Space Plasma Physics: The Study of Solar-System Plasmas (1978)

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PLASMA PROCESSES IN THE EARTH'S RADIATION BELT by L. R. Lyons 973

974 I. Introduction The earth's radiation belts consist of energetic electrons and ions (energies from <1 Kev to >100 MeV) trapped in the geomagnetic field on magnetic field lines with L-values from M..1 to 'VLO. (For the purposes here, L can be defined as the equatorial crossing of a magnetic field line measured in earth radii from the center of the earth.) Here we consider the basic plasma processes affecting trapped particles from VL keV to a few MeV. An excellent and more in depth review of theoretical concepts is 37 given by Schulz , and a more complete review of research over the past 48 few years is given by West . The very high energy (>30 MeV) trapped protons below L=2, not considered here, have been reviewed by White Research of the past five years is emphasized in this report. Readers are referred to the comprehensive treatment of the radiation belts by Schulz OQ and Lanzerotti for earlier research . Trapped particles execute quasi-periodic motion which can be divided into three components, each component associated with an adiabatic invar- iant. The three components of trapped particle motion are schematically illustrated in Figure 1, along with magnetospheric regions mentioned in this section. The particle gyration about field lines is associated with 2 the first invariant M=p. /2mB, where m is rest mass, p is momentum, and subscripts " " and ",." refer to the components of vector quantities nor- mal and parallel to the geomagnetic field ]B. The gyrofrequency (often called the cyclotron frequency) is ^880/L kHz for non-relativistic elec- trons and a factor of ^2000 less for protons. The bounce motion between mirror points is associated with the second invariant J=»p..ds, where the line integral is evaluated between mirror points along the field line about

975 which the particle gyrates. Typical bounce frequencies are M3.3-10 Hz for electrons and ^0.03-3 Hz for protons. The particle drift around the earth is associated with the third invariant $ which is equal to the mag- netic flux enclosed by the drift orbit. Drift frequencies range from M).0001-100 mHz. Particle collisions and interactions with plasma waves can cause vio- lation of these adiabatic invariants. Generally each interaction causes a small perturbation on a particle's trajectory, so that the net result of a large number of such interactions is diffusion in velocity or in po- sition. Violation of the first two invariants results in particle diffu- sion in pitch angle and energy. (The pitch angle a is given by tan a=v./v.., where v is velocity. Thus particles with equatorial pitch angles of 90° are confined to the geomagnetic equator, while particles with equatorial pitch angles approaching 0° and 180° mirror at increasingly higher latitudes. Particles with equatorial pitch angles sufficiently close to 0° and 180° strike the atmosphere and are lost from the radiation belts via collisions. Such particles are said to be in the loss cone. The loss cone is <1° wide in equatorial pitch angle in the outer regions of the radiation belts, and increases to a few tens of degrees wide at L<2.) Pitch angle diffusion into the loss cone is often an important loss process for radiation belt particles, while diffusion in energy can be an important particle energi- zation process. Violation of the third invariant causes particles to diffuse to mag- netic field lines closer or further from the earth. Such radial diffu- sion is an important source for much of the inner regions of the

976 radiation belts. Processes which cause radial diffusion generally con- serve the first and second adiabatic invariants, so that particles which diffuse towards the earth are significantly energized due to the increase 2 in B (conservation of M implies p. 'x-B) , In addition to diffusive particle sources and losses, direct energetic particle injection and loss results from processes such as energetic neutron decay and charge exchange with cold neutrals, and particles can be injected directly into the radiation belts from the geomagnetic tail and cusps by electric fields normal to 13 and from the ionosphere by electric fields parallel to IS. The above plasma processes are all known to affect radiation belt particles. Several of them have been quantitatively studied for specific particle populations, and the results have been directly compared with satellite particle observations. Such studies have generally considered only regions within the location of the plasmapause during periods of relatively low geomagnetic activity. The plasmapause is the outer boundary of the plasmasphere, a region 24-3 of high cold plasma density (10 -10 cm ). extending from the equatorial and mid-latitude ionosphere and terminating relatively abruptly at field lines near L=4 to 6. Cold plasma densities outside the plasmapause are < -3 VL cm . Outside the plasmapause, measured radiation belt particle fluxes vary considerably on time scales on the order of hours to days in response to variations in the level of geomagnetic activity, even during relatively quiet periods. On the other hand, only during large geomagnetic storms (which occur several times a year) does direct injection of particles affect radiation belt fluxes below L=4. Following such storms,

977 particle fluxes within the plasmasphere gradually return to their pre- storm levels. An example of observed variations of radiation belt par- ticles, in this case electrons at L=2, 3, 4 and 5, is shown in Figure 2 for a one month period which included a relatively quiet period (Dec. 9-16), a large storm (Dec. 17), and a storm recovery (Dec. 17-Jan. 10). Notice the variability of the fluxes at L=5, the stability at the fluxes at L=2 and 3, and the storm flux enhancement at L=3 followed by a recovery to pre-storm levels. Within the plasmasphere, well-defined plasma processes proceed essen- tially continually for periods of weeks to months uninterupted by direct injections. This region is thus ideal for comparing radiation belt par- ticle measurements with theoretical discriptions of plasma processes, and fortunately good measurement of the plasma energetic particle distri- bution function are available from the Explorer 45 satellite. Beyond L=5, good measurements of the particle distribution function are not available. This, together with the variability of the fluxes, has resulted in our understanding of the processes controlling radiation belt particles being for more limited outside the plasmapause than within. II. General Comments on Particle Diffusion from Resonant Wave-Particle Interaction Considerable attention has been given to understanding the inter- action of radiation belt particles with plasma waves. Such analyses have generally considered waves with frequencies on the order of the particle gyrofrequency, which can cause significant particle pitch angle diffusion, and waves with frequencies on the order of the particle drift frequency, which can cause significant radial diffusion. Resonance with 35 42 waves on the order of the bounce frequency " ' " has received consider- ably less attention.

