Astronomy & Celestial Mechanics
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Exoplanets

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.

Assumes Atmospheric escape and Radius valley.

15 min read 5 figures Small perturbations, large consequencesMeasuring the unreachableScales that defeat intuition

The rate at which a close-in planet loses its atmosphere is usually estimated in one line. Take the extreme-ultraviolet and X-ray power the planet intercepts, multiply by an efficiency — the fraction that ends up heating gas rather than being radiated away — and divide by the energy it takes to lift a gram of gas out of the planet’s gravitational well. The result is the energy-limited rate,

M˙=ε πFR3GM,\dot M = \frac{\varepsilon\, \pi F R^3}{G M},

proportional to the flux FF, to the cube of the radius, and inversely to the mass. It is the scaling that sorts close-in planets into those that keep their envelopes and those that do not, and it is used for everything from hot Jupiters to the early Earth.

It is an energy budget, and energy budgets are upper limits. It says how much gas the absorbed power could lift if every erg of heat went into lifting. What it does not say is whether anything else takes the heat first, or whether there is enough of anything besides heat — enough photons, enough time — to do the lifting. Both of those happen, at opposite ends of the range of planets, and between them they turn the single efficiency ε\varepsilon into a function of where the planet sits.

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.
Fig. 1 The mass-loss rate of a hot Jupiter against the ionising flux it receives. The dashed line is the energy-limited rate, linear in the flux; the solid curve turns over near 104 erg cm−2 s−110^4\ \mathrm{erg\,cm^{-2}\,s^{-1}} and grows only as the square root of the flux above it. At 10610^6 the energy-limited formula overestimates the loss elevenfold.

Where the heat goes instead

An ultraviolet photon absorbed high in a planet’s atmosphere ionises a hydrogen atom, and the photoelectron carries off the photon’s excess energy — several electronvolts — which it shares with the gas by collisions. That is the heating. The heated gas expands, and if the heating is strong enough the expansion becomes a wind: a flow accelerating outward through a sonic point and escaping, the hydrodynamic escape that makes a hot Jupiter look ten times its size in Lyman-α light.

The ionised protons and electrons do not stay ionised. They recombine, and the rate at which they do goes as the product of their densities — as the square of the density. At low flux the gas at the base of the wind is mostly neutral, recombinations are rare, and nearly all the heat goes into the flow. At high flux the base is almost fully ionised, and there is a balance to be kept: each photon absorbed ionises an atom, each recombination undoes one, and in steady state the two rates are equal. Since recombination goes as density squared and ionisation as density times flux, the density at the base of the wind settles at a value proportional to the square root of the flux.

The wind itself, in this regime, is close to isothermal — for a reason taken up below — and the rate at which an isothermal wind carries off mass is its density at the sonic point times its speed there times the area of the sphere. The temperature fixes the speed, the planet fixes the sonic radius, and the density goes as F\sqrt{F}. So the mass-loss rate goes as F\sqrt{F} too.

This is the recombination-limited regime, worked out by Murray-Clay, Chiang and Murray in 2009 for hot Jupiters. The heat that the energy budget assumed would lift gas is instead radiated away, as the Lyman-α photons emitted when recombined atoms are collisionally excited and decay. For a hot Jupiter of HD 209458 b’s size and mass, the transition happens near 10410^4 erg per square centimetre per second. That is within a factor of a few of the flux such a planet receives today from a Sun-like star; its star’s young self delivered a hundred times more.

The efficiency that is really a flux

The energy-limited formula has one free number, and it is always the one that gets argued about.

The efficiency that is really a function of the flux. The effective heating efficiency — the mass-loss rate expressed as a fraction of the energy-limited rate with perfect efficiency — against the ionising flux, for the same hot Jupiter, with the transition to recombination-limited flow placed at 1000, 10⁴, 10⁵ erg cm⁻² s⁻¹. Below its transition each curve is flat at 0.3, the value put into the energy-limited formula; above it the efficiency falls as the inverse square root of the flux, to 0.027 at 10⁶ for the middle case. The efficiency is the single free number in every energy-limited calculation, and it is usually quoted as one value for a planet. It is not a property of the planet at all but of where the planet sits on this axis — and a young star's flux is a hundred times its older self's, so the same planet has two efficiencies in its own history.
Fig. 2 The effective heating efficiency — the loss rate as a fraction of what perfect heating would give — against flux, with the recombination transition at three different fluxes. Below its transition each curve is flat at the 0.3 put into the formula; above it the efficiency falls as the inverse square root of the flux, to 0.027 at 10610^6 in the middle case.

