A belt drained through its outer door

The outer radiation belt is not a reservoir that fills slowly and drains slowly. Within an hour of the start of a geomagnetic storm the flux of relativistic electrons across the outer belt can fall by a factor of a hundred, and within a day it can be back above where it started or still absent. Where the returning electrons come from was the question the previous calculation answered: some are carried inward from the edge of the magnetosphere, many are accelerated where they are found. Where the vanishing ones go is the other half, and it is the half that makes a storm’s outcome uncertain. A belt that loses more than it gains in a storm ends the storm depleted, however vigorous the acceleration was.

Electrons leave the belt in two directions. They can be scattered down into the atmosphere by waves, which is what keeps the gap between the belts empty and is the loss most often drawn. Or they can be lost outward, through the boundary of the magnetosphere itself — and that loss, which needs no atmosphere and no waves at all, turns out to be what empties most of the outer belt at the start of a storm.

The belt therefore has two doors. The inner one is the atmosphere, and it is nearest to the trapped particles where the field is weakest at the surface — over the South Atlantic, where the offset of the Earth’s dipole lowers every drift shell and draws the inner edge of the inner belt. The outer one is the magnetopause, and it is nearest where the solar wind presses hardest, on the dayside. The inner door is fixed in place by the geometry of the Earth’s field and moves over decades. The outer door is pushed around by the solar wind and moves in minutes, and that difference in speed is why the outer belt’s population can change a hundredfold within a storm while the inner belt’s barely changes in a century.

Where the door is #

The magnetosphere ends at the magnetopause, where the pressure of the solar wind’s flow balances the pressure of the compressed geomagnetic field. Because a dipole’s field falls as the cube of distance and its pressure as the sixth power, the boundary’s distance changes only as the inverse sixth root of the solar wind’s pressure — a boundary that hardly moves in proportion to how hard it is pushed.

In quiet solar wind, five particles a cubic centimetre at four hundred kilometres a second, the pressure is 1.3 nanopascals and the magnetopause stands eleven Earth radii from the Earth’s centre. A storm’s wind is denser and faster: at twenty particles and 650 kilometres a second the pressure is fourteen nanopascals, ten times quiet, and the boundary moves in to 7.5 Earth radii. A sixth root makes a tenfold push into a factor of 1.5 in distance — but a factor of 1.5 at the edge of the outer belt is everything.

A drift shell with an opening in it #

A trapped electron in the outer belt does three things at once: it gyrates around a field line, bounces along it between the hemispheres, and drifts slowly around the Earth. The drift carries it along a closed surface, its drift shell, which on the dipole would be symmetric about the Earth. The real field is compressed on the dayside by the solar wind and stretched on the nightside, so a drift shell that lies at a given distance on the nightside comes closer to the Earth on the dayside — and, conversely, a shell that passes close to the magnetopause on the dayside may lie well inside it on the nightside.

When the magnetopause moves in, shells that used to be closed now intersect it. An electron drifting along such a shell reaches the dayside, crosses the boundary, and is gone into the magnetosheath, swept away by the solar wind. It needs no collision and no wave. The innermost shell that does not intersect the boundary is the last closed drift shell, and in the figures it is taken as 0.85 of the magnetopause distance — the compression of the dayside field makes it smaller than the boundary’s own distance by about that factor. At fourteen nanopascals it lies at 6.4 Earth radii, inside geosynchronous orbit, where many communications satellites and much of the outer belt’s measurements live.

How quickly the opened shells empty depends on how fast their electrons drift.

The drift is caused by the field’s gradient and is faster for more energetic particles: a one-megaelectronvolt electron at L = 6 drifts around the Earth in eleven minutes, a five-megaelectronvolt one in under three. The relativistic electrons of the outer belt — the ones that damage satellites — are therefore swept out of the opened shells within minutes of a compression, faster than the solar wind changes. The low-energy electrons of the plasma sheet, at tens of kiloelectronvolts, take most of a day to drift once round; the solar wind’s pressure may have fallen again before they reach the boundary, and many survive. That difference is one reason the dropouts seen at geosynchronous orbit are sharpest at the highest energies.

The name for this is magnetopause shadowing: the boundary casts a shadow along the drift paths, and every electron drifting into it is lost.

