A magnetic clock read off a butterfly

A sunspot is a place where the field is strong enough to stop the convection that carries a solar-type star’s outer heat, so the gas there cools by a thousand kelvin and looks dark against the surroundings. That is a fact about one spot, and by itself it goes nowhere.

The population of spots is different. Spots do not appear at random latitudes or at random times, and the pattern they make is a clock — not a very good clock, and the only direct record of the Sun’s magnetic behaviour that extends back four centuries.

Spörer’s law #

Within a cycle, the mean latitude at which spots emerge falls. The first spots of a new cycle appear near thirty degrees north and south; the last appear within a few degrees of the equator; and the drift is monotone.

That is Spörer’s law, and its most useful feature is that it makes the cycles separable. The last low-latitude spots of one cycle overlap in time with the first high-latitude spots of the next by a year or two, so a plot of spot number against date shows a minimum that is not a zero, and a plot of latitude against date shows two wings that clearly belong to different cycles.

The equatorward march is the single most important constraint on any theory of the mechanism, and it is not obviously what a dynamo should do. Something is propagating towards the equator over eleven years, and whatever it is has to move at about a metre a second at the base of the convection zone.

Hale’s polarity law, and the period that doubles #

Spots come in pairs, roughly east–west, and the two members have opposite magnetic polarity. Which one leads — the one at the more westerly longitude, in the direction of rotation — is not random.

Within a cycle, every leading spot in the northern hemisphere has the same polarity, and every leading spot in the southern hemisphere has the opposite one. At the start of the next cycle both reverse.

So a magnetic description of the Sun repeats every twenty-two years and a description in terms of spot counts repeats every eleven. The famous number is half the real one, and it is half because the instrument that discovered it — an eye, and later a photographic plate — records a spot’s existence and throws away its sign.

The measurement that established this is the subject of the rung on reading a field off a line that will not split: Hale detected Zeeman splitting in sunspots in 1908, where the field is kilogauss and the splitting is a substantial fraction of the line width, and by 1919 had enough cycles of it to state the polarity rule.

There is a third component that does not appear on the diagram at all. The Sun has a weak large-scale field at its poles, a few gauss, and that field reverses at each activity maximum rather than at each minimum — a quarter of a cycle out of phase with everything the spots do. Its strength at minimum is currently the best single predictor of the size of the following maximum, which is a statement about the mechanism and not merely a correlation.

What the pattern is evidence for #

The standard account has three ingredients, and the diagram constrains each.

Differential rotation winds a poloidal field into a toroidal one. The Sun’s equator rotates in about 25 days and its poles in about 34 — a differential rotation of the kind that winds up a galaxy’s pattern and therefore cannot be what makes one, so a field line frozen into the gas is stretched around the star, and after a few years a weak north–south field has become a strong east–west one. Helioseismology has measured the rotation profile all the way down, and the shear is concentrated in a thin layer at the base of the convection zone. Buoyancy brings the toroidal field back up. A tube of strong field is lighter than its surroundings, rises, and breaks through the surface as a pair of spots with the polarity the toroidal field had — which is why the leading polarity is a hemispheric constant and why it reverses when the underlying field does.

And something regenerates the poloidal field from the toroidal one, with the sign flipped. What that something is remains the open question. The leading candidate is the systematic tilt of spot pairs — leading spots sit slightly closer to the equator than following ones — which lets the following polarity be carried poleward by the surface flow and cancel the existing polar field, then rebuild it reversed.

What the record actually is #

Everything above rests on a series of numbers that deserves to be described honestly, because it is not an instrumental record.

The sunspot number is defined as $R = k(10g + n)$, with $g$ the number of spot groups, $n$ the number of individual spots, and $k$ a factor accounting for the observer and the telescope. It was invented by Wolf in 1848 and extended backwards by him to 1700, and by later workers to 1610, using whatever drawings and notes could be found.

Three features of it matter.

It is not a physical quantity. Ten times the group count plus the spot count is a construction, chosen because it correlated with what could be seen. It has no units and does not measure flux, area, or field.

