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
October 1997
We report on the POLAR/CEPPAD discovery of a trapped,
60 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 were correlated with
increases in the radiation belt electrons. Since electromagnetic
fluctuations of the appropriate resonant frequency and power adequate
to pump up the energy of such a trapped magnetosheath 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 Baker97. In particular, the order-of-magnitude flux
enhancements that have characteristic rise times of 0.5--2 days are
especially difficult to predict. Weak statistical links
Blake97 or neural net ``black boxes'' Koons91 are all
we have found in the ~40 years since their accidental discovery
by Van Allen [1959] \nocite{VanAllen59. What has been lacking are
physical models that direct the statistical investigation into
fruitful correlations that might create better predictors of these 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
Schulz74. 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'' Roederer70. Generally it is the third
invariant that is violated most easily, meaning that the electrons
cannot drift 360 degrees 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
Ingraham94,Paulikas79,Nishida76).
Recent work using POLAR has confirmed what other data sets have been
telling us, that the phase space density, f, at constant magnetic
moment uniformly rises toward higher L-shells, away from the Earth
during quiet conditions Selesnick97. However during more active
times, f can show multiple peaks in the outer zone. Since the
diffusive radial transport rate for these electrons is increasingly
rapid with L-shell, this suggests that a time-variable source/sink
external to the trapping region is consistent with both observations,
whereas an internal source, whether steady or time-variable, is
inconsistent with the quiet-time profile. 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 Li97. Thus until recently we could only say
that the mysterious time-variable source of these electrons lies at
10Li93), yet unlike the shock mechanism,
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 2x9 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.
In Figure \ref{polar} we plot time/energy/roll-angle spectrograms of
phase space density from the CEPPAD/HIST and CEPPAD/IES electron
instrument Blake95,Contos97 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 90 degrees. The color scale displays the logarithm of f
(s^3/km^6) 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
(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.
In Figure \ref{october} we plot counts vs energy of the same channels
displayed in the HIST detector (upper half of Fig. \ref{polar}) on 32
successive passes of the cusp and radiation belts occurring in October
1-24, 1996, during which the MLT of the POLAR orbit passed very close
to noon. The color scale extends from 1 (purple) to 100,000
(red). Each panel is the dayside pass when L<20, with geosynchronous
orbit occurring nearly in the center of each panel, and the slot
region at minimum L~3 occurring in the final third of the
panel. From Figure \ref{polar}, we can identify the cusp as the
non-zero fluxes of MeV electrons observed outside of L=10. Since they
occur on nearly every orbit (on every orbit if we use a more sensitive
singles rate), we infer that the cusp always has a residual trapped
population. Although the correlation is not perfect, one can see that
the intense cusp fluxes observed on Oct 14 precede by 35 hours an
enhancement seen in the radiation belts on Oct 16. The cusp
enhancement on Oct 16, though equally as intense as the 14th, produced
a much smaller but measurable increase on the 18th. However the three
consecutive cusp enhancements of the 18-19 produced a growing
radiation belt enhancement on the 20th. This suggests that the cusp
may fill the radiation belts best when given a few days of constant
cusp activity.
FIGURE 1:
Electrons trapped in the cusp observed by POLAR. See text for
details.
Simulations
Now this trapped cusp population is highly unusual because,
classically speaking, the cusp cannot trap particles
Roederer70, it is not an ``excluded region'' in the St\"ormer
theory of an idealized dipole Stoermer11,Rossi70. However, the
interaction of a magnetic dipole with the solar wind modifies the
topology in a fundamental way; rather than a dipole, the cusp appears
to be quadrupolar. We demonstrate the existence of this particle trap
using the geomagnetic conditions of a nearly minimum latitude cusp and
a nominal 10 Re standoff distance (figure \ref{trap}).
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 degrees around the cusp in
an opposite sense to the trapped radiation belt particles. Our
results show that 5--6000 keV electrons can be trapped in the cusp of
a T96 magnetosphere for tau>300 seconds (i.e., many drift
orbits), though admittedly without an electric field (see Figure
\ref{trap}). 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 a dipole by their pitch angle
and |B| at the crossing of the cusp equator. In figure \ref{trap}
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 |B_m| 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 alpha0~60 degrees.
The limiting 3rd invariant is the maximum value of |B| for which the
``cusp equator'' is still defined over a closed, 360 degree loop.
