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Chapter 5.0 Internal Acceleration Mechanisms:
Testing the Whistler/EMIC Wave Mechanism
5.1 Introduction
The previous chapter demonstrated that during some storms, phase space density
peaks at low L*, indicating local acceleration of electrons. This chapter examines whether
the evolution of electron pitch angle distributions (PADs) is consistent with a particular
local electron acceleration model, the whistler/EMIC wave acceleration model. We focus
on the whistler/EMIC wave model because it is the most thoroughly developed of the
local acceleration models and makes specific predictions concerning PADs. It is distinct
from the other local acceleration models because the acceleration process benefits from a
dawn dusk PAD asymmetry. We begin this chapter by describing the whistler/EMIC
wave model in more detail. We explain how the mechanism affects PADs and compare
observed PADs to the theoretical PADs. We identify characteristics of PADs unique to
storms with electron acceleration by comparing superposed data from storms with and
without electron flux enhancements. These observations are used to test the validity of
the model.
5.2 The whistler EMIC wave model
As the name suggests the whistler/EMIC wave model accelerates electrons
through sequential interaction with two types of Very Low Frequency waves (VLF):
whistler waves, and EMIC waves. Brice [1961] and Kennel and Petschek [1966] first
described how these waves interact with electrons. They showed that, in the non-
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relativistic limit, electrons interacting with whistler and EMIC waves change pitch angle
with little change of energy suggesting these waves were not responsible for accelerating
relativistic electrons. More recently, Summers et al. [1998] expanded the theory of wave
particle interactions to include relativistic effects. They showed that, in the relativistic
limit, electrons interacting with whistler waves diffuse in both energy and pitch angle.
They also found that the effect of EMIC waves on electrons is unchanged by relativistic
corrections. In both the relativistic and non-relativistic regimes, EMIC waves cause
diffusion in pitch angle but not energy. Even though EMIC waves do not change electron
energy they are a necessary part of the electron acceleration process invoked by the
whistler/EMIC wave model. The model relies on EMIC waves to prime electrons by
producing PADs necessary for subsequent electron acceleration by whistler waves.
Following the framework of the authors above, we now describe how electrons
interact with VLF waves in both the non-relativistic and relativistic limit and show how
these waves accelerate electrons in the relativistic limit. The interaction of electrons with
waves of any frequency is described by resonance theory. As discussed in chapter 3, the
radial diffusion model invokes resonance theory to explain the acceleration of electrons
by ULF waves. Now we show how resonance theory governs the interaction of electrons
with VLF waves.
Whistler and EMIC waves propagate at an angle  from the direction of the
magnetic field and have magnetic and electric field perturbations perpendicular to the
direction of propagation. Resonance between the wave and an electron occurs when the
gyrofrequency of the electron equals the Doppler shifted frequency of the wave such that
the following equation is obeyed [Brice, 1964]:
2
  k cosv|| 

g

,
(1)
1
 v2  2
  1  
 c2 


 g  qB / m0
(2)
The parameters are defined as follows:  is the relativistic correction, v is the electron
velocity, c is the speed of light,  is the angle between the propagation direction of the
wave and the magnetic field direction, v|| is the parallel velocity of the particle, uph is the
phase velocity of the wave,  is the frequency of the wave, g is the gyrofrequency of
the electron, q is the charge, B is the magnetic field magnitude and m0 is the rest mass of
the electron. The equation, called the resonance condition, shows that only electrons with
a specific parallel velocity will be accelerated or decelerated by a wave of a given
frequency. In the non-relativistic limit, =1; therefore the resonance condition specifies
only the parallel velocity of electrons affected by the waves and particles of arbitrary
perpendicular velocity are resonant. In the relativistic case, the resonance condition
applies for parallel and perpendicular velocity with a known functional relationship.
Figure 1 gives a schematic representation of the relativistic and non-relativistic resonance
conditions. In the non-relativistic limit, the resonance condition is a straight line when
plotted as a function of perpendicular and parallel velocity. In the relativistic limit, the
resonance condition plots as an ellipse.