The door drains the room behind it #

If shadowing removed only the electrons on opened shells, its effect would be confined to the outer edge of the belt. It is not, because the belt is not static. Electrons random-walk across drift shells all the time, driven by fluctuations of the magnetospheric field on the timescale of their drift — the radial diffusion that, in quiet times, carries electrons inward from the edge and keeps the outer belt supplied. Diffusion carries particles down a gradient. When the outer shells are suddenly emptied, the gradient reverses at the boundary, and diffusion begins carrying electrons outward instead, into the region where they are lost.

The calculation follows the electrons’ phase-space density — the number per unit volume of position and momentum, at fixed values of the adiabatic invariants, which is the quantity diffusion acts on. Before the storm it rises steadily outward, because the belt is supplied from outside. At the moment of compression it is set to zero beyond the new last closed drift shell, here at L = 5, and the diffusion rate is raised to its storm-time value, which scales as the tenth power of L and rises steeply with geomagnetic activity. Nothing else is changed: no waves, no atmosphere, no acceleration.

The density inside L = 5 falls from the outside in. After an hour the depletion has reached a few tenths of an Earth radius inside the door; after twelve hours the density at L = 4.5, half an Earth radius inside, is three per cent of its quiet value. The same diffusion also carries some electrons inward, and below L = 4 the density briefly rises above its quiet value — diffusion only redistributes, and the door decides which way most of the redistribution drains.

This is a loss with no atmosphere in it. The electrons removed from L = 4.5 never came within thousands of kilometres of the atmosphere and never met a wave that scattered them. They were carried outward across drift shells, one random step at a time, until their drift took them through a boundary that had moved.

Why the same diffusion fills the belt and empties it #

Radial diffusion has no preferred direction. It is the accumulated effect of small, random kicks to the electrons’ drift: the magnetosphere rings at frequencies of a few millihertz, in waves driven by the buffeting of the solar wind on the magnetopause and by the cycle of loading and release that powers substorms, and an electron whose drift period matches a wave’s period is pushed slightly inward or outward on each orbit. Over many orbits the pushes add as a random walk across drift shells. The tenth-power dependence on L comes from the field: the outer magnetosphere is weakly held and strongly perturbed, so its fluctuations are large compared with the field itself, while close to the Earth the dipole is strong and the same disturbances barely move it.

A random walk carries particles from where they are dense to where they are sparse. In quiet times the outer belt’s density rises outward, fed from the plasma sheet beyond it, and the walk carries electrons inward: diffusion is how the belt is filled. When the outer shells are opened and emptied, the density near the boundary drops to zero and the gradient reverses there, and the same random walk, with the same physics and a larger rate, carries electrons outward to the boundary. Nothing about the transport changes except the sign of the gradient it acts on. The storm that speeds up the diffusion — because a stronger solar wind drives larger fluctuations — speeds up the loss just as it would have sped up the filling.

Hours outside, days inside #

The tenth-power dependence of the diffusion on distance gives the depletion a strong gradient in time.

A fifth of an Earth radius inside the door, the density halves in about an hour. Half an Earth radius inside, in two. A full Earth radius inside, at L = 4, it takes eleven hours. The depletion front therefore moves fast at first and slows as it moves inward, because the diffusion that carries it weakens by a factor of a thousand between L = 6 and L = 3. A storm whose compression lasts a few hours empties the outer shells and barely touches L = 4; one whose compression lasts a day reaches deep into the belt’s heart.

That makes the duration of the compression as important as its depth. A short, sharp pulse of solar wind pressure — the front of a coronal mass ejection, say — moves the door in for an hour and lets it out again, and the belt loses its outer edge. A long interval of high pressure, as in a slow, dense stream following the ejection, keeps the door in while the diffusion works, and the belt loses far more.

How far inside the door the loss reaches #

The two storm-time effects — the door moving in and the diffusion speeding up — act together, and the combination decides how much of the belt a storm destroys.

With the door at L = 5 and the diffusion of a strong storm, Kp = 6, the density has halved down to L = 3.8 after a day — more than a full Earth radius inside the last shell that was actually opened. A weaker storm, Kp = 4, reaches only 4.3. The more striking feature of the figure is how little the curves depend on where the door is: moving the door from L = 6.5 to L = 4.5 shifts the depth of the loss by only half an Earth radius, because a door farther out is reached by faster diffusion. The diffusion rate, not the door’s position, sets how deep a day of compression reaches.

