The flare that arrives from somewhere else

A solar flare happens on the Sun. The particles it accelerates arrive at the Earth some tens of minutes later, and the ones that arrive in quantity did not come from the flare pointing at the planet. They came from a flare about sixty degrees to the west of the central meridian, and the difference is worth understanding because it is entirely geometric — a matter of where the field goes rather than of where the light comes from, in the way that a magnetopause is a shape rather than a surface.

Charged particles do not travel in straight lines through interplanetary space. They follow the magnetic field, which is not radial, because the wind carrying it comes from a rotating star.

A garden hose, made of nothing #

The construction is elementary once stated. A parcel of plasma leaves the Sun at some longitude and travels radially outward at the wind speed. While it travels, the Sun turns, so the next parcel leaving the same footpoint departs from a longitude that has moved.

The field is frozen into the plasma — the magnetic Reynolds number in the solar wind is enormous — so each parcel drags its own field line behind it, anchored at the footpoint. The line connecting all the parcels that left one footpoint is therefore a curve, and its shape follows from two speeds.

At radius r the parcel there left the Sun a time r over v ago, during which the footpoint has rotated through an angle equal to the rotation rate times that time. So the azimuthal displacement grows linearly with radius, which is an Archimedean spiral.

The pitch angle — the angle between the field and the radial direction — is the arctangent of the rotation speed at that radius divided by the wind speed. At one astronomical unit, with a twenty-five-day rotation and a four-hundred-kilometre-a-second wind, those two speeds are nearly equal and the angle is about forty-five degrees.

Eugene Parker derived this in 1958, in the same paper that predicted the solar wind itself, at a time when the prevailing view was that interplanetary space was essentially empty and the corona was static. The paper was famously nearly rejected. The spacecraft measurements a few years later found both the wind and the spiral.

The wind’s existence had a piece of evidence behind it that is worth recording, because it is a good example of an observation carrying more than it appears to. Comet tails come in two kinds: a curved dust tail lying along the comet’s orbit, and a straight ion tail pointing almost exactly away from the Sun and changing direction on a timescale of hours. Ludwig Biermann argued in 1951 that radiation pressure could not accelerate ions fast enough to produce the second, and that a corpuscular stream must be doing it. The ion tail is a wind sock, and it had been photographed for decades before anyone read it as one.

The same tails also show the spiral, faintly. An ion tail is not exactly antisolar: it is aberrated by a few degrees, in the direction the comet is moving, because the wind’s apparent direction depends on the comet’s own velocity. Measuring that aberration gives the wind speed, and it was done before spacecraft.

The consequence that matters most #

Energetic particles from a flare have gyroradii far smaller than an astronomical unit, so they are tied to field lines and travel along them, scattering as they go.

That means a particle observed at the Earth arrived along the field line that passes through the Earth, and that line does not connect to the point on the Sun facing the Earth. Following the spiral back gives a footpoint about sixty degrees west of the central meridian — because the winding angle, as distinct from the pitch angle, is the rotation angle accumulated over the transit and comes out near one radian.

So the flares that produce large, prompt particle events at the Earth are western-hemisphere flares. This is not a subtlety: it is one of the strongest and oldest correlations in space weather, established from ground-level neutron monitor events long before anyone knew why. An eastern flare can still produce an event, arriving later and more gradually, because the particles have to diffuse across field lines rather than stream along them. The distinction between prompt, impulsive events from well-connected sources and gradual events from shocks driven by coronal mass ejections is largely this geometry.

The practical importance is worth stating plainly. The particles in a large event reach energies of hundreds of megaelectronvolts, they arrive within tens of minutes of the flare’s light, and they are a radiation hazard to anything outside a magnetosphere. Warning of them is therefore a matter of minutes to hours, and the warning depends on knowing which longitude is connected — which depends on the wind speed at the time, which is measured at a spacecraft upstream of the Earth.

The connection is also why forecasting is asymmetric in an unhelpful way. A flare on the eastern limb, whose particles will arrive slowly and weakly, is easy to see coming as the region rotates into view over days. A flare on the western limb, whose particles will arrive promptly and strongly, is on a region that is about to rotate out of sight and has been visible for a fortnight.

What sets the shape, and what does not #

Two speeds set the spiral and neither is the field strength. That is the first thing to notice: the geometry of the interplanetary field is a kinematic result, and a stronger or weaker solar field gives the same shape.

The wind speed is not one number. The slow wind, from the streamer belt near the equator, runs at three to four hundred kilometres a second; the fast wind, from coronal holes, at seven to eight hundred. A fast stream therefore has a more radial field than a slow one, and where a fast stream catches up with slow wind ahead of it the two cannot interpenetrate — the field prevents it — so they compress into a corotating interaction region.