To understand how wave particle resonance affects the energy and pitch angle of
an electron, we require that in the reference frame of the wave, the energy of the particle
does not change. This assumption is written in the wave reference frame as:
Non-relativistic and Relativistic limits:
3
 dv
  v|| dv||  0
v
(3)
 is the particle velocity perpendicular to the magnetic field and v|| is the velocity
where v
parallel to the magnetic field in the wave reference frame. To transform the equation
from the reference frame of the wave to our inertial reference wave requires a Lorentz
transformation.
v|| ' 
v|| u ph
v ' 
2
[1  v|| u ph / c ]
[1  u ph
1
2 2
/c ] v

2
[1  v|| u ph / c ]
(4)
where uph is the phase velocity of the wave .Transforming equation (3) to our inertial
reference frame we find,
Non relativistic:
v dv  (v||  u ph )dv||  0
Relativistic:
(1 
u ph v||
c2
)v  dv   (v||  u ph 
2
u ph v 
c2
)dv||  0
(5)
Integrating the equations over v  and v|| one obtains a function that defines how v  and v||
of the electron change in the interaction with a wave. As the electron velocity changes so
does the pitch angle and energy. The equation plotted as a function of v  and v|| is called a
pitch angle diffusion curve.
Summers et al. [1998] assume parallel propagating waves and use the cold
plasma wave dispersion relation to obtain uph() . The diffusion curves are obtained as
integrals of the above equations. The authors calculate diffusion curves for both EMIC
and whistler waves using a range of plasma densities and magnetic field strengths. The
diffusion curves calculated for EMIC waves show that electrons diffusing along the
curves change pitch angle but not energy during the interaction. The whistler wave
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diffusion curves show that electrons change both pitch angle and energy. Individual
electrons interacting with the waves move in either direction along the diffusion curves to
smaller and larger pitch angles. An electron moving along the whistler diffusion curve
towards 0o pitch angle loses energy. An electron moving toward 90o pitch angle gains
energy.
The net energy change of a distribution of many electrons interacting with
whistler waves depends on whether more electrons diffuse to larger pitch angle and
higher energy or diffuse to smaller pitch angle and lower energy. The initial PAD
establishes whether more electrons move to smaller or larger pitch angle and therefore
determines net energy gain or loss.
Gendrin [1968] developed an intuitive graphical method for determining net
energy gain or loss. The method involves plotting PADs as contours of constant phase
space density along with diffusion curves. PADs are not commonly plotted as contours of
constant phase space density. More often electron flux is plotted as a function of pitch
angle measured at fixed energies. To aid in understanding we show examples that map
the more familiar flux plots to contours of constant phase space density. The examples
show how the mapping varies for data with different pitch angle and energy
dependencies. Figure 2 shows contours based on simulated data with soft energy spectra
and fairly low pitch angle anisotropy. Figure 2 Panel A plots differential flux at fixed
energies with an assumed pitch angle dependence given by j ( )  sin  . Panel B plots
the energy spectra at fixed pitch angles with an assumed energy dependence of j(E)=10-2E
where E is kinetic energy. Panels C and D map the data to contours of constant phase
space density in both momentum and velocity space. The contours plotted as a function
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of velocity are the theoretically useful quantities but we include contours as a function of
momentum for comparison with Summers et al. [1998] who show results in momentum
space. The contours in this example are nearly circular except very near the loss cone.
Figure 3 plots contours from simulated data with the same energy spectra but greater
pitch angle anisotropy. In this figure the simulated data have the same energy dependence
as before but now j()=sin2. Increasing the pitch angle anisotropy modestly changes the
contours of Panel C and D causing them to be slightly peaked at 90o. The change is most
noticeable near the loss cone. Figures 4 and 5 show similar plots for hard energy spectra.
Figure 4 shows data with j ( )  sin  and j(E)=10-0.2E. Hardening the spectra causes the
contours in Panel C and D to become more peaked at 90o. Finally, Figure 5 shows
contours based on data with hard energy spectra and high pitch angle anisotropy. In this
figure j ( )  sin 2  and j(E)=10-0.2E. The contours in Panel C and D are noticeably
peaked at 90o.
We now discuss the Gendrin [1968] method for determining net energy gain that
relies on the phase space density contours. The first illustration of how energy changes,
Figure 6 Panel A, shows a contour of constant phase space density for an isotropic
electron pitch angle distribution (gray trace) and a diffusion curve slightly peaked at 90o
(black trace). Electrons that satisfy the resonance condition diffuse in both directions
along the diffusion curve (black arrows). A diffusion imbalance occurs when there is a
phase space density gradient (green arrow). In the example shown, the gradient causes
more electrons to diffuse towards 90o along the negative gradient of phase space density
and to higher energy (red arrow). Thus, the distribution of electrons gains net energy.
Panel B shows the same diffusion curve with an electron distribution initially peaked at
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90o. In this example, more electrons diffuse towards small pitch angle and lower energy.
