An aurora that is a sixth root inside an arcsine

The first rung of this anchor established the magnetopause as a pressure balance and drew the consequence of the exponents: a dipole’s pressure falls as the sixth power of distance, so a sixty-fourfold gust in the solar wind moves the boundary by a factor of two. That is a statement about a surface nobody can see.

What can be seen is the aurora, and it is the same number.

Eight degrees for a factor of a hundred is a remarkable insensitivity, and it is why the aurora is where it is rather than somewhere else.

It also settles a question that gets asked about the aurora more than any other. The lights are seen from Scotland and from Alberta and hardly ever from Rome, and the reason is not that the wind rarely reaches Rome — it is that the geometry above puts the boundary at seventy degrees of magnetic latitude and keeps it there. Moving it to forty-five degrees, which is what a display over southern Europe requires, needs the magnetopause driven inside about two Earth radii, and that has happened perhaps a handful of times in recorded history.

The word magnetic in that sentence is doing real work, because the magnetic pole is eleven degrees from the geographic one. The oval is a ring about the magnetic pole, so its geographic latitude depends on longitude by up to twenty-two degrees — which is why the aurora is seen more often from northern Canada than from Siberia at the same latitude, and why the geographic latitude of an auroral sighting says nothing without the longitude beside it.

The mapping, which is exact #

A dipole field line is described by $r = L R_p \sin^2\theta$ with $\theta$ measured from the axis, so the line that reaches equatorial distance $L R_p$ comes down to the surface where $\sin^2\theta = 1/L$. There is no approximation in that and no free parameter: it is the shape of a dipole line, and nothing else enters.

So the polar cap — the region threaded by field lines that go out to the magnetopause and beyond rather than returning to the other hemisphere — has an edge at colatitude $\arcsin(1/\sqrt{L})$, which for $L = 10$ is 18.4 degrees, or a latitude of 71.6.

The magnetic flux through that cap follows just as directly. Integrating the radial component of a dipole over the cap gives

$$\Phi \;=\; 2\pi R_p^2 B_s \sin^2\theta \;=\; \frac{2\pi R_p^2 B_s}{L}$$

so the open flux is exactly the reciprocal of the standoff distance. That relation has no exponent to remember and it is the quantity a space-weather forecast is actually about, because it is the reservoir the tail draws on.

It is worth pausing on why the open flux, rather than the standoff or the oval, is the quantity that matters. A magnetosphere does work — it accelerates particles, drives currents through the ionosphere, heats the upper atmosphere — and the energy for all of that comes from the solar wind through the open field lines. A closed line is a line whose two ends are both on the planet, and nothing crosses it; an open line is a wire connecting the planet to the wind. So the number of open lines is the size of the connection, and it is the reciprocal of a sixth root, which is to say a very slowly varying number multiplied by the field.

That last point is easy to miss and it is the reason the planets differ at all. The open flux is $2\pi R_p^2 B_s / L$, so it carries the surface field to the first power and the standoff to the minus first — and the standoff itself carries the field to the one-third. Net: the open flux goes as $B_s^{2/3}$. Jupiter’s field is a hundred and thirty times the Earth’s and its open flux is about twenty-five times larger, which is a factor a magnetosphere can be built on.

The same arithmetic on other planets #

The two inputs to the standoff — the planet’s field and the wind it sits in — vary by orders of magnitude across the solar system, and they vary in opposite directions with distance from the Sun.

The flatness of that curve over most of its range is the essay’s subject. Between Earth and Jupiter the standoff changes by a factor of four and the cap edge by nine degrees, because taking a sixth root and then an arcsine of the result compresses everything.

Mercury is also the only planet whose magnetosphere is small enough for the whole system to reconfigure in minutes rather than hours — the transit time of the wind across it is seconds — so its substorm cycle runs some thirty times faster than the Earth’s. The physics is the same and the clock is not.

The insensitivity fails there for a specific reason: the arcsine’s argument is close to one, where the function is steep. Everything above about a standoff of five planetary radii sits on the flat part of the curve and everything below it does not.

The energy that comes down the funnel #

The aurora is not merely located by this geometry; it is powered through it, and the numbers are worth having because they are surprisingly small.