That is why dropouts are not confined to geosynchronous orbit, where the opened shells are, but are measured down to L = 4 or below in strong storms, by spacecraft that never crossed the magnetopause. The probes that flew through the belts from 2012 to 2019, and the satellites at geosynchronous orbit, saw exactly this sequence repeatedly: a sudden compression, an immediate loss at the outer edge, and a depletion moving inward over the following hours, with no corresponding increase in the electrons precipitating into the atmosphere.

What it means at geosynchronous orbit #

Geosynchronous orbit, 6.6 Earth radii out, is where the outer belt’s edge and the most valuable orbit in the solar system coincide. Satellites there sit inside the outer belt in quiet times and outside the magnetosphere altogether when the solar wind pressure exceeds about thirty nanopascals and the boundary moves inside 6.6. The dropouts drawn here are therefore what those satellites experience first in a storm: a sudden loss of the relativistic electrons that charge their insulators, and sometimes, a day or two later, a recovery to fluxes ten or a hundred times higher than before, as acceleration refills the belt. The second stage is the dangerous one, and its size depends on what the first stage left: a belt drained deep by a long compression has fewer seed electrons for the acceleration to work on.

That connection is why forecasting the belt needs the loss as much as the gain. A storm with intense wave activity that follows a long compression may refill nothing, because the electrons the waves would have accelerated were drained through the door before the acceleration began; a storm with modest waves after a brief compression may produce a large enhancement. The two calculations — what reaches the boundary, and what is accelerated in place — have to be run together.

Planets with no door, and planets with no belt #

The Earth’s arrangement, with its outer belt only a factor of two inside a boundary the solar wind can push, is not general. Jupiter’s magnetopause stands fifty to a hundred Jovian radii from the planet, inflated by plasma from its moon Io, while its intense radiation belts lie within a few radii; a compression that moves the boundary in by half leaves the belts far inside it, and shadowing plays no part in them. Saturn’s belts are cut by its rings and moons, which absorb the particles that drift into them — a sink inside the belt rather than at its edge. Venus and Mars, with no global field, borrow the solar wind’s field and hold it off with their ionospheres, and have no trapped radiation at all.

The Earth sits between the extremes: a field strong enough to trap a belt, too weak to hold its boundary far out, so that the belt’s outer half lives within reach of the solar wind’s pressure. The dropout is a consequence of that proportion.

What the calculation leaves out #

The diffusion is one-dimensional in L, with a rate depending only on L and on the Kp index; real radial diffusion depends on the electron’s energy and on the details of the magnetospheric field fluctuations, and the empirical Kp formula used here is one of several that differ by factors of a few. The last closed drift shell is a fixed fraction of the magnetopause distance, where a real one depends on the field’s full shape and on the electron’s pitch angle. The calculation also leaves out the other storm-time processes — acceleration by chorus waves, scattering into the atmosphere by electromagnetic ion cyclotron waves near the plasmapause — which act at the same time and can dominate in particular storms. And it works in phase-space density at fixed invariants, which is what diffusion acts on; what a detector measures is flux at fixed energy, and during a storm the two do not move together, for a reason that has nothing to do with loss.

What does not depend on the simplifications is the direction: a door that opens the outer shells turns radial diffusion into a sink, and the belt empties from the outside in, through the boundary rather than into the atmosphere.

Still open: how much of a dropout is really gone #

Every dropout begins with a fall in measured flux, and flux is the quantity an instrument counts: electrons of a given energy arriving at a given place. A storm changes the magnetic field the electrons sit in — the ring current that builds up during a storm weakens the field across the inner magnetosphere by tens to hundreds of nanotesla, the same depression a compass needle on the ground records — and electrons respond to a slowly changing field by changing their energy while keeping their adiabatic invariants. A weaker field means less energetic electrons at the same place, and a detector counting at a fixed energy sees fewer of them although none has gone anywhere. When the ring current decays, the field returns and the electrons with it. How much of a measured dropout is the outward drain drawn here, and how much is this reversible shift of the field, decides what a storm actually destroyed — and the two can only be separated in the coordinates diffusion uses, not in the flux the detectors record.