A dropout that is only the field

A geomagnetic storm empties the outer radiation belt’s outer shells through a door the solar wind opens in the magnetopause, and diffusion drains the shells behind it. That is a real loss: the electrons leave the magnetosphere and do not come back. But a storm also does something to the electrons that are not lost, and an instrument cannot tell the two apart by counting.

A storm builds up a ring current: ions of tens to hundreds of kiloelectronvolts, injected from the magnetotail, drifting around the Earth in a current of millions of amperes. The current’s own magnetic field opposes the Earth’s inside it, and a ground magnetometer reads the depression at the equator as the Dst index — minus fifty nanotesla in a moderate storm, minus two hundred in a severe one, nearly minus six hundred in the strongest of the space age. The field the radiation belt sits in is weaker during the storm than before it.

An electron in a slowly changing field does not keep its energy. It keeps three adiabatic invariants, one for each of its three periodic motions, and the first of them ties its momentum to the field strength. When the field weakens, the electron’s momentum falls with it. What follows is about what that does to a measurement, and why the answer can make a storm look far more destructive than it is.

Where the current comes from and how long it lasts #

The ring current is built in the storm’s main phase. Ions from the plasma sheet in the magnetotail are carried towards the Earth by the electric field that the solar wind’s reconnection with the Earth’s field imposes across the magnetosphere, heated as they move into stronger field, and trapped on drift paths around the Earth at three to five Earth radii. Positive ions drift westward and electrons eastward, so the trapped population carries a net westward current, whose field opposes the Earth’s inside it. The more energy the trapped particles hold, the larger the depression — the theorem that lets a magnetometer on the ground weigh them in joules.

The current decays by charge exchange. A trapped proton meeting a neutral hydrogen atom of the Earth’s extended outer atmosphere, the geocorona, takes its electron and becomes a fast neutral atom, no longer bound by the field; it flies off in a straight line and the current loses it. That takes hours for the most energetic ions and days for the rest, which is why a storm’s recovery is slower than its onset and why the adiabatic part of a dropout comes back over a day or two rather than at once.

How much of the field a storm takes away #

The ring current’s depression is largest where the Earth’s own field is weakest. The dipole falls as the inverse cube of distance, while the ring current depresses the field in the inner magnetosphere by roughly the same number of nanotesla at every distance inside it.

At L = 4.5, the heart of the outer belt, the Earth’s field at the equator is 341 nanotesla. A moderate storm removes fifteen per cent of it, a strong one almost thirty, a severe one nearly sixty. At geosynchronous orbit, where the dipole is about a hundred nanotesla, a severe storm removes more than half. The depression is drawn as uniform, which is a simplification: a real ring current is concentrated between L = 3 and 5, depresses the field strongly inside its peak and less outside, and its distribution changes as the storm develops. But the order of magnitude is right, and it is large. The field the belt’s electrons sit in can lose a third of its strength in a few hours.

Fewer at the same energy, with none lost #

An electron’s first invariant is its magnetic moment, $\mu = p_\perp^2 / 2 m B$: the square of the momentum across the field, divided by the field. While the field changes slowly compared with the electron’s gyration — a millionth of a second — $\mu$ stays fixed, so when $B$ falls by a third, $p_\perp^2$ falls by a third too. For a highly relativistic electron, whose energy goes with its momentum, that is a drop in energy of about a sixth; for a slow one, whose energy goes with the momentum squared, closer to a third.

A detector does not follow electrons. It counts those that arrive at a chosen energy. During the storm, the electrons it now counts at one megaelectronvolt are not the ones it counted before; they are electrons that had more momentum before the field weakened and have been slid down to one megaelectronvolt. And the outer belt’s electrons fall off steeply with energy, so there were fewer of them up there to slide down.

The calculation sets the belt’s phase-space density as a power of the magnetic moment, with exponents of two to four, which span the steepness of the outer belt’s spectrum at megaelectronvolt energies. The flux at a fixed energy is the square of the momentum times the phase-space density at the magnetic moment that energy now corresponds to. At Dst = −100 nanotesla the flux at fixed energy and place falls to between a half and a quarter of its quiet value. The steeper the spectrum, the larger the fall, because sliding down a steep spectrum brings fewer electrons into the counting window.

