A peak that says where the electrons were made

The gap between the Earth’s two radiation belts is a race between two clocks: electrons diffusing inward across the drift shells against electrons scattered into the atmosphere by waves. That essay could draw the gap without deciding where the outer belt’s electrons come from, because the gap is set by what happens to them once they arrive. The arrival itself is a separate and longer argument. During a geomagnetic storm the flux of electrons with energies of a million electronvolts and more in the outer belt can rise a hundredfold within hours — enough to damage satellites that fly through it — and there are two ways it could happen.

The electrons could be brought in from outside: carried inward across the drift shells from the outer edge of the magnetosphere, gaining energy as they go because they conserve their magnetic moment while moving into a stronger field. Or they could be accelerated where they are found, by waves in the plasma that give energy to electrons already present in the belt. The two mechanisms predict the same increase in flux at a fixed energy. They predict opposite things about one other quantity, which is the density of electrons in the coordinates their motion conserves — and finding that quantity required a spacecraft designed for the purpose.

Three motions and the coordinates they conserve #

A trapped electron moves in three superposed ways, each on a timescale about a thousand times longer than the last, and the separation of those clocks is what makes the problem tractable.

The three conserved quantities are the natural coordinates of a trapped electron. The first, $\mu = p_\perp^2/2mB$, is the magnetic moment of the gyration; it fixes how the electron’s momentum across the field changes as the field strength changes. The second, $K$, measures the bounce path along the field line and fixes where the electron mirrors. The third fixes which drift shell it is on, and is usually expressed as $L^*$ — the equatorial distance, in Earth radii, at which a dipole field would enclose the same magnetic flux as the electron’s actual drift path.

Phase-space density expressed in those three coordinates has a special property: a process that respects all three invariants does not change it. An electron moved along with the field, compressed, expanded or carried round the planet slowly keeps its three numbers, and so the density of electrons at those three numbers stays the same. The density can change only through a process that breaks an invariant — waves resonating with the gyration, which change $\mu$; waves resonating with the bounce, which change $K$; fluctuations of the field on the timescale of the drift, which change $L^*$ — or through a source or a loss.

How much energy the inward trip gives #

Transport from outside is not only a way of moving electrons; it is itself an accelerator, and the amount of energy it gives follows from the first invariant alone.

An electron that keeps its magnetic moment $\mu = p_\perp^2/2mB$ while it is carried inward into a stronger field must increase its perpendicular momentum in proportion to the square root of the field. In a dipole the equatorial field grows as $L^{-3}$, so $p_\perp^2$ grows as $L^{-3}$ too. An electron of 100 kiloelectronvolts carried from $L = 6.6$, the geostationary orbit, to $L = 4$ sees the field rise by a factor of $(6.6/4)^3 = 4.5$, its perpendicular momentum squared by the same factor, and its kinetic energy — accounting for relativity, since the electron’s rest energy is 511 kiloelectronvolts — rises to about 350 kiloelectronvolts. Carried in from further out, from the edge of the magnetosphere at $L = 10$, the same electron would arrive at $L = 4$ with about nine hundred kiloelectronvolts, and a seed of 150 kiloelectronvolts would arrive with more than one. Radial diffusion is a slow betatron, and it can make MeV electrons from a seed population at the boundary without any wave touching them.

That is why the question was genuinely open. Both mechanisms can produce MeV electrons in the heart of the belt; they draw on different reservoirs — one on electrons already in the belt at lower energy, the other on electrons at the boundary at the same $\mu$ — and only the density at fixed $\mu$ distinguishes which reservoir was drawn on.

Two storms that look alike in flux #

Radial diffusion is the process that breaks only the third invariant. Fluctuations of the magnetospheric field on timescales of minutes — shorter than the drift period, longer than the bounce — kick electrons from one drift shell to another at random while leaving $\mu$ and $K$ intact. It is a diffusion in $L^*$ at fixed $\mu$ and $K$, and like every diffusion it moves density from where there is more to where there is less.

So the two mechanisms leave different shapes. Transport from outside needs the density to be higher at the source, at the outer boundary, than where the electrons arrive; the profile rises outward. Local acceleration raises the density where the waves act, deep inside; the profile has a peak. The test was proposed in the 1990s and it is clean in principle. It is hard in practice, because no instrument measures phase-space density at fixed invariants.

Why a detector’s flux cannot answer #

A particle detector counts electrons arriving within a range of energies and directions, and what it reports is flux at a fixed energy. The flux is the phase-space density times the momentum squared, and at a fixed energy the momentum is fixed — so flux at fixed energy is proportional to phase-space density at fixed energy. But fixed energy is not fixed $\mu$.