978 The conditions for resonance between waves of frequency w and par- ticles of gyrofrequency fl is given by: 0)-k. v =n^ ; n=o ±1, ±2, ±3 where jc and v are the wave vector and the particle velocity. The dis- persion relation for the wave mode under consideration relates Ic to w. The cyclotron harmonic resonances have the doppler shifted wave frequency equaling a harmonic of the particle gyrofrequency and are given by n=±l, ±2, ..., and the n=o Landau resonance has the wave parallel phase velo- city w/kfj.=Vjj. All resonances result in particle diffusion in both pitch angle and energy. The cyclotron resonances can cause significant diffusion into the loss cone, while the Landau resonance results in diffusion solely in v. . so that diffusion is primarily in energy at pitch angles near the loss cone. For simplicity, assume w«fl. Then the resonant parallel velocity for each cyclotron harmonic resonance vf. =n^/k,.aand for the Landau I I y 11 I I' resonance v., =o'/krf« |v. . |. Assuming waves propagate both up and I j o HI I j n down field lines, resonance at each harmonic occurs for both v,, and I I ,n -v. , . Figure 3 illustrates the regions of cyclotron resonance in vel- • ' »11 ocity space for waves distributed over some band of k *s. Relativistic effects are not included. Resonance at each cyclotron harmonic occurs over a band of vf.*s and no cyclotron interaction occurs for v, . less than a minimum v,, . . That no cyclotron resonance occurs for v,, less !I,min I I than a v... does not depend on the assumption OJ«^: however the value I I , tain of v.. . depends upon the band of k..'s over which wave energy is dis- tributed. The simple picture in Figure 3 is modified somewhat in the radiation belts because the geomagnetic field is not homogeneous, and wave energy is generally distributed over some range of geomagnetic latitudes.

979 As a particle moves away from the equator along its bounce trajectory, the increasing geomagnetic field strength causes both the particle pitch angle and the parallel velocities v. for resonance with the waves 11 I • n (for most waves modes) to increase. The effects of off-equatorial interactions can be represented by Figure 3 by removing the upper v.. bounds of the resonant regions and re-labeling the axes as equatorial velocity. Thus cyclotron resonance occurs for all equatorial parallel 2 2 particle energies Er =l/2mvrf greater than some minimum value E.. . , so that particle diffusion into the loss cone occurs for all particle energies E>E.. , , but not for E<E,. . , H, mm' I I ,min As shall be seen in the example discussed later, equatorial par- ticle measurements can show marked effects from cyclotron resonance at £,,>£,, . but not for E,,<E,. . , and such observations have given II I I,min II II,min strong support to specific wave-particle interactions which have been proposed as the dominant loss mechanism for specific populations of trapped particles. Given a proposed distribution of wave energy in the 20 radiation belts, quasi-linear diffusion theory can be applied to 23 24 obtain pitch angle and energy diffusion coefficients " . These diffusion coefficients can then be used to calculate the distribution of 19 22 trapped particles as a function of pitch angle ' , and these calcula- tions can be directly compared with particle measurements. The above approach for study resonant wave-particle interactions re- quires knowledge of the wave distribution either from measurements directly within the radiation belts or from theoretical predictions. The more gen- eral approach of self-consistently calculating the wave distribution and

980 particle diffusion rates is generally unreasonably difficult for the ra- diation belts. Specific cases have been attempted ' , but sufficient sophistication to allow definitive comparison with particle observations has not been achieved. Fluctuations of large-scale magnetospheric electric fields with fre- quencies equal to a harmonic of the particle drift frequency can cause radial diffusion. Such fluctuating electric fields are generally divided 12 into two classes ; 1). Electric fields induced by fluctuations in the geomagnetic field (VxE^O)? 2) Fluctuations of the magnetospheric potential electric field for which VxE,=0. By attempting to make reasonable assumptions about the distribution of fluctuating fields, expressions for radial 33 diffusion coefficients have been derived (e.g. Nakada and Mead ' , for 2 magnetic fluctuations; Cornwall , for electric field fluctuations). Given radial diffusion coefficients and loss rates calculated from pitch angle diffusion coefficients, and including other processes such as those listed in Section I when relevant, radiation belt fluxes as a function of radial distance can be calculated. Such calculations can be compared directly, and as shall be seen, some of these comparisons have been remarkably successful. III. Plasma Processes and Comparisons with Observations 1. Quiet-time electrons Valid measurements of radiation belt electrons within the plasma- sphere are available for electron energies from V30 keV to ^2 MeV. The interaction of these electrons with naturally occurring whistler- mode waves is a well understood example of wave-particle interactions in the radiation belts. During geomagnetically quiet times, the electrons are distributed in two zones, inner zone fluxes peak near L=1.2-2, outer zone fluxes peak near L=4-6, and the region of low

981 fluxes in between is called the electron slot. Electron interactions with whistler-mode waves have been studied in detail to explain the O Q electron losses required to form the slot '' . Satellite measurements have shown that a band of whistler-mode waves (called hiss) centered near a few hundred Hz exists nearly con- tinually and are probably the dominant wave mode throughout the plasma- sphere ~ ' , These waves, which can readily propagate across magnetic 26 field lines , are believed to be generated by radiation belt electrons 45 46 in the outer region of the plasmasphere ' . The generated wave energy then propagates across field lines to all regions of the plasma- sphere. 28 Pitch angle diffusion coefficients were calculated / for electrons resonant with the observed distribution of plasmaspheric hiss. Using the calculated diffusion coefficients, prediction of electron pitch angle distributions and loss rates from diffusion into the loss cone were obtained. Examples of the calculated pitch-angle diffusion coefficients and equatorial pitch-angle distributions at L=4 are shown in Figure 4 for 20 keV, 200 keV, and 2 MeV electrons. Cyclotron resonance occurs for equatorial pitch angles up to V>5° at 20 keV. Cyclotron resonance extends to increasing pitch angles with increasing electron energy, since cyclotron resonance occurs for all E..-E.. . independent of the electron energy. Landau resonance occurs only at pitch angles near 90°, but off-equatorial interactions cause the range of equa- torial pitch angles affected by Landau resonance to overlap the pitch-

