A Sun read through a shield that also changed

Sunspots have been counted for four centuries, and the butterfly they draw is the record of the Sun’s magnetic cycle for that long and no longer. Everything known about solar activity before 1610 — the grand minima, the long cycles of two centuries and more, the claim that the late twentieth century was unusually active — comes from a different record: the radioactive isotopes that cosmic rays make in the Earth’s atmosphere, preserved in tree rings and polar ice.

The chain is short. Galactic cosmic rays, mostly protons of a few GeV, strike nitrogen and oxygen in the upper atmosphere and break them apart; among the products are carbon-14, which is oxidised into carbon dioxide and taken up by trees, and beryllium-10, which attaches to aerosols and falls out in snow within a year or two. The more cosmic rays reach the atmosphere, the more of both is made. And the Sun decides how many cosmic rays reach the Earth’s orbit, because its magnetised wind fills the solar system and turns back the low-energy ones — more strongly when the Sun is active.

So an isotope record is a record of cosmic rays, and a record of cosmic rays is a record of the Sun. That is the inference. The difficulty is a second shield that the inference passes over: the Earth’s own magnetic field, which also turns cosmic rays away, which also varies, and which leaves the same mark in the isotopes.

The spectrum the Sun filters #

The cosmic rays arriving at the edge of the heliosphere have a spectrum that has not changed on any timescale that matters here: its shape is set by the sources that accelerate them and by how fast they leak out of the Galaxy, and both act over millions of years. Inside the heliosphere, the particles have to diffuse inward against the outflowing solar wind, scattering off the irregularities of a field wound into a spiral by the Sun’s rotation, and they lose energy as they do. The standard description reduces all of that to one number. In the force-field approximation the spectrum at the Earth’s orbit is the interstellar spectrum shifted down in energy by an amount $\Phi$, with a factor that conserves the density of particles in phase space:

$$J(T, \phi) = J_{\mathrm{LIS}}(T + \Phi)\,\frac{T(T + 2T_r)}{(T + \Phi)(T + \Phi + 2T_r)} ,$$

where $T$ is the kinetic energy, $T_r = 0.938$ GeV the proton’s rest energy, and for protons $\Phi$ is the modulation potential $\phi$ in GeV. That potential, a few hundred megavolts at solar minimum and about a thousand at maximum, is what isotope reconstructions report as “solar activity”. It is a single number standing in for the whole state of the Sun’s open magnetic field.

Modulation is strongest at low energy. At 1 GeV, solar maximum removes most of what solar minimum lets through; at 10 GeV the difference is less than half; by 100 GeV the spectra converge on the interstellar one. Far above that, the particles energetic enough to arrive from directions that begin to point back at their sources are as indifferent to the Sun’s field as to the Earth’s; the isotopes are made by the bottom of the spectrum, where both shields act.

The same figure carries the second shield. A charged particle approaching the Earth is deflected by the geomagnetic field, and only particles above a cutoff rigidity can reach the atmosphere at a given latitude. For a dipole the vertical cutoff is about $14.9\,(M/M_0)\cos^4\lambda$ GV, where $\lambda$ is the geomagnetic latitude and $M/M_0$ the dipole moment relative to today’s. At the equator it is about 14 GeV in energy, which admits only the part of the spectrum the Sun barely touches. Near the magnetic poles it falls below 0.1 GeV and admits nearly everything.

Where the solar signal enters #

Because the two shields act on different parts of the spectrum, they act on different parts of the atmosphere.

Near the equator the geomagnetic cutoff admits only high-energy particles, production is low, and it barely changes over a solar cycle. Poleward of about 60°, the cutoff falls below the energies where modulation acts, and production there both is highest and swings most with the Sun. The solar signal in the global production rate therefore comes overwhelmingly from high latitudes, from a band whose equatorward edge is set by the dipole.

The yield used in these figures — how many isotope atoms a proton of a given energy makes in the atmosphere below it — is schematic rather than taken from a detailed cascade calculation. It is chosen so that the global production rate behaves as the published calculations do: about twice today’s with no geomagnetic field at all, and about a third lower at solar maximum than at minimum. The argument below depends on those two ratios and on the shape of the spectrum, not on the details of the yield.

A stronger dipole pushes the edge of the high-cutoff region poleward, which shrinks the area where the Sun’s modulation reaches the atmosphere and lowers production everywhere. A weaker dipole does the opposite. The isotope archives receive the sum, with no label on which shield did what.

One production rate, many Suns #

Everything an isotope record measures is the global production rate, averaged over a year or longer. That rate depends on both shields at once.

Each contour in the plane is a production rate, and every point along it is a pairing of Sun and Earth that makes exactly that rate. An isotope measurement says which contour the world was on, and nothing about where along it. A dipole 28 per cent stronger than today’s and a Sun with a modulation potential of 449 megavolts is indistinguishable from today’s dipole and a Sun at 650 — a difference of 200 megavolts, comparable to the difference between an average solar minimum and an average maximum.

