On the Physical Origin of Outer Radiation Belt 1--10 MeV Electrons

R. B. Sheldon, H. E. Spence, J. D. Sullivan,T. A. Fritz and Jiasheng Chen

Boston University Center for Space Physics
725 Commonwealth Av, Boston, MA 02215
18 May 1997

Abstract

We report on the POLAR/CEPPAD discovery of a trapped, 60o < PA <120o pitch angle electron population in the outer cusp (8--11 Re), whose energetic electron component extends from below 30 keV to ~2 MeV. We have carried out test particle simulations using the Tsyganenko 1996 model (T96) to demonstrate theoretically the trapping of these energy electrons in the outer cusp region and the resonant frequencies of its trapped motion. We argue that the large phase space densities observed there are sufficient to fill the outer radiation belts with 1--10 MeV electrons. The origin of these electrons is still unknown despite the ~40 years of work since the discovery of the outer radiation belts. During the equinoxes, the POLAR orbit passed through the outer cusp as well as the radiation belts, and observed increases in the trapped cusp electrons which preceded the occurrence of large fluxes of these electrons at geosynchronous altitude. Since electromagnetic fluctuations of the appropriate resonant frequency and power adequate to pump up the energy of such a trapped population are known to exist in the cusp region, the cusp may possibly be the birthplace of outer radiation belt electrons.

Introduction

 Large fluxes of 1--10 MeV electrons which define the outer radiation belt, may pose a significant environmental hazard to geosynchronous satellites, yet the origin and prediction of these electrons remains a mystery (Baker [1997]). Weak statistical links (Blake [1997]) or neural net ``black boxes'' (Koons [1991]) are all we have found in the ~40 years since their accidental discovery by Van Allen [1959]. What has been lacking are physical models that direct the statistical investigation into fruitful correlations that might create better predictors of MeV electron enhancements.

Where are these electrons and why is their origin a mystery? These MeV electrons are trapped in the Earth's dipole field, at a radial distance from ~3--7 Re at the equator, forming the outer radiation belt. As the Earth's dipole field weakens at larger distances, the fluctuations of the field caused by solar wind disturbances become increasingly important, so that the radial transport rate is faster than one Re per day at distances beyond 6 Re (Schulz [1974]). Beyond 8 Re, the dipole field is so distorted that it cannot trap the electrons at all, that is, they do not possess all three adiabatic invariants of the motion and therefore are at best ``pseudo-trapped'' (Roederer [1970]). Generally it is the third invariant that is violated most easily, meaning that the electrons cannot drift 360o around the Earth without encountering the magnetopause and becoming lost. Thus if the source of the electrons is not in the trapping region itself, 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 extensive searches in the trapping region have shown no acceleration region or population that can be the source of the outer zone electrons (e.g., a ``Nishida recirculation'' type mechanism (Ingraham [1994], Paulikas [1979], Nishida [1976]).

Recent work using POLAR has confirmed what other data sets have been telling us, that the phase space density at constant magnetic moment uniformly rises toward higher L-shells, away from the Earth (Selesnick [1997]). Since the diffusive radial transport rate for these electrons is rapid, this suggests that the source is external to the trapping region. 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 (Li [1997]). Thus until recently we could only say that the mysterious source of these electrons lies at 10Li [1993]), yet remain for many hours while filling the radiation belts.

Another, previously unexamined possibility exists, that there may be a trapped population at large L-shells whose trapping does not depend on the usual dipolar magnetic bottle geometry. If this population exists, it could account for the greater phase space density seen at higher L-shells as well as provide a location for slow, steady acceleration to occur. And that is what we have found. The recent discovery of energetic electrons trapped in the cusp has provided a fortuitous glimpse of what we believe to be a suitable birthplace of these electrons, which then populate the outer radiation belts.

Data Analysis

 POLAR is in a 2 X 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 (8--10 Re), with noticeably reduced magnetic field strengths, having broadband 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, generally during the two seasons per year when the POLAR orbit precesses through this region. 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.

FIGURE 1: Electrons trapped in the cusp observed by POLAR. See text for details.

In Figure 1 we plot time/energy/roll-angle spectrograms of phase space density from the CEPPAD/HIST and CEPPAD/IES electron instrument (Blake [1995], Contos [1997]) 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 fluxes are clearly peaked around 90o. 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 and the fluxes more isotropic, 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 (keV/nT), which involves connecting the two regions by a diagonal line roughly parallel to the isodensity contours of the radiation belts, reveals that the outer cusp has 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.

