Limit on stably trapped particle fluxes in planetary magnetospheres

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi:10.109/009ja01448, 009 Limit on stably trapped particle fluxes in planetary magnetospheres Danny Summers, 1, Rongxin Tang, 1 Richard M. Thorne 3 Received May 009; revised 8 July 009; accepted 0 July 009; published 1 October 009. [1] We reexamine the Kennel-Petschek concept of self-limitation of stably trapped particle fluxes in a planetary magnetosphere. In contrast to the original Kennel-Petschek formulation, we carry out a fully relativistic analysis. In addition, we replace the wave reflection criterion in the Kennel-Petschek theory by the condition that the limit on the stably trapped particle flux is attained in the steady state condition of marginal stability when electromagnetic waves generated at the magnetic equator acquire a specified gain over a given convective growth length. We derive relativistic formulae for the limiting electron integral differential fluxes for a general planetary radiation belt at a given L shell. The formulae depend explicitly on the spectral index pitch angle index of the assumed particle distribution on the ratio of the electron gyrofrequency to the electron plasma frequency. We compare the theoretical limits on the trapped flux with observed energetic electron fluxes at Earth, Jupiter, Uranus. Citation: Summers, D., R. Tang, R. M. Thorne (009), Limit on stably trapped particle fluxes in planetary magnetospheres, J. Geophys. Res., 114,, doi:10.109/009ja01448. 1. Introduction [] Since the discovery in 1958 of the Earth s radiation belts by James Van Allen colleagues using Explorer 1 [Van Allen et al., 1958], radiation belt physics has been an active area of theoretical experimental research. Extensive progress has been made in understing the basic physical processes occurring in the radiation belts of Earth the other magnetized planets [e.g., Roederer, 1970; Schulz Lanzerotti, 1974; Lyons Williams, 1984; Schulz, 1991; Lemaire et al., 1996; Santos-Costa et al., 003; Bolton et al., 004], though many unsolved problems remain. For Earth, current problems include quantifying the variability of the energetic electron flux in the outer radiation belt understing the physical processes that control the transport, acceleration loss of radiation belt electrons [Summers et al., 007a, 007b; Li et al., 007; Baker Kanekal, 008; Hudson et al., 008]. [3] It has been recognized since the very early studies [Dungey, 1963; Cornwall, 1964; Kennel Petschek, 1966; Kennel, 1969; Roberts, 1969] that electromagnetic waves can scatter magnetospheric electrons into the loss cone by pitch angle diffusion hence cause electron precipitation into the atmosphere. Dungey [1963] Cornwall [1964] considered precipitation of radiation belt electrons by externally generated whistler mode waves, for 1 Department of Mathematics Statistics, Memorial University of Newfoundl, St. John s, Newfoundl, Canada. School of Space Research, Kyung Hee University, Yongin, Korea. 3 Department of Atmospheric Oceanic Sciences, University of California, Los Angeles, California, USA. Copyright 009 by the American Geophysical Union. 0148-07/09/009JA01448 instance by lightning-generated whistlers. Kennel Petschek [1966] showed that a trapped anisotropic electron population, if sufficiently anisotropic, will itself generate whistler mode waves which will scatter electrons lead to precipitation. This process in which the pitch angle diffusion is self-excited is inherently nonlinear since the trapped particles act back on themselves by means of the waves. Nevertheless, Kennel Petschek [1966] assumed that the whistler mode turbulence is weak were able to use a linear perturbation technique,, in a seminal paper, obtained the widely quoted result that there exists an upper limit on the integral omnidirectional flux of particles that can be stably trapped in a dipole field; if the limiting flux is exceeded then increased particle precipitation results. Kennel Petschek [1966] argued that the limiting flux is attained in the steady state situation of marginal stability when the wave energy lost due to internal reflection is balanced by the increase in wave energy due to convective wave growth. We provide a brief technical derivation of the Kennel-Petschek flux limit at the end of our paper in Appendix B. The Kennel-Petschek limit is not an actual physical limit on the flux of particles that can be stably trapped in a planetary magnetosphere,, further, the limit applies only to the case of weak diffusion wherein the loss cone is partially empty. While a sufficiently intense particle source will lead to an increase in trapped flux beyond the Kennel-Petschek limit to an increased rate of precipitation, the precipitation rate itself cannot exceed the strong diffusion rate [Kennel, 1969]. Therefore, in principle, if the particle source population continues to increase beyond that required to produce strong diffusion, then the trapped particle flux could increase indefinitely. In general, a trapped particle distribution, in addition to being subject to the selfinduced or nonparasitic pitch angle scattering postulated 1of1

