The heaviest atom a hydrogen wind can lift

A planet can lose its atmosphere in two quite different ways. In the gentle way, the fastest atoms in the hot uppermost layer escape one at a time, and the rate depends steeply on how heavy each atom is: hydrogen leaves, helium barely, everything else not at all. In the violent way, the upper atmosphere is heated so strongly by a young star’s ultraviolet and X-ray light that it expands outward as a continuous wind, and the wind is mostly hydrogen. That second way is the one that strips the envelopes of close-in planets and carves the radius valley, and the one that could have removed any nebular hydrogen the young Earth held.

A wind carries passengers. Every other atom in the escaping gas is jostled by collisions with the hydrogen streaming past it, and if the collisions push it outward harder than gravity pulls it back, it leaves too. Whether they do depends on the atom’s mass, the strength of the wind and the planet’s gravity, and the condition comes out as a single number with a sharp meaning: a crossover mass, below which a gas is carried away and above which it is left behind. The gases a planet keeps, and the proportions of their isotopes, then record how strong the wind was.

A wind with a weight limit #

The argument is a tug of war between two forces on a heavy atom sitting in a stream of hydrogen. Gravity pulls it down with a force equal to its weight. The hydrogen drags it up, through collisions, with a force proportional to how fast the hydrogen streams past it relative to the atom itself and inversely proportional to how easily the atom diffuses through the hydrogen. An atom that diffuses easily slips through the stream and is hardly pushed; one that diffuses poorly is carried along.

Balance the two and the atom’s own outward flux comes out as the hydrogen’s flux reduced by a factor that falls linearly with the atom’s mass. At some mass the factor reaches zero, and that is the crossover:

$$m_c = m_{\mathrm{H}} + \frac{kT\,F_{\mathrm{H}}}{b\,g\,X_{\mathrm{H}}}$$

Here $F_{\mathrm{H}}$ is the hydrogen flux per unit area, $g$ the gravity, $T$ the temperature, $X_{\mathrm{H}}$ the fraction of the gas that is hydrogen and $b$ the binary diffusion parameter, which measures how easily the heavy atom diffuses through hydrogen and is about the same for every heavy partner to within tens of per cent. The crossover rises in proportion to the flux and falls in inverse proportion to the gravity. A gas just below it leaves with a small fraction of the hydrogen’s rate per atom; a gas well below it leaves almost as readily as the hydrogen does.

The fluxes involved are enormous by present standards. The Earth loses hydrogen today at about three hundred million atoms per square centimetre per second, limited by how fast hydrogen can diffuse up through the lower atmosphere, and that flow lifts nothing at all. Carrying neon takes thirty thousand times as much; carrying xenon, two hundred thousand times. Fluxes like those exist only in the first hundred million years or so of a planet’s life, while its star is at its most active and while it still has hydrogen in quantity to lose.

Two limits on one flow #

The same diffusion parameter that sets the crossover also sets the other limit on how fast a planet can lose hydrogen, and the two are worth keeping apart. In the Earth’s present atmosphere hydrogen is a trace gas, carried up in water vapour and methane to the level where the air stops being stirred, and above that level it has to diffuse upward through the heavier nitrogen and oxygen to reach space. However hot the top of the atmosphere, hydrogen cannot leave faster than it can diffuse to the top. That diffusion limit is why the Earth’s present loss is three hundred million atoms per square centimetre per second: it is set by the hydrogen content of the lower atmosphere and the parameter $b$, not by the Sun.

A planet whose upper atmosphere is mostly hydrogen has no such bottleneck. The hydrogen is the background gas, and the rate at which it leaves is set by how much energy reaches it: the energy-limited regime, in which the flux follows the star’s ultraviolet output. The crossover is the diffusion argument turned round. Instead of hydrogen diffusing through a heavy background, a heavy minor gas diffuses through a hydrogen background, and the question is whether the stream of hydrogen past it outruns its tendency to settle. A large $b$ means the heavy atom slips through easily and is left behind; a small $b$ means it is caught.

So a hydrogen-rich planet has both kinds of loss available, at different times. While the envelope is thick and the star bright, the energy-limited wind runs and carries passengers up to the crossover. When the envelope has thinned until hydrogen is no longer the dominant gas at the top, the flow chokes to the diffusion limit, the crossover collapses towards hydrogen’s own mass, and the passengers stop leaving. The pattern a planet’s noble gases record is written in the first stage and frozen at the transition.

Small planets lose heavier gases #

The energy-limited flux — the rate at which the absorbed stellar energy can lift gas out of the planet’s well — scales with the planet’s radius over its mass, because a larger radius intercepts more light per unit of mass and a smaller mass is easier to escape from. The gravity that holds a heavy atom back scales as mass over radius squared. Their ratio, which sets the crossover, therefore falls steeply with a planet’s mass.

The consequence is a strong ordering among the terrestrial planets. Under the same young Sun, the Earth’s wind could carry helium and neon and stop short of argon. Venus, a little smaller and closer to the Sun, could carry argon. Mars, a tenth of the Earth’s mass, could carry krypton. Gases that were safe on the Earth were not safe on Mars, and the reason is not that Mars was hotter or closer to the Sun — it was neither — but that its gravity was weak and its flux per unit of mass large.

