounded by the surface at which the two pulls balance in the rotating frame. For a Jupiter at four days that surface is only two or three planetary radii from the centre — so an atmosphere that expands by a factor of a few is not merely hot, it is outside the planet’s gravitational jurisdiction, and it leaves through the inner Lagrange point rather than by any thermal process at all.

The observed Lyman-α absorption is consistent with exactly that: gas at ten planetary radii, in a comet-like tail swept back by the stellar wind and by radiation pressure on the hydrogen itself.

The effect is included in the energy-limited expression as a correction factor that can be several, and it is the reason the closest-in planets lose mass faster than the flux scaling alone predicts.

The population this carves

The energy-limited expression scales as one over the density, which means it does very little to a gas giant and everything to a low-mass planet with a light envelope. That asymmetry has a fingerprint.

A rocky core of a few Earth masses with a hydrogen envelope of a few per cent by mass has a radius of two and a half Earth radii or so, because a mass and a radius together give a density and nothing more, because a small mass of hydrogen is very fluffy. Strip the envelope and the same core is 1.5 Earth radii. There is no stable configuration in between: the envelope either survives or it does not, and the transition is fast.

So a population of such planets, exposed to a range of stellar fluxes, should split into two groups with a gap between them — and the gap is observed, at about 1.8 Earth radii, sloping in period exactly as the escape argument requires.

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. 8 The same curves over the inner two tenths of an astronomical unit, which is where the whole of the transiting population lives. At this magnification the two behaviours separate cleanly: the gas giant’s curve is flat and near zero across the entire range, and the mini-Neptune’s crosses from total loss to no loss over a factor of two in distance. That sharpness in orbital distance is what produces a sharp gap in radius, because a planet’s fate is decided by which side of a steep curve it sits on rather than by how far from it.

What is not settled

The efficiency is a fitted number. The η\eta in the energy-limited expression covers everything about how absorbed extreme-ultraviolet becomes bulk flow, and it is somewhere between a few per cent and about a third depending on the planet, the flux and whose simulation is consulted.

The early flux history is inferred from other stars. The Sun’s extreme-ultraviolet output four billion years ago is estimated from observations of young solar analogues, and the scatter among stars of the same age is large. Since the total fluence is dominated by the first few hundred million years, that scatter propagates directly into how much any given planet lost.

The same wind strips a mini-Neptune inside 0.07 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 10 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 5.2·10¹⁵ J m⁻² at one astronomical unit and is some 4 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 0.0094 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 0.07 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. 9 The same integral run to ten billion years rather than five, which is the check on that claim. The boundary moves from six hundredths of an astronomical unit to seven — a fourteen per cent change for a doubling of the time — because the saturated phase lasts a hundred million years and the decay afterwards is steep enough that the fluence has converged long before a gigayear. So the scatter in the early flux propagates into the answer and the age of the system barely does, which is why the radius gap is a fossil and why measuring its dependence on age is such a demanding test.

The exobase itself is a fiction that has to be placed somewhere. The whole Jeans calculation is a boundary condition applied at the altitude where the mean free path first exceeds the scale height, and that altitude is not sharp: the transition from collisional to collisionless takes a scale height or two, and the temperature is changing across it. Kinetic calculations that drop the boundary entirely and follow the molecules through the transition get loss rates a factor of two or three below the Jeans formula, consistently and in the same direction, because a molecule in the tail is more likely to be knocked out of the tail on its way up than the sharp-boundary picture allows. That is a small error on a quantity that spans twelve orders of magnitude, which is why the formula survives — but it is a systematic one, and it is the reason a measured escape rate is never quoted as a confirmation of the formula.

And the two accounts of the radius gap are not distinguished. Photoevaporation and core-powered mass loss make the same gap in the same place with the same slope; separating them needs the dependence on stellar type, since one is driven by the star and the other by the planet, and the samples are not yet large enough.

The escape that needs no heat

The essay has treated one thermal channel carefully and mentioned the others in a clause. They deserve a section, because for the terrestrial planets today they dominate.

Photochemical escape produces fast atoms without heating anything. A molecular ion recombining with an electron dissociates, and the energy released — a few electron volts, which is the difference between the molecule’s binding energy and the ion’s — goes into the kinetic energy of the fragments. An oxygen atom produced that way from an ionised oxygen molecule has a speed of several kilometres a second, which on Mars is above the escape speed. The population producing the escape is not a Maxwellian tail; it is a monoenergetic product of a chemical reaction.

Sputtering is the same idea driven from outside. An ion from the solar wind, picked up by the planet’s own field and accelerated, strikes the upper atmosphere and knocks neutrals out of it in a collision cascade. The energy comes from the wind rather than from the atmosphere’s temperature.