in the Kennel Petschek [1966] scenario, could also be subject to parasitic scattering by externally generated waves. For instance, energetic electrons in the slot region can be scattered by plasmaspheric hiss [Lyons et al., 197] while those in the Earth s outer zone can be subject to intense precipitation due to scattering by electromagnetic ion cyclotron waves, under appropriate conditions [Thorne Kennel, 1971; Lyons Thorne, 197; Summers Thorne, 003]. The integral flux of stably trapped particles is expected to be much less than the Kennel-Petschek limit if significant precipitation losses occur due to parasitic pitch angle scattering. Other particle loss processes not accounted for in the Kennel Petschek [1966] formulation include loss due to sweeping by planetary satellites or to absorption by dust clouds each of which occurs, for example, at Saturn [e.g., Santos-Costa et al., 003]. Jovian radiation belt particles likewise suffer losses due to satellites rings [e.g., Santos-Costa Bourdarie, 001; Sicard Bourdarie, 004]. [4] The self-limiting flux concept has been addressed in various studies. Schulz [1974] incorporated the Kennel- Petschek limit on the trapped flux in a nonlinear phenomenological model of particle saturation in the Earth s outer radiation belt. Schulz Davidson [1988] extended the Kennel-Petschek theory to derive a limiting form of the particle differential flux, which at large energy E varies asymptotically as 1/E. Comparisons of the Kennel-Petschek integral flux limit with observed particle fluxes in the Earth s outer zone have been made by Baker et al. [1979], Davidson et al. [1988], other authors, as well as by Kennel Petschek [1966]. Observed values of the electron flux in the region 5 L 10, in particular at geosynchronous orbit L = 6.6, were found to be generally below, often near, the Kennel-Petschek limit. Typically, observed values were within a factor of or 3 of the Kennel-Petschek limit, though some of the data suggest that electron fluxes exhibit greater temporal variability than can be accounted for by the steady state Kennel-Petschek theory. Baker et al. [1979] present data at Earth s geosynchronous orbit that provide evidence for an experimental upper limit to the electron integral flux that well exceeds the Kennel-Petschek limit. Barbosa Coroniti [1976], assuming an ultrarelativistic electron population, calculated the Kennel-Petschek flux limit for Jupiter s inner magnetosphere (4 L 1), concluded that within theoretical experimental uncertainties observed fluxes were near the stably trapped limit. Using data from the LECP instrument on the Voyager spacecraft, Mauk et al. [1987] found that measured electron integral fluxes in the inner magnetosphere of Uranus near L = 4.73 were in excess of the Kennel-Petschek limit by an order of magnitude. At the same time, intense whistler mode wave activity was observed by the Voyager plasma wave experiment [Gurnett et al., 1986]. [5] The Kennel-Petschek concept of self-limitation of radiation belt particle fluxes remains important in magnetospheric physics, though the conditions relating to its validity are imperfectly understood warrant further investigation. Almost all previous investigations of flux limiting, including the original work by Kennel Petschek [1966], have used the nonrelativistic approximation. Here for the first time, we present an exact, fully relativistic analysis of the Kennel-Petschek trapping limit. When relativistic effects are included, wave growth properties can be significantly different from those derived from a nonrelativistic treatment, for electron thermal energies above 100 kev, especially in regions of low plasma density [e.g., Xiao et al., 1998]. We present in section our derivation of the limiting electron integral flux in a fully relativistic regime. To achieve this, we derive the temporal growth rate for field-aligned whistler mode waves in a relativistic plasma modeled by our adopted distribution function (equation (1)), we apply the condition that waves generated