Panel C shows a diffusion curve that lies on a circle of constant energy with an initial
electron distribution peaked at 90o. In this example, more electrons diffuse along the
curve to smaller pitch angles but the energy remains constant. Diffusion occurs until the
interaction reaches marginal stability and the contour of constant electron phase space
density lies along the diffusion curve (Panel D).
The Summers [1998] model predicts that electrons gain energy by a two step
process involving both types of waves. First, EMIC waves, acting predominantly on the
dusk side of the magnetosphere [Jordanova et. al, 2001; Braysy et al., 1998], scatter
particles towards small pitch angles while conserving electron energy. This interaction is
similar to the illustration of Figure 6 Panel C. Next, the more isotropic distribution drifts
to the dawn side where whistler waves are commonly present [Meredith et al., 2001,
Lorentzen et al., 2001; Burtis and Helliwell, 1976; Koons and Roeder, 1990]. The
whistler waves scatter electrons to larger pitch angles and higher energies as in the
illustration of Figure 6 Panel A. After multiple drift periods, electrons may gain
significant energy.
5.3 PADs Predicted by the Whistler/EMIC wave model
The PADs predicted by the whistler/EMIC wave electron acceleration model
depend on local time and the relative power of the two types of waves. The PADs depend
on local time because they are shaped by whistler waves on the dawnside of the
magnetosphere and EMIC waves on the duskside. The power of the waves determines
how quickly electron distributions are altered. Relativistic electrons drift about the earth
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in less then ten minutes. The rapid drift limits the interaction time of electrons with each
type of wave. If wave power is high, the interaction will bring the electron distribution to
marginal stability even in this limited time. At marginal stability the electron distributions
lie along the diffusion curves. Therefore, if both the EMIC and whistler wave power are
high, the PADs will have a dawn-dusk asymmetry. At dusk the PADs will be isotropic
and lie on curves of constant energy like the EMIC wave diffusion curve, while at dawn
the PADs will be peaked at 90o like the whistler wave diffusion curves. Exactly how
peaked the diffusion curves and PADs are depends on the local magnetic field strength
and density. The diffusion curves calculated by Summers [1998] become increasingly
peaked as density drops and magnetic field strength increases. To describe how peaked
these curves are, we give the ratio of p/m0c at 90o and 45o. In the regime where
(e/p)2=1 the ratio is ~1.3, while in the regime where (e/p)2=10 the ratio is ~1.75.
There is some evidence that whistler wave power is not high enough to achieve
marginal stability. Measurements of the two types of waves during storms suggest that
whistler wave amplitudes are an order of magnitude lower than EMIC wave amplitudes.
Meredith et al. [2001] determined whistler wave amplitudes from CRESS data over the
15 month mission and found amplitudes of ~.1-1mV/m during geomagnetically active
periods with AE>300 nT. Braysy et al., [1998] using data from the Freja satellite,
determined EMIC wave amplitudes of 1-50 mV/m during one storm. If the imbalance of
wave power favors EMIC waves as the observations suggest, PADs will be defined by
the EMIC waves and only slightly modified by the whistler waves. In this case,
predominantly isotropic PADs are expected. The opposite imbalance of wave power with
whistler waves dominating is currently not supported by wave observations. But if
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whistler waves dominated, they would create electron distributions peaked at 90o and
they would be only slightly modified by the EMIC waves.
5.4 Previous Investigation of Storm-time Pitch Angle Evolution
Blake et al. [2001] investigated the evolution of pitch angle distributions during
storms using the HIST instrument as well. The study investigated pitch angle
distributions in a unique way. Pitch angle distributions were determined by comparing
electron flux measured at high latitude to the electron flux measured along the same field
line at low latitude. The high latitude measurements represent electrons with small pitch
angles and the low latitude measurements represent electrons with larger pitch angles.
The authors used this method to study the pitch angle distributions for 4 storms at the L
value of peak flux. The study showed a similar evolution of pitch angle distributions in
each storm. At energies >2.2 MeV the pitch angle distributions became more peaked at
90o after the storm main phase. The peaked distributions slowly became more isotropic
after periods of 3-6 days. The authors concluded that the peaked distributions resulted
from an initial increase in radial diffusion followed by a secondary acceleration due to
processes that violated the first adiabatic invariant. We expand this study by investigating
pitch angle distributions using a superposed epoch analysis including many storms and
covering L values ranging from 2-6. We also compare PADs of electrons during storms
with large flux enhancements to those without. In addition, we investigate local time
asymmetries to specifically test the whistler/EMIC wave model.
