Download 1.1 Motivation - the Institute of Geophysics and Planetary Physics

Chapter 1
Introduction
“It has occurred to me recently that saying that physics is an exact science is like saying
that Shakespeare was a precise speller. This he perhaps was – but he was a great deal
more. Likewise, we try to be precise spellers when the occasion demands in science, but
the exploratory phases of science are intuitive, subjective, controversial, and often
inconclusive – in short, anything but exact.”
- Dr. Van Allen
1
1.1 Motivation
On January 31st, 1958 the Explorer 1 spacecraft was launched into earth orbit with
a cosmic ray experiment designed by Dr. J. Van Allen. The experiment was intended to
measure cosmic rays but instead led to the discovery of bands of high energy charged
particle radiation encircling the earth now known as the Van Allen radiation belts. The
unintentional discovery of the radiation belts prompted many questions, the most
fundamental being, ‘How are particles in the earth’s magnetosphere accelerated to
relativistic energies to form the radiation belts?’. Since the discovery of the radiation
belts much descriptive knowledge has been gathered but this fundamental question
remains unanswered.
In the past four decades more than 20 spacecraft have probed the earth’s
magnetosphere providing the following qualitative picture of the radiation belts. The belts
consist of protons and electrons trapped in the earth’s magnetic field that form a torus
shaped region extending from ~1.5 Re to ~10 Re as shown in Figure 1. The protons form
only a single belt. The electrons, in contrast, form two belts separated at ~2.5 Re by a
region of enhanced losses known as the slot region [Lyons and Thorne, 1973]. This slot
region separates the inner (1.5-2.5 Re) and outer (2.5 – 10 Re) belts. The name “radiation
belts” implies high energy but no exact energy differentiates these particles from the rest
of the magnetospheric plasma. The name applies to those particles with energies high
enough to penetrate solid material and cause radiation damage to spacecraft and humans
alike. Generally, particles with energies >~0.1 MeV fall into this category. Electrons with
2
such high energies and small mass have velocities approaching the speed of light. At
these velocities relativistic effects become significant, therefore the electrons of the
radiation belts are often referred to as relativistic electrons. The flux of particles at fixed
energy in the belts is structured and depends on the radial distance from the earth. The
observed fluxes increase with decreasing radial distance as demonstrated in Figure 2
[Vette, 1991]. As seen in the figure, the energy range and spatial characteristics of the
protons and electrons are quite different and suggest different formation mechanisms. For
this reason they are usually studied separately. This thesis focuses only on the formation
of the outer electron radiation belt.
The static description given above hides the most surprising and interesting aspect
of the electron belts, which is their extreme temporal variability. Observations, such as
those plotted in Figure 3, show that the fluxes of relativistic electrons increase by orders
of magnitude on short timescales of about 1 day [Baker et al., 1994b]. These rapid flux
increases are of great interest because spacecraft, on which our society is increasingly
dependent, have failed during periods of high electron flux [Baker et al., 1994a]. The data
plotted in Figure 3 are taken from Las Alamos National Laboratories (LANL) spacecraft
that reside at geosynchronous altitude (6.6 Re). A satellite orbiting at this altitude will
remain at constant geographic longitude above the earth. The region is therefore heavily
populated by communication satellites. As Figure 3 demonstrates, these spacecraft are
often subjected to high fluxes of relativistic electrons and are at risk of being damaged.
The rapid flux increases evident in Figure 3 are surprising because common
sources of energy in the magnetosphere do not yield MeV electrons. For example,
electrons drifting in the electric field imposed by the solar wind across the magnetosphere
3
typically gain only ~100 keV in energy. Substorms, which transfer energy from the solar
wind to the earth’s magnetosphere, often inject electrons with 10-100 keV energy. But
these injections are not usually coincident with increases in > 1MeV electron flux [Birn et
al.,1997; Li et al., 1998]. Even assuming large or multiple substorms could create MeV
electrons as one simulation suggests [Kim and Chan, 2000], it is difficult to transport the
electrons from large radial distances (e.g. 10Re to 4 Re) on the observed short timescales.
Although accelerating electrons to relativistic energies appears difficult, theorists
have proven worthy of the challenge. Many theoretical models of relativistic electron
acceleration have been proposed. However, determining which, if any, of these
theoretical mechanisms causes electron acceleration in the magnetosphere has been
problematic.
The goal of this thesis is to investigate two models of relativistic electron
acceleration; the enhanced ultra low frequency (ULF) wave driven radial diffusion model
and the whistler/electromagnetic ion cyclotron (EMIC) wave acceleration model. The
models are tested using electron data obtained by the High Sensitivity Telescope (HIST)
instrument [Blake et al., 1995] onboard the Polar spacecraft described in Appendix A and
Chapter 2. Appendix A describes how the instrument functions and difficulties with the
data. Chapter 2 describes the transformation of the data to a more useful dataset of phase
space density of electrons with constant adiabatic invariants. The remainder of the thesis
is devoted to testing the two models. Below we introduce observations of typical
relativistic electron acceleration events followed by a brief description of some of the
acceleration models proposed to explain these events.
4
1.1 Observations: Anatomy of a Relativistic Acceleration Event
Figure 4 provides a more detailed look at a typical relativistic electron acceleration
event. Figure 4, panel A, shows fluxes of relativistic electrons with energies from 0.7-7
MeV measured by the LANL spacecraft for a 12 day time period. Panel B shows the Dst
index for the same time period that indicates geomagnetic activity. This figure serves as
an example of how the fluxes of relativistic electrons change with time. Changes in
relativistic electron fluxes often occur during geomagnetically active periods known as
geomagnetic storms.
