The origin of the MeV electrons in the Earth's ``outer radiation belt'' (ORBE) has been mysterious since their discovery 40 years ago by Van Allen ([,,]). Despite their understandably deletrious effects on space electronics, neither military nor civilian spacecraft operators are able to predict their sudden enhancements, and must rely on passive shielding and in situ monitors of these hazardous particles. Statistical models are valuable for post mortems, but are too imprecise for real-time prevention. There is still debate on whether real-time prevention is cost-effective, but the debate is clouded by the poor success rate of present predictors. By analogy to the weather bureau, preventive measures have become much more common as predictions improved. That is, we have a 40-year database on the climate of space, but have very little ability to predict the weather.
Generally it is the third invariant that is violated most easily, meaning that the electrons cannot drift 360° around the Earth without encountering the magnetopause and becoming lost. Thus the source of these ~ 1 day rising enhancement of MeV electrons might be argued to be in the trapping region itself, for otherwise it would appear to be a transient in the outer regions, having a lifetime of several tens of minutes, the time it takes for electrons to drift around the Earth. Yet up until POLAR, extensive searches in the trapping region have shown no acceleration region or population (observable by a peak in phase space density) that can be the source of the outer zone electrons (e.g., a ``Nishida recirculation'' type mechanism [,,]).
Recent work using the high spectral resolution MeV electron spectrometer on POLAR, CEPPAD/HIST, have confirmed what other data sets have been telling us, that during quiet times, the phase space density at constant magnetic moment uniformly rises toward higher L-shells, away from the Earth . (This is always true for quiet periods, but strictly true only for E < 1MeV electrons during stormy periods. Selesnick in his most recent work with improved instrument calibration functions, [private communication, 1999] states that the data are most consistent with an external source. Since the existence of external source is undisputed, we refer to this source alone.) Furthermore, since the diffusive radial transport rate for these electrons during storms is quite rapid, this suggests that the source is external to the trapping region, but populates the ORBE rapidly during a storm. However, the solar wind (at infinite L-shell) has a lower phase space density than the magnetosphere whether we compare at constant energy or constant magnetic moment  and cannot be the putative source. Thus until recently we could only say that the mysterious source of these electrons lies at 7 < L < ¥.
There are several basic observational constraints that any predictive model of ORBE enhancements must address:
The theory of radial transport was well developed from the 1960's, where magnetic and electric impulses were shown to account for the radial diffusion and transport of radiation belt particles. This is consistent with smaller radial gradients during times of enhanced diffusive transport, and give transport rates well in excess of 1 Re/day. Thus one can show that field fluctuations measured during a magnetic storm give transport times for ORBE is on the time scale of hours (J. Wygant, 1997, private communication).
This leads us to the possibility that an electron could perhaps take many small steps that prolong the acceleration process and incrementally provide the energy, say, from a series of several intense substorms (e.g, [,]). Then time between these acceleration events would allow transport and diffusion, so that some particles would get accelerated many times, others only once. Thus this random acceleration would appear to us to be a diffusion in energy space, or a 2nd order, stochastic acceleration. The two basic requirements on any stochastic acceleration are a sufficiently energized ``seed'' population to start the process, and a sufficiently ``trapped'' population that will undergo multiple acceleration steps with some probability.
When we examine the LANL geosynchronous satellites, we find that spectra taken before and after an ORBE enhancement often have identical spectral indices with merely enhanced fluxes, exactly as predicted by a stochastic process (private communication, J. Freeman98). However, we must be clear that the amplitude change cannot extrapolate to infinite energy, since the highest energies take the longest time to accelerate, longer, perhaps, than the temporal extent of the storm. Thus if stochastic processes indeed energize the spectra, there must exist some threshold energy, above which the spectra remain unchanged. Closer examination of the higher spectral resolution POLAR/HIST data show exactly this characteristic, a sharp bend in the spectral index at the higher energies, E > 3MeV. This break is consistent with the stochastic acceleration time becoming longer than the duration of the event. Furthermore, during the fastest ORBE enhancement of the last 3 years, January 10, 1997, the spectral index displayed a lower than usual break in the spectral index, at E < 2 MeV.