at the magnetic equator acquire a specified convective gain over a given growth region. It is important to note that we apply the latter condition to replace the condition used by Kennel Petschek [1966] involving wave reflection. The idea of Kennel Petschek [1966] that wave packets generated near the equator are subsequently reflected back from higher latitudes has not been borne out by observations. In section 3 we briefly examine in the context of the present study the aforementioned limiting energy spectrum derived by Schulz Davidson [1988]. In section 4, we apply our trapping limits to Earth, Jupiter Uranus, compare our results with observed energetic electron fluxes. Finally, in section 5 we summarize the results state our conclusions.. Stably Trapped Flux Limit.1. Particle Distribution Function [6] To model the (equatorial) electron phase-space density we use the distribution function J? ðp Þ p l f p k ; p? ¼ ðsin aþ s ; p p ð1þ p p with p = p k + p?, where p k = gm e v k p? = gm e v? are the components of relativistic momentum p = gm e v, m e is the electron rest mass, v is the electron velocity with components v k v? respectively parallel perpendicular to the ambient magnetic field, g =(1 v /c ) 1/ = (1 + p /(m e c) ) 1/, with v = v k + v?, c is the speed of light; a = tan 1 (p? /p k ) is the electron pitch angle; l is the spectral index s is the pitch angle index; p is a minimum value of the momentum to be specified below. Since the electron differential number flux J is given by J ¼ p f ; then J? (p ) in equation (1) is the perpendicular (a = p/) differential number flux at p = p. We introduce the electron kinetic energy E, given by E/(m e c )=g 1, which satisfies the relation p ¼ m e c E m e c E m e c þ ðþ : ð3þ Then, by using equations (1) (), the electron integral omnidirectional flux, Z 1 I 4p E > ¼ 4p 0 dðcos aþ JdE ð4þ of1

whistlers in Jupiter s magnetosphere, Schulz Lanzerotti [1974] used a nonrelativistic form of equation (1) in their determination of the limit on trapped particle flux in the Earth s magnetosphere. In the nonrelativistic case, g =1, p = m e v, E =(m e v )/, then substitution of distribution (1) into equation (4) gives E I 4p ðe > E Þ¼p 3= Gð s þ 1 J Þ G s þ 3? ðp Þ ; ð7þ ðl 1Þ where = p /(m e ). [8] Taking into consideration the form of the distribution function (1) relation (7), we restrict our attention to the physically feasible parameter values l > 1 s > 0. In the top panel of Figure 1, for l = s = 0.5, we show typical contours of constant phase-space density for distribution (1); in the bottom panel we illustrate the pitch angle variation of the distribution for different values of s... Wave Growth Rates [9] In Appendix A we derive the temporal growth rate w i for electromagnetic R-mode waves propagating parallel to a uniform magnetic field, corresponding to the anisotropic electron distribution (1) in the presence of a cold background electron population. In the fully relativistic case, we find w i p l jw e j ¼ p ð1 xþ J? ðp Þ ðm N 0 e cþh i me c 1 þ axð1 xþ n I ðx; yþ x o y I 1ðx; yþ ð8þ where N 0 is the cold electron number density, a ¼jW e j =w pe ð9þ is a cold plasma parameter where jw e j is the electron gyrofrequency w pe is the plasma frequency (each defined in Appendix A), Figure 1. (top) Contours of constant phase space density f = constant, where f is given by equation (1), for s = 0.5, l = ; the contours are labeled by values of l where [(p k /(m e c)) + (p? /(m e c)) ] l+s+1 = l[p? /(m e c)] s. (bottom) The pitch angle variation (at a fixed energy) for distribution (1), for the indicated values of the pitch angle index s. is given by p I 4p E > ¼ p 3= Gðs þ 1Þ J G s þ 3? ðp Þ me c Z m e c 1 dz where mec l ðz þ zþ l ; ð5þ p E ¼ m e c m e c m e c þ ; ð6þ G is the gamma function. [7] Barbosa Coroniti [1976] used a particle distribution of the form (1) in their study of relativistic electrons x ¼ w=jw e j; y ¼ ck=jw e j; ð10þ where w is the (real) wave frequency k is the (real) wave number. I 1, I are the integrals given by dz z sþ1 ðl þ 1Þz sp R I 1 ðx; yþ ¼ 0 D R ðz þ p RÞ lþsþ ; ð11þ where I ðx; yþ ¼ 0 dz D R g R sð p R Þz sþ1 z þ p lþsþ1 ; ð1þ R g R ¼ x þ y ½ð y x Þð1 þ z Þþ 1Š 1= y x ; ð13þ p R ¼ g Rx 1 ; ð14þ y D R ¼ 1 x ð g Rx 1Þ g R y ; ð15þ 3of1