5.5 Method: Identifying PADs Associated with Relativistic Electron Acceleration
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We identify characteristics of PADs associated with relativistic electron
acceleration using superposed epoch analysis of 82 storms. The storms occurred between
January 1997 and July 2000 and were identified by minima in Dst <-50 nT. The criterion
for identifying storms required that Dst increase above –15 nT before the next storm was
counted.
Our goal is to identify pitch angle changes unique to electron acceleration.
However, chapter 3 established that some storms result in significant electron
acceleration while others do not. Therefore, we divide the storms into two groups; those
with relativistic electron flux enhancement and those without. We use the total flux of
electrons per orbit measured by HIST with energy >2.0 MeV to identify electron flux
enhancement events. The flux is measured at 4 L ranges; 2.75<L<3.25, 3.75<L<4.25,
4.75<L<5.25 and 5.75<L<6.25. We define relativistic electron flux enhancement events
as storms where the average > 2.0 MeV HIST electron flux 1-3 days after minimum Dst
is more than twice the average > 2.0 MeV flux 1-3 days prior to minimum Dst at more
than 2 of the 4 L ranges. The criterion separates the storms into 40 relativistic electron
flux enhancement events and 42 non-events. The storms are plotted for each year in
Figures 7-10 with events marked by asterisks and non-events marked by circles.
We compare the PADs of events and non-events at L values from 2-6 and energy
from 0.67-7.2 MeV using an anisotropy indicator. An anisotropy indicator gives a
quantitative description of the shape of a PAD. We are interested in comparing contours
of constant phase space density to diffusion curves predicted by whistler and EMIC
waves. Therefore, we wish to know how the flux near 90o pitch angle compares with flux
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at smaller pitch angles. Thus we define our anisotropy indicator, A, based on contours of
constant phase space density as
An 
p (60     90  , n)
p (30     60  , n)
(6)
where p is the average momentum corresponding to a particular phase space density
contour level (n) over the specified range of  where  is pitch angle. The anisotropy in
momentum can be compared to diffusion curves calculated by Summers [1998].
We calculate An for 14 contours of constant phase space density from the HIST
data as shown in Figure 11. The contour levels are chosen so that the momentum at 90o
corresponds to one of the 14 energies measured by HIST. For example, the magenta
contour has p/m0c=7.3 at 90o and corresponds to an energy of 3.3 MeV which is the 10th
energy channel of HIST. The anisotropy ratio of the magenta contour is 1.02 and nearly
isotropic. An isotropic distribution has an anisotropy ratio, A=1.0. A distribution peaked
at 90o has A>1.0 and a butterfly distribution has A<1.0.
The pitch angle ranges used in our anisotropy indicator were chosen so as not to
include electrons in the loss cone. Loss cone data are excluded because HIST measures
electrons in 22o pitch angle bins and does not resolve the loss cone. In general, we do not
expect electrons near the loss cone to be accelerated by the whistler/EMIC wave
acceleration mechanism. The loss of electrons to the atmosphere forms a steep phase
space density gradient near the loss cone. The gradient is such that electrons
predominantly scatter to smaller pitch angles and lower energy.
In analyzing the dependence of PADs on local time, we seek to avoid effects
arising from magnetospheric geometry such as drift shell splitting. Drift shell splitting
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modifies pitch angle distributions predominantly in the noon/midnight region [Selesnick
et al., 2002]. It produces butterfly distributions on the nightside and pancake distributions
on the dayside. For this reason we compare PADs only in the dawn/dusk sectors of the
magnetosphere and calculate anisotropy ratios only for Polar orbits in local time sectors
from 4-8 and 16-20 MLT.
5.6 Results: PADs Associated with Relativistic Electron Acceleration
5.6.1 Events
Anisotropy indicators for electron event cross-sections in the Y-Z plane
(geomagnetic coordinates) are sorted relative to the time of minimum Dst and plotted in
Figure 12. The anisotropies are plotted as superposed epoch for several days including
the day of the storm. This type of presentation is not possible with most other spacecraft
data that do not have the extensive latitude, radial distance and local time coverage of
Polar. The figure shows anisotropy of electron flux enhancement events for contours
corresponding to E=3.3 MeV at 90o pitch angle. The figure shows that A increases after
the storm between L=3-6 and at all latitudes. An increase of A signifies PADs more
peaked at 90o during storms.
