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Transcript
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* 2M 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. 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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