Two exponents a solar cycle cannot pull apart

Of all the ways a planet loses the gas it cannot keep, the loss of ions to the solar wind is the one a magnetic field is supposed to stop. Whether the Earth’s field protected its atmosphere when the Sun was young turned out to depend on what limits the leak from a planet that has no field. If the loss of ions from an unmagnetised planet is limited by how many ions the Sun’s ultraviolet makes in its upper atmosphere, then the young Sun — twenty times brighter in the ultraviolet — raised the loss twentyfold, and a field that kept the wind away saved a great deal. If the loss is limited by how much power the solar wind delivers to the planet, then the young Sun’s wind, perhaps a hundred times stronger, raised it a hundredfold, and the comparison with the Earth comes out differently. The two answers differ by the ratio of two numbers that are not known well, and the ratio decides the sign of the conclusion.

The obvious way to decide is to measure. Mars and Venus are unmagnetised planets that lose ions today, spacecraft at both have counted the losses for years, and the Sun’s ultraviolet and its wind both vary. A fit of the measured loss to both causes gives two exponents — how steeply the loss rises with each — and the exponents say which limit applies. The trouble is that the two causes do not vary independently.

Two causes that move together #

The Sun’s ultraviolet follows its eleven-year cycle, rising by about a factor of two from minimum to maximum. Its wind’s pressure varies far more from day to day, rising several times in a storm when a burst of fast or dense wind arrives — the same bursts whose arrival at the Earth a compass needle can weigh — and the storms are more frequent and stronger near solar maximum. Averaged over months, the two rise and fall together.

The model series is built to look like the measurements, and it has the property that makes separation hard: a correlation of about six-tenths between the logarithms of the two causes. When both rise together and the loss rises too, a fit cannot tell how much of the rise to give each. It can give the ultraviolet a large exponent and the wind a small one, or the reverse, and fit the data almost equally well, because along most of the data the two causes are moving in step.

The opening figure shows what that does to a fit. The two exponents are drawn as a region of the plane — every pair consistent with the data at two standard deviations — and the region is an ellipse whose shape depends on the correlation. With weak correlation it is round and small, and the fit could tell a loss set by the ultraviolet, at the corner (1, 0), from a loss set by the wind, at (0, 1). As the correlation rises, the ellipse stretches along a line running from upper left to lower right, the line on which raising one exponent and lowering the other leaves the fit unchanged. At a correlation of 0.9 the ultraviolet’s exponent is uncertain by more than half its value. That is the confusion the earlier comparison ran into: the spacecraft see the losses rise in storms and rise with the cycle, and the two observations cannot be separated cleanly.

The ellipse also shows what a single number would hide. A fit reported as two exponents with independent error bars would quote the ultraviolet’s as uncertain by a half and the wind’s by a quarter, and a reader multiplying those through to a young-Sun prediction would conclude that the prediction is uncertain by a factor of ten. The errors are not independent; they are strongly anticorrelated along the ellipse’s long axis, and the prediction depends on a combination along its short axis. Reporting the full covariance, rather than two marginal uncertainties, is the difference between a prediction that looks hopeless and one that is precise.

A prediction that survives the confusion #

The surprise comes when the fit is used for the question it was meant to answer. The young Sun was brighter in the ultraviolet and stronger in its wind at the same time. A prediction of the young Sun’s loss asks the fit to extrapolate along a direction in which both causes increase — and that is the direction along which the data are best constrained.

The ellipse is long in the direction in which one exponent rises and the other falls, and narrow in the perpendicular direction — the direction in which both rise together. A prediction for a Sun that is brighter in both outputs depends on the exponents only through a weighted sum of the two, and the weighted sum is exactly what a correlated data set measures well. So as the correlation grows from zero to 0.95 and the individual exponents dissolve, the prediction’s uncertainty barely moves: about 0.15 to 0.19 decades, a factor of one and a half in the predicted loss.

That is a reassuring result, and an unusual one. Collinearity in a regression is usually treated as fatal, because it makes the individual coefficients meaningless. Here the individual coefficients were never the goal. The question was how much more the young Sun stripped from Mars, and a cycle that varies both causes together answers it as well as one that varied them separately would — for a young Sun that differs from today’s in the same proportions the cycle samples.