Figure 13 plots the anisotropy of electron flux enhancement events measured
along the equator for 12 energies measured by HIST. Thick red traces plot the mean
value three days before the storm minimum Dst and thick black traces plot the mean
value 3 days after minimum Dst. Thin black and red traces plot the standard error of the
means. Only values prior to minimum Dst appear in panels A-I that plot anisotropies of
0.7-2.7 MeV electrons. For these lower energy channels, we exclude values measured
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after minimum Dst because flux levels become high and the data are suspect as discussed
in Appendix A. Shaded gray areas mark the approximate region of the inner electron belt.
For this analysis we are concerned only with the outer belt. Figure 13 shows that
anisotropies of contours corresponding to E=>4 MeV at 90o all respond similarly during
storms with relativistic electron flux enhancements. The anisotropies increase with PADs
becoming more peaked at 90o after storm minimum Dst.
To lend support to this result we show pre- and post-storm raw pitch angle data
and energy spectra for an example event that contributed to the statistical study. Figure
14 panels A-F plot differential flux versus pitch angle at fixed energy before and after the
storm that occurred on Feb 18, 1999. In this figure we present only data taken when Polar
is within 2o of the magnetic equator and between L=4-4.5. Figure 14 panel G plots the Dst
index. Vertical lines mark the times of the equatorial crossings. The figure shows the flux
peaking at 90o pitch angle in panels E and F taken 3-4 days after the storm minimum Dst.
Figure 15 plots the energy spectra in the same format throughout the storm.
Panels E and F show the spectra becoming harder in the 3 to 4 days after the storm. Both
the changing pitch angles and energy spectra contribute to the increased post-storm
anisotropy demonstrated by the statistical study.
Figure 16 plots the contours of constant phase space density constructed from the
data shown in the previous two figures. Panels E and F show the anisotropy increase 4-5
days after the storm.
5.6.2 Non-Events
Figure 17 shows anisotropy during storms without electron flux enhancement
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plotted in the same format as Figure 18. The figure shows that even in these non-events
the anisotropy of contours corresponding to E>4.0 MeV at 90o increases. To confirm the
result we show pre- and post-storm raw pitch angle data and energy spectra for an
example event that contributed to the statistical study. Figure 18 panels A-F plot
differential flux versus pitch angle at fixed energy before and after the storm non-event
that occurred on August 8, 1998. We show only data taken when Polar is within 2o of the
magnetic equator between L=4-5. Figure 18 panel G plots the Dst index with vertical
lines marking the times of each equatorial crossing. (Figures 18-20 will be added shortly)
Figure 19 plots the energy spectra for the same event. Both the changing pitch angles and
energy spectra contribute to the post-storm anisotropy increase shown in the statistical
study. Figure 20 plots the contours of constant phase space density constructed from the
data shown in the previous two figures.
5.6.3 Event and Non-Event Comparison
Figure 21 compares event and non-event anisotropy. The red traces plot the
difference of event and non-event anisotropies pre-storm (Anon-event-Aevent). The green
traces plot the difference of event and non-event anisotropies post-storm. There is no
systematic difference between event and non-event anisotropy.
5.6.4 Dawn-Dusk Comparison
Next we compare dawn and dusk anisotropy during the storms. Figure 22
compares dawn and dusk anisotropy for storms with electron flux enhancements. Red
traces plot dusk anisotropies and blue traces plot dawn anisotropies. The figure shows no
differences between dawn and dusk greater than the standard error. Figure 23 compares
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dawn and dusk anisotropy for storms without electron flux enhancements. No dawn/dusk
differences exist in either events or non-events.
5.7 Discussion: Comparison of data with the whistler/EMIC wave model
We now evaluate the relevance of whistler/EMIC electron acceleration by
comparing the measured anisotropies with the model predictions. The superposed epoch
analysis of PADs revealed three features.
1) PADs associated with >3.3 MeV electrons become more peaked at 90 degrees
during the recovery phase of storms whether or not the flux is enhanced.
2) No dawn/dusk PAD asymmetry is observed.
3) No systematic difference between event and non-event anisotropy is observed.
The observations relate to the whistler/EMIC wave acceleration as follows. The
mechanism predicts that dawnside whistler waves make PADs more peaked at 90o and
duskside EMIC waves make more isotropic PADs. The first observation is consistent
with PADs predicted by interaction with whistler waves. However, restoration of isotropy
on the dusk side by EMIC waves is necessary for the mechanism to continue to be
effective. Observation 2 shows no tendency towards greater dusk-side isotropy.