Every one of those electrons is still in the belt. They are counted at a lower energy now, or not counted because the detector’s window has moved past them; the phase-space density at any fixed set of invariants is exactly what it was. There has been no loss, only a relabelling of energies by the field.

This is called the Dst effect, and it has been known since the 1960s. What makes it more than a footnote is that it is the same size as the real losses, happens at the same time, and looks identical in flux.

A storm in which part of the dropout comes back #

A storm’s ring current grows over hours and decays over days. The adiabatic part of a dropout follows it: deepening with the main phase, recovering as the ring current fades.

The figure follows a storm with a ring current deepening to −150 nanotesla over eight hours and decaying with a one-day e-folding. With no real loss at all, the flux at one megaelectronvolt and L = 4.5 falls to eighteen per cent of its quiet value at the storm’s peak — a fivefold dropout — and comes back completely over the following days. Add a real loss of sixty per cent of the phase-space density during the main phase, such as a compressed magnetopause and outward diffusion would cause, and the flux falls to seven per cent and recovers only to thirty-nine.

During the storm the two curves are almost the same shape. A detector watching them would see a fivefold and a fourteenfold dropout, both beginning within the hour, and nothing in the flux at the time distinguishes the part that will return from the part that will not. The distinction appears days later, in the recovery — when it is too late to have been a forecast.

In the coordinates the electrons keep #

The way to separate them during the storm is to stop measuring in energy and place and to measure in the quantities the electrons conserve.

In phase-space density at fixed invariants, the adiabatic response vanishes. When the field weakens, each electron moves to a lower energy and keeps its magnetic moment; the density at that magnetic moment is unchanged. Only real losses and real gains move it. The same storm, re-expressed this way, shows a flat line for the field’s effect and a step for the real loss — the distinction the flux could make only after the recovery is made at the time.

The price is a model. Turning a measured flux at an energy into a phase-space density at an invariant requires knowing the field along the electron’s whole orbit at that moment: its strength at the equator for the first invariant, its shape along the field line for the second, and the magnetic flux enclosed by the drift shell for the third. During a storm the field is exactly what is changing. The empirical field models used for the conversion are fits to many storms’ worth of spacecraft measurements, and their errors at the height of a storm are large enough to move the inferred density by tens of per cent. The conversion that removes the field’s illusion depends on knowing the field — and the coordinates that showed where the electrons were made carry the same dependence.

The conversion has been made, storm by storm, for the long record the twin probes that crossed the belts every nine hours from 2012 to 2019 returned. It shows both kinds of dropout. In some storms the phase-space density at the outer shells fell as steeply as the flux and stayed down, and the losses were real — the outward drain through the magnetopause in most of them. In others the density barely moved while the flux at fixed energy fell several-fold, and the dropout was mostly the field. The two kinds are not separate classes of storm but ends of a range, and where a given storm falls depends on how far its solar wind pushed the boundary in and how deep its ring current grew. The same distinction matters below the outer belt: the gap between the belts is measured in flux, and its apparent widening in a storm is partly the same relabelling.

The mirror points move too #

The first invariant does a second thing. An electron spiralling along a field line towards the stronger field near the Earth converts its motion along the line into gyration until, where the field is strong enough, it turns back: its mirror point. Where that point lies is set by the same invariant, through the ratio of the field at the mirror point to the field at the equator. When the ring current weakens the equatorial field, the electron’s pitch angle there changes, and so does where along the field line it mirrors.

For electrons that mirror far from the equator the effect is small. For those mirroring close to the atmosphere, it can move the mirror point lower, into the atmosphere, and those electrons are then truly lost — not by any wave or boundary, but because the field they were trapped in changed shape under them. That is the same geometry that makes the inner belt’s lower edge dip into the atmosphere over the South Atlantic, where the field at the mirror altitude is weakest. The adiabatic effect is reversible for almost every electron, but for the few near the edge of the loss cone, it is a door of its own.

