The gap between the belts is a race between two clocks

The Earth’s radiation belts are usually drawn as two nested doughnuts with an empty ring between them. The ring is the slot, and it sits between two and three Earth radii from the planet’s centre, measured where a field line crosses the magnetic equator. The electrons of the outer belt come from outside: they are carried inward from the edge of the magnetosphere, and they gain energy on the way, because the first adiabatic invariant is conserved as the field they sit in grows stronger. They are also scattered by waves in the dense cold plasma near the Earth, and a scattered electron that ends up in the loss cone hits the atmosphere within a bounce.

So each electron is subject to two clocks. One is the time it takes to be carried inward by one Earth radius. The other is the time it survives before the waves remove it. Where the first clock runs faster, the belt is fed. Where the second runs faster, electrons are lost before they can be carried any further, and nothing replaces them.

That much is qualitative and has been understood since the early 1970s. The shape of the answer comes from the numbers. The transport clock is not a gentle function of distance. It changes as the tenth power of it, so over the three Earth radii between the outer belt and the slot it changes by a factor of about ten thousand. The loss clock changes by a factor of a few. Two curves with those slopes cross at one sharp place, and that crossing is the inner edge of the outer belt.

A random walk across drift shells #

A trapped electron drifts around the Earth on a drift shell labelled by $L$, the equatorial distance of its field line in Earth radii. If the field were perfectly steady, the electron would stay on its shell forever, because the third adiabatic invariant — the magnetic flux enclosed by the drift — would be exactly conserved. The field is not steady. The solar wind pushes on the magnetopause and the pressure fluctuates. Ultra-low-frequency waves with periods of minutes, comparable to the drift period itself, shake the field lines. Each fluctuation moves the electron to a slightly different shell, and the sum of many uncorrelated nudges is a random walk in $L$.

A random walk in one coordinate is described by a diffusion coefficient, and for the radiation belts it is written $D_{LL}$, in units of $L^2$ per day. The equation for the density of electrons in phase space, at fixed first and second invariants, is

$$\frac{\partial f}{\partial t} = L^2 \frac{\partial}{\partial L}\left( \frac{D_{LL}}{L^2} \frac{\partial f}{\partial L} \right) - \frac{f}{\tau},$$

with the last term standing for losses on a timescale $\tau$. The factors of $L^2$ come from the geometry of the drift shells, and they are what makes the equation conserve particles rather than simply smoothing a profile.

The steepness comes from the coefficient. A fluctuation of given size in the solar wind’s pressure moves the magnetopause by a given amount, and its effect on a drift shell grows rapidly with distance from the Earth, because distant field lines are weaker and more easily distorted. Theory for a compressed dipole gives $D_{LL}$ proportional to $L^{10}$. Fits to measured particle fluxes agree, and the one used here, from Brautigam and Albert, is

$$D_{LL} = 10^{\,0.506\,\mathrm{Kp} - 9.325}\, L^{10} \ \text{per day},$$

where Kp is the planetary index of geomagnetic activity, running from 0 in the quietest conditions to 9 in the most disturbed. At Kp 2 that coefficient gives an electron at $L = 5$ about three weeks to move one Earth radius, at $L = 3$ ten years, and at $L = 2$ more than five centuries.

The waves that scatter #

The loss clock is set by plasmaspheric hiss. The plasmasphere is the torus of cold dense plasma, a few electron-volts per particle and hundreds to thousands per cubic centimetre, that corotates with the Earth and is fed from the ionosphere. Its outer boundary, the plasmapause, sits at $L \approx 4$ to $5$ in quiet times and is pressed inward during storms, when the electric field the solar wind imposes across the magnetosphere strengthens, strips its outer layers away and carries them sunward. An empirical relation of Carpenter and Anderson puts it at $L \approx 5.6 - 0.46\,\mathrm{Kp}$.

Inside the plasmasphere there is a persistent incoherent whistler-mode emission between about a hundred hertz and a few kilohertz, which on a loudspeaker attached to a receiver sounds like static; hence the name. An electron whose gyration is Doppler-shifted into resonance with a wave of the right frequency has its pitch angle nudged, one way or the other, at random. Pitch-angle diffusion carries it eventually into the loss cone, and it precipitates into the upper atmosphere, where precipitating particles are also what lights the aurora.

The rate depends on the wave intensity and on how many electrons at a given energy can resonate at a given place, and scattering calculations with measured hiss spectra put the lifetime of megaelectronvolt electrons at a few days between $L = 3$ and $3.5$, where the resonance is most effective, rising to weeks at $L = 2.5$ and to years inside $L = 1.5$, where the resonance requires more energy than these electrons have and only Coulomb collisions with the thin atmosphere remain. The figures use a table of that shape. Outside the plasmapause the model lumps every other loss — scattering by chorus waves, escape through the magnetopause — into a single lifetime of thirty days.

Where the two clocks cross #

The crossing in the first figure is not a threshold that someone chose. It is where the two timescales happen to be equal, at $L \approx 4.8$ for quiet conditions. Just outside it an electron is delivered faster than it is removed, and the belt is nearly as dense as at its source. Just inside it an electron has a few days to live and more than a month to wait before diffusion carries it a further Earth radius. Most never make it. Nothing forbids the journey; it is simply slower than the loss, and the outcome is decided by which clock runs faster rather than by any barrier.