At the other extreme are the close-in sub-Neptunes whose envelopes the same light strips over a few hundred million years. At a hundred times the Earth’s flux the crossover for a five-Earth-mass core is near three hundred: the wind carries every gas there is, and nothing about the envelope’s heavier constituents is held back. The radius valley’s bare cores lost not just their hydrogen but, in proportion, everything that came with it. A temperate Earth-mass planet is in the opposite regime, where the wind is selective and the selection is a record.

What the bare cores kept #

The planets below the radius valley are the extreme case. A core of a few Earth masses that began with a few per cent of its mass in hydrogen and lost all of it lost, with it, every gas the wind could reach — which at their fluxes was every gas. The valley is a gap between planets that kept an envelope and planets that kept none, and on the bare side of it the cores have no primordial gas left at all.

Whatever atmosphere such a core has now must therefore be its own: gas released from its interior by volcanism, or delivered later by impacts, after the wind stopped. That is a prediction with a measurable consequence. A secondary atmosphere outgassed from rock is made of water, carbon dioxide, nitrogen and sulphur compounds, with noble gases only in the small amounts the interior retained; a residual primordial one keeps the nebula’s proportions. For the smaller planets round smaller stars whose valleys sit at lower radii, the same arithmetic applies at lower fluxes and lower crossovers, and the boundary between planets that lost everything and planets that lost only their lightest gases moves accordingly. A spectrum of a bare core’s atmosphere, if it has one, is a spectrum of what came out of the rock.

A clock for each gas #

A young star does not stay at its brightest. The X-ray and ultraviolet output of a Sun-like star is saturated, at about a three-thousandth of its total light, for a period that depends on how fast it was born spinning, and then falls as the star’s magnetic wind brakes its rotation. The crossover falls with it.

So each gas has a closing date. Neon could leave the Earth with a hydrogen wind only while the Sun was saturated, and the date that window closed depends on the Sun’s rotation, which is not known. Argon, krypton and xenon could not leave by this route at all, at the Earth’s mass and distance, under any plausible young Sun. If the Earth’s air is depleted in any of those, something other than a wind of neutral atoms driven by sunlight did it.

The calculation is deliberately simple. It puts the base of the flow at the planet’s surface and its temperature at a thousand kelvin, uses one diffusion parameter for every gas and holds the efficiency of heating fixed. Each of those moves the crossover by tens of per cent, and none of them moves argon’s crossover from the Earth within reach. Earlier work that did lift xenon from the Earth by this route needed ultraviolet fluxes hundreds of times the present Sun’s, sustained, and hydrogen inventories large enough to feed the wind throughout — which is possible in principle and leaves a specific pattern behind.

The ramp the survivors draw #

That pattern is the most distinctive thing a hydrodynamic wind leaves. A gas below the crossover leaves at a rate per atom that is the hydrogen’s rate reduced by its distance from the crossover. As the hydrogen inventory falls, each gas falls with it, by the hydrogen’s own depletion raised to a power that runs linearly from one, for hydrogen itself, to zero at the crossover.

On a logarithmic scale the survivors form a ramp: a straight line rising with mass until it meets complete retention at the crossover. The light gases are lost almost as completely as the hydrogen; the gases just below the crossover are barely touched; the gases above it are entirely kept. The slope of the ramp says how much hydrogen went, and the point where it meets the top says how strong the wind was.

Neither of the other ways a planet loses gas draws that shape. Evaporation atom by atom removes only the lightest gases and leaves everything from neon upward untouched. A giant impact that blows off part of an atmosphere removes every gas in proportion to its abundance and sorts nothing. Solar-wind sputtering, the slow erosion of an unmagnetised planet’s upper atmosphere by ions of the wind, removes gas from above the level where the atmosphere stops being mixed, and favours the lighter isotopes of each element in its own characteristic way. The noble gases are the right tracers precisely because they do nothing else: no chemistry, no rocks to dissolve into in quantity, no life to cycle them. What a planet has of them is what it was given, minus what escaped and how.

The same sorting on every isotope #

The isotopes inside each element are sorted too. Two isotopes of one gas differ in mass by one or a few units, so they sit at slightly different distances from the crossover, and the lighter one leaves slightly more readily.

That independence is the surprising part, and it follows from the linearity. The power to which each isotope’s survival is raised changes by the same amount per mass unit whatever the element, so the ratio of two neighbouring isotopes changes by the same factor. Neon, argon and krypton, if all three are below the crossover, are fractionated by the same per-mass-unit amount. The gas nearest the crossover is the least depleted and exactly as fractionated as the rest. A planet whose noble gases came through a hydrogen wind therefore carries one isotopic signature, shared by every gas the wind could reach, and the size of that signature measures how much hydrogen was lost.