That would be harmless if the dipole were constant. It is not. The Earth’s dipole moment is currently falling by about five per cent a century, and has been for as long as it has been measured directly. Reconstructions from the magnetisation of lava flows, fired clay and lake sediments find it about a third stronger than today roughly three thousand years ago and markedly weaker seven or eight thousand years ago. The band across the plane — between about 6 and 11 in units of $10^{22}$ ampere square metres — is where the dipole has been since the end of the last ice age, and it spans several contours.

So a solar reconstruction from isotopes is always a pair of inversions. The isotope concentration, corrected for its transport into the archive, gives a production rate. The production rate, at a dipole moment taken from a separate paleomagnetic reconstruction, gives a modulation potential. The second step is where the Sun is separated from the Earth, and the Sun inherits every error the paleomagnetic record makes.

The error a geomagnetic error makes #

How large that inheritance is depends on how well the dipole is known and on how active the Sun was.

A dipole overestimated by 10 per cent makes the reconstruction attribute too much of the shielding to the Earth and too little to the Sun, and the inferred modulation potential comes out low by 60 to 100 megavolts. The error grows with the potential, because a heavily modulated spectrum has lost most of its low-energy particles already and changes less for each further megavolt, so a given change in production corresponds to a larger change in $\phi$.

The paleomagnetic reconstructions themselves differ. Independent compilations of the Holocene dipole, built from different sets of lavas, archaeological artefacts and sediment cores and with different ways of weighting them, disagree by ten to twenty per cent over parts of the last ten thousand years, and the uncertainty is larger where the data are sparse — before about seven thousand years ago, and in the Southern Hemisphere. A 20 per cent disagreement in the dipole is worth 100 to 200 megavolts in the reconstructed Sun: the whole range of a solar cycle.

These errors are not random from year to year. The dipole changes over millennia, so an error in it is a slow, smooth error, and it lands in the reconstructed Sun as a slow, smooth trend. The rapid features of the solar record — the eleven-year cycle where the archive resolves it, the two-century cycles, the sharp onset of a grand minimum — survive a dipole error intact, because the dipole could not have changed that fast. The slow features do not.

Ten thousand years with the shield held still #

The clearest way to see what this does is to make a record whose answer is known.

The true Sun in the figure has no long-term trend at all: its potential oscillates around 620 megavolts with cycles of 208 and 88 years, the periods found in real isotope records, and drops into five grand minima of a few decades each. The dipole follows a history of the shape the paleomagnetic reconstructions find, rising to about a third above today’s three thousand years ago. From the production that results, the potential is reconstructed with the dipole held at today’s value — the simplest possible correction, and the wrong one.

The reconstruction keeps everything fast. Every grand minimum is there, at the right date and very nearly the right depth, and the centennial cycles are there with the right periods. But the long-term level has acquired a millennial wave of its own. Around two and a half thousand years ago, where the real dipole was strongest, its extra shielding is read as an active Sun, and the smoothed reconstruction runs more than 250 megavolts too high; around seven and a half thousand years ago, where the dipole was weak, the Sun appears quiet by almost as much. A reader of the reconstruction would conclude that the Sun was unusually active in the first millennium BCE and unusually quiet in the early Holocene. Neither is true of the Sun in the figure.

That is not a hypothetical worry. A reconstruction of sunspot numbers from carbon-14 published in 2004 concluded that the Sun had been as active as in the second half of the twentieth century for only about a tenth of the past eleven thousand years. Later reconstructions using different paleomagnetic models, and combining carbon-14 with beryllium-10, found the late twentieth century high but less exceptional, and much of the difference between the conclusions was traced to the geomagnetic correction. The question of whether the modern Sun is unusual is, to a degree that is not always stated, a question about the history of the Earth’s dipole.

Two archives that cannot check this #

The natural defence is to compare two isotopes, carbon-14 in tree rings and beryllium-10 in polar ice, which have entirely different routes from the atmosphere to the archive. Where they agree, the agreement is evidence that both are recording production rather than something about their transport. They do agree, broadly, over the Holocene. But they do not test the same thing at every period.

Carbon-14 enters the global carbon cycle. A few years after it is made it has been mixed through the atmosphere and has begun exchanging with the ocean, which holds some sixty times as much carbon. A production change that lasts a decade is spread through that reservoir before it has built up, so the eleven-year cycle — a swing in production of about a fifth either way — appears in tree rings as a few parts per thousand. Changes lasting centuries are diluted less, but still by a factor of twenty. Reading production from carbon-14 therefore means running the carbon cycle backwards, which amplifies noise at short periods.

Beryllium-10 has no such reservoir. It is removed from the atmosphere within a year or two, so the ice records production nearly as it happened, cycle included. But what it records is local deposition, which depends on the year’s snowfall and atmospheric circulation, and how much of the beryllium falling on Greenland or Antarctica was made in polar air — where, as the latitude figure shows, the solar signal is strongest — and how much in the global stratosphere is itself debated. The two records disagree in detail at exactly the level where those differences matter.