The first invariant is $\mu = p_\perp^2/2mB$, and the equatorial field strength falls as $L^{-3}$. At a fixed energy, $\mu$ grows as $L^3$: the same one-million-electronvolt electron has a first invariant eight times larger at $L = 6$ than at $L = 3$. A detector that reports flux at one energy along its orbit is sampling electrons of different $\mu$ at different $L$, and the phase-space density usually falls steeply with $\mu$ — there are many more low-energy electrons than high-energy ones.

That is the reason a peak in flux proved nothing, and the reason early studies of the outer belt disagreed. A flux peak inside the belt at MeV energies was observed long before its cause could be established, and both mechanisms could produce it. The transformation from energy to invariants, which needs the local magnetic field strength along the whole drift path and not just at the spacecraft, is what separates them — and the size of the effect depends on how steeply the density falls with $\mu$, which is itself measured.

Computing the invariants from a model field #

Turning a flux measurement into phase-space density at fixed $\mu$, $K$ and $L^*$ requires the electron’s momentum and pitch angle, which the detector measures, and the magnetic field along its entire drift path, which no spacecraft measures. $\mu$ needs the field at the spacecraft, which the spacecraft’s own magnetometer supplies. $K$ needs the field along the whole field line through the spacecraft, between the mirror points. $L^*$ needs the magnetic flux enclosed by the whole drift shell — an integral over a surface that wraps the Earth.

Both of the last two come from an empirical model of the magnetospheric field, driven by the measured solar wind and geomagnetic indices, and the models are least accurate during storms, when the field is most distorted and the answer matters most. The quantity that settles the question is the one most dependent on a model, and every published phase-space density profile carries a systematic uncertainty from the field model that is often larger than its statistical error. The standard defence is to compute the profile with several field models and see whether the conclusion survives. A peak that appears with every model is believed; a peak that appears with one is not.

The order in which shells fill #

The shape of the profile is one test. The timing of its changes is another, and it uses the same logic in time rather than space.

Two spacecraft on different drift shells see the order directly. A single spacecraft on an elliptical orbit crossing the whole outer belt twice every nine hours, as the Van Allen Probes did, sees it by building a profile on each crossing. The two tests — a peak in space, a sequence in time — are not independent, since a peak inside requires the inner density to rise before the outer one, but they are vulnerable to different errors. A field-model error distorts the $L^*$ axis and can move or blur a peak; it is much less likely to reverse the order in which two fixed shells fill.

What the probes found #

NASA’s two Van Allen Probes were launched in August 2012 into elliptical orbits through both belts, carrying detectors that measured electrons from a few electronvolts to ten megaelectronvolts and magnetometers and wave instruments to go with them. Within weeks a storm on 9 October 2012 raised the flux of multi-MeV electrons in the outer belt by more than a factor of a hundred in about twelve hours.

The phase-space density profiles computed from the probes’ data, at fixed $\mu$ corresponding to MeV energies, showed a peak at $L^* \approx 4$ to 4.5 that grew during the storm, with the density lower at larger $L^*$ — the shape local acceleration predicts and diffusion from outside cannot make. The same result held with several field models. At the same time the wave instruments recorded intense whistler-mode chorus — bursts of rising tones in the kilohertz range, generated by lower-energy electrons injected from the magnetotail — at the location of the peak. Chorus resonates with the electrons’ gyration, breaking the first invariant and giving them energy, and simulations driven by the measured wave intensities reproduced the growth of the peak.

The result, published in 2013, settled a long argument for that storm: the most intense MeV electrons were made inside the belt, from a seed population of electrons of tens to hundreds of kiloelectronvolts injected from the magnetotail during substorms, accelerated in place by waves they themselves had helped to generate. Later storms showed both mechanisms, sometimes in the same event — local acceleration building a peak, radial diffusion then spreading it — and the balance between them varies from storm to storm, with no single answer for all.

A peak left behind #

There is one way to make a peak without local acceleration, and it matters for how the test is applied.

When the solar wind compresses the magnetosphere, the magnetopause moves inward and electrons on the outermost drift shells find their drift paths intersecting it; they are lost to the solar wind within one drift period. That magnetopause shadowing, together with outward radial diffusion into the depleted region, lowers the density at the outer boundary, and a density that was rising outward can be left with a maximum inside. Such a peak is not evidence of local acceleration. It is distinguished by where it sits and when it appears — near the boundary, as the outer density collapses — and by the fact that the density at the peak does not rise. The test the probes applied was not simply “is there a peak” but “is there a peak that grows, deep inside, while the boundary is steady or falling”.

The boundary that moves #

The outer boundary of the calculation is a physical surface, and its motion is part of every storm. The magnetopause is where the pressure of the solar wind balances the pressure of the Earth’s field, normally about ten Earth radii out on the day side; its distance varies only as the sixth root of the solar wind pressure, so it hardly moves under ordinary changes, and a storm-time compression large enough to bring it inside the geostationary orbit needs the solar wind pressure to rise twenty-fold or more.