The observer factor is the whole problem. Two observers looking at the same Sun on the same day report different numbers, and the ratio between them is not constant with the level of activity. Reconstructing a four-century series means chaining observers, and the chain has been recalibrated twice — most recently in 2015, which changed the amplitudes of eighteenth-century cycles by tens of per cent.

And the sparsest part of the record is the part that matters most. Between about 1645 and 1715 the number is near zero: the Maunder minimum, seventy years in which almost no spots were reported. That is either a genuine near-shutdown of the cycle or a shortage of people looking, and the difference is exactly the kind of thing an eyeball index cannot settle.

The one number the diagram gives that nothing else does #

There is a quantity in the butterfly pattern that no single measurement contains, and it is the reason the diagram is drawn rather than tabulated: the two hemispheres are not synchronised.

The northern and southern wings of a given cycle reach their maxima at different times, sometimes by more than a year, and the asymmetry alternates in sign without an obvious rule. A hemispheric spot count shows it immediately; a whole-disc count averages it away and produces a smooth cycle that conceals two rougher ones.

That matters because the dynamo is usually modelled as a single global oscillator, and a global oscillator has one phase. Two hemispheres that lead and lag each other by a substantial fraction of a year are two coupled oscillators with a weak coupling, and how weak the coupling is turns out to be one of the few numbers a model can be tested against.

What is actually predictable #

Very little, and the honest statement of it is short.

The length of a cycle varies between about nine and fourteen years, and the variation is not random noise on a fixed period: long cycles tend to be weak, and a cycle’s rise time is anticorrelated with its amplitude — steep rise, big cycle — which is the Waldmeier effect and which is at least partly a definitional artefact of how the rise is measured.

The amplitude of the next cycle is predicted best by the polar field at the preceding minimum, which is a physical precursor rather than a curve fit, and which has one cycle of lead time and no more.

Anything beyond one cycle ahead has no demonstrated skill. Dozens of predictions are published before each maximum and their spread is comparable to the range of all recorded cycles, which is the polite way of saying that the ensemble contains no information.

The events the tree rings found #

The carbon-14 record was introduced above as a way of checking whether the Maunder minimum was real. It turned out to contain something nobody was looking for, and the discovery is a good illustration of what a well-dated proxy can do that a direct record cannot.

Measuring carbon-14 in single annual tree rings, rather than in the decadal averages the earlier work used, revealed a sharp spike: a rise of about a per cent in a single year, an order of magnitude faster than anything the solar cycle produces, in a ring dated to the eighth century. A second spike of similar size was found three centuries earlier, and several more have since been identified in rings and in ice cores, which record the same event through a different cosmogenic isotope.

Two things make these events interesting. The first is that they are global and simultaneous: the same spike appears in trees from both hemispheres, in the same ring, which rules out a local contamination and confirms that the atmosphere mixes within a year. The second is the size. Producing that much carbon-14 in a year requires a particle flux far above anything measured by any instrument since instruments existed — an event perhaps an order of magnitude beyond the largest solar particle event of the space age.

The leading explanation is an extreme solar eruption, which places the Sun’s ceiling well above what four centuries of watching it had suggested. The competing explanations — a nearby supernova, a magnetar giant flare — are disfavoured because the spikes are too fast and too frequent.

That matters for the reading of the sunspot record. Four hundred years of counting spots samples about thirty-six cycles, and the largest event in that window is not the largest the Sun can produce. The isotope record samples ten thousand years and finds several events with no counterpart in the telescopic era at all — so the amplitude distribution the direct record supports is truncated at the top by nothing more than how long anybody has been looking.

A record that measures the wrong quantity at the right cadence has said something the right quantity could not, which is the same lesson as the polarity rule, arrived at from the other direction.

Two of the three laws the diagram carries are worth redrawing at settings the historical record itself covers, because both of them are claims about a sample rather than about any single cycle.

The Sun as a variable star #

There is a photometric consequence, and it is smaller than almost anyone expects.

Spots are dark and remove light. Faculae — bright magnetic regions, much more extensive and much less conspicuous — add it. Over a cycle the two nearly cancel, and what is left is a total solar irradiance that varies by about 0.1 per cent from minimum to maximum, with the faculae winning so that the Sun is very slightly brighter when it is spottiest — an amplitude far below that of a star that tells its distance by blinking.