The solar wind delivers about 10¹³ watts to the whole cross-section the Earth’s magnetosphere presents, of which a few per cent is coupled in — a hundred gigawatts or so during a moderate storm, and up to a terawatt in a severe one. That is deposited into a ring a few degrees wide, so the power per unit area is comparable to full sunlight over a small region, which is why the aurora is bright enough to read by and why it heats the upper atmosphere enough to raise satellite drag measurably — one of the reasons an atmospheric density model is wrong by a factor of two at the altitudes where it matters most.

Where the energy is stored between delivery and release is the tail’s magnetic field. The lobes of the tail hold a field of about twenty nanotesla over a cross-section of forty Earth radii, which is an energy of order 10¹⁵ joules — a few hours of the coupling rate. A substorm is the discharge of that store, which is why the aurora brightens abruptly rather than in proportion to the wind, and why an auroral display is a poor instantaneous measure of anything.

What the standoff is doing underneath #

There is a fourth planet worth mentioning and it is the one with no field at all. Venus and Mars have no internal dynamo, so there is no dipole, no standoff in the sense used here, and no polar cap. The wind is stopped instead by the ionosphere’s own pressure, at a few hundred kilometres rather than at ten planetary radii, and the field it meets is the interplanetary field draped around the obstacle rather than a field the planet made. Everything in this essay’s chain is absent, and what replaces it is a boundary a hundred times closer whose position depends on the ionosphere’s density rather than on a sixth root of anything.

Where the picture stops being right #

The dipole mapping is exact and the magnetosphere is not a dipole, so it is worth naming the three places the answer moves.

The tail. Field lines dragged back by the wind — frozen into the flowing plasma, which is what lets the wind carry them at all — are stretched into a tail that extends hundreds of planetary radii, and a line at the boundary of the polar cap does not go to the magnetopause at the nose — it goes down the tail. That makes the real polar cap larger than the dipole calculation gives, by a few degrees, and it makes it variable in a way the dipole calculation cannot capture, because the tail’s flux content changes on a timescale of hours.

Reconnection. Open flux is created at the dayside magnetopause and destroyed in the tail, and the two do not balance instantaneously. A substorm is precisely the failure to balance: flux accumulates in the tail for an hour or two, the cap grows, and then the tail reconnects and the cap shrinks abruptly. So the oval’s latitude at any moment is a statement about the recent history of that cycle rather than about the present wind.

The aurora is not the cap edge. Auroral emission comes from particles precipitating out of the plasma sheet, which maps to the outermost closed field lines rather than to the open–closed boundary itself. The oval is therefore a few degrees equatorward of the polar-cap edge, and by an amount that depends on how the plasma sheet is behaving.

All three push the same way — they make the observed oval larger and lower than the dipole calculation — and the shaded bands in the figures above show the size of the discrepancy: the calculation gives 71 degrees for the Earth and the oval is observed between 67 and 75.

There is a fourth and larger caveat that applies to none of the drawn planets and to most of the interesting cases. The dipole approximation assumes the field’s higher multipoles are negligible at the magnetopause, which is true when the standoff is many planetary radii — a quadrupole falls as the fourth power of distance against the dipole’s cube, so it is gone by five radii. At Mercury, where the standoff is 1.5, the higher multipoles are not gone, and the field’s own offset from the planet’s centre matters as much as its strength. The northern and southern polar caps at Mercury are not the same size, for that reason, and neither is centred on a geographic pole.

What is actually measured #

The oval is imaged, from above, in the ultraviolet, by spacecraft in polar orbits, and the imaging is the cleanest part of the whole business — a bright ring against a dark planet, with an edge that can be traced to a fraction of a degree.

The particles that make the light are measured too, and they are not what most descriptions suggest. The aurora is not the solar wind hitting the atmosphere: the wind’s own particles carry far too little energy per particle to produce the observed emission, and almost none of them reach the atmosphere directly. What precipitates is plasma-sheet electrons, accelerated through a field-aligned potential drop of a few kilovolts a few thousand kilometres above the ground, in a region where the current the magnetosphere demands is larger than the available charge carriers can supply without help. The wind supplies the energy, through the open flux; the acceleration is local, and it is why the emission is a curtain rather than a glow.