Such compressions happen during the strongest storms, and when they do the outer belt empties from outside in. Electrons whose drift paths cross the compressed magnetopause are lost in a single drift period, a few tens of minutes, and outward radial diffusion then drains the shells inside them towards the depleted edge. A storm therefore often begins with a dropout — the outer belt’s MeV flux falling by a factor of a hundred within hours — before any enhancement, and the phase-space density during the dropout has exactly the left-behind shape the model draws. The belt is emptied from the edge and refilled from the middle, and the two stages of a storm are separated by looking at where the density changes first.

The same boundary sets the losses of other trapped particles. Electrons that are not lost outward can be scattered into the atmosphere, where they deposit their energy at altitudes of sixty to a hundred kilometres and add to the auroral precipitation whose geometry is set by the field; and where the Earth’s field is weakest, over the South Atlantic, the inner belt reaches down into the atmosphere and loses particles there continuously.

Belts round other planets #

Every magnetised planet has radiation belts, and the question of transport against local acceleration has been asked of each.

Jupiter’s belts are the most intense in the Solar System, and they were first detected from the Earth, in the 1950s, by the radio emission of their electrons — synchrotron radiation from electrons of tens of MeV spiralling in Jupiter’s strong field, a spectrum with no temperature in it because it comes from a power-law distribution of electron energies rather than a thermal one. The Galileo and Juno spacecraft showed that radial diffusion inward from the outer magnetosphere, driven by the planet’s rapid rotation and fed by plasma from the volcanic moon Io, supplies much of Jupiter’s inner belt, with the moons sweeping out gaps where their orbits cross it. Saturn’s belts are interrupted by its rings, which absorb any particle drifting through them, so its innermost belt lies between the rings and the planet and is fed by a different route: cosmic rays striking the rings and producing neutrons that decay into protons inside the shielded region.

The Earth’s outer belt is unusual in being refilled so often and so fast, and in being close enough to fly through with instruments able to measure the phase-space density directly. The same analysis, applied to the Juno data at Jupiter, has begun to find evidence of local acceleration there too.

Why it matters in orbit #

Which mechanism fills the belt is not only a question of plasma physics. The satellites most exposed to MeV electrons fly in or through the outer belt — navigation satellites in medium orbit at about $L = 4$, communications satellites at geostationary height near $L = 6.6$ — and electrons of a few MeV penetrate spacecraft shielding and build up charge inside dielectrics until it discharges. Forecasting the outer belt’s flux days ahead — a space-weather forecast of the kind that already turns satellite orbits into measurements of the upper atmosphere — requires knowing where the electrons will come from, because the two mechanisms respond to different drivers: radial diffusion to fluctuations of the solar wind pressure and field, local acceleration to the intensity of chorus waves, which depends on substorm activity. A forecast built on diffusion alone misses the fastest enhancements, which are the ones that do the damage, and operational models now include wave acceleration explicitly.

What the model leaves out #

The model is one-dimensional. It follows the density at a single pair of first and second invariants along $L^*$; real acceleration by chorus changes $\mu$ and $K$ together and moves electrons between invariant pairs, and a full model is three-dimensional in invariant space.

The source is a Gaussian standing for wave acceleration. Its position, width and strength are chosen to reproduce the shape of a local-acceleration event, not computed from wave intensities; the physics that sets them — the location of the plasmapause, outside which chorus is strongest, and the supply of seed electrons — is not in the model.

The loss is a single lifetime. Real losses depend strongly on $L$, energy and the location of the plasmapause, and inside it scattering by plasmaspheric hiss dominates with lifetimes of days to weeks.

And the flux figures assume a power-law spectrum at the boundary and an equatorial field falling exactly as $L^{-3}$. The conclusion — that a monotonic density can make a peaked flux — needs only that the density falls with $\mu$ and that the field weakens outward, both of which hold; the peak’s position does not survive a change of spectrum or field.

Still open: how much each mechanism contributes, storm by storm #

The probes established that both mechanisms operate and that local acceleration dominates the most dramatic enhancements. What they did not establish is the balance in a typical storm, or why some storms that bring intense chorus produce no enhancement while others with modest chorus produce a large one. The answer seems to depend on the seed population, the location of the plasmapause, and losses running at the same time — a storm is a competition between acceleration and loss, and small changes in either change the result from a hundredfold increase to a net depletion. Forecasting the belt, rather than explaining it afterwards, needs the competition modelled in advance from measurements of the solar wind alone, and that is where the work now is: in turning a result about which process fills the belt into a prediction of how much it will fill, a day before the electrons arrive.