For a spacecraft the illusion is real #

Everything above argues that a fall in flux at fixed energy overstates a storm’s destruction of the belt. For a satellite in the belt, though, the flux at fixed energy is the hazard. Energetic electrons that bury themselves in a spacecraft’s insulators and charge them until they discharge do so according to how many arrive at each energy, not according to the phase-space density at fixed invariants. During the adiabatic phase of a storm, there really are fewer electrons at the damaging energies near a satellite, and the hazard really is lower — temporarily.

So the same dropout is an illusion for one purpose and a fact for another. The belt’s physics, which asks what the storm did to the population, needs the invariant coordinates and finds the field’s part reversible. The satellite’s engineering, which asks what is hitting the spacecraft now, needs the flux and finds a genuine, if temporary, respite. The difference matters most in the recovery: a satellite operator who reads the dropout as a destroyed belt will be surprised when the adiabatic part returns, before any acceleration has happened, and the surprise is larger the stronger the storm. Geosynchronous satellites, which sit at the belt’s outer edge, see both the largest adiabatic swings and the real losses through the door, and need the two separated to know which will come back.

The bigger the storm, the larger the illusion #

The real loss and the adiabatic effect both grow with the storm. Their ratio decides how much of a measured dropout is real.

Because the two effects multiply the flux, their shares are compared on its logarithm. With a real loss of sixty per cent of the electrons, the adiabatic part is about half of the measured dropout at Dst = −100 nanotesla and three quarters at −200. Even if ninety per cent of the electrons are truly lost, half of the dropout at −200 is the field. The stronger the storm, the larger the field’s share, for a reason that is almost circular: the ring current that deepens the dropout is the same current whose decay will give most of it back.

That has a practical consequence and a scientific one. Practically, a storm’s measured dropout at fixed energy overstates its destruction of the belt by a factor that grows with the storm’s strength, so a flux record used to judge what a storm did to a satellite’s environment over the following week needs the adiabatic part removed first. Scientifically, every claim about how much of the belt a storm destroyed — and so every comparison of loss mechanisms — has to be made in invariant coordinates, or it is partly a measurement of the ring current.

The drift shells move too #

The figures keep each electron on its drift shell and change only its energy. A full treatment also moves the shell. The third invariant is the magnetic flux enclosed by the electron’s drift orbit, and when the ring current weakens the field inside the orbit, the electron must move outward to keep enclosing the same flux. An electron that drifted at L = 4.5 before the storm drifts a little farther out during it, where the field is weaker still, and loses more energy than the first invariant alone implies.

The outward motion deepens the adiabatic dropout at fixed position and energy, so the figures understate the effect. It also matters for the real losses: an electron moved outward by the third invariant is closer to the magnetopause, and if the door is open, a shell that would have been safe at its quiet position may now intersect it. The adiabatic and the real effects are not entirely separate even in principle — the field’s reversible motion of the electrons can carry them into an irreversible loss.

What the model leaves out #

The ring current’s depression is uniform rather than concentrated, the electrons mirror at the equator rather than at a spread of latitudes, the phase-space density is a single power of the magnetic moment, and the drift shells are held fixed. The real loss in the storm figures is a step chosen for illustration rather than computed from a compression. Each simplification changes the numbers by tens of per cent; none changes the conclusion that a fall in flux at fixed energy during a storm is partly or largely a reversible consequence of the field’s weakening, and that its size grows with the storm.

Still open: whether the belt’s response to a storm can be forecast #

The belt’s state after a storm is the difference between three things that happen at once: electrons lost through the door, electrons accelerated in place, and the field’s reversible relabelling of energies. The third is the most predictable, because it follows the ring current, which follows the solar wind; the first two depend on the waves and the boundary in ways that still differ from storm to storm. A forecast that predicted flux at fixed energy without separating them would be right about the dip and wrong about the recovery. The forecasting models now being built work in invariant coordinates for exactly this reason, and convert to flux only at the end, using a field model that is itself forecast. Whether that chain — solar wind to ring current to field to invariants to flux — can be made accurate enough to predict the belt a day ahead, or whether the field’s own uncertainty during a storm sets a limit on it, is what the next decade of radiation-belt measurements will decide.