982 angle range of cyclotron resonance. Note that the pitch-angle dif- fusion rates are relatively low at values of E( ( just below Er( . This slow diffusion manifests itself in the equatorial pitch-angle distributions as a region of large slope. Thus the equatorial distribu- tions develop bumps surrounding 90° pitch angles, and this bump de- creases in size and pitch angle extent with increasing electron energy. O O / Since E ^ B /N ^ L'~'' (for an equatorial electron density N ^ L ), > ' ,mm these bumps are also expected to decrease with increasing L-value for a constant electron energy. Such pitch-angle distributions, with the expected variation with 49 energy and L, were first observed on OGO-5 , and are now known 29 to be a general feature of quiet-time radiation belt electrons - Examples of West et al.'s observations are shown in Figure 5 (from 28 Lyons et al. * . The theoretically predicted pitch angle distributions are shown as solid lines at the appropriate L-values and energies. The calculations overestimate the magnitude of the 90° pitch angle bumps, but this could be corrected by reasonable alterations of the distribu- tion of wave energy. However, no reasonable alteration of the wave dis- tribution could change how the bumps vary with L and energy. Since E.. , increases with decreasing L, cyclotron resonance it I , rom with the band,-limited hiss does not occur within the inner zone for the energies considered here. Coulomb collisions with the cold plasma- spheric plasma thus become the dominant pitch angle diffusion mechanism within the inner zone, becoming dominant at L-3 for 50 keV electrons and L~2 for 350 keV electrons. Equatorial pitch angle distributions obtained by Explorer 45 clearly show the transition from pitch angle

983 diffusion dominated by wave-particle interactions at higher L to domina- 29 tion by Coulomb collisions at lower L Using the calculations of diffusion rates for interactions with the hiss, and including the effects of Coulomb collisions in the inner zone, we have realistic predictions of particle loss rates as a function of electron energy and L throughout the plasmashphere. 27 Lyons and Thome balanced the calculated electron loss rates with radial diffusion to obtain an equilibrium structure of radia- tion belt electrons throughout the plasmasphere. Radial diffusion was assumed to be driven by fluctuations of the magnetospheric po- 2 tential electric field as modeled by Cornwall The resulting particle distribution function f for equator- ially mirroring electrons as a function of L at constant first adiabatic invariant M is shown in the left-hand panel of Figure 6. Curves for the various values of M were all normalized to the same value at the plasmapause, here taken to be L=5.5. Since f must vanish at L=l, all the curves monotonically decrease with decreasing radial distance, and there is no sign of a two-zone structure. How- ever, the curves at constant M were then used to obtain the differential 2 unidirectional particle flux j=fp (p is momentum), as a function of L at fixed electron energy, A flux versus energy spectrum at L=5.5 based on observations was used as the required boundary condition. The results are shown in the right panel of Figure 6, and a two-zone structure appears as a consequence of radial diffusion from the plasmapause to a sink at L=l and of the increase in energy as an electron 3 diffuses inwards (energy ^1/L for non-relativistic,

984 equatorially-mirroring electrons). The calculated slot moves in- ward and becomes increasingly pronounced with increasing energy and the outer zone peak mover inward within the plasmasphere above 1 MeV in agreement with observation. A comparison between the calculated equilibrium electron structure and quiet-time observations of Pfitzer 34 et al. is given in Figure 7. During large geomagnetic storms, electrons are injected within the quiet time location of the plasmapause, and the equilibrium pitch angle distributions and radial structure are destroyed. How- ever, Explorer 45 observations have shown that the pitch angle dis- tribution and radial profiles gradually return to their pre-storm 30 structure over a period of a few weeks " . It thus appears that the dominant, long-term-averaged, quiet-time source and loss processes for radiation belt electrons within the plasmasphere have been correctly identified and quantitatively evaluated. Variations of waves distribu- tions and resulting diffusion rates over time scales of a day or less are not as well understood. 2. Quiet-time ions Ions undergo cyclotron resonance with ion-cyclotron waves in much the same manner as do electrons with whistler-mode waves. While whistler waves have frequencies below the electron gyrofre- quency and generally above the proton gyrofrequency, ion-cyclotron wave frequencies are below the proton gyrofrequency. However, our knowledge of the distribution of ion cyclotron waves within the plasmasphere is far less than our knowledge of the distribution of whistler-mode wave. This is because satellite instrumentation has been less sensitive at ion-cyclotron wave frequencies (generally ~10 Hz) than at whistler-mode frequencies (generally ~100 Hz).