The ultimate cause of geomagnetic storms can be traced back to the sun. Although
the visible photosphere of the sun reaches only to 4.6X10-3 AU (1AU=distance from the
sun to earth), the influence of the solar magnetic field extends to ~90-120 AU,
encompassing the planets. From early on it was clear that a connection existed between
the sun and geomagnetic activity at the earth. During several harrowing expeditions to
study the aurora of the northern polar regions, Birkeland [1908] observed a 27 day
geomagnetic activity cycle similar to the 27 day solar rotation period. He surmised that a
continuous stream of particles streaming from the sun was responsible for much of the
geomagnetic disturbances observed at earth. Parker [1958] presented fluid theory that
predicted a fast solar wind emanating from the suns corona. In 1962, Mariner 2 made insitu measurements of the solar wind definitively confirming the theory [Neugebauer et
al., 1962]. The solar wind measured at 1 AU travels at speeds of ~412 km/s carrying with
it a magnetic field ( ~5.2 nT ) and plasma (density ~7 cm-3) that continuously buffet the
magnetic field of the earth [Ness et al., 1971]. Energy is transferred from the sun to the
earth via the solar wind connection.
5
Figure 5 depicts how the solar wind magnetic field influences the earth’s
magnetosphere through a multi-step process that causes a geomagnetic storm. The earth’s
magnetosphere is separated from the flowing plasma and magnetic field of the solar wind
by a thin boundary called the magnetopause as labeled in Figure 5. When the magnetic
field of the solar wind and earth are oriented in opposite directions, the field lines merge
allowing efficient energy transfer from the solar wind to the earth across the
magnetopause boundary [Dungey, 1961; Gonzalez et al., 1974]. This first step is labeled
‘Step 1’ in Figure 5. Gonzalez and Tsuratani [1987] demonstrated that large storms occur
when the solar wind Bz component remains southward and magnetic field merging
continues for more than 3 hours. The interaction imposes a strong dawn to dusk
convection electric field as well as impulsive electric fields across the earth’s
magnetosphere [Boonsiriseth et al, 2001; Weygant et al., 1998] that modify particle drifts
(Step 2). The enhanced electric fields cause electrons and protons in the geomagnetic tail
of the earth to drift earthward (Step 3). The relative importance of the two types of
electric fields on the particle population is still debated [Daglis, 2001]. The gradient and
the curvature of the earth’s magnetic field causes eastward drift of electrons and
westward drift of protons. Some of these particles end up on roughly circular drift paths
forming a ring current about the earth (Step 4) [Fok et al., 2001; Korth et al., 2000].
Others continue drifting sunward eventually encountering the magnetopause boundary
where they are lost from the magnetosphere [Kawasaki and Akasofu, 1971, Grafe, 1999].
These particles that do not complete a full drift about the earth form what is the called the
partial ring current. The ring and partial ring currents make a magnetic field directed
6
southward at the earth’s surface. An enhancement of these currents is a geomagnetic
storm.
Geomagnetic storms are identified by the storm time disturbance index or Dst
index shown in Figure 4, panel B. The Dst index is the average of the horizontal north
south component of the earth’s magnetic field with a baseline subtracted. It is measured
at 4 ground stations whose geographic positions are given in Figure 6. A negative
excursion in the Dst index indicates enhancement of the ring and partial ring current and
thus the occurrence of a storm.
Storms are divided into two activity phases, the main phase and the recovery
phase. During the main phase the Dst index becomes more negative as the ring and
partial ring currents intensify. The timescale of the main phase is approximately one day.
The recovery phase begins as the currents diminish and the Dst index moves slowly back
to zero over a period of about 5 to 6 days.
The changes of the relativistic electron fluxes can also be described in terms of
the main and recovery phase of a storm. At times the changes in the flux of relativistic
electrons track the changes in the Dst index, as seen most clearly in Figure 4 during the
main phase of the storm. Typically, during the main phase, the decrease in Dst is
accompanied by a decrease in the flux of relativistic electrons. During the recovery phase
the electron flux increases and sometimes surpasses the pre-storm flux levels by orders of
magnitude. The flux increase during the recovery phase is not consistent at all energies.
The low energy electron flux increases most rapidly while the high energy electron flux
increases more slowly [Li et al., 1999].
7
Although geomagnetic storms require enhanced energy input from the solar wind
to the magnetosphere not all storms result in a relativistic electron flux increase. Flux
increases occur only when various solar wind parameters are favorable. For example,
several studies have noted that electron flux increases are well correlated with increases
in solar wind velocity [Paukilas and Blake,1979; Baker, 1996; Baker et al., 1997].
O’Brien et al. [2000] went further and identified several more solar wind features that
accompany relativistic electron flux increases. The study compared geomagnetic storms
with electron flux increases at geosychronous orbit to those without. The authors found
that those storms with no relativistic flux increases had solar wind density increases prior
to the time of minimum Dst and sharp northward turnings of the solar wind Bz
component within 12 hours after minimum Dst. Along with these observations have
come models hoping to explain them as detailed below.
1.2 Models
The many models of relativistic electron acceleration will be described here in
terms of particle adiabatic invariants. Adiabatic invariants are quantities that describe the
general characteristics of the complicated motion of particles in the magnetosphere.