The sharpness follows from the slopes. Because the diffusion time changes by a factor of ten thousand between $L = 5$ and $L = 2.5$, the region where the two clocks are comparable — where the outcome is in doubt — is only a fraction of an Earth radius wide. Inside it the losing clock is ahead by factors of hundreds and then thousands. That is why the inner edge of the outer belt is an edge rather than a gradual fade, and why the slot is a gap rather than a dip.

What a steady source builds #

The equation has a steady solution for any fixed source. Hold the density at the outer boundary, geosynchronous orbit at $L = 6.6$, at a fixed value, hold it at zero at the ground, and let the profile settle.

The dashed curve is the belt diffusion alone would build, and it is almost a flat line. That is not obvious. A diffusion equation with an absorbing wall at the ground might be expected to give a smooth ramp from the source down to zero. It gives something else because the coefficient falls so steeply inward that almost the whole drop is squeezed into a thin layer against the atmosphere. The flux that diffuses inward and is absorbed at the ground must pass through every shell in turn, and at small $L$, where $D_{LL}$ is tiny, a small flux needs a steep gradient; at large $L$ the same flux needs almost none. Without loss, the Earth would be wrapped in radiation at geosynchronous intensity all the way down.

The solid curve is the belt with hiss. It falls off inside the plasmapause and is empty by $L = 3$. The slot in this picture has not been carved out of a full belt by something reaching in from outside. It is the region the source never reaches. Particles are not being removed from a population that exists there; the population never forms, because every electron that starts inward across $L \approx 4$ has a lifetime of days and an inward journey of years.

That distinction matters for how the slot responds to change. A gap carved into a full belt would be refilled by the full belt next to it the moment the carving stopped. A gap that is simply the far end of a supply line refills only when the supply is pushed harder.

Pushing harder #

The obvious lever is geomagnetic activity. At higher Kp the solar wind is gustier and the magnetosphere shakes more, so the diffusion coefficient rises; the fit multiplies it by $10^{0.506}$, about 3.2, for every unit of Kp, and at every $L$ alike.

The step size can be predicted without the solver. The edge sits where the diffusion time equals a lifetime of a few days, and the lifetime changes slowly with $L$. If the coefficient everywhere is multiplied by 3.2, the same diffusion time is found at an $L$ smaller by a factor of $3.2^{1/10}$, which is 1.123 — twelve per cent. Three steps compound to 1.42, and 4.7 divided by 1.42 is 3.3; the solver gives 3.4. The tenth power turns a threefold change in the driving into a modest shift of the edge, and it does so at every step, which is why the outer belt’s inner edge sits in a fairly narrow range for most of the time despite large swings in the solar wind.

The plasmapause moves too, by half an Earth radius per step, and it is natural to expect that the shrinking of the region where hiss operates is what moves the edge. The model says otherwise. Holding the plasmapause fixed at its Kp 1 position and raising only the diffusion moves the edge from 4.7 to 3.5, nearly the whole shift. In steady conditions it is the transport that sets the edge, and the loss region simply decides how fast the density falls once the edge has been crossed.

What the steady states never do, even at Kp 4, is carry the belt inside $L = 3$. The slot survives any steady level of activity that the Earth normally sees. Filling it takes something that is not steady.

A storm as a change of both clocks at once #

A great geomagnetic storm, driven by a coronal mass ejection with a strong southward magnetic field, changes both clocks at the same time. The diffusion coefficient rises by several powers of ten. The plasmasphere is eroded down to $L \approx 2$, and with it goes the hiss, so the loss clock outside the new plasmapause slows to the long lifetime of the outer region.

The crossing moves inward by almost two Earth radii, not by the twelve per cent per step of the steady states, because this time both clocks have moved. For as long as the storm lasts, the electrons of the outer belt are free to diffuse across what had been the slot, and nothing is waiting there to remove them.

The storm fills the slot almost to the density of the outer belt; at $L = 2.5$ the density rises by a factor of ten trillion in two days. Then the storm ends, the plasmasphere refills from the ionosphere over a few days, and the hiss returns. The electrons in the slot are now on the wrong side of the crossing: their lifetime is days to weeks, and the diffusion that brought them is back to a decade per Earth radius. They decay in place at the local hiss lifetime, and the figure’s e-folding time at $L = 2.5$, fourteen days, is the lifetime the table assigns there, because nothing is replacing them.

The decay is fastest where the lifetime is shortest, between $L = 3$ and $3.5$, and slower on either side. Outward of the minimum, diffusion restores the outer belt from its source. Inward of it, the lifetime lengthens to months, and the electrons carried to $L \approx 2.2$ by the storm linger. A hundred days after the storm the model has two belts — a remnant peaking near $L = 2.2$ and the restored outer belt — with an empty slot between them at $L = 3$, the shell where hiss is most effective.