985 Low altitude (several hundred km) satellite observations within the plasmasphere have found ions precipitating into the ionosphere during quiet times and during storms, implying that trapped ions are pitch angle diffused into the loss cone '' . However, the importance of such diffusion in determining radiation belt structure has been a sub- ject of continuing research for the past several years. A significant advance in our understanding of the quiet time distribution of trapped protons has recently been achieved by Spjeld- 43 3 vik . He followed the approach of Cornwall and balanced radial 2 diffusion as modelled by Cornwall with proton losses from charge exchange with the ambient neutral hydrogen geocorona and Coulomb col- lisions. The outer boundary condition was taken from the observed ion fluxes from ATS-6 at L=6.6. His results are displayed in Figure 8 as energy spectra at various L-values for equatorially mirroring ions. The circles and triangles at L~5.25 are equatorial ion observations from Explorer 45. Unfortunately, the lack of valid data prohibits comparison below 25 keV at L=4 to 5 and below 150 keV at L<4. However, the remark- able agreement between the calculations and the observations implies that the dominant processes affecting the protons have been identified over the energy range of valid observations, and that wave-particle interactions are probably not important for equatorially mirroring protons at these energies. Spjeldvik considered only equatorially mir- roring protons, and whether or not his results holds for non-equatorially mirroring protons is yet to be determined. Since the ion measurements can not distinguish between protons and other ions, the agreements in Figure 8 imply that the measured

986 ion fluxes were indeed dominated by protons, since the calculated radial diffusion and loss rates were obtained by assuming the par- ticles were protons and the rates are significantly different for other ion species. However, the calculated proton spectra at L<4 fall off markedly below 100 keV due to the fast charge exchange loss rates below 100 keV. It is unfortunate that valid measurements are not available below 100 keV, since Spjeldvik's calculations im- ply that any significant ion fluxes below 100 keV may not be domi- nated by protons. This range of energies at L~4 is discussed in more detail in the following section on the decay of the stormtime ring current. 44 + Spjeldvik and Fritz performed a similar calculation for He I j and He . They compared their results to the very limited Ex- I i plorer 45 He observations at 1-3 MeV. The comparison was fairly good, but sufficient observations for a definitive comparison are simply not yet available. 27 43 Note that Lyons and Thome , Spjeldvik , and Spjeldvik and 44 2 Fritz all used radial diffusion rates as formulated by Cornwall for fluctuations of the magnetospheric potential electric field. In addition, the magnitude of Cornwall's diffusion rates appear to be valid for the > 30 MeV protons below L=2 ' . It thus appears that Cornwall's formulation, which was based on very limited elec- tric field observations, is remarkably accurate over a wide range of particle energies, L-values, and particle species. Without such a valid estimate of radial diffusion rates, attempts to explain the quiet-time structures of radiation belt particles within the plasma- sphere would have been far less successful.

987 3. Decay of stormtime ring current ions During storms, electrons and ions are injected to L-values as low as 2 so that trapped particle fluxes become significantly enhanced over their quiet-time equilibrium values within the location of the quiet-time plasmapause. The drift of these particles around the earth forms a current, called the ring current, which causes a significant C50-400y) depression of the magnetic field at the earth's surface. While the decay of the electrons is understood in terms of interactions with whistler-mode waves, the decay of the stormtime ring current ions is presently an active area of research. Even the composition of these ions, until recently believed to be mostly protons, is now being seriously questioned. Enhanced ion fluxes are observed from VL keV to ^200 keV during storms, and fortunately these fluxes are above the Explorer 45 back- ground levels so the valid measurements of equatorial pitch angle distributions and energy spectra are available over most of this energy range from L=2.5 to 5. Figure 9 shows Explorer 45 observations of the distribution function f of equatorially mirroring ions throughout the period of the large storm on Dec. 17, 1971. The quantity 2mf=j/E is shown ver- sus time for L=2.5, 3.0, ...5.0 for 8 representative ion energies where j is the measured ion flux. Dst, a measure of the magnetic de- pression on the ground, is also shown versus time. Every valid data point is shown from both inbound and outbound portions of Explorer 45 orbits that are within ^8° of the geomagnetic equator. The pre- storm intensities at energies ~26 keV are upper limits to the true intensities because of an undetermined background count rate; however,

988 these data are included to emphasize the stormtime increase in par- ticle intensities. In association with the December 17 storm main phase (the large decrease in Dst), the data show intensity increases at the lower ion energies and, at L-4, intensity decreases at the higher energies. Some response to the sudden commencement on December 16 is also evi- dent. Following the storm, the particle intensities gradually re- turn towards their pre-storm values. By December 21, Dst had nearly recovered, though some of the particle intensities remained near their stormtime values (e.g. 164 keV protons at 3.5 R ). Small intensity variations at low E and high L occurred on December 22 along with a small decrease in Dst. The decrease in f at the higher energies simply results from the conservation of M (M=E/B for equatorially mirroring ions). During the storm, B decreases so that particle energies decrease in order to conserve M. The decrease in particle energies, together with the over- all decrease in f with increasing E, results in a decrease in f when mea- sured at a fixed value of E. However no actual loss of high energy 31 particles occurs ~ . On the other hand, the observed increase in f at the lower energies represents a net increase in the trapped ion popula- tion, and this increase is a significant part of the stormtime ring current. Two processes are believed to be important in the decay of the 9 stormtime ring current: charge exchange with neutral hydrogen and resonant interactions with ion-cyclotron waves . The importance of

989 both of these processes has been tested by using Explorer 45 measure- ments of equatorial pitch angle distributions. 3.1 Interaction with ion-cyclotron waves Figures 10 and 11 show the equatorial pitch angle distri- butions, obtained VL6 hours (orbit 103) and ^24 hours (orbit 104) after the minimum Dst of the Dec, 17, 1971 storm. Observations from selected ion energy channels are shown every 0.4 in L from L=3.0 to 5.0. Notice the transition from relatively isotro- pic pitch angle distributions at the lower energies to rounded distributions, peaked at 90° pitch angle, at the higher energies. The isotropic distributions show significant flux decreases when- ever the pitch angle scan of a measurement reaches the loss cone, implying the loss cones are nearly empty of particles. Such isotropic distributions with empty loss cones indicate a stably trapped particle populations undergoing negligible pitch angle diffusion into the loss cone. As the storm recovery progresses, the transition from flat to rounded pitch angle distributions shifts to lower energies, so that flat distributions become rounded. Note, for example, the pitch angle distributions in Figure 10 and 11 for 14 keV at L=3.8, 26 keV at L=4.2, and 42 keV at L=4.6. Such rounding of the pitch angle distributions represents a loss of non-equatorially mirroring particles, and rounded distribu- tions with fluxes monotonically increasing towards 90° pitch angle are expected under conditions of pitch angle diffusion into the loss cone.