Figure 7 panel A shows an example of the motion of an electron in a dipole magnetic
field. The complex motion can be understood more simply by separating it into three
types of motion that occur on different time scales: the gyro-motion of a particle about a
field line, the bounce motion of a particle along a field line and the azimuthal drift of a
particle about the earth (Figure 7) [Roederer, 1970; Schulz and Lanzerotti, 1974]. When
changes in the magnetic field are slow compared with the period of the motion three
8
quantities corresponding to each type of motion are conserved. These quantities are
called the adiabatic invariants and are defined as follows. The first invariant of the
motion is associated with the cyclotron motion of the electron about a field line (Figure 7
panel B) and is given by:

First invariant:
p 2
2 m0 B
(1)
Here p  is the relativistic momentum in the direction perpendicular to the magnetic
field, m0 is the rest mass of the electron, and B is the field magnitude.
The second invariant corresponds to the bounce motion of a particle along a field
line (Figure 7 panel C) and is given by:
J   p||ds
Second Invariant:
(2)
where p|| is the particle momentum parallel to the magnetic field and ds is the distance a
particle travels along the field line. It is convenient to rewrite the second invariant in
terms of only the magnetic field geometry by the following manipulation. If no parallel
forces act on a particle then momentum is conserved along a bounce path and J=2pI
where p is momentum and I is given as
'
sm
1
B( s )  2

I   1 
 ds
Bm 
sm 
(3)
Here sm is the distance of the particle mirror point, B(s) is the field strength at point s and
Bm is the mirror point magnetic field strength. If the first invariant is conserved then K, as
defined below, is also conserved.
9
K
J
2 2 m0 
'
sm
1
 I Bm   Bm  B( s) 2 ds
(4)
sm
where m0 is the rest mass of an electron. Throughout this thesis we will refer to K when
speaking of the second invariant.
The third and final invariant corresponds to the drift motion of a particle about the
earth (Figure 7 panel D) and is given by:
Third invariant:
   A dl
(5)
In this equation A is the magnetic vector potential and dl is the curve along which lies
the guiding center drift shell of the electron. Using Stokes theorem the third invariant can
be written as,
   (  A)dS   BdS
(6)
where B is the magnetic field and dS is area. Therefore, conservation of this invariant
requires that an electron always enclose the same amount of magnetic flux as it drifts
about the earth. In a dipole field this is equivalent to saying that the electron remains at
fixed radial distance. The Roederer L parameter, commonly written as L*, is another
useful form of the third invariant and is written as
L* 
2M
R E
(7)
10
where M is the magnetic moment of the earth’s dipole field. The L* parameter is the
radial distance to the equatorial location where an electron would be found if all external
magnetic fields were slowly turned off leaving only the internal dipole field.
An irreversible increase of relativistic electron flux can occur only when one of
the above invariants is violated. Therefore, acceleration mechanisms are described by
whether they perturb the gyro-motion, bounce motion, or drift motion of an electron
violating the first, second, or third adiabatic invariant. Below is a list of proposed
acceleration mechanisms categorized by the invariants that each violates.
Violation of the third invariant (External Source Acceleration Models):
-Enhanced radial diffusion driven by ULF waves [Falthammer, 1965; Elkington et
al.,1999].
-Radial diffusion with substorm accelerated electron seed population [Kim and Chan,
2000].
Violation of the first and second invariants (Internal Source Acceleration Models):
-Whistler/EMIC wave acceleration [Summers et al., 1998].
-Non-linear whistler wave acceleration [Albert, 2000].
-ULF/VLF wave recirculation [Liu et al., 1999].
-ULF fast mode wave acceleration [Summers et al., 2000].
-Multi-mode diffusion simulated by the Salammbo code [Beutier et al., 1995; Bourdarie
et al., 1996; Bourdarie et al., 1997].
11
The two categories of acceleration mechanisms above are described as “external
source” and “internal source” mechanisms. The external source mechanisms perturb the
radial distance of an electron and violate the third invariant. These mechanisms cause
flux increases in the inner magnetosphere by transporting a source of electrons at large L
radially inward and concurrently accelerating them. The internal source mechanisms
cause flux increases by locally accelerating a source of electrons already present in the
inner magnetosphere.
The broad goal of this thesis is to establish whether relativistic electron flux
increases are caused by an internal or external source acceleration mechanism. We
achieve this goal using electron data from the Polar HIST instrument. We transform the
HIST data of electron flux measured at constant energy into a more applicable dataset of
phase space density (chapter 2). In this form, the nature of allowed acceleration
mechanisms can be more readily identified. We begin by examining the relevance of one
external acceleration model, the ULF wave enhanced radial diffusion model (chapter 3).
The results show that this external acceleration model does not explain enhancements at
low L*. Next we analyze radial profiles of electron phase space density (chapter 4). All
external acceleration mechanisms require a population of electrons that begin at large L*
values. The observations of electron phase space density radial profiles are not consistent
with external acceleration models. Therefore, the final chapter focuses on differentiating
between internal acceleration models (chapter 5). The internal acceleration models all
violate the first adiabatic invariant and thus cause pitch angle anisotropy. We use
observations of pitch angle anisotropies to evaluate the relevance of several internal
12
acceleration mechanisms. Below we briefly describe each of the models and how they
will be addressed by this thesis.
1.2.1 ULF Wave Enhanced Radial Diffusion Mechanism
The ULF wave enhanced radial diffusion mechanism is one model tested by this
thesis. The mechanism accelerates electrons by a drift resonance with ULF waves.