Only along the diagonal #

The qualification is important. The prediction is precise only for a young Sun that lies along the direction in which the cycle’s variations lie.

The cycle samples a particular ratio of change: in decades, the wind’s pressure changes by about one and a half times as much as the ultraviolet does, averaged over the cycle. A young Sun whose wind and ultraviolet were both raised in about that proportion is predicted well. One whose wind rose much faster — a young Sun with a ferocious wind and a modestly brighter ultraviolet — or much slower lies off the diagonal, in the direction the fit cannot constrain, and its predicted loss inherits the full width of the ellipse.

The young Sun’s ratio is itself uncertain. Its ultraviolet is estimated from young Sun-like stars observed in X-rays and the ultraviolet, and is known to a factor of two or three at a given age. Its wind is estimated from the astrospheres of a handful of nearby young stars, whose hydrogen walls absorb a little of their Lyman-alpha light, and from models of how a wind’s mass loss scales with activity; those estimates span more than an order of magnitude. Whether the young Sun lay on the cycle’s diagonal is not something the cycle can say.

There is a further reason to doubt that it did. Young Sun-like stars flare far more often and more energetically than the Sun does now, and their largest flares are thought to be accompanied by coronal mass ejections that would raise the wind’s pressure at a planet by factors of hundreds for a day or two. A young Sun’s wind may therefore have differed from today’s not only in its average but in its distribution — a background a few times stronger, punctuated by extreme events — and an extrapolation of a mean pressure misses the events that might dominate the loss. A present-day cycle samples storms up to a few times the mean pressure; the young Sun’s extremes lie far beyond them, in the direction where the fit is least constrained.

How many storms it takes #

Separating the exponents individually is still possible in principle. It needs enough independent samples to beat the correlation down, and the samples that do it are the ones in which the two causes moved apart.

The uncertainty falls as the inverse square root of the number of samples, and the correlation inflates it by one over the square root of one minus its square. To pin the ultraviolet’s exponent to a tenth — enough to tell a loss set by the ultraviolet from one set by the wind — needs a few hundred independent samples at modest correlation and more than a thousand at high. A storm lasts a few days, and a solar cycle holds a few hundred, so one cycle of continuous measurement is enough only if the correlation can be kept modest.

It can, by choosing which samples to use. Storms at solar minimum, when the ultraviolet is low and steady and the wind’s pressure still jumps, vary the wind alone; quiet days at solar maximum vary the ultraviolet alone. A fit weighted toward those decorrelated samples narrows the ellipse much faster than one that uses every day equally. The spacecraft at Mars has now been measuring through more than a full solar cycle, including the minimum between cycles 24 and 25, and analyses that exploit those periods have begun to separate the two dependences. They report a loss that responds to both — which is what a planet between the two limits, or near the switch between them, would show.

The switch that exponents cannot see #

There is a deeper difficulty, which no number of samples removes. The two limits in the comparison of a shield and its absence are not two contributions that add. They are two ceilings, and the loss is set by whichever is lower. A planet whose loss is limited by the wind’s power today can become limited by the ultraviolet’s supply of ions if the wind’s power rises faster — and then the wind no longer matters at all.

The figure takes a planet sitting just on the wind-limited side of the switch. A cycle measured there finds a loss that follows the wind almost exactly — local exponents of about 0.1 for the ultraviolet and 0.9 for the wind — and the fit is perfect within the range sampled. Extrapolated to the young Sun, it predicts a loss eighty-six times today’s. But the power limit rises faster than the supply limit as the Sun gets younger, so somewhere along the way the supply limit becomes the lower ceiling, and the true loss stops following the wind and follows the ultraviolet instead. It reaches twenty-nine times today’s. Exponents that describe the present perfectly overestimate the past by a factor of three.

The point is general. A local power law fitted to data spanning a few tenths of a decade describes the function’s slope there and nothing about its curvature. An extrapolation over two decades — from today’s Sun to the young one — crosses whatever transitions lie in between. The funnel that a magnetic field opens at the poles of a magnetised planet is one such transition; the switch between supply and power limits at an unmagnetised one is another. Neither shows up as a misfit in the present-day data. The only sign of a switch in a local fit is the fit’s position: exponents that sum to about one and split between the two causes are what a planet near a switch would show, and also what a planet that genuinely depends on both would show. Which it is can be told only by data that reach further along the path.