x, y satisfy the cold plasma R-mode dispersion relation, y ¼ x x þ að1 xþ : ð16þ [10] In general, the integrals I 1 I cannot be found in closed analytical form so must be evaluated numerically. We evaluate I 1 I by employing the MATLAB function quadgk which uses an adaptive Gauss-Kronrod quadrature method [Shampine, 008]. In addition, it is efficient to evaluate the integral occurring in equation (5), also the similar integral in equation (46) below, by using quadgk even though these integrals can be evaluated analytically for certain values of the parameter l. [11] The nonrelativistic approximation to the wave growth rate can be recovered by setting D R = 1 g R = 1 in formulae (11) (1) for the integrals I 1 I. In this special case, the integrals can be obtained analytically we find I 1 ðx; yþ ¼ Gðs þ 1ÞGðÞ l y l; Gðl þ s þ 1Þ 1 x I ðx; yþ ¼ sgðs þ 1ÞGðÞ l y l 1: ð17þ Gðl þ s þ 1Þ 1 x Substitution of results (17) into equation (8) yields the nonrelativistic growth rate, w i p l jw e j ¼ p ð1 xþ J? ðp Þ ðm N 0 e cþh i me c 1 þ axð1 xþ Gðs þ 1ÞGðÞ l y l 1 n s x o ; ð18þ Gðl þ s þ 1Þ 1 x 1 x which can be shown to agree with the growth rate formula (.65) given by Schulz Lanzerotti [1974]. We find it useful to provide, in Appendix A, an equivalent derivation of expression (18) in which the parameter s is specifically identified as the electron pitch angle anisotropy in the nonrelativistic formulation (see equation (A15)). [1] In Figure we select typical cases to illustrate the dependence of the relativistic nonrelativistic profiles of the whistler mode wave growth rate on the values of the parameters s l; in the top panel we keep s fixed vary l, in the bottom panel we keep l fixed vary s. The top panel indicates that a change in l value does not significantly change the frequency b for wave growth but does have some effect on the frequency at which the wave growth maximizes. The bottom panel shows that the value of s can significantly influence the frequency b over which wave growth occurs, consequently also the value of the frequency at which the wave growth maximizes. The dependence on s l of the frequencies at which the wave growth rate vanishes maximizes is illustrated in detail for the relativistic case in Table 1, discussed below..3. Limiting Electron Flux [13] We assume that a stably trapped electron flux can be maintained close to its limiting value by whistler mode waves of sufficient power generated at the magnetic equator. The power flow P w of an electromagnetic wavefield is proportional to the magnitude of the Poynting vector Figure. Relativistic nonrelativistic whistler mode wave growth rates (given by equations (8) (18)) for the case a = jw e j /w pe = 0.05 the indicated values of the pitch angle index s the spectral index l. hence to the square of the wave amplitude B w [e.g., Swanson, 1989]. We write P w / B w / eg ; ð19þ where G is the wave gain defined by Z wi ds G ¼ : ð0þ v g The path integral (0) is taken along a flux tube with element ds; w i is the wave growth rate v g = jdw/dkj is the group speed. [14] If we apply condition (0) to a region in the vicinity of a planet with a dipole or dipole-like magnetic field, then we shall require a gain of 3 e-foldings in wave power (i.e., G = 3/) over a distance LR P / where R P is the radius of the planet. Then equation (0) takes the approximate form LR P w i ¼ 3v g : ð1þ 4of1

Table 1. For the Relativistic Case, Values of x, x m, (kev) for the Given (s, l) Values; x = w /jw e j is the Normalized Frequency at Which the Whistler Mode Wave Growth Rate Vanishes; x m = w m /jw e j is the Normalized Frequency at Which the Growth Rate Maximizes; (kev) is the Minimum Electron Resonant Energy for Interaction With Waves of Frequency w s\l 1.1 1. 1.5 1.5.5 3 4 5 6 7 8 9 10 0.1 x 0.051 0.05 0.053 0.056 0.060 0.06 0.064 0.066 0.067 0.067 0.068 0.068 0.068 0.069 x m 0.04 0.07 0.08 0.03 0.039 0.044 0.048 0.053 0.056 0.059 0.060 0.061 0.06 0.063 0. 179.4 173.6 171.1 161.5 150.6 144.6 140.9 136.7 134.3 13.8 131.8 131.0 130.4 130.0 x 0.10 0.13 0.14 0.19 0.135 0.138 0.140 0.14 0.143 0.144 0.144 0.145 0.145 0.145 x m 0.058 0.063 0.065 0.075 0.091 0.101 0.108 0.118 0.14 0.18 0.130 0.13 0.134 0.135 0.5 66.41 64.40 63.55 60.36 56.89 55.11 54.04 5.86 5. 51.83 51.57 51.37 51.3 51.1 x 0.153 0.156 0.157 0.163 0.169 0.17 0.174 0.176 0.177 0.178 0.178 0.179 0.179 0.179 x m 0.073 0.080 0.083 0.096 0.115 0.18 0.137 0.148 0.155 0.159 0.16 0.165 0.167 0.168 47.4 46.00 45.4 43. 