Simulations

 Now this trapped cusp population is highly unusual because, classically speaking, the cusp cannot trap particles (Roederer [1970]), it is not an ``excluded region'' in the Stoermer theory of an idealized dipole (Stoermer [1911], Rossi [1970]). 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 (Figure 2).

FIGURE 2: Trajectories of trapped 1 MeV electrons in the the Earth's outer cusp, projected into the GSM X-Z and Y-Z planes. Dashed lines are field lines from the T96 magnetic field model (Dipole: June 21, 1996, 1300UT; Solar Wind: +10nT Bz, 1/cc, and 1000km/s V_SW). Black lines are contours of |B| in nT. Blue trajectory completes a full drift orbit; red trajectory escapes poleward after a half-drift, green trajectory (started near the cusp center) never completes a bounce.

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 360o around the cusp in an opposite sense to the trapped radiation belt particles. Our preliminary results show that 5--1700 keV electrons can be trapped in the cusp of a T96 magnetosphere for t > 300 seconds, though admittedly without an electric field (see Figure 2). 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 2 one can see two 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 grad-B drift away from the cusp. From the pitch angle distribution, this value appears to be PA ~60o. The limiting 3rd invariant is the maximum value of |B| for which the ``cusp equator'' is still defined over a closed, 360o loop.

Are these trapped particles topologically connected to the nightside trapped particles that have drifted into the bifurcated dayside minimum? Yes, they are physically in the same region of space, but separated in phase space by very different second invariants. Take for example a 50 keV 90o pitchangle particle in the outer cusp, mirroring at 25 nT. For it to maintain the same magnetic moment while drifting, it must find a region of the magnetosphere with <25 nT fields. The only other such region is deep in the tail, and topologically disconnected from the cusp, so that the particles remain trapped in the cusp and cannot drift away without destroying their first invariant. Conversely a 50 keV 90o pitchangle particle trapped at midnight in a 50nT field can mirror through the cusp, but its pitchangle when at the 25 nT level must be 30o. Thus the faint background level inside the wide cusp loss cones could be understood as dipole-trapped particles, but the peak at 90o can only be locally trapped.

A Theory of Cusp Acceleration

 How would this cusp trap accelerate electrons? As others have shown (Delcourt [1992]), the magnetic moment of these trapped electrons need not be conserved as they pass near the minimum B-field point. Now in the absence of an electric field, such a ``scattering'' event leads only to a change in pitch angle, or a circular constant energy surface in v_perp-v_|| space. However when a DC or an AC electric field is present, energy may be gained or lost. One example is so called magnetic pumping. If the cusp has a DC electric field of about 1 mV/m (typical values as inferred from flows measured by POLAR/TIDE), then a ~30 kV potential exists around the cusp. If the electron drifts 180o, it can gain 30 keV of energy from the global electric potential which shows up mostly as perpendicular energy. If then a scattering event changes the pitch angle, this energy can be converted into parallel energy so that it is not lost on the ``return trip'' of 180o drift, ready to begin the process over again. Note that the full 1 MeV potential is not needed to accelerate the electrons in a single step, rather a recirculating, multi-step acceleration can reuse the same potential many times. Eventually these MeV electrons escape the trap (perhaps by becoming too energetic to remain trapped) and diffuse into the radiation belts, adiabatically gaining energy from ~1 MeV to ~5 MeV in the process.

Now the crucial feature of the magnetic pumping example above (in addition to an electric field) is that there exist a ``scattering'' mechanism with a period resonant with the drift period. One can invoke a whole plethora of resonant mechanisms, each based on one of the three frequencies associated with adiabatic invariants (e.g., Bohm diffusion is resonant with the gyromotion). Even with this resonance, it takes many steps, uncorrelated with each other, to accelerate the particles, so that the electrons change their energy in a random fashion, diffusing in energy space by ``stochastic'' acceleration (Fermi [1949]). 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 Table 1) leading to a ``Arnol'd web'' in phase space, and rapid stochastic acceleration (Arnol'd [1964]). The correlation of MeV electrons with high speed solar wind (Blake [1997]) may then be due to the higher fluctuation power in such a solar wind delivering more power at the resonant frequencies of the electrons.