Electrons drifting about the earth with the frequency of a ULF wave present in the
magnetosphere experience an electric field that moves some of the electrons radially
inward, violating the third invariant. The electron moves radially inward to a higher
magnetic field region while conserving the first and second invariants and thus results in
acceleration. The model has a lengthy history and will be described in greater detail in
chapter 3. The analysis of chapter 3 shows that this model does not explain electron
acceleration at low L* values.
1.2.2 Substorm Acceleration
In the substorm electron acceleration model, electrons are accelerated in the
earth’s geomagnetic tail by induced electric fields resulting from substorm dipolarization.
The accelerated electrons create an energetic seed population which is transported to
small L values by radial diffusion. Kim et al. [2000] have explored the feasibility of
substorm electron acceleration by tracing particle orbits in the magnetohydrodynamic
(MHD) model developed by Birn and Hesse [1996]. The MHD model simulates the
magnetic and electric fields of the magnetotail that result during a substorm. The study
showed that an electron with initial energy of 20 keV located near the substorm
13
reconnection area at X=-20 RE can be transported to X=-10 RE with an increased energy
of ~400 keV. The electron acceleration in the dipolarization region beyond X=-10 RE
does not conserve the first or second invariant. The substorm makes a seed population of
accelerated electrons at X=-10. It is then assumed that the electrons are transported from
X=-10 to X=-6 by processes such as radial diffusion which conserve the first and second
invariant. The final energy of the electron at X=-6 is ~1MeV.
Assuming all electrons in the magnetotail have a similar energy gain, the
substorm acceleration model can easily account for the fluxes of 1 MeV electrons
observed at geosynchronous orbit. However, tracing test particle trajectories shows that
not all electrons are accelerated to high energies. Electrons that spend a substantial
amount of time in the dipolarization region where electric fields are large gain the most
energy. Test particle trajectories show that electrons that make circular trajectories
around local magnetic field enhancements spend substantial time in the dipolarization
region and gain the most energy. The circular trajectories result from magnetic gradient
drifts. The study estimates the percentage of particles that gain significant energy and
find that the substorm of the simulation only provides 2 percent of the relativistic
electrons observed in the radiation belts. However, the authors note that the estimate is
conservative. The model substorm studied extended only 3 RE in the Y direction and was
located at X=-15 RE. A larger substorm such as might be expected to occur during storm
time might accelerate a larger volume of electrons. Likewise, multiple substorms would
increase the number of accelerated electrons.
Ingraham et al. [2001] tests whether the substorm acceleration model is
responsible for relativistic flux enhancements at geosynchronous altitude using multiple
14
satellite data during the March 10, 1998 storm. The LANL data show a continuous
increase of >1 MeV electron flux throughout this storm. Flux increases of >1 MeV
electrons occur coincident with dispersionless injections at lower energies. Dispersionless
injections are used to identify substorms in this study.
The authors propose two possible explanations of the fluctuations and flux
increase of high energy electrons associated with substorms. The first possibility is that
the high energy fluctuations result from magnetic field changes that simply move
electrons. They discount the hypothesis because fluctuations are seen even on the dayside
which should be unaffected by substorm dipolarization and changing tail magnetic fields.
The alternative hypothesis is that each substorm injects high energy electrons. The study
compares the time of electron flux changes seen on the nightside to those on the dayside
for one substorm. There is an energy dependent delay between nightside and dayside flux
increases. The authors argue that the delay is consistent with injected electrons drifting
from the nightside to the dayside magnetosphere. Comparisons between GPS data that
measures off equatorial fluxes and LANL data suggest that acceleration begins first at 90
degrees. The observation is also consistent with expectations of electron acceleration by
substorm injection [Baker et al., 1978].
The authors conclude that in the March 10, 1998 storm frequent substorms
contributed to relativistic electron flux increases. However, the authors remark that some
storms have relativistic electron flux increases but little substorm activity. Investigation
of more storms is required to determine whether substorms frequently contribute to
relativistic electron acceleration.
15
The substorm acceleration model will not be directly tested by this thesis.
However, chapter 4 establishes that the radial profile of electron phase space density is
not consistent with external acceleration mechanisms.
1.3 Internal Acceleration Models
Chapters 3 and 4 of the thesis establish that electrons are not accelerated by the
external source acceleration mechanisms described above. The final chapter is devoted to
differentiating between several internal acceleration mechanisms. All internal
acceleration mechanisms violate the first invariant and cause pitch angle anisotropy.
Therefore, the internal acceleration models are tested by comparing observed pitch angle
anisotropies to predicted anisotropies.
1.3.1 Whistler/EMIC wave electron acceleration mechanism
Our examination of internal acceleration mechanisms will focus mainly on the
Summers et al. [1998] whistler/EMIC wave model. This model is the most complete and
makes testable predictions of pitch angle anisotropy. The model asserts that electrons
gain energy by a two step process involving electron interaction with two types of waves.
First, EMIC waves, acting predominantly on the dusk side of the magnetosphere
[Jordanova et. al, 2001; Braysy et al., 1998], scatter electrons towards small pitch angles
while conserving electron energy. 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]. Whistler waves scatter
16
electrons to larger pitch angles and higher energies. After multiple drift periods, electrons
may gain significant energy. The model is discussed in greater detail in chapter 5.