This is the model’s most interesting result, and it reverses the usual order of explanation. The inner electron belt, at these energies, is not a steady population fed from outside that the slot separates from the outer belt. It is what a storm deposited, decaying at a rate set by where it happens to lie, and the slot is where it decays fastest. The inner edge of the slot is not set by a source at all; it is set by the last storm that reached that far.

A storm that stops short #

The same recovery after a smaller storm shows the other side of the argument.

A Kp 6 storm is common; there are dozens in a solar cycle. It moves the crossing to $L = 3$, carries electrons a short way inside it, and leaves the slot empty of anything that survives. A Kp 8 storm is rare, a few per cycle, and only at that level is the crossing pushed far enough inward, for long enough, to deposit electrons beyond the hiss minimum where they can outlive it. The slot is filled by particular events, not by a level of activity, and the model makes that a matter of arithmetic on the two clocks rather than of classification.

The observations that test it #

The account is testable because both of its predictions — that the slot fills in extreme storms and that the inner remnant then persists — are specific.

The Halloween storms of October 2003, among the most intense of the space age, filled the slot with megaelectronvolt electrons within a day. The plasmapause was pushed in to $L \approx 1.5$ to $2$, and the new electron belt centred near $L = 2.5$ persisted for weeks and decayed away over months, as a pure hiss-loss decay would require. The satellites that observed it, SAMPEX in low orbit among them, recorded the filling as a sudden appearance of intense flux at shells normally almost empty, with no gradual inward approach from the outer belt.

The Van Allen Probes, launched in 2012, gave the first measurements with enough energy resolution and coverage to see the structure change in detail. Within days of launch they recorded a third belt — a narrow ring of ultra-relativistic electrons left at $L \approx 3$ to $3.5$ as the outer belt behind it was emptied — that persisted for four weeks until an interplanetary shock destroyed it. That structure is exactly the kind of remnant the storm figure shows, separated from the restored belt by a region of fast loss.

The same spacecraft found an inner edge to the ultra-relativistic electrons, near $L = 2.8$, that no storm in their seven-year mission pushed past. It was described as an impenetrable barrier, and the explanation that has held up is the one in the first figure: at those energies diffusion is slow, hiss scattering at $L \approx 2.8$ is fast, and the crossing never reaches further in. The same measurements showed that above about 1 MeV the inner electron belt is essentially empty most of the time, which the model predicts as the natural state between rare great storms.

What the model leaves out #

The figures follow one family of electrons, at one fixed magnetic moment corresponding to about 1 MeV at $L = 4$. Real electrons are spread over energy, and the lifetime to hiss depends on energy in a way that moves the minimum: lower-energy electrons are scattered most efficiently at larger $L$, higher-energy ones further in. Electrons of a hundred kiloelectronvolts fill the slot often, and their inner belt is a permanent feature.

The diffusion coefficient is an empirical average over many events, and during a single storm it can differ from the Kp fit by a factor of ten either way. The loss outside the plasmapause is a single number in the model, where in reality chorus waves and the loss of electrons through a compressed magnetopause can empty the outer belt in hours; if those losses were included as the fast processes they sometimes are, the plasmapause would matter more than the steady figures suggest, which is consistent with the observed tendency of the outer belt’s inner edge to follow the plasmapause from storm to storm.

Most importantly, the model accelerates electrons only by radial diffusion. Chorus waves outside the plasmapause can also accelerate electrons in place, without transport, and measurements since 2012 show that local acceleration often dominates in the heart of the outer belt. That changes where the source is — at $L \approx 4$ to $5$ rather than at geosynchronous orbit — but not the arithmetic of the slot, which still depends on how quickly electrons can be moved inward from wherever they are made compared with how quickly the hiss removes them.

Why the answer matters in orbit #

Satellites cross the slot on every transfer from low orbit to geostationary orbit, and electric-propulsion spacecraft, which spiral outward under continuous low thrust over months, spend weeks in it. An empty slot means a manageable dose. A filled slot after a great storm can deliver a year’s dose in weeks, and the model says the slot’s filling is a threshold event, tied to the rare storms at Kp 8 and above, rather than something that scales smoothly with activity. The Sun’s magnetic cycle sets how often those storms come, and their arrival is governed by the same solar wind whose pressure moves the magnetopause. The slot inherits their statistics, much as the energetic particles that reach the Earth along the spiral of the interplanetary field inherit the statistics of flares.

Still open: the barrier and the source #

Two questions remain. The first is whether the impenetrable barrier at $L \approx 2.8$ is really held by hiss alone or by something the models do not include, such as the pitch-angle scattering by powerful ground-based radio transmitters, whose signals leak into the magnetosphere at low $L$, or a steeper inward decline of the diffusion coefficient than any fit has used. The two explanations predict different behaviour in a storm larger than any measured since 2012, and none has occurred.

The second is where the electrons come from. The slot model needs only a source outside the plasmapause; whether that source is inward diffusion from geosynchronous orbit or local acceleration by chorus at $L \approx 4.5$ changes the storm response but not the gap. Separating the two requires measuring how phase-space density at fixed invariants peaks — a peak inside the outer boundary means local acceleration, a monotonic rise outward means diffusion — and the measurements show both, in different storms, with the balance between them still being worked out.