Two possibilities explain the lack of observed dawn-dusk asymmetry. One possibility
is that an asymmetry exists that is below the detection level of our method. The standard
error of the plotted mean values gives an estimate of our detection threshold. The
standard error is ~  1 %. As discussed previously, the relativistic electrons have drift
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periods of less than ten minutes and spend only a limited time interacting with each type
of wave. The interaction time may be short enough that only small modification of the
pitch angle distributions occurs. However, in this model the amount of acceleration is
defined by the dawn-dusk pitch angle asymmetry. An asymmetry less than 1 % which
would not be detectable, limits the rate of acceleration per electron drift orbit. The other
possible explanation of the absence of dawn-dusk asymmetry is that such asymmetry
does not exist and the whistler/EMIC wave mechanism is not responsible for accelerating
electrons.
Observation 3 states that there is no systematic difference between event and nonevent anisotropy. The result is surprising because one might expect anisotropy to be
affected only in those storms accompanied by electron acceleration. One explanation of
the result is that our method of separating storms does not correctly differentiate between
storms with and without electron acceleration. Although separating storms into those with
and without flux enhancements seems straightforward, the criterion is subjective. As
described in section 5.5 our definition of acceleration events was based on a comparison
of pre- and post-storm flux levels. But there are other ways to define acceleration events
and non-events. For example, O’Brien et al. [2001] performed a superposed epoch
analysis using relativistic electron flux measured at geosynchronous orbit that compared
storms with electron flux enhancements to those without. The study did not consider prestorm flux levels when defining electron flux enhancement events and non-events.
Instead, flux enhancement events were defined based solely on post-storm flux levels. An
event was defined as a storm with post-storm flux levels exceeding a threshold value. It
may seem counterintuitive to measure electron flux enhancements without considering
16
the pre-storm flux level. However, McAdams et al. [2001] examined electron flux at L
values from 4-7 measured by GPS, LANL, SAMPEX and Polar and found a poor
correlation between pre- and post-storm flux levels. One interpretation of the result is that
most electrons in the radiation belts are lost during the main phase of the storm. The poststorm flux level is then independent of the pre-storm levels.
To test whether the similarity of event and non-event anisotropy in our study results
from poor selection criterion we repeated the study using 4 alternate definitions of storm
events and non-events. One definition was similar to that used by O’Brien et al. [2001b]
and relied only on post-storm flux levels. We defined events using LANL > 2MeV
electron flux measured at geosynchronous. We used noon-reconstructed fluxes obtained
from the PEER method [O’Brien et al., 2001a] to remove the local time asymmetry of the
data. We defined relativistic electron flux enhancement events as storms with LANL
>2MeV average electron flux greater than 0.5 /cm2-s-str-keV two to three days after
minimum Dst. The events and non-events determined using this new definition are shown
in Figure 22. We repeated the HIST anisotropy study using the new set of events and
non-events. The final results were unchanged. PADs became more peaked at 90o poststorm in both events and non-events. We also repeated the study using events and nonevents defined by flux increases measured by Polar at L=3,4 and 5 separately. Events
were defined by post-storm fluxes two times higher than pre-storm flux. None of the
event selection criteria resulted in event and non-event anisotropy differences.
The consistent lack of anisotropy difference between events and non-events suggests
that the result is not due to a poor selection criterion. An alternative explanation for the
similar response is that some acceleration process occurs in both events and non-events
17
that modifies anisotropy but is less effective at accelerating electrons in the non-events.
In the case of the whistler/EMIC wave mechanism the amount of acceleration depends on
the strength of the waves and the resulting dawn-dusk asymmetry. If the asymmetry is
below our detection threshold we would observe no difference between event and nonevent anisotropy.
5.8 Summary
Enhanced anisotropy is relevant to various mechanisms that rely on pitch angle
scattering to accelerate electrons in the inner magnetosphere. In particular, the
EMIC/whistler wave mechanism would yield A>1.0 if whistler wave scattering is more
effective than EMIC wave scattering. In the extreme case of intense whistler and EMIC
wave scattering the mechanism predicts a dawn-dusk asymmetry. Pitch angle
distributions of electrons corresponding to energy > =4.0 MeV measured by Polar show
distributions with A>1.0 but no dawn-dusk asymmetry greater than the standard error.
The lack of an observed asymmetry does not exclude the whistler/EMIC wave
mechanism. However, an asymmetry less than 2% limits the rate of acceleration provided
by the whistler/EMIC wave mechanism. This limit may be useful to theorists modeling
electron acceleration by this mechanism. The evolution of PADs during storms is
consistent with but does not provide conclusive support for the whistler/EMIC wave
method.