1.4.2 Non-linear Whistler Wave Acceleration
The non-linear whistler wave acceleration mechanism proposed by Albert [2000]
also relies on whistler waves to accelerate electrons. This model applies to the interaction
of electrons with whistler waves when wave amplitudes are large. The author uses a
Hamiltonian formulation to describe the resonance between electrons and waves. He
calculates the interaction of the electron in the wave fields for whistler waves propagating
along the magnetic field in a slab geometry. The interaction of electrons with the wave
breaks down into two regimes. In the first regime, the electron passes through the
resonance quickly. In this regime the particle energy and pitch angle change in a diffusive
manner. In the second regime, the electron moves along the field with the resonance. In
this regime, the change in energy and pitch angle is not random. If the wave propagates
away from the equator the electrons lose energy. If the wave propagates towards the
equator the electrons gain energy. It is not clear how frequently whistler waves reach
amplitudes large enough to invoke this theory. At present the model does not make
specific predictions of pitch angle anisotropy. This thesis establishes that electrons are
accelerated by an internal mechanism but does not specifically test whether this internal
mechanism is responsible for accelerating electrons.
1.4.3 ULF/VLF Wave Recirculation Mechanism
17
As the name suggests the ULF/VLF wave recirculation model accelerates
electrons through the combined interaction of ULF and VLF waves. Figure 8 is a
schematic demonstrating how the mechanism accelerates electrons. An electron (marked
as a circle with an e in the figure) is pushed radially inward by the azimuthal electric field
of a ULF wave (labeled (1)). The electron gains energy according to the following
equation
 
W   E  vd dt
(8)


where W is energy, E is the electric field of the wave, v d is the drift velocity of the
electron and t is time. Next the electron scatters to smaller pitch angle (labeled (2)). The
drift velocity decreases because of the change to smaller pitch angle. Finally the electron
is pushed radially outward by the electric field of the ULF wave (labeled (3)). The

electron loses energy during this step. However, v d is pitch angle dependent and is
smaller for the small pitch angles of step 3 than for the initial pitch angles so the energy
loss is less than the energy gain. The process results in a net energy gain as described by
the equation below,
 

W   E  (vd1  vd 2 )dt
v d1  v d 2
(9)


where E is the wave electric field and v d is the drift velocity. For a single electron this
process may result in a net energy loss if scattering occurs such that vd2>vd1. Liu et al.
[1999] claim that a statistical ensemble of particles will always gain energy because the
electron energy has a lower limit of 0. They equate this acceleration process to a random
walk problem in one dimension. They argue that a random walk along an infinite line has
18
equal probability of ending up either to the right or left of the starting position. However,
if the line is not infinite but instead bounded by a reflective point at one end the
probability that the random walk ends on the unbounded end asymptotes to 1 as time
goes to infinity. In an analogous sense the statistical ensemble of electrons will move to
higher energy because there is a reflective boundary at energy=0.
Liu et al. [1999] demonstrate the process with a simplified simulation. In the
model, electrons are moved at each step by 0.25L either inward or outward to simulate
the effect of the ULF wave. Periodically during this motion, the pitch angle is perturbed
using a random number generator. The simulation begins with two different initial
electron distributions determined by Cayton et al. [1989]. One initial distribution has
n0=5x10-3 cm-3 and 25 keV temperature and the other has n0=0.1 cm-3. The simulation
initiated with a low density electron population accelerates electrons to flux levels above
103 /cm2-s-sec at 1 MeV after 5.2 hours. The simulation beginning with a high density
electron population accelerates electrons to flux levels above 103 /cm2-s-sec at 1 MeV
after 4.2 hours.
The simulation produces the desired electron acceleration but the authors
recognize that unphysical assumptions have been made. The prescribed pitch angle
diffusion is unrealistic and the 0.25L radial motion of electrons due to ULF waves is
arbitrarily assigned. More accurate representation of the wave particle interactions is
necessary for specific predictions of pitch angle distributions. Chapter 5 of this thesis
determines the type of pitch angle distributions observed during relativistic electron
acceleration events. These observations will be relevant once the model is refined.
19
1.4.4 Fast Mode Wave Electron Acceleration Mechanism
The electron acceleration proposed by Summers and Ma [2000] accelerates
electrons through a gyroresonance with ULF fast mode waves. In general electrons will
be in resonance with a wave and accelerated when the following equation is obeyed,
  k|| v|| 
n e

 
1
1
v
(10)
2
c2
where  is the wave frequency, k|| is the wave number parallel to the magnetic field, v|| is
the particle velocity parallel to the magnetic field, n is the harmonic, e is the
gyrofrequency of the electron and  is the relativistic factor. The fast mode acceleration
mechanism applies when n=0 and the resonance condition is =k||v||. Plugging in the
dispersion relation of a fast mode wave gives v//=vA/cos where vA is the Alfven velocity
and  is the propagation angle relative to the magnetic field. Therefore, resonance and
acceleration occur when the particle velocity is greater than the Alfven speed.
Summers and Ma [2000] calculate the time required to accelerate electrons from
~100 keV to 1 MeV for waves with amplitudes from 10-50 nT at L=3-6.6 and frequencies
from 2-22 mHz. The calculations assume isotropic pitch angle distributions. The
acceleration timescales decrease with increasing wave amplitude and L value. To
accelerate ~100 keV electrons to 1MeV in 10 hours requires wave amplitudes from 129427 nT at L= 3 and 26-87 nT at L=6.6. Several studies have correlated ULF waves with
relativistic electron flux enhancements. However, no attempt has yet been made to
specifically link fast mode compressional waves with relativistic electron flux
enhancement events.
20
Like the last mechanism described this mechanism does not make specific
predictions of pitch angle distributions. A more complete description of the wave particle
interaction is required. The pitch angle distributions during acceleration events described
in chapter 5 will be a useful test of this model as it is more fully developed.