The message in this data is that the ORBE is not directly driven, or else there should be a better correlation with the driving energy. Or mathematically, we might say that the response is non-linear. Using our discussion of stochastic acceleration as a guide, we might argue that, say, reconnection Ey is the energy driver, but thermalized Vsw is the seed-population energy. Thus higher velocity may lead to both a higher energy driver and a higher temperature seed population in a way that gives more correlation with velocity than with energy alone.
What is perhaps not expected is the sometimes better correlation of MeV ions with Bz northward. This suggests that there is a non-electrical aspect to the stochastic acceleration process, which is in some sense, independent of Ey. In addition, the subset of the fastest 1/3 Vx possess nearly all the positive correlations, with the slowest 1/3 showing a negative correlation. Clearly a non-linear effect is coupling solar wind velocity into ORBE.
With the above discussion, we have motivated a stochastic acceleration
mechanism for ORBE enhancements occurring somewhere at large
distances, L > 7 Re from the Earth though probably not down tail. This
mechanism must take both mechanical and electrical energy from the
solar wind in a non-linear fashion. The acceleration times must be on
the order of hours to days. Based on these well-accepted
observational constraints, we outline a model that has 3 parts:
Rather than piping all these constraints into a multi-layer neural
net and hoping for the best, we learn from Chen's work on neural
net predictors of magnetic storms (),
that neural nets work best when constrained by physical laws. Thus we
attempt to write a differential equation expressing the dependence of
the acceleration region to each of these effects, weighting each input
with a coupling constant that will be determined from fitting to the
data. For example, if Vsw is the ``seed'' population for a
stochastic acceleration, then the rapidity with which electrons reach
a certain energy is given by the Fermi mechanism equation :
Likewise, if we say that the time rate of change of ORBE in some energy
bounded subset is proportional to their density, N, divided
by some characteristic loss time, t,
This theory predicts that the production of MeV electrons is exponentially related to the energy of the seed population that begins the process. This is an important constraint that goes into the coupled differential equations describing the ``efficiency'' of the stochastic acceleration box. Similarly there are arguments for the trapping time t that depend on IMF Bz, dipole tilt, and solar wind pressure, among others. However, there may be more than one physical model for how the trapping time depends on, say, solar wind pressure. As we discuss later, direct observations by POLAR of the trapping region will be crucial in narrowing down the possibilities.
We want to emphasize that the above model can be developed quite completely without ever observing the trapping region. Such models have been developed for Jupiter, for example. Clearly, however, direct observations of the trapping region will advance the model more rapidly than incremental iterative fits. For example, if the trapping time is time variable, as gross comparisons of the spectral indices of ORBE enhancements suggest, we must apply additional constraints to the model beyond what we can learn from simple spectra. Fortunately, POLAR is in a high latitude orbit that precesses around the magnetosphere in 6 months and therefore made a complete sampling (at high latitude) the 3 < L < ¥ region. The discovery of trapped, energetic electrons at L ~ 12 on the dayside was serendipitous, providing simultaneously a trapped population of electrons and a measurement of the energy drivers.
Thus we are confident that the in situ constraints, combined with the new high-resolution data sets are sufficient to construct an empirical model of a, the energy diffusion rate, and t, the trapping time, so as to model a time-dependent accelerator for the source boundary of ORBE. Since these observations have only recently been published [,,], we review them briefly.
We have observed energetic electrons (30 < E < 2000 keV) trapped in the outer cusp, Figure , a region which can be unambiguously identified by the well-instrumented POLAR spacecraft.