990 For each L-value shown in Figures 10 and 11, the energy of the highest Explorer 45 energy channel showing a nearly isotropic pitch angle distribution was assumed to be E. . 1 I I,min (the minimum parallel ion energy subject to cyclotron reson- ance with a band of waves), and the pitch angles corresponding to the chosen value of E.. was calculated for all higher energy channels. These pitch angles have been indicated by vertical ticks on the distributions. Distributions with ticks at 0° and 180° give the chosen value of E. . , though these distributions i' I ., in i n are not shown at all L-values in the figures because of the use of only selected energy channels. Notice that to within the o accuracy of the pitch angle measurements (which is 22 for -104 keV and 33 p for E>104 keV), the pitch angle distributions are nearly isotropic between the ticks, i.e. for £,.<£,, Mil,min At larger E.. (pitch angles approaching 0° and 180°), the dis- tributions become rounded. For all L-values shown, the ticks quite accurately separate regions of nearly isotropic distribu- tions at lower values of E. from regions of rounded distribu- tions at higher values of E... In order for an E., , to be evident in the observed pitch I I,min angle distributions, pitch angle diffusion driven by resonant wave-particle interactions must have been the dominant loss process responsible for the rounding of the distributions. Such interactions are governed by the parallel particle velocity, while other loss processes are not organized by the parallel

991 velocity. Thus another loss process, if dominant, would have masked the ability to determine Eii . . In addition, the ob- 11,min served values of Eii . satisfy the condition for cyclotron I I,min resonance with ion-cyclotron wave when realistic cold plasma den- sities are assumed for the outer region of the plasmasphere during •I Q C 1 CO a storm recovery phase ' ' ' t Landau resonant diffusion of the ions is probably of little importance for these waves, since wave energy at large wave normal angles is needed for such diffu- sion and such wave energy is probably damped by low energy electrons . 51 52 Williams and Lyons ''"" concluded that the ion-cyclotron waves were amplified by the ring current particles as proposed by Cornwall et al. , and that the evolution of isotropic distributions to rounded distributions occurred as the cold plasma density in- creased during the storm recovery phase. (The parallel energies for cyclotron resonance with ion-cyclotron waves E decrease with I I ,r increasing cold plasma density, and during storms the cold plasma in the outer regions of the plasmasphere is severely depleted. This plasma is replenished from the ionosphere over a several day period following storms.) However, Williams and Lyons' argument does not explain how the isotropic distributions, which apparently were not undergoing diffusion, could become resonant with unstable waves, A sufficient pitch angle anisotropy is required to grow ion-cyclotron waves, and the isotropic distribution, even with the empty loss cones, have too small an anisotropy to grow waves.

992 18 Joselyn and Lyons suggested a resolution of this difficulty by by showing that waves growing off the equator can propagate towards the equator and resonate with the isotropic distributions. E. , increases markedly away from the equator along field lines, so that we can find a frequency which resonates with rounded distributions off the equator (where E.. . >E ) but which resonates with isotropic distributions near the equator (where E. <E- . ^. II,r ||,min Joselyn and Lyons showed that as the plasma densities continually increase during the storm recovery, wave growth will occur at frequencies so that the minimum E at the equator will continu- ally decrease at the equator. Thus their calculations imply that E., , should continually decrease as is observed. lI,min 3.2 Inconsistency with proton charge exchange 47 Tinsley has noted that the charge exchange lifetimes for equatorially mirroring protons at energies -30 keV should be on the order of hours. The loss rates from charge exchange should increase significantly with increasing mirror latitude for particles of a given energy on a given L-shell. This results from the decrease in mirror altitude with increasing mirror latitude together with the increase in neutral hydrogen density with decreasing altitude. 25 Lyons and Evans have investigated the question of how the nearly isotropic pitch angle distributions at the lower energies in Figures 10 and 11 can remain isotropic when charge

993 exchange with neutral hydrogen should cause the loss rates to increase markedly for equatorial pitch angles increasing or de- creasing from 90°. In Figures 12 and 13 we compare the observed pitch angle distributions at L=3.0 and 3.5 with those expected to evolve from proton charge exchange with neutral hydrogen. Observa- tions are shown from the inbound portion of orbits 101, 102, ..., 106, which are 'vS hours apart, with the observations from orbit 101 being less than one hour following the minimum of Dst. The evolving pitch angle distributions predicted from charge exchange were obtained by neglecting possible sources and assuming an isotropic proton distribution at the time of the orbit 101 observations, which is taken to be t=0. Fluxes j at subsequent times were calculated from j^exp(-t/T ), 47 where the charge exchange lifetimes T are given by Tinsley . These lifetimes were obtained from a recent neutral hydrogen density model using parameters for Dec., 1971. At both L-values, the charge exchange calculations predict that the pitch angle distributions for 2 and 10 keV protons will become greatly anisotropic in <8 hours. This dramatic rounding of the pitch angle distribution is not observed! At L=3.5, the observed distributions remain essentially isotropic, while at L=3.0 some rounding of the distributions occurs. This rounding is apparently the result of resonant interactions with ion-cyclo- tron waves. T ^ 1 hour at L=3.5 for 20° pitch angle. Thus in order for there to be essentially no rounding of the pitch angle distribution in 40 hours, as is observed, a life- time 'v4O hours is required.