1.4.5 Multi-mode Diffusion
The last mechanism to be discussed is multi-mode diffusion. Multi-mode
diffusion incorporates diffusion in pitch angle and radial position. The Salambo code has
been used to analyze how electrons are accelerated by multi-mode diffusion. The
Salambo code simulates relativistic electron flux enhancements with both types of
diffusion by solving the Boltzmann equation. The model includes radial diffusion from
both magnetic and electric field fluctuations with L dependent diffusion coefficients.
Pitch angle diffusion is confined to the plasmapause. The location of the plasmapause is
varied with the level of geomagnetic activity described by the Kp index. Pitch angle
diffusion coefficients are determined from the measurements of Thorne et al. [1973]. The
model includes frictional terms that describe loss and scattering due to Coulomb
collisions and the newest version of the model incorporates the Volland-Stern potential
electric field.
Analysis of the model results shows that electrons are accelerated by radial
diffusion as well as a combined pitch angle and radial diffusion recirculation mechanism
[Boscher et al., 2000]. The recirculation is described by Figure 9. First radial diffusion
pushes electrons with large pitch angle radially inward while conserving the first and
second invariants causing acceleration (Step 1). Inside the plasmapause the electrons are
21
scattered to smaller pitch angle at constant energy (Step 2). This step changes the
particles first and second invariant. Because there were previously no particles at this new
 and J a radial gradient forms that causes electrons to diffuse radially outward. Once
again the particle scatters towards larger pitch angle and the process repeats. The net
energy gain through one cycle is positive. This mechanism has been criticized because
the confinement of pitch angle scattering to within the plasmapause boundary is artificial.
Observations show that waves that interact with relativistic electrons are present outside
the plasmapause boundary and are enhanced during geomagnetic storms [Meredith et al.,
2001, Lorentzen et al., 2000].
The multi-mode diffusion model makes predictions of pitch angle distributions
near the acceleration region at L=4. The model predicts nearly isotropic pitch angle
distributions at energy <=2.0 MeV. At high energy (E=5.0 MeV) the model predicts pitch
angle distributions highly peaked at 90o. Comparisons with the Polar HIST data can only
be made with the highest energy channel. The characterization of pitch angle
distributions during acceleration events discussed in chapter 5 will be relevant to this
model. However, a more physical description of wave particle interactions must be
incorporated into the model before it can be accurately tested.
1.5 Goal of This Thesis
As discussed above, the broad goal of this thesis is to establish whether relativistic
electron flux increases are caused by an internal or external source acceleration
mechanism. The goal is achieved using data form the Polar HIST instrument. The details
of the instrument and problems associated with the data are documented in Appendix A.
22
Chapter 2 describes how HIST data of electron flux measured at constant energy is
transformed into a more applicable dataset of phase space density that will be used
throughout the thesis. Chapter 3 tests the enhanced ULF wave acceleration mechanism by
examining correlations between ULF wave power and relativistic electron phase space
density. Chapter 4 examines electron phase space density as a function of radial distance
and compares to predictions of external and internal acceleration models. Finally, chapter
5 tests one internal acceleration model by comparing measured pitch angle distributions
to those predicted by the whistler/EMIC wave acceleration model.
23
References
Albert, J.M., Gyroresonant interactions of radiation belt particles with a monochromatic
electromagnetic wave, J. Geophys. Res., 105, 21191, 2000.
Baker, D. N., Solar wind-magnetosphere drivers of space weather, J. Atmos. Terr. Phys.,
58, 1509, 1996.
Baker, D.N., P.R. Higbie, E.W. Hones, Jr., R.D. Belian, High resolution energetic
particle measurements at 6.6 RE, 3 low energy electron anisotropies and short term
predictions, J. Geophys. Res., 83, 4863, 1978.
Baker, D. N., S. Kanekal, J.B. Blake, B. Klecker, and G. Rostoker, Satellite anomalies
linked to electron increases in the magnetosphere, Eos Trans. AGU, 75, 404, 1994a.
Baker, D.N., J.B. Blake, L.B. Callis, J. R. Cummings, D. Hovestad, S. Kanekal, B
Klecker, R. A. Mewaldt, and R. D. Zwickl, Relativistic electron acceleration and decay
time scales in the inner and outer radiaion belts: SAMPEX, Geophys. Res. Lett., 21, 409,
1994b.
Baker, D. N., X. Li., N. Turner, J. Allen, L. Bargatze, J.B. Blake, R. Sheldon, H. Spence,
R. Belian, G.D. Reeves, S. Kanekal, Recurrent geomagnetic storms and relativistic
electron enhancements in the outer magnetosphere: ISTP coordinated measurements, J.
Geophys. Res., 102, 14141, 1997.
24
Beutier, T., and D. Boscher, A three dimensional analysis of the electron radiation belt by
the Salammbo code, J. Geophys. Res., 100, 14853, 2000.
Birkeland, K., The Norwegian Aurora Polaris Expedition 1902-3 vol 1, On the Cause of
Magnetic Storms and the Origin of Terrestrial Magnetism, H. Aschehaug and Co.,
Christiania, 1908.
Birn, J. and M. Hesse, Details of current disruption and diversionii in simulations of
magnetotail dynamics, J. Geophys. Res., 101, 15345, 1996.
Blake, J. B., et al., CEPPAD: Comprehensive energetic particle and pitch angle
distribution experiment on Polar, Space Sci. Rev., 71, 531, 1995.