TABLE 1: Periods of the Motion
Energy Mu B_0 PA_0 T_0 T_1 T_2
MeV keV/nT nT degree sec sec sec
1000 16.6 26.4 41 0.004 1.0 77
1000 15.8 21.2 35 0.006 1.1 67
1000 21.5 12.8 32 0.009 0.6 28
1000 150 6.7 88 0.016 0.1 1.3
95 8.0 4.8 30 0.007 0.2 10
5 1.2 1.1 85 0.040 0.4 14


Does this trap hold the electrons long enough for such an indirect acceleration process to raise them up to MeV energies? Ideally we would tag some cusp electrons and observe their trapping times, but since all electrons look the same, we turn to solar wind ions as a ``tracer'' of particle trapping. On May 29, 1996, the POLAR/CAMMICE instrument observed solar wind O6+ ions deep in the cusp, nearly 2 hours after a brief interlude of Bz southward in the midst of a strongly Bz northward solar wind stream (Grande [1997]). We then scale the 500 keV oxygen ion to a 5 keV electron (since at the same rigidity the ions and electrons follow the same trajectory, only the timescale changes), and conclude that electrons are trapped in the cusp for at least 30 minutes. Using the Bohm diffusion rate as an upper limit on stochastic acceleration, we calculate that this same 5 keV electron will reach 1 MeV energies in the presence of a resonant 10 mV/m cusp electric field (a typical cusp AC field as observed by POLAR/EFI) in about 8 seconds. Thus we appear to have plenty of time for stochastic acceleration to operate on this trapped population.

Then why doesn't the cusp trap keep the outer radiation belt constantly full? Most probably because the efficiency of the mechanism has large time variations. We list several factors that control the efficiency with which the cusp can fill the outer radiation belts, which can be further classified as ``trap efficiency'' and ``accelerator efficiency.''

Trap Efficiency

 The volume of phase space in the trap is limited by the range of pitch angles that mirror around the cusp. The minimum pitch angle is determined by the ratio of the magnetic field strength at the cusp equator to that at the magnetopause, sin(PA)/sin(PA0) = |B/B0|. Since the magnetopause Chapman-Ferraro currents are stronger near the nose, we expect the phase space volume and efficiency of the trap to increase with increasing dipole tilt. This tilt might be geometric, during the summer and winter solstices for example, or caused by dayside reconnection and erosion that tilts the cusp toward the nose.

Conversely, reconnection electric fields can distort the cusp third invariant drift orbits, causing them to move beyond the radial extent of the ``cusp equator'' and so lose their cusp second invariant. Thus a DC electric field in the cusp reduces the volume of phase space in the trap by extracting the lower energy particles.

The trap may also be capable of positive feedback so that sufficient trapped plasma deepens the diamagnetic cavity and enhances the trapping time, which we surmise to be the case for the May 29 or Aug 27, 1996 events (Chen [1997]). Such positive feedback can generate large variations from small perturbations such that trapping efficiencies should depend strongly on the magnetic geometry.

Accelerator Efficiency

 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 \grad-B drifts dominate over EXB 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.

Since the nightside trapped population overlap the cusp, substorm injections may also provide a seed population that must be pitchangle scattered to become trapped in the cusp. Thus 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. The 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.

With so many degrees of freedom, it is difficult to make theoretical progress without empirical data. In a further study, we will compare several MeV electron enhancements observed at geosynchronous to this model.

Conclusions

 We have shown that the POLAR spacecraft observed trapped MeV electrons in the Earth's cusp, and that these distributions are consistent with particles trapped in the outer cusp simulated using the Tsyganenko 96 model. Although this trapping geometry is quite different than the standard dipole geometry, we show that an analogous three invariants of the motion exist for this trapped population as well. We show that trapping alone or electric fields alone are not sufficient to produce an energetic electron enhancement, but in concert may be very effective at accelerating electrons to MeV energies. Thus the efficiency of the acceleration mechanism can be highly time variable, producing a typical 2-day time delayed response with respect to the arrival of a high speed solar wind stream, or an 8 hour response as observed on January 10, 1997, or even no response at all (Blake [1997]). If cusp trapping and energization could be established as the origin of the 1-10 MeV outer radiation belt ``killer'' electrons and as the explanation for the variable efficiency of the solar wind drivers, it would be a major breakthrough in space weather and permit the specification and prediction of a major natural hazard to Earth orbiting satellites.

Acknowledgements

 This study was supported by NASA contract NAS5-97147. We gratefully acknowledge the POLAR/CEPPAD data provided by B. Blake and the magnetic field data provided by C. Russell.

References

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