The efficiency ε\varepsilon is meant to absorb everything the one-line estimate leaves out: the fraction of the photon’s energy carried by the photoelectron, the fraction the photoelectron wastes on further ionisation and excitation, the energy radiated away. Values between 0.1 and 0.6 are quoted, and a planet’s inferred mass-loss history can move by the same factor. Written as the ratio of the true loss rate to the energy-limited one with perfect efficiency, the recombination ceiling makes ε\varepsilon a declining function of flux: flat at low flux, falling as F−1/2F^{-1/2} past the transition, and an order of magnitude smaller by the time the flux is a hundred times the transition.

The efficiency is therefore not a property of the planet. It is a property of where the planet is on the flux axis, and a single planet occupies a long stretch of that axis over its life: a young star’s ultraviolet output is a hundred to a thousand times its older self’s, because the star spins down and its magnetic activity fades. An efficiency fitted to a planet observed losing mass today — the only kind of measurement there is — is the efficiency for today’s flux, and applying it to the planet’s youth overestimates the early losses by the factor the ceiling removes. The figure also shows the transition flux moving by a factor of ten either way: the fall is the same shape, displaced. Where the transition sits depends on the planet’s gravity and on the spectrum of the star, and the full calculation has to be done for each.

A thermostat set by one spectral line

The reason the wind is nearly isothermal is the second half of the same physics, and it is what makes the ceiling a ceiling.

Five decades of flux, and a temperature that barely moves. The temperature at which photo-heating of an escaping hydrogen atmosphere balances cooling by collisionally excited Lyman-α emission, in a one-zone estimate, against the ionising flux. Cooling depends on the temperature through the Boltzmann factor for exciting the n = 2 level, whose excitation temperature is 118,348 K, and on the square of the density, which rises as the square root of the flux; so each factor of e in heating is absorbed by a small change in temperature. Across five decades of flux the balance temperature moves only from 8,371 to 14,121 K. That is why the escaping gas of every strongly irradiated planet sits near 10⁴ K, and why extra flux stops buying extra escape: it is radiated away as Lyman-α before it can lift anything. The same line is the one in which the escaping gas is seen, as a planet ten times its own size.
Fig. 3 The temperature at which photo-heating balances Lyman-α cooling in the escaping gas, in a one-zone estimate, against flux. Five decades of flux move it only from about 8,400 to 14,000 K: the Boltzmann factor for exciting hydrogen’s second level absorbs each factor of ee in heating in a small step of temperature.

The gas cools by collisions exciting neutral hydrogen from its ground state to its second level, from which it decays by emitting a Lyman-α photon that escapes. The rate of that excitation depends on temperature through a Boltzmann factor, exp⁡(−118,348 K/T)\exp(-118{,}348\ {\rm K}/T), whose excitation temperature is ten times the gas temperature. An exponential that steep means that a factor of ee more heating is balanced by a small rise in temperature — about ten per cent near 10410^4 K. The one-zone estimate in the figure, which includes the rise of the base density with flux, moves the balance temperature by less than a factor of two across five decades of flux.

That is a thermostat, and its set point is written into the hydrogen atom. Every strongly irradiated escaping atmosphere sits near 10410^4 K, and extra flux beyond the transition does not heat it further; it is radiated away as Lyman-α, in the same line in which the escaping gas is observed transiting its star. It is also why a wind in this regime is limited by its density rather than its temperature: the temperature is fixed, the speed of an isothermal wind is fixed by the temperature, and only the density — held down by recombination — is left to grow.

The photons run out first

At the other end of the range of planets, the energy-limited formula fails in the opposite direction. Consider how many atoms one absorbed photon can lift. The heat it deposits is εhν\varepsilon h\nu — with an efficiency of 0.3 and a typical ionising photon of 20 electronvolts, about 6 eV. The energy needed to lift one hydrogen atom out of a planet’s well is mHGM/Rm_H GM/R: about 9 eV for a hot Jupiter, 1.3 eV for a sub-Neptune of five Earth masses and two and a half Earth radii, 0.65 eV for the Earth.

For the hot Jupiter the photon’s heat lifts less than one atom, and the energy is the limit. For the Earth it could lift nine — but the photon only ionised one. The escaping gas is the gas that was heated, and heating comes from photoionisation; a flow cannot carry off atoms faster than photons arrive to ionise and heat them. In a shallow well the energy argument over-counts, and the loss rate is set by the photon count instead: one atom per ionising photon. This is the photon-limited regime, identified by Owen and Alvarez in 2016, and in it the loss grows linearly with the flux but is independent of the planet’s gravity entirely.