What the factor of three is worth in air #

The difference between the two predictions in the last figure, a factor of three in the young Sun’s loss, is not a detail, and Mars shows why. Its present ion loss is a couple of kilograms a second, which over four and a half billion years at today’s rate would remove about ten times the mass of its present atmosphere — a sixtieth of a bar or so, interesting but not decisive. Most of the planet’s atmospheric history happened under a younger, more active Sun, so the integrated loss is dominated by the first billion years, when the ultraviolet was several to twenty times today’s and the wind perhaps a hundred.

Integrated over that history, an enhancement that follows the ultraviolet removes a few tenths of a bar; one that follows the wind at the exponents a present-day fit would suggest removes several times more, approaching the bar of carbon dioxide that the valley networks and lake beds of early Mars are often taken to require. The difference between a planet that lost a thick early atmosphere to space and one that must have lost it some other way — into carbonate rocks, or never had it — is of the same size as the extrapolation error the figure exhibits. Isotope ratios in the present atmosphere, which escape enriches in the heavy species, record the total lost and agree that most of it went to space; they do not record when, and so do not decide which limit applied.

The Earth’s case is the one the question began with, and it is subtler, because the Earth has a field and loses ions through it in a different way. But the comparison between a magnetised planet and an unmagnetised one rests on the unmagnetised planet’s young-Sun loss, and that is the number the switch makes uncertain by a factor of three.

Two planets, one Sun #

Mars and Venus together help with both difficulties. They sit at different distances from the Sun, so the same solar output reaches them with different ratios of ultraviolet flux to wind pressure — both fall as the inverse square of distance, but the wind’s interaction with each planet depends on the size of its ionosphere and its gravity, which differ greatly. A combined analysis gets two different directions of variation from one Sun, which is partly what the decorrelated samples supply, and its exponents can be compared with each planet’s own.

They also bracket the switch. Venus, with a dense ionosphere close to the Sun, is plausibly limited by the wind’s power; Mars, small and with a thin ionosphere far out, by the supply of ions or by its gravity. If the two planets show different exponents in the present-day data, the switch is real and lies between their conditions, and the young Sun’s loss at each can be predicted on the correct side of it. The comparison has been made qualitatively; doing it with matched methods and simultaneous solar monitoring is one of the motivations for keeping spacecraft at both planets through the same cycle.

What the regression leaves out #

The fits drawn here treat each sample as independent and the scatter as random. Real losses respond to the solar wind with delays of hours to days, as the planet’s ionosphere adjusts, and consecutive days are not independent; the effective number of samples is the number of independent episodes, not of days. Losses also depend on the planet’s season and on where its crustal magnetic fields point — at Mars, the southern highlands’ fields form small magnetospheres of their own that rotate with the planet — and those variations add scatter that does not average away over a single cycle.

And the measurement counts ions above a threshold energy leaving through the spacecraft’s field of view, scaled to a planet-wide rate. The scaling assumes the loss’s spatial pattern is the same in every condition, which storms violate. Each of these broadens the ellipse; none changes its orientation, and none changes the conclusion that the young-Sun prediction is better constrained than either exponent.

Still open: on which side of the switch the young planets sat #

The two-exponent question looked like the hard part and turned out to be the easy one. A solar cycle that varies the ultraviolet and the wind together cannot say which of them drives the loss today, but it can predict the young Sun’s loss to a factor of one and a half — provided the young Sun differed from today’s Sun in the proportions a cycle samples, and provided no switch between limits lies between them.

Both provisos are about the young Sun rather than about the planets, and both can be tested on stars. The young Sun’s ultraviolet and wind are being measured in its analogues — Sun-like stars of a few hundred million years — and if their wind-to-ultraviolet ratios lie along the present Sun’s cycle diagonal, the first proviso holds. The second needs a planet observed near the switch, which Mars and Venus together may already supply. If they do, what the young Earth kept and what its field did for it can be read from a regression on today’s data. If they do not, the answer depends on physics a solar cycle at Mars never reaches.