A tenth of a per cent is a hundredth of what would be needed to explain any climatic swing of interest, which is worth saying plainly because the Maunder minimum coincided with a cold period in Europe and the coincidence has been asked to carry more than it can. What the cycle does modulate strongly is the ultraviolet, by several per cent, and the solar wind, by a great deal — and those affect the upper atmosphere and the cosmic-ray flux rather than the surface energy budget.

For a distant observer the same star would be an unremarkable one. A 0.1 per cent variation at a period of eleven years is at the edge of what photometry can detect on another star, which is why stellar activity cycles are monitored in chromospheric emission lines — where the contrast is tens of per cent — rather than in brightness, where the error bar comes from counting and a tenth of a per cent is hard-won.

How large a cycle can be, measured on other stars #

The Sun’s own record is short, and the way to lengthen it is to observe many stars at once and read the distribution rather than the sequence. That has been done photometrically, and the result is contested in an instructive way.

Surveys monitoring hundreds of thousands of solar-type stars for years find that a small fraction of them produce flares releasing $10^{33}$ to $10^{36}$ ergs — up to ten thousand times the largest flare ever observed on the Sun. The stars producing them are, in the main, rapid rotators and therefore young and magnetically active, which is unsurprising. The contested part is that a handful appear to be slow rotators resembling the Sun.

If that is right, the Sun is capable of an event far outside its observed record, and the isotope spikes of the previous section are the local evidence for it. If it is wrong — if the apparent slow rotators are unresolved binaries, or misclassified subgiants, or stars with a close companion supplying the energy — then the Sun’s ceiling stays where four centuries of watching put it.

Resolving it is a matter of characterising the host stars rather than of observing more flares, and it has been going the sceptical way: follow-up spectroscopy and astrometry have removed a substantial fraction of the original candidates from the sample, mostly as binaries or as stars rotating faster than the photometry implied. The rate for genuinely Sun-like stars keeps falling as the samples are cleaned, and it has not fallen to zero.

The two lines of evidence are worth holding together because they are independent and they point the same way. One is a chemical record of this star; the other is a photometric census of stars like it. Neither is decisive alone, and both say the observed cycle is a sample from a distribution with a longer tail than the sample shows.

A four-century record of one star is a small sample of a process whose interesting behaviour is rare, and the two ways around that — a longer record by proxy, and a larger sample by looking elsewhere — are the same two moves this collection makes whenever a single object is being asked to represent a class.

What this cannot say #

Spots are not the field. They are where the field is strongest and they cover about a thousandth of the surface at maximum. Most of the Sun’s magnetic flux is in a small-scale field that has no cycle worth speaking of and that a spot count is blind to.

The diagram is a surface phenomenon. Everything in it is what emerged, and the mechanism is at the base of a convection zone two hundred thousand kilometres below. Nothing in this essay is a measurement of the dynamo; it is a measurement of its output.

And eleven years is not a period. It is a mean interval between maxima whose standard deviation is more than a year, and a series with that much scatter is not periodic in any useful sense — a fact that matters because the whole appeal of the cycle is as a clock, and a clock that is wrong by a year in eleven is a bad clock.

And two readings that separate the eleven-year quantity from the twenty-two-year one.

Where this ladder goes next #

This rung has taken a four-century list of counts, sorted it by latitude and by sign, and got out of it a twenty-two-year magnetic period that the counts themselves cannot show.

The rung above is the output that reaches the Earth. The solar wind, its field, and the modulation of the cosmic-ray flux are what turn the cycle into a measurable quantity elsewhere in the solar system — and into the carbon-14 and beryllium-10 records that extend the series ten thousand years past where any telescope reaches.

Beside it lies the same cycle on other stars, where the rotation rate can be varied and the Sun cannot: a family of activity cycles across the main sequence, and the spin-down relation that makes a rotation rate into an age.

And below it, the habit this rung is the cleanest example of in the collection: a record made by an instrument that discards a sign has half the period of the thing it is recording. Nobody was wrong about the eleven years. The number was simply the period of the quantity that could be seen, and the quantity that could not had twice it.