The magnetopause distance is measured by spacecraft crossing it, which happens whenever the boundary moves past a spacecraft rather than whenever anybody wants a measurement. Thousands of such crossings have been catalogued and fitted into empirical models, and those models agree with the sixth-root scaling to within the scatter, which is about ten per cent.

The open flux is not measured directly at all. It is inferred, either from the area of the polar cap in an auroral image — which is this essay’s chain run backwards — or from the magnetic field measured in the tail lobes multiplied by the tail’s cross-section. The two methods agree to about twenty per cent, which is the honest accuracy of the quantity that governs how much of the wind’s energy reaches the planet.

The solar wind’s own pressure, which is the input to everything above, is measured by a spacecraft at the first Lagrange point about an hour upstream — a place chosen for exactly this — and that hour is the whole of the warning time for a geomagnetic storm. It is also a single-point measurement of a structured medium: the wind arriving at the Earth is not necessarily the wind that passed the monitor, and the discrepancy is one of the larger error terms in space-weather forecasting.

Two things the geometry gets right that it should not #

It is worth ending the physics on the successes, because a chain with this many idealisations in it has no business working as well as it does.

The first is the shape. The oval is an oval and not a circle: it is displaced towards midnight by a few degrees, and it is wider on the night side than the day. The dipole calculation gives a circle, and the displacement follows from the tail — the open flux is not distributed symmetrically because the tail is on one side. That correction is computed from the same field model and it matches.

The second is the conjugacy. The two hemispheres’ ovals are connected by the field lines that bound them, so they should be mirror images and they are: an auroral feature in the north has a counterpart at the magnetically conjugate point in the south, and simultaneous imaging from two spacecraft has shown them brightening together. That is a strong test of the mapping, and it is one the dipole picture passes even though the field is not a dipole, because both hemispheres are mapped through the same distorted field and the distortion cancels.

The generalisation #

The structure worth extracting is that composing flat functions makes something flatter, and that this is why so many astronomical quantities cluster.

A sixth root turns four orders of magnitude into a factor of four. An arcsine near its middle turns a factor of four into ten degrees. Neither operation is remarkable and their composition produces a quantity that is nearly the same for every magnetised planet in the solar system — not because the planets are similar but because the function is flat.

The same shape is why a sunspot’s darkness is set by a pressure ratio rather than by the field’s absolute value, and why almost every scale height in this collection is within a factor of a few of every other. A quantity reached through several fractional powers is a quantity that will not surprise anybody, and one reached through an exponential will.

There is a second reading about what a flat relation is good for. Precisely because the oval’s latitude barely depends on the wind, it depends strongly on the planet — and a measurement of an oval is therefore a good measurement of a planetary field. That is how the fields of the outer planets were first estimated before any spacecraft arrived: radio emission from the auroral regions, a rotation period from its modulation, and a latitude from the beaming geometry.

The corollary is a warning that runs the other way. Flat means insensitive, and insensitive means uninformative: the auroral latitude is a poor measurement of the solar wind precisely because it barely responds to it. Anything measured through this chain inherits the flatness, and inverting it amplifies the errors by the same factors that suppressed the signal.

Where the ladder goes next #

The next rung follows the flux rather than the geometry. The Dungey cycle — open at the nose, carried back over the poles, closed in the tail, returned to the dayside — is the circulation that converts solar wind energy into everything a magnetosphere does, and its rate is set by a reconnection efficiency that is the least well determined number in the subject. The oval’s size is the integral of that rate; the oval’s motion is its derivative.

Further rungs on this anchor: the trapped population on the closed lines, whose three clocks and nothing to fall onto is the belts’ whole dynamics; the induced magnetospheres of Venus and Mars, which have no internal field and whose obstacles are their own ionospheres; the ring current, and why a storm’s strength is quoted as a magnetic depression measured at the equator; the radiation belts as the trapped population on the closed lines this essay is about; and the magnetospheres of the giant planets, which are driven by rotation rather than by the wind and where nearly everything in this essay has the wrong cause. The Sun’s own field, whose eleven-year cycle is read off a butterfly, modulates all of it from the other end.