Boonsiriseth, A., R. M. Thorne, G. Liu, V. K. Jordanova, M. F. Thomsen, D. M. Ober, A.
J. Ridley, A semiempirical equatorial mapping of AMIE convection electric potentials
(MACEP) for the January 10, 1997 magnetic storm, J. Geophys. Res., 106, 12903, 2001.
Bourdarie, S., D. Boscher, T. Beutier, J.-A. Sauvaud, and M. Blanc, Magnetic storm
modeling in the Earth’s radiation belt by the Salammbo code, J. Geophys. Res., 101,
27171, 1996.
Boscher, D., S. Bourdarie, R. Thorne, B. Abel, Influence of the wave characteristics on
the electron radiation belt distribution, Adv. Space Res., 26, 163, 2000.
25
Bourdarie, S., D. Boscher, T. Beutier, J.-A. Sauvaud, and M. Blanc, Electron and proton
radiation belt dynamic simulations during storm periods: A new asymmetric convectiondiffusion model, J. Geophys. Res., 102, 17541, 1997.
Braysy, T., K. Mursala, and G. Marklund, Ion cyclotron waves during a great magnetic
storm observed by Freja double-probe electric field instrument, J. Geophys. Res, 103,
4145, 1998.
Burtis, W. J., and R. A. Helliwell, Magnetospheric chorus: Occurrence patters and
normalized frequency, Planet. Space Sci., 24, 1007, 1976.
Cayton, T. E., R. D. Belian, S. P. Gary, T. A. Fritz, D. N. Baker, Energetic electron
components at geostationary orbit, Geophys. Res. Lett., 16, 147, 1989.
Elkington, S.R, M.K Hudson, A.A. Chan, Acceleration of relativistic electrons via driftresonance interaction with toroidal-mode Pc5 oscillations, Geophys. Res. Lett., 26, 3273,
1999.
Falthammar, C.-G., On the transport of trapped particles in the outer magnetosphere, J.
Geophys. Res., 71,1487, 1966.
Fok, M.-C., R. A. Wolf, R.W. Spiro, T.E. Moore, Comprehensive computational model
of the Earth’s ring current, J. Geophys. Res., 106, 8417, 2001.
26
Gonzalez, W. D. and F. S. Mozer, A quantitative model for the potential resulting from
reconnection with an arbitrary interplanetary magnetic field, J. Geophys. Res., 79, 4186,
1974.
Gonzalez, W. D., and Tsurutani, B. T., ‘Criteria of interplanetary parameters causeing
intense magnetic storms (Dst<-100 nT)’, Planetary Space Sci., 35, 1101, 1987.
Grafe, A., Are our ideas correct about Dst?, Ann. Geophys., 17, 1, 1999.
Ingraham, J.C., T. E. Cayton, R. D. Belian, R. A. Christensen, R. H. Friedel, M.M. Meier,
G. D. Reeves, M. Tuszewski, Substorm injection of relativistic electrons to
geosynchronous orbit during the great magnetic storm of March 24, 1991, J. Geophys.
Res., 106, 25759, 2001.
Jordanova, V. K., C. J. Farrugia, R. M. Thorne, G. V. Khazanov, G. D. Reeves, M. F.
Thomsen, Modeling ring current proton precipition by electromagnetic ion cyclotron
waves during the May 14-16, 1997, storm, J. Geophys. Res, 106, 7, 2001.
Kawaski, K. and S.-I. Akasofu, Low latitude DS component of geomagnetic storm field,
J. Geophys. Res., 76, 2396, 1971.
27
Kim, H. J., A.A Chan, R. A. Wolf, and J. Birn, Can substorms produce relativistic outer
belt electrons?, J. Geophys. Res., 105, 7721, 2000.
Koons, H. C., and J. L. Roeder, A survey of equatorial magnetospheric wave activity
between 5 and 8 RE, Planet. Space Sci., 3 ,1335, 1990.
Korth, A., R. H. W. Friedel, C. G. Mouikis, J. F. Fennell, J. R. Wygant, H. Korth,
Comprehensive particle and field observations of magnetic storms at different local times
from the CRRES spacecraft, J. Geophys. Res., 105,18729, 2000.
Li, X., D. N. Baker, M. Temerin, T. E. Cayton, G. D. Reeves, R. S. Selesnick, J. B.
Blake, G. Lu, S. G. Kanekal, H. J. Singer, Rapid enhancement of relativistic electrons
deep in the magnetosphere during the May 15, 1997, magnetic storm, J. Geophys. Res.,
104, 4467, 1999.
Lorentzen, K., J. B. Blake, U. S. Inan, J. Bortnik, Observations of relativistic electron
microbursts in association with VLF chorus, J. Geophys. Res, 106, 6017, 2001.
Lyons, L.R. and R. M. Thorne, Equilibrium structure of the radiation belt electrons, J.
Geophys. Res., 78,2142, 1973.
Liu, W.W., G. Rostocker, D.N. Baker, Internal acceleration of relativistic electrons by
large-amplitude ULF pulsations, J. Geophys. Res.104, 17391-17407, 1999.
28
Meredith et al., R. B. Horne, R. R. Anderson, Substorm dependence of chorus
amplitudes: Implications for acceleration of electrons to relativistic energies, J. Geophys.
Res, 106, 13165, 2001.
Ness, N.F., A. J. Hundhausen, S. J. Bame, Observations of the interplanetary medium:
Vela 3 and Imp 3, 1965-1967, J. Geophys. Res., 76, 6643, 1971.