None of this touches the geomagnetic term. The dipole acts on production, before either isotope begins its journey to an archive, and both records inherit the same factor. Carbon-14 and beryllium-10 agreeing on a millennial trend says that the production rate had that trend. It says nothing about whether the trend was the Sun’s or the Earth’s.

What the record says regardless #

The degeneracy limits what the isotopes can say about slow change. It leaves untouched almost everything they say about fast change, and that is most of what they are known for.

The grand minima are the clearest case. The Maunder minimum of the late seventeenth century, when sunspots almost vanished, appears in both isotopes as a rise in production lasting several decades, and the isotope records find similar episodes throughout the Holocene — the Spörer minimum two centuries earlier, the Wolf minimum before that, and some two dozen more over the last eleven thousand years. Each lasts fifty to a hundred years, far too short for the dipole to have produced it, so each is a solar event whatever the geomagnetic correction. Counted together they say the Sun spends something like a sixth of its time in such states, which is a statement about the statistics of a cycle with one step of memory that the four-century telescopic record could not make.

The same holds for the long cycles. A periodicity of about 208 years in carbon-14, named after de Vries and Suess, and the Gleissberg cycle of about 88 years both appear in records from both hemispheres and in both isotopes, with periods that a dipole changing over millennia cannot fake. So do the single-year spikes in production found in annually resolved tree rings, which no change in either shield could produce at all. What remains in doubt is the level around which all of these happen — whether the Sun’s average activity in one millennium differed from another — and that is the one quantity the dipole correction controls.

The calibration point is the present. Since the 1950s, ground-based neutron monitors have recorded the cosmic-ray flux at the Earth’s surface directly, at stations spread across cutoff rigidities, and since the 1990s the spectrum itself has been measured above the atmosphere. Over that interval the modulation potential is known from the cosmic rays rather than from isotopes, and the dipole from magnetometers, and the production model can be checked against both. The check is good. It is also made at a single dipole moment, the modern one, and it says nothing about how well the model handles the dipole of three thousand years ago.

What does break the degeneracy #

The dipole history has to be measured independently, and the measurements are the paleomagnetic ones: the direction and strength of the field frozen into rocks and fired clay as they cooled. Their quality has improved steadily, but they are sparse, unevenly spread over the globe, and dated with errors of decades to centuries.

The isotopes contain one piece of information that helps. The geomagnetic field shields equatorial latitudes and leaves polar ones open, so a change in the dipole changes production more at middle latitudes than at the poles, while a change in the Sun changes it most at the poles. A beryllium-10 record from polar ice, if it reflects mostly polar production, weights the two shields differently from carbon-14, which is mixed globally. In principle the ratio of the two separates them. In practice the transport uncertainties of beryllium are large enough that this has given a check on the geomagnetic correction rather than a replacement for it.

The other help is the solar cycle itself. Where an archive resolves individual eleven-year cycles, as annually resolved carbon-14 now does for several long stretches of the Holocene, the cycle’s amplitude in production depends on how wide the high-latitude window is — on the dipole — while its existence and period do not. The amplitude of the cycle thus carries some information about the dipole at that time, independent of the paleomagnetic data.

What the figures leave out #

The dipole is treated as a centred, axial dipole, which today’s field is not: the dipole axis is tilted by about ten degrees, and non-dipole components cause regional anomalies, the largest being the weak spot over the South Atlantic through which trapped particles drain. Averaged over centuries the tilt and the anomalies largely cancel for a global production rate, which is why reconstructions use a centred dipole, but they matter for any record that samples one region.

The cutoff ignores the atmosphere’s own response to a geomagnetic storm, when the ring current weakens the field at the surface and lets particles reach lower latitudes for a day or two. The modulation of the cosmic-ray spectrum is described by one number, which is known to be an approximation: the spectrum is modulated differently during alternate cycles, as the polarity of the Sun’s polar field reverses every eleven years, and the force-field model has no way to represent that. And the carbon cycle in the figures is two boxes, where a real one has the deep ocean, the biosphere and the soils, each with its own timescale.

None of these changes the central point. As long as a single production rate is the observable and two shields act on it, the solar record is only as good as the geomagnetic record, and the errors are slow.

Still open: whether the modern Sun is unusual #

The second half of the twentieth century had the highest sunspot numbers of the telescopic era. Whether it was also high on a ten-thousand-year scale — a grand maximum of a kind the Sun reaches rarely — is a question the isotopes were expected to settle and have so far bounded rather than answered. The answer depends on the dipole moment over the Holocene to about ten per cent, which the paleomagnetic reconstructions do not yet agree on, and on the transport of beryllium to the poles, which the climate models do not yet agree on.

The polar field that predicts each cycle is measured directly only since the 1970s. Every claim about how often the Sun enters a grand minimum — which depends on where the threshold for one sits against the slowly varying level — about whether there is a cycle longer than two thousand years in solar activity, and about whether the present Sun is near the top of its range, passes through the dipole correction first. Annually resolved isotope records across the Holocene, now being assembled, will resolve individual cycles over many more centuries, and the amplitudes of those cycles may provide a dipole history independent of rocks. Whether the reconstructed Sun’s slow trends survive that test is not yet known.