A shield that depends on what limits the leak

The planet with a magnetic field loses its atmosphere to space about as fast as the two without one. The Earth’s ions stream out of its polar regions along open field lines at around a kilogram a second, Venus and Mars lose theirs to the field they borrow from the wind at rates within a factor of a few of that, and the comparison, scaled to planets of the same size, leaves the magnetised planet losing slightly more. The textbook picture of a magnetic field as a shield does not survive the measurements.

What the measurements describe is today’s Sun. The Sun at a few hundred million years old rotated several times faster, its corona was hotter and its activity higher, and both its wind and its ultraviolet light were tens of times more intense. The question the Earth’s atmosphere actually depends on is the comparison under that Sun, integrated over its history — and the essay on the borrowed field ended by asking it. This essay works it out in the simplest terms that keep the physics, and finds that the answer turns on a single property of escape that the present cannot measure.

The young Sun, in two outputs #

Two outputs of the young Sun matter, and they are known in different ways and to different precisions. The extreme ultraviolet — the light short of 912 ångströms that ionises atmospheres and heats their upper layers — is inferred from Sun-like stars of known ages, observed in X-rays and the far ultraviolet, and it rises into the past as the age to roughly the minus five-fourths. At a few hundred million years old the Sun emitted tens of times its present ultraviolet; at a hundred million, a hundred times. The history is tied to the Sun’s spin, which set its activity, and the young Sun’s spin is itself uncertain by a factor that moves the ultraviolet by several.

The wind is harder. Stellar winds of Sun-like stars are too thin to see directly; their mass-loss rates are inferred from the absorption that the hydrogen wall at the edge of each star’s astrosphere adds to its Lyman-α line, for a handful of nearby stars. Those measurements suggest a mass-loss rate rising steeply with activity, as the age to about the minus two and a third, up to an activity level beyond which the most active stars seem to drive weaker winds rather than stronger ones. The figure holds the wind at its value at 0.7 billion years before that, eighty times today’s pressure, which is a guess at where the turnover lies. The wind’s history is the least certain input in this essay, and it matters most.

A magnetosphere pressed in and opened up #

A stronger wind does two things to a magnetosphere, and they pull in opposite directions.

It presses the magnetosphere inward. The magnetopause stands where the dipole’s magnetic pressure balances the wind’s dynamic pressure, and since the dipole’s pressure falls as the sixth power of distance, the boundary hardly moves: a wind eighty times stronger brings it in by the sixth root of eighty, from nine and a half Earth radii to four and a half. The magnetosphere’s cross-section to the wind — the area of solar wind it deflects — shrinks by a factor of four.

And it opens the polar caps. The field lines that close inside the magnetosphere are those whose tops lie inside the magnetopause; those that would reach beyond it are open, connected to the wind, and their feet on the ground form the polar caps, whose edge is where the auroral oval sits. A smaller magnetosphere leaves more field lines open: the caps widen from nineteen degrees of colatitude to twenty-eight, and the area of open field — the outlet through which ions flow out of the ionosphere into space — doubles.

So under the young Sun the Earth’s field presented a smaller target to the wind and a larger door to space. Which of the two matters more depends on what limits the escape.

A field is a large net #

The surprising feature of the present-day comparison is explained by area. A magnetosphere is a target seventy-five times larger in cross-section than the planet inside it, so it intercepts seventy-five times as much of the wind’s power. Most of that power is deflected around the magnetopause, but a fraction — a few per cent, through reconnection at the dayside boundary — enters, drives the magnetosphere’s convection, heats the polar ionosphere and lifts ions out along the open field lines. A few per cent of seventy-five times as much is comparable to all of the power an unmagnetised planet’s small obstacle intercepts. That is why the measured losses are comparable today.

Under the young Sun the factor shrinks, from seventy-five to eighteen at 0.7 billion years, because the wind’s pressure compresses the magnetosphere while the bare planet’s obstacle — its ionosphere, held off by the pressure of its own gas just above the surface — stays the same size. If both planets’ losses follow the power they intercept, the field’s disadvantage shrinks as the wind strengthens.

How an ion leaves a magnetised planet #

The scalings are compressions of real processes, and it is worth seeing what each compresses. On a magnetised planet, ions leave from the high-latitude ionosphere in a sequence of steps. The ultraviolet ionises the upper atmosphere and keeps the ionosphere supplied; the wind’s energy, entering through reconnection at the magnetopause and released in the tail in the bursts that time substorms, drives currents through the polar ionosphere and heats it, by friction between the ions and the neutral gas; the heated ions are pushed upward, accelerated by waves and electric fields along the field lines, and those on open field lines that reach escape speed are carried into the tail and lost.