Neugabauer, M. and C. W. Snyder, The mission of Mariner II: Preliminary observations,
Science, 138, 1095, 1962.
O’ Brien, T. P., R.L. McPherron, D. Sornette, G. D. Reeves, R. Friedel, H. J. Singer,
Which magnetic storms produce relativistic electrons at geosynchronous orbit?, J.
Geophys. Res., 106, 15533, 2001.
Parker, E.N., Dynamics of the interplanetary gas and magnetic field, Astrophys. J., 128,
664, 1958.
Paulikas, G. and J. B. Blake, Effects of the solar wind on magnetospheric
dynamics:energetic electrons at synchronous orbit, in Qualitative Modeling of
Magnetospheric Processes, Geophys. Monogr.Ser., vol. 21, edited by W. Olson., pp 180,
AGU, Washingotn D.C., 1979.
29
Rostoker, G., S. Skone, D. N. Baker, On the origin of relativistic electrons in the
magnetosphere associated with some geomagnetic storms, Geophys. Res. Lett., 25, 3701,
1998.
Roederer, J. G., Dynamics of Geomagnetically Trapped Radiation, Cambridge Univ.
Press, New York, 1970.
Schulz, M. and L. J. Lanzerotti, Particle Diffusion in the Radiation Belts, SpringerVerlag, New York, 1974.
Summers, D., R.M.Thorne, F. Xiao, Relativistic theory of wave-particle resonant
diffusion with application to electron acceleration in the magnetosphere, J. Geophys.
Res.,103, 1998.
Summers, D. Ma, Chun-yu, Rapid acceleration of electrons in the magnetosphere by fastmode MHD waves, J. Geophys. Res., 105, 15,887, 2000.
Thorne, R. M., E. J. Smith, R. K. Burton, R. E. Holzer, Plasmaspheric hiss, J. Geophys.
Res., 78, 1581, 1973.
Vette, J. I., The AE-8 Trapped Electron Model Environment, NSSDC WDC-A-R&S 9124, 1991.
30
Wygant, J., D. Rowland, H. J. Singer, M. Temerin, F. Mozer, M. K. Hudson,
Experimental evidence on the role of the large spatial scale electric field in creating the
ring current, J. Geophys. Res., 103, 29527, 1998.
31
Figure Captions
Figure 1: Schematic of the radiation belts. Panel A shows the single proton radiation belt.
Panel B shows the two electron radiation belts. The inner belt is shown in green and the
outer belt in purple. The slot region separates the two belts.
Figure 2: Plot of omnidirectional differential electron flux (#/(cm2-s-str-keV)) versus L
taken from the AE8MAX model [Vette, 1991]. The model is based on data from more
than 20 satellites flown in the early 1960’s to 1970’s. The AE8MAX model averages the
electron flux in L bins measured during solar maximum. The data were obtained from the
website at http://nssdc.gsfc.nasa.gov/space/model/models/trap.html.
Figure 3: Six months of relativistic electron flux measured by LANL satellite at
geosynchronous orbit (6.6 Re). Differential energy flux in units of #/(cm2-s-sr-keV) from
four energy channels which range from 0.7–6.0 MeV are plotted versus day of year for
1997. The plot demonstrates the extreme temporal variability of the relativistic electron
flux.
Figure 4: Plot demonstrating how the flux of relativistic electrons changes during the
course of one storm. Panel A shows differential electron flux measured by LANL
satellites in units of #/cm2-s-sr-keV versus time. Measurements from four energy
channels are shown which range from 0.7-7.8 MeV. Panel B shows the Dst (nT) index
plotted over the same time period.
32
Figure 5: Schematic of the earth’s magnetosphere showing the processes that cause a
magnetic storm as described in the text.
Figure 6: Geographic positions of the ground stations used to create the Dst index given
in
geographic
and
geomagnetic
coordinates
(http://swdcdb.kugi.kyoto-
u.ac.jp/dst2/onDstindex.html). The ring current is depicted by the gray shaded region. An
enhancement of the ring current makes a negative perturbation in the north south
component of the magnetic field measured at the ground. Storms are identified by a large
negative excursion of the Dst index.
Figure 7: Schematic of the adiabatic invariants of particle motion. Panel A shows the full
particle motion in a dipole field which consists of gyromotion around a field line, bounce
motion along a field line and drift motion around the earth. Panel B shows only the gyromotion of an electron about a field line associated with the first invariant. Panel C shows
the bounce motion of an electron along a dipole field line associated with the second
invariant as viewed from the sun. Panel D shows the drift motion around the earth in the
equatorial plane associated with the third invariant as viewed from above looking down
on the equatorial plane.
Figure 8. Schematic showing the acceleration of electrons in the outer radiation belts by
the combined interaction of ULF and VLF waves as proposed by Liu et al. [1999]. At
step (1) the electron is pushed radially inward by the electric field of a ULF wave. At step
(2) the electron is scattered to smaller pitch angle through the interaction with VLF
33
waves. At step (3) the electron is pushed radially outward by the ULF wave. The electron
gains energy by this process as explained in the text.
Figure 9. Schematic demonstrating the cycle of an electron accelerated by multi-mode
diffusion described by the Salambo code [Boscher et al., 2000]. Radial diffusion pushes
the electron radially inward (Step 1) until it reaches the plasmapause. Inside the
plasmapause the electron is scattered to smaller pitch angle (Step 2). The electron is
pushed outward by radial diffusion (Step 3). The electron is then scattered to larger pitch
angle (Step 4) and the cycle repeats.
34