994 In addition to the disagreement between the shapes of the pitch angle distributions, charge exchange predicts that even the 90° pitch angle fluxes should decay much more rapidly than is observed. This discrepancy is particularly dramatic at L=3. The comparisons presented here show that the ring current ions do not decay in a manner consistent with proton charge exchange with neutral hydrogen. Charge exchange decay rates are far too rapid, especially for I>3.5 and for non-equatorially mirroring particles. Some form of pitch angle diffusion cannot account for the discrepancy in the shapes of the pitch angle distributions, since this would imply total loss rates even greater than those from charge exchange alone. 25 Lyons and Evans concluded that neither a strong, continual proton source between L=3 and 4 during the storm re- covery nor a large error in the neutral hydrogen densities were a reasonable explanation for the observed discrepancy be- tween the observations and the charge exchange predictions. They concluded that the most likely explanation was that the ring current at particle energies VjO keV and IA-4 was dominated by some ion other than protons during the storm recovery phase. Such ions must have much longer lifetimes for charge exchange with hydrogen than do protons (a factor 'xAO appears to be re- quired) in order to resolve the gross disagreement between the predicted and observed pitch angle distributions. A candidate ion, which has sufficiently long charge exchange lieftimes, is He . Additional arguments that He may be the dominant ring cur- rent ion during recovery phase, and calculations of charge ex- change lifetimes using recent hydrogen models, are given by

995 47 > Tinsley . At energies ^50 keV, charge exchange lifetimes for He become longer than for H , so that the above argument does not apply above 50 key, Lyons and Evansi arguments also imply that quiet time ring current ions below 50 keV at 1/^4 should not be dominated by protons. 39 40 Sharp et al. * have presented mass spectrometer observa- tions from a low altitude satellite showing significant 0 and H precipitating from the outer region of the ring current (IAi4) during the recovery of the Dec. 17, 1971 storm; however, precipitating fluxes below L=4 were generally undetectable. In addition, Smith 41 and Bewtra have found that charge exchange can account for some features of the decay of equatorially-mirroring, ring current ions if a multi-component (H , He , and 0 ) plasma is assumed. However, the relative composition of the ring ions, and the relative roles of wave-particle interactions and charge exchange as loss mechanisms, is far from being resolved as a function of par- ticle energy, L-value and time throughout a storm recovery. IV. Summary and Future Directions ' While the dominant particle source and loss processes have yet to be understood throughout most of the magnetosphere, significant quantitative understanding of the quiet-time structure of radiation belt electrons O30 keVl and equatorially-mirroring protons CVLOO keV) within the plasmasphere has been obtained over the past several years. The dominant plasma processes affecting these particles apparently have been correctly identified and quantitatively evaluated, though exact quantitative agreements have yet to be obtained. In fact, for

996 the electrons we can now explain both the quiet-time structure and the post-storm decay be invoking the same processes, though the cause of the stormtime injections is still unknown. Such understanding has only been possible because of the availability of valid measurements of the equatorial pitch angle distributions as a function of radial distance and particle energy. Most of these observations have come from the equatorially orbiting Explorer 45 satellite, with some useful observations also obtained from OGO-5. Electrons with energies <30 keV have not yet been adequately studied, and the suggestion that harmonic waves from ground power distribution networks may affect these electrons needs to be evaluated, The Explorer 45 equatorial pitch angle distributions of ions, in conjunction with some low altitude observations of precipitating ions, have also given us significant information on the post-storm decay of ring current ions. Effects of wave-particle interactions have been identified and strong evidence has been obtained that protons are not the dominant ion species throughout the ring current. However, pre- sently available observations will probably not allow the ring current decay problem to be completely solved. The distribution of wave energy is unknown from either observations or theory, since instrumentation has not been sufficiently sensitive to measure most of the wave energy below the proton gyrogrequency and we do not know the relative ion com- position of the ring current. Calculations of wave growth rates assum- ing a single ion species have yielded useful information; however the relative densities of different ion species are needed since one species may be unstable to wave growth over a range of frequencies while another

997 4 species may damp the same wave frequencies . Even the effects of charge exchange, in principle a simple process to analyze, is currently a subject of debate. An extremely serious lack in our understanding of the radiation belts exists outside the plasmasphere at IA-5, The most intense ion and electron precipitation into the ionosphere occurs outside the quiet time plasmasphere at energies from M. to VLOO keV (e.g. Hultqvist ). Some of this precipitation is associated with bright auroral features and may result from acceleration by low altitude electric fields parallel to _B . However, the vast majority of the observed particle precipita- tion outside the plasmasphere appears to have been pitch angle scattered out of a trapped particle population . However, no equatorial measure- ments of the pitch angle distributions as a function of particle energy and radial distance exi.st for these particles. Energy spectra at one pitch angle and pitch angle distributions at one energy have been obtained at one L-value, 6.6, from the ATS satel- lites at synchronous orbit. These observations have shown that the a radiation belt fluxes vary in association with geomagnetic activity and that electric fields along IJ may at times accelerate particles out 32 of the ionosphere into the radiation belts ~ . However, identification and quantitative evaluation of the dominant source and loss processes outside the plasmapause will be impossible until equatorial pitch angle distributions as a function of particle energy and radial distance be- come available. Hopefully, some of the concepts that have been success- fully applied to the inner region of the radiation belts, particularly

998 those concepts associated with wave-particle interactions, will also be useful in understanding the region beyond L=5. However, available wave observations from outside L=5 indicate that more attention will have to be given to non-linear interactions. Detailed non-linear phenomena, such as the VLF wave-electron interactions observed by Hel- 14 liwell and Katsufrakes may be important. The processes discussed here are probably not unique to the Earth's radiation belts, and attempts have been made to apply them to the Jovian radiation belts, We now have made crude measurements of Jovian radiation belt fluxes. However, only measurements of the total particle flux above several fixed energies are available (which necessitates numerical dif- ferentiation of data to obtain a crude energy spectra). Until pitch angle distributions and good energy spectra become available from Jupiter, we will be unable to definitively identify which processes govern the distribution of Jovian trapped particles and to quantitatively analyze these processes.

