Generator

The butterfly-diagram generator

4 cycles of wings, and the polarity reverses at every boundary

4 cycles of wings, and the polarity reverses at every boundary. Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in appearance repeats only every 22 in magnetism. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded.

Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in *appearance* repeats only every 22 in *magnetism*. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded.

3 essays call butterfly-diagram. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

4 cycles of wings, and the polarity reverses at every boundary. Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in appearance repeats only every 22 in magnetism. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded.

A magnetic clock read off a butterfly

Plot sunspot latitude against date and the marks form wings that open at thirty degrees and march to the equator. The polarities reverse between wings, so the magnetic period is twenty-two years and the famous eleven is an artefact of counting spots rather than fields.

$The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.$ Stars

A fluid that turns as one piece

The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that.

One step of memory, and it is kept at the poles. Two predictors of a solar cycle's amplitude, each tested against the seven cycles for which both quantities exist, with amplitudes scaled so that cycle 21 is one. Left: the polar field measured at the minimum before the cycle begins, which correlates with what follows at r = 1.00. Right: the amplitude of the previous maximum, which correlates at r = 0.34 — that is, not at all. The cycle is therefore not a pendulum with momentum; it is a process with exactly one state variable, and the variable is the poloidal field that the decay of the previous cycle's spots leaves behind at high latitude. That is the observational core of the flux-transport picture: spots emerge tilted, their trailing polarity drifts poleward, and what accumulates there is the seed the next cycle's shear will wind up. It is also the only prediction in solar physics with a lead time of years, and it was what said, correctly and unpopularly, that cycle 24 would be the weakest in a century.

One step of memory kept at the poles

The amplitude of a solar maximum tells almost nothing about the next one. The polar field measured at the minimum in between tells a great deal — so the cycle is not a pendulum with momentum but a process with exactly one state variable, and the variable is a field of a few gauss at latitudes nobody can see well.

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