Each step has its own dependence. The ion supply follows the ultraviolet; the heating follows the wind’s power; the area through which the ions can leave follows the polar caps’ size; and the fraction that escape rather than return depends on how far down the tail they are carried before the field lines close. A loss proportional to the intercepted wind power is the right scaling when the heating is the bottleneck — when there are always more ions available than the wind’s energy can lift. Observed outflow does rise steeply with the energy entering the polar ionosphere, which is why the wind-power scaling is the usual choice for the magnetised side.

Two ways an unprotected planet can lose its air #

The hero figure’s two curves are two assumptions about what limits the escape of an unmagnetised planet, and both are physically reasonable.

In the first, escape is limited by energy: the bare planet loses ions in proportion to the solar-wind power its obstacle intercepts, just as the magnetised one does through its larger, compressible obstacle. Then the ratio of the two losses scales as the ratio of the intercepted powers, which falls into the past as the wind’s pressure to the minus one-third. Today’s ratio of two becomes 0.46 at 0.7 billion years: the young Earth with its field lost half what it would have lost without it, and on balance the field protected it before about two billion years.

In the second, escape is limited by supply. The ions an unmagnetised planet loses are the ones the Sun’s ultraviolet creates above its exobase, picked up by the wind’s electric field as soon as they are made; the wind can carry away whatever is ionised, and the loss follows the ultraviolet rather than the wind. The ultraviolet rises into the past more slowly than the wind’s pressure does, so the magnetised planet’s losses, driven by the wind, outrun the bare planet’s: the ratio climbs to 3.7 at 0.7 billion years, and the field cost the young Earth nearly four times what it would have lost without one.

These are the two limiting regimes of any escape process — one where the energy supply sets the rate and one where the reservoir of escaping particles does — and real escape sits somewhere between them, nearer one or the other depending on the species, the planet and the epoch. The present-day measurements cannot tell which, because today the two scalings give the same ratio by construction: both are normalised to it.

How an ion leaves a bare planet #

On a planet without a field the steps are fewer. The wind meets the ionosphere almost at the top of the atmosphere, and the wind’s magnetic field drapes around the planet into an induced magnetosphere, with a tail behind it. Neutral atoms of the extended upper atmosphere, reaching above that boundary, are ionised by the ultraviolet, by collisions with the wind’s electrons or by exchanging charge with its protons, and once ionised they are caught in the wind’s electric field and swept away — picked up. Ions from the ionosphere itself are dragged into the tail and escape down it.

The pickup step is the one that decides between the scalings. If every atom ionised above the boundary is picked up and lost, the loss is the production rate of ions there, and that follows the ultraviolet: the supply-limited case. If the wind cannot remove ions as fast as they are made — if its electric field, or its momentum, limits how much mass it can accelerate — the loss follows the wind: the energy-limited case. At Mars, measurements through a solar cycle show the escape rising with the ultraviolet by a factor of several and responding to the wind’s pressure during storms, which suggests both limits matter at different times, and the balance under a wind eighty times stronger is not a measurement anyone can make.

Counted over the Sun’s whole history #

What an atmosphere keeps is the integral, not the rate, and most of the integral is early. In both scalings more than half of everything lost since the Sun was a hundred million years old was lost in the first billion years, when both the wind and the ultraviolet were strongest. Counted from the start, the wind-power scaling gives a magnetised Earth that lost 0.61 of what a bare Earth would have: the field preserved about forty per cent of what would otherwise have gone. The ultraviolet scaling gives 1.9, close to today’s ratio: the field cost more at a billion years, where the wind’s pressure had risen most against the ultraviolet, and less in the first few hundred million years, where the wind had stopped rising and the ultraviolet had not, and the two epochs nearly cancel.

So the integral is less sensitive to the scaling than the instantaneous ratio — the field’s net effect over the whole history lies between protecting forty per cent and costing ninety — but it is more sensitive to the input that is least known. Both totals depend almost entirely on the Sun’s first billion years, on the wind’s saturation and on the ultraviolet’s, and moving the wind’s turnover earlier or later by a few hundred million years changes both totals substantially.

A stronger field is a larger net #

In the wind-power scaling, the strength of the field enters in an unexpected direction. A stronger dipole holds the magnetopause further out — as the cube root of its strength — so it presents a larger target and intercepts more of the wind: the magnetised loss grows as the two-thirds power of the field. A planet with a tenth of the Earth’s field, under today’s Sun, loses less than half of what it would lose bare; under the young Sun, a tenth. In this scaling the best shield is a weak one: enough field to hold the wind off the atmosphere, not so much as to sweep up power from a large area around it.