40.87 39.69 38.99 38.3 37.83 37.58 37.41 37.9 37.0 37.13 0.5 x 0.91 0.94 0.96 0.30 0.308 0.311 0.313 0.315 0.315 0.316 0.316 0.317 0.317 0.317 x m 0.141 0.154 0.160 0.187 0.3 0.45 0.59 0.76 0.85 0.91 0.96 0.99 0.301 0.303 1.0 15.19 14.76 14.58 13.95 13.3 13.03 1.87 1.70 1.61 1.56 1.5 1.50 1.48 1.46 x 0.470 0.473 0.475 0.480 0.484 0.486 0.487 0.488 0.489 0.489 0.489 0.489 0.490 0.490 x m 0.40 0.64 0.76 0.34 0.38 0.41 0.49 0.448 0.458 0.464 0.469 0.47 0.474 0.476 3.996 3.891 3.849 3.706 3.577 3.53 3.493 3.464 3.449 3.440 3.434 3.49 3.46 3.44.0 x 0.651 0.653 0.654 0.657 0.659 0.660 0.660 0.661 0.661 0.661 0.661 0.66 0.66 0.66 x m 0.361 0.403 0.43 0.495 0.566 0.596 0.61 0.69 0.637 0.64 0.645 0.648 0.649 0.651 3.0 0.835 0.817 0.809 0.786 0.767 0.760 0.756 0.75 0.750 0.749 0.748 0.748 0.747 0.747 x 0.740 0.74 0.74 0.744 0.746 0.746 0.746 0.747 0.747 0.747 0.747 0.747 0.747 0.747 x m 0.436 0.489 0.513 0.597 0.667 0.694 0.707 0.71 0.78 0.731 0.734 0.736 0.737 0.739 0.303 0.97 0.95 0.88 0.83 0.81 0.80 0.79 0.76 0.78 0.78 0.78 0.78 0.78 4.0 x 0.793 0.795 0.795 0.796 0.797 0.797 0.798 0.798 0.798 0.798 0.798 0.798 0.798 0.798 x m 0.487 0.549 0.575 0.665 0.731 0.754 0.765 0.776 0.78 0.785 0.787 0.789 0.790 0.791 5.0 0.143 0.140 0.139 0.137 0.135 0.134 0.133 0.133 0.133 0.133 0.133 0.133 0.133 0.133 x 0.89 0.89 0.830 0.831 0.831 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 x m 0.56 0.593 0.6 0.713 0.774 0.794 0.804 0.814 0.818 0.81 0.83 0.84 0.85 0.86 0.078 0.077 0.077 0.075 0.074 0.074 0.074 0.074 0.074 0.074 0.073 0.073 0.073 0.073 10.0 x 0.908 0.908 0.908 0.908 0.908 0.909 0.909 0.909 0.909 0.909 0.909 0.909 0.909 0.909 x m 0.633 0.716 0.748 0.833 0.875 0.887 0.893 0.898 0.901 0.903 0.904 0.904 0.905 0.905 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 If we apply equation (0) to a region with a stretched magnetic field or current sheet, then we specify a wave gain G = 3/ over the distance sr P, where the thickness of the current sheet is sr P s is a scalar constant independent of L shell. In this case, we replace equation (1) by the condition sr P w i ¼ 3v g =: Since the power gain in decibels (db) is Z wi ds gdb ð Þ ¼ 10 log 10 exp v g ðþ ; ð3þ then setting G = 3/ is equivalent to a power gain of 30 log 10 e 13 db over the specified region of convective wave growth. In both equations (1) () we evaluate w i v g at the frequency w = w m corresponding to the maximum value of the wave temporal growth rate. [15] From equation (16) we find the group speed of a whistler mode wave is given by v g c ¼ dx dy ¼ ay ð 1 x Þ 1 þ axð1 xþ : ð4þ [16] We emphasize that the wave gain conditions (1) or () replace the original condition used by Kennel Petschek [1966] involving wave reflection. The idea of wave packets bouncing back forth along a field line used by Kennel Petschek [1966] has not been validated since observations have established that the Poynting flux of the waves appears to be always directed away from the equatorial source region. As well, it has been recently found that waves are not well guided by the magnetic field lines so, at least locally, a wave generated on one flux tube would not be fed back along the same flux tube via reflection. This makes the global modeling problem all the more difficult. Regarding the gain required to ensure self-sustaining waves, we have elected to make our formulation equivalent to that of Kennel Petschek [1966] by setting G = 3/, corresponding to a net wave gain of 13 db, as we have stated. In fact, there is some latitude in the specification of G. Analysis of whistler mode wave growth using data from the Combined Release Radiation Effects Satellite (CRRES) [Li et al., 008] the Time History of Events Macroscale Interactions during Substorms (THEMIS) satellite [Li et al., 009] suggests that strong waves require a gain of about 50 db to grow from background noise level. This corresponds to G 6 would consequently increase our estimate of the limit on stably trapped flux by a factor of 4. It can be argued that substantial whistler mode wave growth, at least at Earth, requires a gain in the range G 3 to 6. The comparisons below of our limiting solutions with observed electron fluxes at Earth should be viewed with this caveat. [17] Application of condition (1) or () to result (8) for the growth rate w i serves to determine the limiting value of J? (p ) hence (by equation (5)) the upper limit on the integral omnidirectional flux I 4p (E > ). We find the limiting values are J? ðp Þ¼ 3B 0 4p 3 e 1 1 H P ðm e cþ m e c l Kx p m ; y m ð Þ ð5þ 5of1