POLAR is in a 2 × 9 Re orbit that on October 14, 1996, passed through the nominal outer cusp before traversing the radiation belts. The outer cusp is defined to be a region inside and adjacent to the magnetopause (7-10 Re), with noticeably reduced magnetic field strengths, having broadband electromagnetic wave power, and generally within some radial distance (2-3 Re) of the topological minimum B point. We do not define the outer cusp with respect to a particle population for the same reason that the plasmasphere, radiation belts, and ring current define overlapping regions in the dipole magnetosphere. We observe a trapped electron population in the outer cusp on this orbit, and more generally during the two seasons per year when the POLAR orbit precesses through this region (with over 90% probability of observation when orbits are favorable). Data from TIMAS and HYDRA on this day show that the magnetopause was first crossed at 0100, at which time EFI showed an abrupt increase in broadband noise. HYDRA showed brief bursts of sheath electrons between 0100-0230 that appeared to be anti-correlated with IES and HIST trapped electrons. These short magnetopause crossings ceased by 0230 along with most of the EFI wave power.
In Figure we plot time/energy/roll-angle spectrograms of phase space density from the CEPPAD/HIST and CEPPAD/IES electron instrument [,]on the POLAR spacecraft. The vertical stripes in the upper panels are an instrument artifact caused by mode switching of the HIST telescope. Successive panels are logarithmically spaced in energy where each panel displays the roll modulation (pitch angle) of the particles. The inset plots the average count rate from 0-20 cts/s in 16 pitchangle sectors, summed between 0100 and 0330 UT showing that the fluxes are clearly peaked around 90°. The color scale displays the logarithm of f (s3/km6) from 0.00001 (purple) to 100 (red). The left half of the plot shows 30-1000 keV electrons with trapped pitch angle distributions located in the outer cusp at L > 10. The right half of the plot is an outer radiation belt traversal. Comparing the radiation belt and cusp loss cones, we see that the cusp's is much wider, which is characteristic of a ``leaky magnetic bottle''. It also appears that the wide loss cone of the cusp is filled at a very low, isotropic level. Comparing the phase space densities at equal magnetic moment (black dots at 7.4 keV/nT), reveals that the outer cusp has equal or higher densities than the outer radiation belts, allowing the possibility of inward diffusion at constant first invariant. Note that the radiation belt pass is at high latitude so that the electron flux would map into the wide loss cones of the outer cusp, suggesting that the second invariant is not conserved if the cusp is the source of these electrons. These observations are very suggestive, and led us to simulate these particles to understand these signatures.
Now this trapped cusp population is highly unusual because, classically speaking, the cusp cannot trap particles at all , it is not an ``excluded region'' in the Störmer theory of an idealized dipole [,]. However, the interaction of a magnetic dipole with the solar wind modifies the topology in a fundamental way that has not been adequately considered in the theory of trapped particles; rather than a dipole, the cusp appears to be quadrupolar. We demonstrate the existence of this particle trap using the somewhat extreme geomagnetic conditions of a nearly minimum latitude cusp and a high speed solar wind, producing a nominal standoff distance of ~ 11 Re.
When we trace particles through this region we find trapping to occur when the electrons mirror around the local minimum of the field line found at the center of the cusp. The orbits take the shape of a lily, with a locally outward magnetic gradient instead of the typical inward gradient so that the particles drift 360° around the cusp in an opposite sense to the trapped radiation belt particles. Our results show that 5-5000 keV electrons can be trapped in the cusp of a T96 magnetosphere for t > 300 seconds, though admittedly without an electric field (see Figure ). Examination of particle trajectories in this region shows that although they lack a dipolar second and third invariant, since they never cross the dipole magnetic equator, we can find an analogous second and third ``cusp'' invariants of the motion if we define the ``cusp equator'' to be the surface of minimum |B| along a field line that approaches the cusp. Thus we can uniquely identify these invariants in analogy to B-L space by their pitch angle and |B| at the crossing of the cusp equator. In Figure one can see three nested ``cusp-shells'' analogous to L-shells of the dipole. The limiting 2nd invariant of these trapped orbits occurs when the mirror point |Bm| approaches the dayside equatorial field strength, at which point the electrons join the dipolar pseudo-trapped population and ÑB drift away from the cusp. From the pitch angle distribution, this value appears to be a0 ~ 60°. The limiting 3rd invariant is the maximum value of |B| for which the ``cusp equator'' is still defined over a closed, 360° loop.