The isotopes carry the information more robustly than the elements do. An elemental abundance depends on what the planet started with, which is uncertain; an isotope ratio starts out the same in every body made from the same nebula and is shifted only by processes that sort by mass. That is why the comparison that matters is between a planet’s air and the primordial ratios measured in the Sun’s wind and in primitive meteorites.

The xenon that should not have gone #

Measured against those primordial ratios, the Earth’s air is strange. Its xenon is depleted, by more than an order of magnitude relative to its krypton when both are compared with the pattern in primitive meteorites, and its xenon isotopes are fractionated by three to four per cent per mass unit, the heavy ones enriched. Its krypton is barely fractionated at all. This combination has been called the missing-xenon problem, and the arithmetic of the hydrogen wind shows why it is a problem rather than a detail.

A wind of neutral atoms sorts by mass and by nothing else, so it always takes more of the lighter gas than the heavier. It can fractionate krypton and leave xenon alone. It can fractionate both equally. It cannot fractionate and deplete xenon while leaving krypton untouched, at any strength and for any amount of hydrogen lost. The heavier gas is the one that was taken, and mass alone cannot choose it.

The explanation now most favoured turns on a property xenon has and krypton lacks: xenon is easier to ionise than hydrogen. Its first ionisation energy is 12.1 electronvolts, below hydrogen’s 13.6, while krypton’s, at 14.0, is above it. In an upper atmosphere bathed in the young Sun’s ultraviolet, xenon would be ionised where hydrogen and krypton mostly were not. An ion is coupled to other ions by electric forces, far more strongly than a neutral atom is coupled by collisions, and an outflow of hydrogen ions along the open field lines over the Earth’s poles — the route by which a magnetised planet loses ions today — could drag ionised xenon with it while leaving neutral krypton behind. The mass sorting would then come from the drag of a wind on ions, with a crossover of its own, and only xenon would feel it.

The idea has a test the Earth has kept. Rocks that trapped samples of the atmosphere when they formed preserve its xenon, and the xenon in the oldest such samples, from more than three billion years ago, is less fractionated than today’s. The fractionation grows through the Archean and reaches the modern value by about two billion years ago. The xenon was not lost in the first hundred million years, in a burst driven by a young Sun; it was lost slowly, over more than a billion years, as hydrogen escaped steadily from an atmosphere that had a great deal of it — probably as methane, broken apart high in the atmosphere by sunlight. Hydrogen escaping for a billion years leaves behind the oxygen it was bound to, and the slow loss of xenon may be a record of the Earth’s surface being oxidised by the same escape.

Three planets, three records #

The neighbouring planets carry the same kind of record, and each records something different. Mars, whose weak gravity raised its crossover highest, has an atmosphere whose argon is strongly enriched in its heavy isotope, the signature of an upper atmosphere stripped from above — mostly by the solar wind, in Mars’s case, after its own magnetic field died. Venus has far more primordial neon and argon than the Earth relative to its mass, which says that whatever removed the Earth’s early inventory either did not act on Venus or acted on a larger stock. Each planet’s noble gases are a mixture of what it captured, what it was given later by impacts of comets and asteroids, and what escaped, and separating the three needs more than the elemental pattern.

The isotopes are where the separation is made. A hydrogen wind leaves one per-mass-unit signature on every gas below its crossover. Sputtering from above leaves a different one, larger for the lighter element. Late delivery by comets adds gas with its own primordial ratios. A spacecraft that measured all the noble-gas isotopes of the Venusian atmosphere precisely, as the probes planned for the coming decade are designed to do, would say which of these processes shaped that atmosphere and when.

What the arithmetic leaves out #

The crossover is computed for a single temperature, a single diffusion parameter for every heavy gas, a hydrogen mole fraction of one and an energy-limited flux that begins at the planet’s surface. A real flow starts higher, where the light is absorbed, is cooler or hotter than a thousand kelvin depending on how it is heated and cooled, and carries some heavy gas whose own presence slows it. The Rayleigh arithmetic behind the ramp and the isotopes holds the crossover fixed while the hydrogen drains, when in fact it falls as the star fades and as the hydrogen thins, so the true pattern is a superposition of ramps. Each of these moves the numbers by tens of per cent. None of them changes the two structural results: that the survivors form a ramp ending at the crossover, and that every gas below the crossover is sorted by the same amount per mass unit — which is what rules out a neutral wind for the Earth’s xenon.

Still open: how fast the Archean Earth lost its hydrogen #

If the xenon left as ions over the Archean, its fractionation is a running record of the hydrogen escape rate through the planet’s middle age — a record with dated points, from the trapped atmospheres of rocks of known age. Turning it into a rate needs the efficiency with which ionised xenon is dragged by an ion outflow, which depends on the magnetic field’s geometry, the composition of the upper atmosphere and the ultraviolet that ionised it, and is computed rather than measured. The measurement it would be checked against is the oxidation history of the surface, read independently from the chemistry of ancient sediments. Whether the two agree — whether the hydrogen that took the xenon is the hydrogen whose loss let oxygen accumulate — is a question about the Earth that the noble gases are unusually well placed to answer.