The Earth’s own field was weaker at times in its early history, by a factor of two or so in the record preserved in ancient crystals, and the inner core, which now helps drive the dynamo, did not form until much later. A weaker early field, in this scaling, would have protected the young atmosphere better than today’s would have. In the ultraviolet scaling the field’s strength matters less, because the bare planet’s loss does not depend on it and the magnetised planet’s is driven by the wind either way.

Why the measurements today cannot decide #

The two scalings make different predictions about present-day escape, and in principle those differences are measurable. The wind-power scaling predicts that the bare planets’ losses should rise in proportion to the wind’s pressure when a storm hits, and the ultraviolet scaling that they should follow the solar cycle’s ultraviolet and hardly notice a storm. The spacecraft at Mars and Venus have seen losses rise during storms and with the solar cycle, by factors of a few in each case, and the separation of the two effects is confounded because storms and ultraviolet both peak near solar maximum.

The comparison is also among three planets that differ in more than their fields. Venus has a massive carbon-dioxide atmosphere and a slow rotation; Mars is small, cold and has crustal magnetic fields in its southern highlands that form mini-magnetospheres of their own. Scaling their measured losses to the Earth’s size and gravity introduces assumptions as large as the effects being compared, and the “factor of two” taken here as today’s ratio is uncertain by a factor of two itself.

Planets of other stars #

The same question decides something larger than the Earth’s history. Most of the rocky planets in the habitable zones of other stars orbit M dwarfs, close in, where the stellar wind’s pressure is hundreds to thousands of times the Sun’s at the Earth and the ultraviolet, for the first billion years of a red dwarf’s active youth, is tens of times higher again. A planet there with the Earth’s field would have a magnetopause at one or two planetary radii, barely above its atmosphere, and polar caps reaching the mid-latitudes.

In the wind-power scaling, such a field would be compressed to a target hardly larger than the planet, and it would protect it from a wind that would otherwise strip it. In the ultraviolet scaling, the wind is not the limit, the field’s compression changes little, and its enlarged polar caps would open the atmosphere to the sky. The habitability of the commonest planets in the galaxy — whether they kept atmospheres at all — depends on which regime their escape is in, and the planets of the Solar System, measured through a solar cycle, are the only laboratory in which that can be found out.

Whether such planets have fields at all is not known. A planet’s field has been detected outside the Solar System only indirectly, if at all — candidate radio bursts from star–planet interaction, and the shapes of a few hot Jupiters’ escaping atmospheres that may require one. For rocky planets there is nothing. A dynamo needs a liquid, convecting, conducting core, which an Earth-sized planet probably has for billions of years, and a planet locked to its star rotates once per orbit, slowly enough to weaken the dynamo but not obviously to stop it. Until a field is measured on a temperate rocky planet, the question of whether it helped or hurt is asked about a hypothetical object, and the answer still turns on the same distinction between the wind’s power and the ultraviolet’s supply.

What the scalings leave out #

The magnetised loss is taken to follow the intercepted power with a fixed coupling efficiency. Reconnection at the dayside magnetopause depends on the orientation of the wind’s own field, and its efficiency in a compressed magnetosphere under a strong wind may differ from today’s. Ion outflow from the polar caps also depends on the ultraviolet, which supplies the ionospheric ions the wind’s energy lifts, so the magnetised loss is partly supply-limited too; a mixed scaling sits between the two curves.

Neither scaling includes thermal escape — Jeans escape and hydrodynamic outflow — which a field cannot affect at all, and which under the young Sun’s ultraviolet may have dominated the loss of hydrogen for the first few hundred million years. The isotopic record of the Earth’s atmosphere, the xenon whose isotopes were sorted by a hydrogen wind, says that such an outflow happened, and its rate depended on the ultraviolet alone.

And the comparison is between ions. Neutral atoms escape by other routes — sputtering, photochemical reactions that give atoms escape speed, charge exchange with the wind — some of which the field affects and some of which it does not. The total loss of an atmosphere is a sum of channels, and the field’s effect on the total depends on how the channels were shared, which is itself a function of the epoch.

Still open: what limited the escape when it mattered #

The question whether a magnetic field protects an atmosphere has been argued as if it had a single answer. It has two, and the young Sun separates them: under its strong wind the field either protected the Earth substantially or cost it substantially more than it does today, depending on whether the escape of ions from an unmagnetised planet is limited by the wind’s power or by the supply of ions the ultraviolet makes.

That is a question about the present that can be asked of the present. Measuring how Mars’s and Venus’s ion losses respond separately to the wind’s pressure and to the ultraviolet — through a solar cycle, across storms of different strengths, with the ultraviolet monitored at the same time — would place them on one curve or the other, or between. Applied to the young Sun’s history, with a better-determined wind from more astrospheres of young stars, it would say whether the Earth’s field, a few hundred million years into the Sun’s spin-down, was keeping its atmosphere or helping to give it away.