999 CUSP MAGNETO- PAUSE GEOMAGNETIC TAIL FIGURE 1 Schematic illustration of the three components of trapped particle motion and relevant mag- netospheric regions.

1000 DECEMBER 1971 13 17 21 25 29 JAN. 1972 Elec. Flux Energy Mpy'd (keV) by 35-70 I03 75-125 I02 120-240 10' 240-560 10° 35-70 10s 75-125 I02 120-240 10' 240-560 10° 35-70 I03 75-125 I02 120-240 10' 240-560 10° 35-70 I03 75-125 I02 120-240 10' 240-560 10° FIGURE 2 Fluxes of equatorially mirroring electrons versus universal time for the period Dec. 9, 1971- Jan. 9,1972. All available data points from both inbound and outbound portions of the Explorer 45 are shown. Each panel shows the observations at the indicated L-value for the four energy channels. The 120- 240 keV, 75-125 keV, and 35-70 keV fluxes have been multiplied by 10i,102,103, respectively. Dst is also shown (from Lyons and Williams, 1975b).

1001 CYCLOTRON RESONANCE |n|*2 - PARALLEL VELOCITY, v-- II'min FIGURE 3 Regions in the (vx, v || >plane of reso- nance at the | n | = 1, 2, and 3 cyclotron harmon- ics are indicated by different shadings. Wave en- ergy is assumed to be distributed over a bank of k|'s with uKI2 for all frequencies in the wave dis- tribution. The minimum parallel velocity for cy- clotron resonance v( mIn is indicated. 3O' 60* EquotorIol PItch-Angle x /' 2000 K. 2OOKfV 0° 30« 60* 90° FIGURE 4 Bounce-orbit averaged cyclotron and Landau resonant pitch-angle diffusion coefficients as a function of equatorial pitch angle at L=4 for 20, 200, and 2000 keV electrons. For each energy, the line coding used for the Landau resonant diffusion coeff1cients corresponds to that for the sum of the cyclotron- harmonic resonances, where the particle energy is indicated. Equatorial pitch-angle distributions, calculated using the diffusion coefficients shown in the f1gure, are displayed on the right-hand side (from Lyons et al., 1972).

1002 L'35 L-37 r ISO" o- 9cr EQUATORIAL PITCH-ANGLE I80- o- L-4.0 822 Ke' 90- I0' LU U. u_ o ISC' FIGURE 5 Comparison of calculated equatorial pitch-angle distributions with equatorial distributions ob- served within the electron slot. The data points (electrons/cm2-sec-ster-keV) are plotted as a function of pitch angle at the Lvalues and energies indicated on the figure. The vertical placement of the corresponding calculated distributions, shown by solid lines, are arbitrary on a logarithmic scale and have therefore been adjusted so as to best fit the data (from Lyons et al., 1972). FIGURE 6 Equilibrium distribution function f of equatorially mirroring electrons versus L at constant first adiabatic invariant M, with each curve normalized to the same value at L=5.5 (left panel), and equilibrium differential flux j versus Lat constant electron energy using the curves in the left panel and a prescribed en- ergy spectrum at L=5.5 (right panel). (From Lyons and Thome, 1973 as modified by Schulz, 1975.)

1003 I 290-690 K 12345671234567 10 FIGURE 8 Comparison of theoretical and observed proton energy spectra in the kiloelectronvolt energy range at L-values of L=2,3,3.5,4,4.5, 5, 5.25, 6.6, and 7. The spectrum at L=6.6 constitutes the adopted boundary condition on the theo- retical calculations and is taken from ATS-6 observations. The data at the lower L-values are from Explorer 45 or- bits 97 (circles) and 667 (triangles) (from Spjeldvik, 1976). r "' U) K>" 'o S •' <VJ i0' 'E ,0i 0) •_ a) FIGURE 7 Comparison between calculated equilibrium electron structure (dashed lines) and quiet time observations of Pfitzer et al. (1966) (from Lyons and Thome, 1973). L-2 L=3 L=35 L-7 i0 I00 i000 I i0 i00 i000 I Energy (keV

1004 ION DISTRIBUTION FUNCTION vs TIME 2mf 2mf CONSTANT ENERGY —r^> wf >s- -*S*~4 CONSTANT ENERGY - ...... . ^. , R-3.S CONSTANT ENERGY I I 16 '7 II I) 20 21 22 23 1C IT II II 20 21 22 23 1C 1T II 19 20 21 22 23 382- CONSTANT ENERGY -_ . . . R-4.5 E(kev) 2.7— ONSTANT ENERGY I : I Elkev) 382- 599~ CONSTANT ENERGY J I I I I II IT II 19 ?0 21 22 23 IT II 19 20 21 22 23 II IT II II 20 21 22 23 100' -200r 16 IT II 19 20 21 22 December 1971 23 II IT II 19 20 21 December 1971 22 23 II IT II 19 20 21 22 tl December 1971 FIGURE 9 Distribution function, multiplied by 2 times the ion mass, for equatorially mirroring ions as a function of time throughout the period of the December 17, 1971 storm for 8 Explorer 45 energy channels. Dst is also shown. Radial distances R=2.5, 3.0,..., 5.0 are shown. Note that the pre-storm intensities for energies < 26 keVare upper limits to the true intensities (from Lyons and Williams, 1976).