Three regimes of escape, mapped by flux and by how deep the planet's well is. Which process limits the escape of a heated hydrogen atmosphere, by ionising flux (across) and by the depth of the planet's gravitational potential, GM/R (up), both on logarithmic axes. Below a potential of 5.74·10¹² erg g⁻¹ one photon's share of heat, 0.3 of 20 eV, is more than enough to lift an atom, so at modest flux the escape is photon-limited: every ionising photon removes one atom and the energy argument over-counts. Above that potential, low flux gives energy-limited escape and high flux recombination-limited escape, with the boundary held at the hot Jupiter's 10⁴ erg cm⁻² s⁻¹ — in the full calculation it moves with the potential, and the map does not pretend otherwise. At high enough flux recombination wins everywhere. The dotted line is where the photon limit would sit for a heating efficiency of 0.1 instead, 1.91·10¹² erg g⁻¹ — so whether a sub-Neptune is photon-limited depends on the same efficiency the energy-limited formula needs as an input. The dashed lines mark the potentials of the Earth, a five-Earth-mass sub-Neptune of 2.5 Earth radii and a hot Jupiter. A planet's history is a leftward walk along its line as its star fades, and the three cross different boundaries on the way.
Fig. 4 Which process limits escape, by ionising flux and by the depth of the planet’s potential. Below 5.7×1012 erg g−15.7\times10^{12}\ \mathrm{erg\,g^{-1}} one photon’s heat can lift more than one atom and escape is photon-limited; above it, energy-limited at low flux and recombination-limited at high. The dotted line is the photon limit for an efficiency of 0.1. The dashed lines are the Earth, a sub-Neptune and a hot Jupiter.

The photon limit is not an exotic case confined to small bodies; it is where the water of rocky planets has been decided. A steam atmosphere on a young Venus or on a planet inside its star’s eventual habitable zone before the star had settled loses its hydrogen after ultraviolet photons split the water high up, and the potential of an Earth-mass planet is an order of magnitude below the boundary. Estimates of how many oceans such a planet could lose, made with the energy-limited formula, are therefore upper bounds on a process that the photon count limits more tightly — and the Earth’s own loss under the ultraviolet of a young, rapidly rotating Sun was governed by the number of photons it received rather than the energy they carried.

The three regimes fill a map of flux against potential. The boundary between energy- and photon-limited escape is a horizontal line, independent of flux, at the potential where one photon’s heat just lifts one atom. The boundary between the energy- and recombination-limited regimes is, in this drawing, a vertical line at the hot Jupiter’s transition flux; in the full calculation it moves with the potential, and the figure says so rather than drawing a slope it has not computed. And at high enough flux, recombination wins even in shallow wells: photon-limited escape grows linearly with flux and recombination-limited as its square root, so the second always overtakes the first eventually.

The dashed lines put three planets on the map. The hot Jupiter sits just above the photon boundary, in the energy-limited regime at modest flux and the recombination-limited regime at high flux — the case Murray-Clay’s calculation was built for. The Earth sits well below it: its ancient escape, under a young Sun, was photon-limited until the flux was high enough for recombination to take over, and the energy-limited formula applied to it overestimates the loss nearly tenfold. The sub-Neptune sits in between, and here the map makes a point about its own assumptions. With an efficiency of 0.3 the sub-Neptune is photon-limited; with an efficiency of 0.1, the dotted line, the photon boundary drops to 1.9×10121.9\times10^{12} erg per gram and the sub-Neptune is just above it. Whether the energy-limited formula even applies to a sub-Neptune depends on the value of the efficiency that the formula needs as an input. It is the planets near that line whose envelopes carve the gap in the radius histogram, which is why the rate law used for them matters beyond any one planet.

When the loss happens

The recombination ceiling changes not only how much gas a planet loses but when. A Sun-like star’s ultraviolet output is saturated at its maximum for its first hundred million years or so and then falls as a power of the age, and in the energy-limited picture the loss simply follows the flux.

When the atmosphere was lost, under two rules. The fraction of a hot Jupiter's total five-billion-year atmospheric loss that has happened by each age, on a logarithmic time axis, with the ionising flux saturated at 300 times its present 1000 erg cm⁻² s⁻¹ for the first 100 million years and then decaying as the 1.23 power of the age. Under the energy-limited rule the loss follows the flux, and 28 per cent of it has happened by 100 million years. With the recombination ceiling the saturated phase is capped — its flux is far above the transition — and only 14 per cent has happened by then; the loss is spread into the long decline. The total is smaller too, but the shift in timing is the point: a planet's envelope is decided less completely by its star's youth than the energy-limited arithmetic says.
Fig. 5 The fraction of a hot Jupiter’s total five-billion-year loss that has happened by each age, with the flux saturated at 300 times its present value for 100 million years and then decaying. Energy-limited, 28 per cent is gone by the end of saturation; with the recombination ceiling, 14 per cent. The ceiling moves the loss out of the star’s youth.