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Figure Captions
Figure 1. Schematic of resonance conditions in the non-relativistic (Panel A) and
relativistic (Panel B) limit. In the non-relativistic limit particles with parallel velocity
equal to the wave phase velocity are in resonance with the wave. Perpendicular velocity
is not constrained. This resonance condition plotted as a function of perpendicular and
parallel velocity is a straight line. In the relativistic limit particles must have specific
parallel and perpendicular velocity to be in resonance. This condition plotted as a
function of parallel and perpendicular velocity is a semi-ellipse.
Figure 2. Illustration of mapping between electron flux and contours of constant phase
space density. The plots are made using simulated data with low pitch angle anisotropy
and a soft energy spectrum. Panel A shows differential electron flux at fixed energies
versus pitch angle. Panel B shows differential electron flux at fixed pitch angle versus
energy. Panel C plots the data as contours of constant phase space density as a function of
perpendicular ( v / c ) and parallel velocity ( v|| / c ). Panel D plots contours of constant
phase space density as a function of perpendicular ( p  / m0 c ) and parallel momentum
( p|| / m0 c ).
Figure 3. Illustration of mapping between electron flux and contours of constant phase
space density. The plots are made using simulated data with high pitch angle anisotropy
and a soft energy spectrum. Panel A shows differential electron flux at fixed energies
versus pitch angle. Panel B shows differential electron flux at fixed pitch angle versus
energy. Panel C plots the data as contours of constant phase space density as a function of
22
perpendicular ( v / c ) and parallel velocity ( v|| / c ). Panel D plots contours of constant
phase space density as a function of perpendicular ( p  / me c ) and parallel momentum
( p|| / c ).
Figure 4. Illustration of mapping between electron flux and contours of constant phase
space density. The plots are made using simulated data with low pitch angle anisotropy
and a hard energy spectrum. Panel A shows differential electron flux at fixed energies
versus pitch angle. Panel B shows differential electron flux at fixed pitch angle versus
energy. Panel C plots the data as contours of constant phase space density as a function of
perpendicular ( v / c ) and parallel velocity ( v|| / c ). Panel D plots contours of constant
phase space density as a function of perpendicular ( p  / m0 c ) and parallel momentum
( p|| / m0 c ).
Figure 5. Illustration of mapping between electron flux and contours of constant phase
space density. The plots are made using simulated data with high pitch angle anisotropy
and a hard energy spectrum. Panel A shows differential electron flux at fixed energies
versus pitch angle. Panel B shows differential electron flux at fixed pitch angle versus
energy. Panel C plots the data as contours of constant phase space density as a function of
perpendicular ( v / c ) and parallel velocity ( v|| / c ). Panel D plots contours of constant
phase space density as a function of perpendicular ( p  / me c ) and parallel momentum
( p|| / c ).
23
Figure 6. Schematics of diffusion curves and contours of constant phase space density
illustrating electron distribution energy gain (Panel A), energy loss (Panel B), and no
energy change (Panel C). Diffusion curves are plotted in black and contours of constant
phase space density (PSD) are plotted in gray. Diffusion of electrons along the diffusion
curves is represented by black arrows. Green arrows show the gradient in phase space
density and red arrows show the dominant direction of diffusion.
Figure 7. Plot showing identification of storms with (events) and without (non-events)
electron flux enhancements. The top panel plots >2MeV electron flux (#/cm2-s-str MeV)
measured by the Polar HIST instrument during 1997 at L=3 (green), 4 ( blue), 5 (black)
and 6 (red). The bottom panel plots the Dst (nT) index. Events are marked with red
asteriks and non-events are marked with red circles.
Figure 8. Plot showing identification of storms with (events) and without (non-events)
electron flux enhancements. The top panel plots >2MeV electron flux (#/cm2-s-str MeV)
measured by the Polar HIST instrument during 1998 at L=3 (green), 4 ( blue), 5 (black)
and 6 (red). The bottom panel plots the Dst (nT) index. Events are marked with red
asteriks and non-events are marked with red circles.
Figure 9. Plot showing identification of storms with (events) and without (non-events)
electron flux enhancements. The top panel plots >2MeV electron flux (#/cm2-s-str MeV)
measured by the Polar HIST instrument during 1999 at L=3 (green), 4 ( blue), 5 (black)
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and 6 (red). The bottom panel plots the Dst (nT) index. Events are marked with red
asteriks and non-events are marked with red circles.