Those regions sweep past the Earth every twenty-seven days, in step with the Sun’s rotation, and are the main driver of moderate geomagnetic activity during the quiet part of a solar cycle. Their period is the Sun’s rotation period as seen from the moving Earth, which is a neat instance of a solar quantity measured entirely from disturbances at a planet — the same distinction between a sidereal and a synodic period that governs every repeating configuration in the sky.

The rotation rate is also not one number, because the Sun rotates differentially, but the field’s footpoints in coronal holes rotate more nearly rigidly than the photosphere does — an observation that is itself evidence about how the coronal field is anchored.

There is a further complication that spacecraft measurements make unavoidable: the actual field at any instant is not a smooth spiral but a spiral with a great deal of structure on it. Turbulence in the wind produces field fluctuations comparable to the mean, and the observed spectrum of those fluctuations follows a power law over several decades of scale, remarkably close to what incompressible turbulence theory predicts. The solar wind is, for that reason, the most thoroughly measured turbulent plasma in existence, and much of what is believed about magnetised turbulence anywhere was checked there first.

The fluctuations matter for the particle transport too. A particle streaming along a nominal spiral is scattered by the fluctuations, so it random-walks rather than travels ballistically, and the arrival time profile of an event is a diffusion problem rather than a flight time. The same scattering off field irregularities is what confines cosmic rays in the galaxy for fifteen million years, operating here over an astronomical unit rather than a kiloparsec.

The sector structure #

Superimposed on the spiral is a division of the interplanetary field into sectors of alternating polarity, discovered in the earliest spacecraft data and initially baffling.

The explanation is that the Sun’s large-scale field is roughly a dipole, tilted with respect to the rotation axis. The wind drags the field out, and the surface separating the outward-pointing hemisphere from the inward-pointing one becomes a warped sheet — the heliospheric current sheet, tilted and therefore wavy as it rotates past a fixed observer.

A planet sitting near the ecliptic passes above and below that sheet as it rotates by, so it sees the field reverse polarity two or four times per solar rotation. That is the sector structure, and its number of sectors is a direct readout of the tilt and complexity of the Sun’s global field. The tilt varies through the cycle, from a few degrees at minimum to essentially ninety at maximum, and that variation drives one of the more elegant effects in the whole system.

Before that, one detail about the sheet deserves recording, because it is a rare case of a structure named before it was understood. The sectors were mapped in 1965 as a pattern of four alternating regions repeating every twenty-seven days, and were treated for years as a property of the Sun’s surface — as though the star had four magnetic quadrants. The reinterpretation as a single warped surface, seen from a point that passes above and below it, came a decade later and dissolved the puzzle entirely: there are not four sectors, there is one sheet and an observer with a vantage point.

Cosmic rays, modulated by a spiral #

Galactic cosmic rays entering the heliosphere have to cross the spiral field to reach the inner solar system, and how easily they do so depends on the field’s structure.

At solar minimum, with a flat current sheet and a quiet, well-ordered spiral, the crossing is comparatively easy and the cosmic-ray intensity at the Earth is high. At maximum, with a warped sheet and turbulent field, it is hard and the intensity falls — by around twenty per cent for the higher energies and far more at low ones. The modulation is anticorrelated with the sunspot number and lags it by several months, which is the transit time for the change in field structure to propagate out to the modulation boundary and back in its effect. So a neutron monitor on a mountain is measuring the state of the entire heliosphere with a delay.

The effect has a component that depends on the sign of the field, which is one of the more surprising results in the area. Positively charged particles drift along the current sheet in one polarity state and in from the poles in the other, so the shape of the modulation cycle alternates between successive eleven-year cycles — the polarity being set by the poloidal field the surface builds each cycle — — peaked in one, flat-topped in the next — with a full period of twenty-two years. That has been seen in the record, and it is one of the few twenty-two-year signals anywhere in the solar system’s behaviour. The record it is seen in extends beyond instruments: cosmogenic isotopes in ice cores and tree rings are produced by cosmic rays striking the atmosphere, so the modulation is written into terrestrial archives going back ten thousand years.

What a spacecraft measured on the way in #

The spiral has been checked from the inside, and the checking was the point of a mission.

A probe flying to within a tenth of an astronomical unit of the Sun passes inside the region where the wind is still being accelerated and, at its closest, inside the Alfvén radius itself — the surface within which the field is strong enough to enforce corotation and the wind is not yet free.

What it found on crossing that surface was the wind behaving as the theory requires and doing so untidily: the boundary is not a sphere but a ragged surface crossed several times in one pass, because the corona’s structure varies with longitude. That is the same lesson the magnetopause teaches — a pressure balance defines a surface, and the surface is corrugated by whatever varies on either side of it.