Under the energy-limited rule, about 28 per cent of the planet’s total loss over five billion years happens during those first hundred million years of saturation, and the young star dominates the history. With the ceiling included, the saturated flux sits far above the transition, where the loss grows only as the square root of the flux, and the same interval accounts for only 14 per cent. The total is smaller, and it is spread into the long decline. For a planet whose envelope was stripped, the ceiling makes the stripping slower and later than the one-line estimate says; for a planet near the boundary between keeping and losing its envelope, it can decide which side it ends on.

That matters for the argument that the radius valley is a fossil of the first few hundred million years. For the sub-Neptunes that define the valley the ceiling is weaker than for a hot Jupiter — their transition fluxes differ — but the direction is the same: the more a history is dominated by high fluxes, the more of it lies above the ceiling, and the less completely the early star decides the outcome. It also bears on the age dependence of the valley, which is one of the observations expected to separate starlight-driven stripping from stripping powered by a planet’s own cooling core.

What a transit measures, and what it assumes

None of the rates in these figures is observed directly. What is observed is absorption: a planet transiting its star blocks a few per cent more light in the Lyman-α line of hydrogen, or in the 1083-nanometre line of helium atoms held in a long-lived excited state, than it does in the continuum, because it is surrounded by an extended cloud of escaping gas. Turning that extra absorption into a mass-loss rate requires a model of the outflow — its density profile, its speed, its temperature — and the model used almost universally is an isothermal Parker wind.

The thermostat is what justifies that choice. A wind that sits near 10410^4 K because the Lyman-α cooling holds it there is close to isothermal over the region the absorption samples, and the fitted temperature, usually five to twelve thousand kelvin, agrees with it. But the same choice builds the regime into the answer. An isothermal wind at a fixed temperature, fitted to an absorption depth, returns a density and hence a rate; it cannot say whether that rate was set by the energy supply, the recombination balance or the photon count, because it has already assumed the temperature that the recombination regime enforces. To learn which regime a planet is in from its transit, the same planet has to be observed at more than one flux — which, for a planet on a nearly circular orbit around a steady star, means observing different planets and assuming they are alike.

There are two exceptions, and both are being pursued. A planet on an eccentric orbit receives a flux that changes by the square of the ratio of its farthest to its nearest distance, a factor of two for an eccentricity of 0.17, so its outflow can in principle be compared with itself at two fluxes a few days apart — if the wind responds faster than the orbit, which near the planet it does. And a star that flares multiplies its ultraviolet output tenfold for an hour; an escaping atmosphere caught during a flare and again in quiescence has been driven at two fluxes without changing anything else. On the linear branch the loss would follow the flare; on the square-root branch it would barely notice.

What the three formulas leave out

Each regime is an idealisation, and the real flows mix them. The transition between the energy- and recombination-limited regimes is smooth, and the figures join the two rates harmonically so that the smaller dominates; the true shape of the turnover depends on the stellar spectrum, because X-rays penetrate deeper than extreme-ultraviolet photons and heat denser gas, where recombination is faster. Many planets receive enough X-rays that the X-ray-driven flow and the ultraviolet-driven flow have to be computed together.

The planet is not a point with a radius. The wind is launched from a height set by where the ionising photons are absorbed — often a large fraction of a planetary radius above the optical surface — and the energy-limited formula’s R3R^3 is really a product of that absorption radius squared and the planet’s radius. For a puffy planet the difference is a factor of several. The star’s tidal field matters too: the Roche lobe reduces the energy needed to escape, and for the closest planets that correction is large.

And the magnetic field of the planet can confine the ionised outflow, suppressing it near the equator where field lines are closed and letting it out along open field lines near the poles — a shield that is also a funnel. None of these changes the existence of the three regimes, but each moves the boundaries between them, and the boundaries are where the planets that matter most sit.

Still open: which regime the planets near the valley were in

The three regimes are well understood as physics. What is not known is which of them governed the planets whose envelopes decided the shape of the observed population. The sub-Neptunes near the radius valley sit close to the photon-limited boundary and, in their youth, far above the recombination transition; the efficiency that determines which regime applies is itself uncertain by a factor of several; and the stellar ultraviolet histories that drive them vary by an order of magnitude between stars of the same mass. Measurements that would narrow it exist in principle — escaping helium and hydrogen observed around young planets, whose fluxes put them above the transition, compared with the same measurements around old ones below it. A handful of such planets have been observed. The question the map poses is whether their loss rates fall on the square-root branch or the linear one, and until enough of them are measured across the transition, every energy-limited calculation of a planet’s history carries an efficiency that is really a guess about which side of the ceiling the planet was on.

Where this goes next

The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in. A planet ten times larger in one colour A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving. 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. 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.

About the same objects

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

The objects this essay names

Each one links to every other essay that touches it.

Atmospheric escapeEnergy-limited escapeHeating efficiencyHot jupiterLyman-alphaParker windPhoton-limited escapeRecombination-limited escapeSub-neptuneXUV flux