Figure 10. Plot showing identification of storms with (events) and without (non-events)
electron flux enhancements. The top panel plots >2MeV electron flux (#/cm2-s-str MeV)
measured by the Polar HIST instrument during 1997 at L=3 (green), 4 ( blue), 5 (black)
and 6 (red). The bottom panel plots the Dst (nT) index. Events are marked with red
asteriks and non-events are marked with red circles.
Figure 11. Schematic illustrating the definition of the anisotropy indicator as explained in
the text. The figure shows contours of constant phase space density plotted as a function
of perpendicular ( p  / m0 c ) and parallel momentum ( p|| / c ).
Figure 12. Superposed epoch analysis of anisotropy using data from all electron flux
enhancement events shown in Figure 8. Each panel shows anisotropy plotted in dipolar
coordinates in the Y-Z plane. Anisotropy is indicated by the color scale. The anisotropies
are plotted for electrons with energy equal to 4.0 MeV at 90o. The thin blue traces plot
dipole magnetic field lines. Panels A-C show anisotropy 3 days prior to minimum Dst.
Panels D-I show anisotropy 6 days after the storm.
Figure 13. Equatorial anisotropy during storms with relativistic electron flux
enhancements. Each panel plots equatorial anisotropy versus Y in dipole coordinates.
Thick red traces plot mean anisotropy 3 days prior to minimum Dst. Thin red traces plot
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the standard error of the mean. Black traces plot mean anisotropy 5 days after minimum
Dst. Thin black traces plot the standard error of the mean.
Figure 14. Panels A-F plot differential flux (#/cm2-s-str MeV) versus pitch angle
measured at fixed energy throughout the storm event that occurred on February 18, 1999.
The data are measured at the equator between L=4-4.5. Panel G plots the Dst index (nT).
Vertical lines mark the time of the equator crossings.
Figure 15. Panels A-F plot energy spectra throughout the storm event that occurred on
February 18, 1999. The data are measured at the equator between L=4-4.5. Panel G plots
the Dst index (nT). Vertical lines mark the time of the equator crossings.
Figure 16. Panels A-F plot contours of constant phase space density throughout the storm
event that occurred on February 18, 1999. The data are measured at the equator between
L=4-4.5. Panel G plots the Dst index (nT). Vertical lines mark the time of the equator
crossings.
Figure 17. Equatorial anisotropy during storms without relativistic electron flux
enhancements. Each panel plots equatorial anisotropy versus Y in dipole coordinates.
Thick red traces plot mean anisotropy 3 days prior to minimum Dst. Thin red traces plot
the standard error of the mean. Thick black traces plot mean anisotropy 5 days after
minimum Dst. Thin black lines plot the standard error.
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Figure 18. Panels A-F plot differential flux (#/cm2-s-str MeV) versus pitch angle
measured at fixed energy throughout the storm event that occurred on February 18, 1999.
The data are measured at the equator between L=4-4.5. Panel G plots the Dst index (nT).
Vertical lines mark the time of the equator crossings.
Figure 19. Panels A-F plot energy spectra throughout the storm event that occurred on
February 18, 1999. The data are measured at the equator between L=4-4.5. Panel G plots
the Dst index (nT). Vertical lines mark the time of the equator crossings.
Figure 20. Panels A-F plot contours of constant phase space density throughout the storm
event that occurred on February 18, 1999. The data are measured at the equator between
L=4-4.5. Panel G plots the Dst index (nT). Vertical lines mark the time of the equator
crossings.
Figure 21. Comparison of anisotropies during storms with (events) and without (nonevents) electron flux enhancements. Red traces plot the subtraction of event anisotropies
from non-event anisotropies in the 3 days prior to minimum Dst. Green traces plot the
subtraction of event anisotropies from non-event anisotropies in the 5 days after
minimum Dst.
Figure 22. Comparison of dawn and dusk anisotropies during events. Red traces plot dusk
anisotropies and blue traces plot dawn anisotropies.
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Figure 23. Comparison of dawn and dusk anisotropies during non-events. Red traces plot
dusk anisotropies and blue traces plot dawn anisotropies.
Figure 24. Plot showing identification of storms with (events) and without (non-events)
electron flux enhancements. Panels 1A-4A plot LANL geosynchronous 1.8-3.5 MeV
electron flux (#/cm2-s-str-keV). Panels 1-4B plot the Dst (nT) index. Events are marked
with red asteriks and non-events are marked with gray asteriks.
28