1005 EXPLORER 45 ORBIT I03 INBOUND = 30 34 3.8 4.2 I--*'-] f.L, 1 I ..' t r ,,fI. •^ ;.I',. ' 't :_4-u. VTT 46 50 keV :l i * f- M n 242 ;1 104 42 26 14 , .1»' »•'—Vr nfsr A Il 0 M " K 0 N IV ' EquotorkJl PIich Angle (Degrees) 1H H lI'n ' ' 'r.' FIGURE 10 Equatorial ion pitch-angle distributions observed on Explorer 45 orbit 103 inbound, ~16 h after the minimum Dst of the December 17, 1971 storm main phase. Distributions are shown every 0.4 in L from L=3 to L=5, and selected ion energy channels are stacked vertically at each L. Elevated fluxes at pitch angles of 90-180° for energies <14 keV are due to reflected sunlight. Ticks are at constant values of E| for each L, with the chosen value of EB at each L being equal to the energy of one of the Explorer 45 channels (from Joselyn and Lyons, 1976).

1006 EXPLORER 45 ORBIT I04 INBOUND M U o |. .-1,. - IH; 1l I! 'rf .1! 34 -,ru - -*- ' 38 •1t ll Id I,' r,-. 42 46 '••-i .1 'It JrnV-1—Vr= 50 k£V 1104 42 26 14 0 H IN 0 N IN EquatorIal Piich Angle (Degrees) FIGURE 11 Same as Figure 10, except for Explorer 45 orbit 104 inbound, ~24 h after the minimum Dst of the storm main phase (from Joselyn and Lyons, 1976).

1007 L=3.0 PITCH ANGLE DISTRIBUTIONS ORBf f = 101 102 103 104 105 106 r= 0HR 7.84 HR 15.68 HR 23.52 HR 31.36 HR 39.20 HR j M U 2 iJ, ft 4< 42 2 91, 3 24 14 III, £) :: 41 ( M, f CS « ? «. p 4J S» 2 91, | C i i I 1 i 1 V | ^^ i , . *=" i ! 4 1 T 1 ^^ i 4 4 . . i . . i . i "! I I.I."1 .II i "! I I I ' I I I I I I i i 1 I I • •I . . • . •+— 2 I 1 4 1 • • - •I 1 ! 4 ! - •1 «/i t 4 I • -~ ~- • I -T- I i i i _ I . . I . . I ._ . . i . . i . . i . . . . r. i .n 1,I.I.. o •otons/ n M 24 2 5» i i T " " "^ i « - -, i T ' ~ 1 Q. • "^ ~" - ^ T i - ~~T i 4 i « o III . iii i i i . . : T ' • n III! sf _o _. •1 '1 •1 i - i - .—^ i i I • 1 • __ — -_ • • -1 1 ! - ~ i _— _ - 1 - r ' - [ •4 i - ( - •4 _- - 1 • I I . . I . . f It ll fn ' fl '* M ' fit ' fi 0 90 18 r. . I . rTr . I t H fN 'III ft »' ' f» ' fll ) it )i fn fi ) 90 1 'AH ' fB ' fl 0 90 1! _ 1 30 10 0BSERVED 10 keV PREDICTED 0BSERVED 2 keV PREDICTED Pitch Angle (Degrees) FIGURE 12 Equatorial pitch angle distributions at L=3 observed during the recovery phase of the Dec. 17, 1971 storm are compared with those expected to evolve from proton charge exchange with neutral hydrogen. Observations are shown from the inbound portion of Explorer 45 orbits 101, 102,..., 106 with the obser- vations from orbit 101 being less than 1 hour following the minimum of Dst. The appearance of elevated fluxes over the pitch angle range of 90° to 180° is an instrumental error resulting from reflected sunlight. The pitch angle distributions expected to evolve from charge exchange were obtained by neglecting possible sources, assuming an isotropic proton pitch angle distribution at the time of the orbit 101 observations (t=0), and using the charge exchange lifetimes given by (Tinsley, 1976). The initial fluxes for the calculations were arbitrarily normalized to approximately sX 10~i (from Lyons and Evans, 1976).

1008 L=3.5 PITCH ANGLE DISTRIBUTIONS ORBIT 1 < I i I I02 7.84 HR I03 I5.68 HR I04 105 I06 23.52 HR 31.36 HR 39.20 HR •= OHR j n 41 j u 5 33 10 3. *L J l« «J J U 5 !? !* i 41 I * is n in, - . - :— H I • H • • • " i • > 4 O) • . I 1 1 • 1 1 1 < - 1 8 _- -_ • • _— _ OT •» • _- — -. - ~ • i - 1 • _~ l - 1 - - • ~ 1 1 1 . ' - -•"-- . . I . . I . . I . . -411- j , Protons 1 47'lt I 49 1 FHTi >.n I • T T 1 • •! I - - • • H n i • ^ • I • - 1 ! - • 1 1 1 ! • 2 . - I 1 1 1 1 1 - 1 1 1 n 1 ' \ • O1 5 - . _ • .~ • • 1 - •I J - - 1 - 1 • _— _ 1 '- • _-~ ~-_ . _ I , . l . . I . . - .1 - . 1 - ,J 1 4.' J.' J 'I ,.I .j.1,.1... l • i!1 jj jj- j 1 If II In III W 99 ft Ifl II r n H In Ii ) 0 90 I8 0 0 90 I80 0 90 18 OBSERVED lOkeV PREDICTED OBSERVED 2 keV PREDICTED Pitch Angle (Degrees) FIGURE 13 Same as Figure 12, except for L=3.5 (from Lyons and Evans, 1976).

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