Can these particles have come from the tail, are they 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 90 degree 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
90 degree 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 30 degree. Thus the faint background level inside the
wide cusp loss cones could be understood as dipole-trapped particles,
but the peak at 90 degree can only be locally trapped.
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/cm^3, 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.
Discussion
How would this cusp trap accelerate electrons? As others have shown
Delcourt92, 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-parallel space. However when a DC or an AC electric field
is present, energy may be gained or lost. One example is anomalous or
Bohm diffusion. 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 across the cusp. If the electron gyrates
180 degree, and scatters by 180 degrees successively, it can gain 30
keV of energy from the global electric potential as it crosses the
cusp, which being perpendicular energy, modifies its first
invariant. If the particle is then allowed to drift without
scattering, returning to its original position, it can undergo the
same process repeatedly, gaining energy up to the trapping limit of
the cusp. 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. Naturally, both drift and scattering are occurring simultaneously,
so that the energy changes by small steps rather than in
the large steps described above.
Now the crucial feature of the Bohm diffusion example above (in
addition to an electric field) is that there exist a ``scattering''
mechanism with a period resonant with the gyration period. One could
invoke a whole plethora of resonant mechanisms, each based on one of
the three frequencies associated with adiabatic invariants (e.g.,
radiation belt acceleration is resonant with the drift motion). Yet even
with a 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 Fermi49.
In addition to resonant stochastic acceleration, if two (or three) of
the adiabatic periods nearly 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. If the cusp supports chaotic
motion, then particle acceleration can occur even more rapidly than
the simple diffusion theory described above. In the dipole trap, these
periods are separated by timescales of 100-1000, and chaotic
behavior does not appear. However the cusp trap is an ``inside-out''
dipole, so that near the minimum field point, the time scales of the
adiabatic invariants converge, (see Table 1), allowing the possibility
that large enough fluctuations will generate chaotic motion, an
``Arnol'd web'' in phase space with rapid chaotic (also sometimes
called stochastic) acceleration Arnol'd64. The correlation of
MeV electrons with high speed solar wind Blake97 may then be
due to more than just the higher fluctuation power in such a solar
wind, which delivers more power at the resonant frequencies of the
electrons, but also due to the onset of non-linear chaotic behavior.
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 O^6+ 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 Grande97. 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, and assuming the presence of a resonant 10
mV/m cusp electric field (a typical cusp AC field as observed by
POLAR/EFI) we calculate that this same 5 keV electron will cross the
cusp in a few seconds, gaining ~30 keV. If it then must drift back to
the top of the cusp for a second pass, that adds ~30 seconds, (see
Table 1.), depending on cusp L-shell. Thirty such traversals are
needed to produce an MeV, which would take approximately 15--20
minutes. Now electrostatic acceleration is most effective for lower
energy electrons, whereas betatron acceleration is more effective for
the higher energies, so that invoking a purely electrostatic mechanism
for 1 MeV electrons may give too small an energy diffusion rate,
nevertheless we appear to have sufficient time for stochastic
acceleration to operate on a trapped magnetosheath population to
produce radiation belt energies.
How could the cusp trap fill the dipole trap? If the cusp trap has a
higher phase space density, as we show above, then diffusion into the
radiation belts is allowable. The simulations show that the cusp
trapping volume shrinks with increasing energy, so that above ~6
MeV, the electrons are no longer trapped. Energization processes would
then cause a continual leakage of MeV particles out of the
cusp. Simulations also show that some fraction, ~50\%, are lost
equatorward into the pseudo-trapping region of the dipole, where they
could conceivably diffuse radially inward and appear as radiation belt
electrons. Other mechanisms are possible, including pitchangle scattering
in the cusp itself causing the electron to exit the cusp and appear in the
trapping region of the subsolar dipole trap. And as we argue in a later
paper, an outward motion of the magnetopause will weaken the subsolar
B-field maximum, effectively widening the ``cusp loss cone''.
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.''
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 |
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^2\alpha/sin^2\alpha_0
= |B/B_0|. 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, much as a DC electric
field reduces the size of the plasmasphere.
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 Chen97. 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. The accelerator
efficiency may be non-linear in fluctuation power, since larger
fluctuations may drive the chaotic and produce an Arnol'd web.
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
Blake97. 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.
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