Are these trapped particles accelerated in situ? Because the phase space density is higher in the trap than either in the solar wind (shocked magnetosheath) or in the neighboring dipole trap, the conclusion that acceleration is local appears inescapable. In fact, the |B| minima seen at the cusp are so low, a 90° particle at this location cannot leave the trap without destroying its 2nd invariant. Now the faint background level inside the wide cusp loss cones could be understood as dipole-trapped particles that have drifted into the bifurcated dayside minima in their drift orbit around the earth, but the peak at 90° can only be locally trapped, cusp particles. One might invoke multistep processes that violate entropy, but simple physics dictates that the higher phase space densities observed in this trap are the result of a local acceleration process.
How would this cusp trap accelerate electrons? First we might ask how acceleration occurs in the dipole trap. Generally speaking, the dipole trap accelerates particles when one or more of the adiabatic invariants of the motion are violated by disturbances that have shorter timescales than the associated period of the motion. For example, random magnetosphere compressions from the solar wind with a typical 8-minute period have no effect on the first or second adiabatic invariants, but the violate the third invariant, causing radial diffusion and the energization of the ring current .
In addition, if two resonances overlap, then the phase space density changes even more chaotically, such that stochastic acceleration is most effective when the frequencies associated with each adiabatic invariant are nearly commensurate. That is, near the minimum field point, the time scales of the adiabatic invariants converge, (see Fig. ) meaning that the level of turbulence, DB/B needed to initiate chaotic trajectories in phase space (producing an ``Arnol'd web'' and rapid chaotic acceleration is much lower. Hence the cusp is inherently more energy diffusive than the dipole trap.
Yet a third factor makes the cusp a more effective accelerator than the dipole: the observation of cusp diamagnetic cavities . In hindsight, the weak field region of the central cusp adjacent to the high plasma densities of the sheath might have led theorists to predict such cavities. The important point in this discussion, is that such cavities have large magnetic fluctuations, the largest in the magnetosphere, simultaneous with low magnetic field strengths. These conditions are optimal for coupling energy from the fluctuations into a chaotically broadened resonance of the trapped electrons, thereby increasing by an order of magnitude, the fluctuation power available.
Stochastic acceleration is dependent upon a minimum energy ``seed population'' that can diffuse in energy space. Since in these processes, the energy gain is often proportional to the initial energy, a seed population with lower energy will take considerably longer to accelerate, perhaps longer than the trapping time. Since the trapping time can be a strong function of energy as well, this produces a sharp cutoff in the lowest energy that can be accelerated by the mechanism.
This minimum energy seed population may not always be available in the shocked magnetosheath plasma. That is, when electric fields are superposed on the cusp trap there exists a minimum energy electron above which ÑB drifts dominate over E×B and permit trapping, in complete analogy to the dipolar plasmapause. Thus slight variations in the temperature of the seed population, or in the DC electric field of cusp could result in large variations in the density of the ``seed population'' and therefore in accelerator efficiency. Both of these processes are controlled by reconnection (), making IMF BZ an important controlling factor. Finally, the nightside trapped population also overlaps the cusp as it bifurcates on its drift from the nightside (), making substorm injections also important as a seed population. All these putative sources must be pitchangle scattered to become trapped in the cusp (with its very small |B|), so that the presence or absence of waves resonant with the gyrofrequency can strongly affect the seed population and accelerator efficiency.
The fluctuation power driving the acceleration mechanism may also be highly time-variable depending on reconnection rates or variations in the solar wind pressure. For example, the well-known 27-day recurrence of MeV electron enhancements has been tied to high speed solar wind streams, which are known to have higher fluctuation power as well. The only solar wind parameter better correlated to ORBE than Vsw, is DVsw with a R=75 linear correlation factor (private communication, M. Temerin, 1997). Sorting out the importance of all these effects will require a better understanding of the cusp dynamics.