Ion pickup removes the ions directly. An atom ionised above the exobase finds itself in the solar wind’s magnetic field, is accelerated by the motional electric field, and is carried away with the flow. Nothing about the planet’s gravity enters if the pickup energy is large enough, which it usually is.

And a polar wind operates on a magnetised planet. Ions along open field lines over the poles are accelerated outward by an ambipolar electric field set up by the lighter electrons’ tendency to escape first, and they leave along the field.

The list matters because the channels scale differently. Thermal escape depends on the exospheric temperature and on the planet’s gravity; the non-thermal ones depend on the stellar wind, on the ionising flux and on the magnetic geometry, and they are largest for the lightest species only weakly.

A planet’s atmosphere is not lost by one process with a threshold, and the reason the retention diagram works as well as it does is that thermal escape is the fastest of the several whenever it operates at all.

The list also explains why an unmagnetised planet and a magnetised one lose comparable amounts: the field closes some channels and opens others.

That last sentence is the one worth testing rather than asserting, because it contradicts the intuition the word shield carries. A magnetic field is supposed to be protective; the three planets whose escape rates have actually been measured by a spacecraft in orbit around them say something less tidy.

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. 10 Measured ion escape rates against surface magnetic field for the three planets where both numbers exist. Venus and Mars have no dynamo and Earth has one thirty times stronger than the noise floor, and the three rates agree to within a factor of ten — with the magnetised planet at the top of the range rather than the bottom.

The field does close the pickup channel, which is the dominant loss at Venus and Mars. What it opens in exchange is the polar cap: a magnetosphere is not a closed shell but a bottle with two holes in it, and along the open field lines over each pole the ambipolar electric field lifts ions out of the ionosphere and hands them to the solar wind directly. The Earth loses of order a kilogram of oxygen every second that way. The exchange is close enough to even that the measured rates cannot distinguish the two arrangements, which means the argument that a planet needs a dynamo to keep an atmosphere is an argument nobody has yet made stick with a number.

The planet with the measurement and the discrepancy

Mars is the case where both the present rate and the integrated loss have been measured, and the two do not obviously agree.

The present rate comes from a spacecraft in Martian orbit carrying instruments that count escaping ions directly — measuring the flux of oxygen, carbon dioxide and hydrogen leaving along the tail, and watching it rise by an order of magnitude during a solar storm. The total comes to a few hundred grams a second, which integrated over four billion years is a modest fraction of an atmosphere.

The integrated loss comes from isotopes. Escape is mass-selective, so the lighter isotope leaves preferentially and what remains is enriched in the heavier one. Mars’s atmosphere is enriched in deuterium relative to hydrogen by a factor of five or six over terrestrial values, and in the heavy isotopes of argon and nitrogen by comparable amounts — and inverting that enrichment implies that most of the original atmosphere is gone.

The two are not straightforwardly reconcilable at the present rate. Either the loss was far faster early on — which it should have been, since the young Sun’s extreme-ultraviolet output and wind were both much stronger — or a substantial part of the atmosphere did not escape at all but was sequestered into the crust as carbonate.

Distinguishing the two matters for the planet’s climate history, because an atmosphere that escaped is gone and one that is locked in rock could in principle be released.

The instrument that would settle it is a measurement of how much carbonate the crust contains, which has been attempted from orbit and is hard: the mineral’s spectral signature is weak, and the deposits found so far account for a small fraction of what the isotopes require.

A rate measured now and an integrated total inferred from isotopes are two measurements of the same history, and where they disagree the disagreement is the result.

Where this ladder goes next

This rung has taken a threshold that looks like a speed and shown it is a place on an exponential, sorted the solar system with it, and then watched it fail entirely in the regime that produced most of the known planets.

The rung above is the composition that survives. Escape is mass-selective — light gases go first — so a planet that has lost an atmosphere is left with an isotopically and chemically altered one, and the deuterium-to-hydrogen ratio of Venus and Mars, both enriched by roughly a factor of a hundred over the Earth’s, is the direct record of how much water each has lost.

Beside it lies the magnetosphere question: whether a planetary magnetic field protects an atmosphere from the solar wind or, by opening polar cusps, helps it escape. The answer is not settled, and the Earth and Venus lose oxygen at comparable rates despite one having a field and the other not.

And below it, the habit: a threshold in a variable that appears in an exponent is not a threshold. Nothing changes at λ=36\lambda = 36. What changes is that everything below it happened long ago and everything above it will not happen at all, and the width of the transition is a factor of two in one number.

Where this goes next

Circular and escape speed. Orbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception. The speed that does not come back, and the √2 that separates it Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

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

What links here

The 8 of 10 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Energy-limited escapeExobaseExospheric temperatureExtreme-ultravioletHydrodynamic escapeJeans escapeJeans parameterMaxwell boltzmann distributionNon-thermal escapePhotoevaporationRoche lobe overflowScale height