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
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.
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 ). Weak statistical links
(Blake ) or neural net ``black boxes'' (Koons )
we have found in the ~40 years since their accidental discovery
by Van Allen . What has been lacking are
physical models that direct the statistical investigation into
fruitful correlations that might create better predictors of MeV
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 ). 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 ). 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 , Paulikas , Nishida ).
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 ). 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 ). Thus until
recently we could only say that the mysterious source of these
electrons lies at 10Li ), 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.
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
Electrons trapped in the cusp observed by POLAR. See text for
In Figure 1 we plot time/energy/roll-angle spectrograms of
phase space density from the CEPPAD/HIST and CEPPAD/IES electron
instrument (Blake , Contos ) 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
Now this trapped cusp population is highly unusual because,
classically speaking, the cusp cannot trap particles
(Roederer ), it is not an ``excluded region'' in the Stoermer
theory of an idealized dipole (Stoermer , Rossi ). 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
A Theory of Cusp Acceleration
How would this cusp trap accelerate electrons? As others have shown
(Delcourt ), 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 ). 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 ). The correlation of
MeV electrons with high speed solar wind (Blake ) 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
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 ). 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
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 ). Such positive feedback can generate
large variations from small perturbations such that trapping
efficiencies should depend strongly on the magnetic geometry.
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
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
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 ). 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.
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.
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