I 4p ðe > Þ¼ 3cB 0 p 3= e where Kx; ð y 1 G ð s þ 1Þ H P G s þ 3 Kx ð m ; y m Þ mec dz ðz þ zþ l ð6þ Þ ¼ y = ½yI ðx; yþ xi 1 ðx; yþš; ð7þ x m ¼ w m =jw e j; y m ¼ ck m =jw e j; k m ¼ kðw m Þ; ð8þ H P is the convective growth length given by 8 < LR P = for a dipolar region H P ¼ : sr P for a current-sheet region : ð9þ We identify p (or ) as the minimum electron momentum (or energy) for which gyroresonance can occur with a wave having a positive value for the growth rate. From equations (8) (16), we see that w i > 0 for 0 < w < w where x = w /jw e j y = ck /jw e j satisfy I ðx ; y Þ¼ x y I 1 ðx ; y Þ y ¼ x þ x að1 x Þ : ð30þ ð31þ Numerical values for x, y are determined from the simultaneous solution of equations (30) (31). By setting w = w, k = k, v k = v, v? = 0 in the relativistic gyroresonance condition, w kv k ¼jW e j=g; ð3þ we can show that p g v m e c ¼ c ; m e c ¼ g 1 ð33þ where! 1= g ¼ 1 v c ð34þ 1= v x y 1 þ y c ¼ x 1 þ y : ð35þ [18] Result (6), with given by equations (33) (35), gives the limiting omnidirectional electron flux for E > in the fully relativistic case, is a key new result in this paper. [19] In Table 1 we present values of x, x m in the relativistic case for the specified array of (s, l) values, we set the cold plasma parameter a = jw e j /w pe = 0.05 which typically represents the Earth s trough region outside the plasmasphere near L = 4[Sheeley et al., 001]. We deduce from Table 1 that for any fixed l value, the values of x, x m depend sensitively on s, with x x m increasing decreasing as s increases. For any fixed s value, x are relatively insensitive to l, while x m increases moderately as l increases. [0] The nonrelativistic forms of results (5) (6), which are obtained from equations (18), (1), () (7) (or equations (5), (17) (7)), are J? ðp Þ¼ 3B 0 4p 3 e 1 1 me c H P ðm e cþ p l Mx ð m ; y m Þ; ð36þ l 3B 0 1 1 me c I 4p E > ¼ p 3= e H P ðm e cþ p E Gðs þ 1Þ ðl 1Þ G s þ 3 Mx ð m ; y m Þ ð37þ with Mðx; yþ ¼ Gðl þ s þ 1Þ ð1 x Gðs þ 1ÞGðÞ l Þ l 1 y ðl 1Þ ½sð1 x Þ xš ð38þ where p assume nonrelativistic values of the minimum momentum minimum energy; the function M(x, y) is the nonrelativistic limit of the function K(x, y). The nonrelativistic growth rate (18) is positive for frequencies such that 0 < w < w, where w is given by x ¼ w jw e j ¼ s s þ 1 ; ð39þ at which, by equation (16), the corresponding wave number is y ¼ ck " # jw e j ¼ sð1 þ s 1= Þ þ as að1 þ sþ : ð40þ Then, setting w = w, k = k, v k = v in the nonrelativistic gyroresonance condition w kv k ¼jW e j ð41þ leads to v /c = (1 x )/y, which, using equations (39) (40), yields the required nonrelativistic values p m e c ¼ a 1= sð1 þ sþ þ as 1= ; ð4þ m e c ¼ a : sð1 þ sþ ð43þ þ as [1] Result (37), with p defined by formulae (4) (43), gives the limiting omnidirectional electron flux in the nonrelativistic case. Kennel Petschek [1966], Schulz Lanzerotti [1974] other authors have given various versions of formula (37). In Appendix B we derive the formula for the stably trapped flux limit originally given by Kennel Petschek [1966]. 6of1

[] Our results (5) for the limiting perpendicular differential flux J? (p ) (6) for the limiting integral omnidirectional flux I 4p (E > ) can be readily extended. For, from relations (1) () it follows that J? ðpþ ¼ J? ðp Þ p l ; p p p : ð44þ Result (44) is therefore the upper limit for the perpendicular differential flux, where in the fully relativistic case J? (p )is given by equation (5) p by equation (33), in the nonrelativistic approximation J? (p ) is given by equation (36) p by equation (4). Further, since the integral omnidirectional flux of electrons exceeding the energy E 0 is I 4p ðe > E 0 Þ ¼ 4p 0 dðcos aþ JdE; ð45þ E 0 where we take E 0 >, then, using equations (1), () (5), we find I 4p ðe > E 0 Þ ¼ 3cB 0 1 G ð Z s þ 1Þ 1 dz p 3= e H P G s þ 3 Kx ð m ; y m Þ E 0 ðz þ zþ l : mec ð46þ Expression (46) gives the limiting integral omnidirectional electron flux for energies exceeding E 0, where E 0 >. The nonrelativistic approximation to result (46), obtained by substituting the nonrelativistic form of equations (1) () into equation (45), by using equation (36), is I 4p ðe > E 0 Þ¼ 3cB 0 1 m e c l 1 1 Gðs þ 1Þ 4p 3= e H P E 0 ðl 1Þ G s þ 3 Mx ð m ; y m Þ: ð47þ [3] In Figure 3 we plot x, x m as functions of the parameter a = jw e j /w pe in both the relativistic nonrelativistic cases, for the pitch angle indices s = 0.1, 0.5,.0, the fixed spectral index l =. Since a / B 0 /N 0, then for a given background magnetic field, a large value of a corresponds to a small value of the electron density N 0. Thus we observe from Figure 3 that, as expected, the differences between the relativistic nonrelativistic profiles of x, x m become more pronounced as a becomes large or N 0 becomes small. The sensitive dependence of x, x m on the pitch angle index s,asis evident from Table 1 (in the relativistic case) as noted earlier, is also illustrated in Figure 3. 