The other finding from close in was that the field is not smoothly spiral at all near the Sun but full of sharp reversals — switchbacks, in which the field doubles back on itself over minutes and returns. They are ubiquitous inside about forty solar radii and largely absent at the Earth, so something makes them and something erases them, and neither is settled.

None of that overturns the spiral. The spiral is the mean field, and it is the mean field that determines where particles come from and how cosmic rays are modulated. The switchbacks are a large fluctuation about a mean that remains exactly what Parker wrote down.

Where the spiral ends #

Followed far enough, the spiral becomes almost purely azimuthal: at ten astronomical units the pitch is about eighty degrees, and beyond that the field is nearly perpendicular to the flow.

The wind itself ends where its pressure can no longer push back the interstellar medium. That happens at a termination shock near eighty to ninety astronomical units, crossed by two spacecraft, beyond which the wind is subsonic and turbulent, and then at a heliopause near a hundred and twenty where the solar plasma stops and the interstellar plasma begins. The wind’s own speed profile, measured on the way out, is the same kind of measurement as a speed read off the edge of a line profile in a stellar wind — one done in situ and the other from a spectrum.

Both crossings were detected largely by what happened to the particles and the field. At the termination shock the wind slowed abruptly and the field strengthened; at the heliopause the density of thermal electrons jumped by an order of magnitude, measured from plasma oscillations, and the cosmic-ray intensity rose to its unmodulated value.

That last measurement is the cleanest confirmation of the whole modulation picture: the cosmic-ray flux outside the heliosphere is the flux the galaxy supplies, and it is higher than anything measured inside by a factor that grows steeply toward low energies.

What the spiral is a special case of #

The construction generalises immediately and it is worth noting where it goes, because the same geometry appears wherever a rotating body drives an outflow.

Every star with a wind has one, with a pitch set by its own rotation and wind speed. A rapidly rotating young star has a tightly wound spiral close in, which matters because the field’s lever arm out to the Alfvén radius is what carries its angular momentum away — and inside the Alfvén radius the field is strong enough to enforce corotation, so the spiral only begins where the wind wins. Pulsars have the same structure in an extreme form: the wind is relativistic and the rotation is fast, so the spiral is wound tightly within a light cylinder’s distance, and the resulting striped field is what much of pulsar wind nebula physics is about.

And the same geometry decides where a stellar flare’s particles go in an exoplanetary system, which is one of the inputs to arguments about whether close-in planets around active stars can retain atmospheres at all — and whether a magnetic field helps them do so is a separate and less encouraging question.

The construction is a hundred and fifty words of kinematics with two speeds in it. What it fixes is which flares matter, what the sector structure is, why cosmic rays vary with an eleven-year period and a twenty-two-year alternation, and where the solar system’s outer boundary sits. That is a considerable return on an Archimedean spiral. There is one closing point about method. Everything in this essay follows from freezing plus two speeds, and none of it required a field strength. That is unusual in magnetic astronomy, where the difficulty is almost always that the strength is unmeasurable. Here the geometry carries the argument, the geometry is fixed by kinematics, and the strength — which is measured routinely at spacecraft, at a few nanotesla — enters only when the question turns to how much momentum or energy the field is carrying.

There is a second methodological point in that, and it is the reason the spiral is worth a whole essay rather than a paragraph. Most of what a magnetic field does in this collection depends on its magnitude: a pressure, a tension, a growth rate, a splitting. Here the field’s shape is the whole content, and the shape is determined by two speeds that are measured directly and continuously.

That makes the interplanetary field the one place in astronomy where a magnetic geometry is predicted rather than fitted. Every other large-scale field in this collection — a galaxy’s, a cluster’s, a star’s interior — has a geometry inferred from projections and modelled with free parameters. The Parker spiral has none, and its pitch angle at any radius can be computed from a solar wind measurement taken an hour earlier and checked against a magnetometer.

The check has been made continuously for sixty years, at a dozen spacecraft, over four decades of heliocentric distance. It holds. That is an unusual amount of confirmation for a result derived in three lines, and it is why the same construction is applied without apology to stars nobody will ever fly a magnetometer to.

The one thing the spiral does not survive is a wind that is not steady, and that limitation decides where it can be used. A coronal mass ejection drives a shock through the ambient wind and drags its own field with it, so for a day or two the field at a planet is the ejecta’s rather than the spiral’s — which is exactly when the space-weather questions are being asked. The spiral is the background against which the disturbance is measured, and a background is not nothing: the ejecta’s field is identified precisely by its departure from what the spiral requires. A smoothly rotating field over many hours, at a strength well above ambient, is the signature — and it is a signature only because the ambient is predictable. Take away the spiral and the identification becomes a matter of judgement rather than of arithmetic.