3. Limiting Energy Spectrum [4] In the formulation of Kennel Petschek [1966], also in the present work, a limiting form for the trapped particle flux is calculated for a specified particle distribution, namely the energy spectrum the pitch angle distribution are assumed a priori. Accordingly, for a particle distribution of the form (1), the limiting flux strictly depends on the spectral index l the pitch angle index s. In a modification of the Kennel-Petschek theory, Schulz Davidson Figure 3. (top) Values of the whistler mode wave frequency x = w /jw e j at which the growth rate vanishes, as a function of a = jw e j /w pe, in the relativistic nonrelativistic cases, for the indicated values of s, l. (middle) Corresponding to the top panel, values of the wave frequency x m = w m /jw e j at which the growth rate maximizes. (bottom) Values of the minimum resonant energy for whistler mode wave-electron interaction corresponding to the wave frequency x given in the top panel. [1988] avoided an explicit assumption concerning the energy spectrum took a particle distribution of the form, f p k ; p? ¼ ðsin aþ s gðpþ : ð48þ 7of1

The corresponding omnidirectional differential particle flux, where J = p f,is J 4p ðeþ ¼ 4p J 4p ðeþ ¼ p 3= G ðs þ 1Þ G s þ 3 p gðpþ: 0 Jdðcos aþ; ð49þ ð50þ In a nonrelativistic analysis, Schulz Davidson [1988] calculated the limiting form of the function g(p), for an energetic particle population of given pitch angle anisotropy s, that corresponds to marginal stability for parallelpropagating electromagnetic cyclotron waves over all frequencies w such that 0 < w/jw e j < s/(s + 1). The limiting omnidirectional differential flux due to Schulz Davidson [1988], for large energy E, can be written ~J 4p ðeþ ¼ cb 0 4p 3= e s þ 1 Gðs þ 1Þ s G s þ 3 ln ð 1=R Þ 1 LR P E ; ð51þ which corresponds to the limiting perpendicular particle differential flux, ~J? ðeþ ¼ cb 0 s þ 1 lnð1=rþ 1 8p 3 e s LR P E : ð5þ To derive result (51), in which R(<1) is a coefficient of wave reflection, Schulz Davidson [1988] employed a wave gain condition similar to that used by Kennel Petschek [1966]; see condition (B1) below. The condition requires that in the limiting state the loss of wave energy due to imperfect reflection at the ionosphere is balanced by path-integrated wave growth along the field line. Expression (51) is not strictly dependent on the form of wave gain condition actually used by Schulz Davidson [1988]. If alternatively one specifies a wave gain of ln(1/r) over the convective growth length LR P, where R P is the planetary radius L denotes shell parameter, then the limiting energy spectrum (51) likewise follows. In the following section, where appropriate, we incorporate the limiting spectrum (5) in comparisons of the limiting solutions obtained in the present paper with observational data. 4. Comparison of Stably Trapped Flux Limits With Data [5] Relativistic expressions (5), (6), (44) (46) for J? (p ), I 4p (E > ), J? (p) I 4p (E > E 0 ), respectively, may be evaluated for any given planetary radiation belt. These results depend partly on B 0 = B 0 (L), the equatorial magnetic field strength at a given L shell, the wave convective growth length H P. The values of the minimum momentum p, the minimum energy, the function K(x m, y m ) each in general depends on the pitch angle index s, the spectral index l, the cold plasma parameter a = jw e j /w pe / B 0 (L)/ N 0 (L), where N 0 (L) is the electron number density at a given L shell. In the relativistic formulation, for any particular case, the parameters s, l a are specified then x x m must be found; x corresponds to the frequency at which the growth rate (8) vanishes, for which we must solve equations (30) (31), while x m corresponds to the frequency of maximum wave growth, for which we must maximize the function (8) with respect to x. Once the values of x x m are determined (which may take nontrivial computing effort), then all the remaining quantities p,, J? (p ), I 4p (E > ), J? (p) I 4p (E > E 0 ) are readily found. [6] In the nonrelativistic formulation, calculations are much more straightforward require less labor. Here x = s/(s + 1) is known (result (39)), p, likewise take simple forms (results (4) (43)). Determination of x m in the nonrelativistic case requires maximizing the function (18) with respect to x. Oncex m is found, the nonrelativistic quantities J? (p ), I 4p (E > ), J? (p) I 4p (E > E 0 ) easily follow from equations (36), (37), (44), (47) respectively. 4.1. Earth [7] For Earth, we set B 0 (L)=B E /L 3 H P = LR E /, with B E = 0.31 gauss R E = 6.4 10 8 cm. Then the limiting formulae (6), (5), (44) (47) can be expressed in the form J? 1:64 10 10 I 4p E > ¼ Al; ð s; aþ cm sec 1 ; ð53þ :88 10 6 J? p ¼ ðpþ ¼ where L 4 :88 106 L 4 E L 4 Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; ð54þ l ðe = ðm e c ÞÞþ ðe= ðm e c ÞÞþ Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; p > p ; ð55þ 1:64 1010 I 4p ðe > E 0 Þ ¼ L 4 Dl; ð s; a; E 0 Þ cm sec 1 ; ð56þ A ¼ G ðs þ 1Þ G s þ 3 Kx ð m ; y m Þ mec! l dz ðz þ zþ l ; ð57þ Kx ð m ; y m Þ C ¼ E l E l ; ð58þ m e c m e c þ D ¼ G ðs þ 1Þ G s þ 3 Kx ð m ; y m Þ E 0 mec dz ðz þ zþ l : ð59þ In Figure 4 we show two-dimensional (l, s) plots of the parameters A C the corresponding minimum electron resonant energy in the case a = jw e j /w pe = 8of1

[8] In the nonrelativistic approximation, results (53), (54) (56) remain valid except that the coefficients A, C D are then given by A ¼ sð1 þ s Þ þ as l 1 G ð s þ 1Þ ðl 1Þa l 1 G s þ 3 Mx ð m ; y m Þ; ð60þ C ¼ sð1 þ sþ þ as a l l Mx ð m ; y m Þ; ð61þ 1 m e c l 1 Gðs þ 1Þ D ¼ ðl 1Þ E 0 G s þ 3 Mx ð m ; y m Þ: ð6þ Figure 4. Two-dimensional (l, s) plots of the parameters A(l, s, a) C(l, s, a), the corresponding electron resonant energy, in the relativistic case with a = jw e j /w pe = 0.05. 0.05. Figure 4 demonstrates that the values of A C can be either moderately or strongly dependent on the values adopted for l s, subject to the particular region of (l, s) parameter space considered. Figure 5. Relativistic nonrelativistic values of the parameters A(l, s, a) C(l, s, a) as functions of the pitch angle index s, for a = jw e j /w pe = 0.05 the indicated values of l. 9of1

Figure 6. Relativistic nonrelativistic values of the parameters A(l, s, a) C(l, s, a) for l = the indicated values of s. In the left panels, A C are plotted as functions of L shell, assuming the electron density model N 0 = 14(3/L) 4 cm 3 [Sheeley et al., 001] a dipole magnetic field in the Earth s trough region outside the plasmasphere. In the center right panels, A C are plotted as functions of a = jw e j /w pe, assuming the full whistler mode dispersion equation (16) in the center panels, the simplified dispersion equation (B) in the right panels. The nonrelativistic version of equation (55) is J? ðpþ ¼ :88 106 m e c l L 4 Mx ð m ; y m Þcm sec 1 sr 1 kev 1 ; E p > p : ð63þ [9] Complementary to Figure 4, in Figure 5 we show the variation of A C (given by equations (57) (58)) as a function of s, for 0.1 s 5, in the cases l = 1.1,.0 4.0, with a = jw e j /w pe = 0.05. Also, in Figure 5 we show the corresponding nonrelativistic forms of A C (given by equations (60) (61)). Differences between the relativistic nonrelativistic forms of A C are relatively small if s 1, though may become more significant if s 1. [30] A typical range of s values measured in Earth s outer radiation belt is 0 < s < 1.5; see Table 1 in Thorne et al. [005] which was reproduced from an unpublished report by A. Vampola. Analysis of electron data measured by CRRES [Li et al., 008] THEMIS [Li et al., 009] during nightside electron injection events when intense whistler mode chorus is present yields peak values of s in the range 1 < s < 1.5. [31] In the left panels of Figure 6, for Earth we show the relativistic nonrelativistic forms of A C as functions of L shell for the spectral index l = the pitch angle indices s = 0.1, 0.5.0. We assume a dipole magnetic field the electron number density variation N 0 = 14(3/L) 4 cm 3 (due to Sheeley et al. [001]) in the trough region outside the plasmasphere; see Figure 7. In the relativistic case, A is a weakly decreasing function of L if s > 0.5 a Figure 7. Electron number density N 0 at Earth (from the Sheeley et al. [001] trough density model) as a function of L shell, together with the corresponding profile of the cold plasma parameter a = jw e j /w pe. 10 of 1

Figure 8. Measured electron differential spectrum J? at L = 5 L = 6 from the MEA experiment on CRRES (orbit 19) during the geomagnetic storm of 9 October 1990, together with corresponding limiting profiles (55) for the indicated values of s, l, a = jw e j /w pe. Also shown are limiting energy spectra due to Schulz Davidson [1988]. moderately decreasing function of L if s < 0.5 (approximately), while C is an increasing function of L for all s values. In the nonrelativistic case, C is likewise an increasing function of L, while A remains independent of L to a good approximation, for all s values. [3] In the center right panels of Figure 6 we plot the relativistic nonrelativistic forms of A C as functions of the cold plasma parameter a = jw e j /w pe, for l = s = 0.1, 0.5,.0; we assume the full whistler mode dispersion relation (16) in the center panels the simplified dispersion relation (B) used by Kennel Petschek [1966] in the right panels. In the relativistic case (both center right panels) for s 0.5, if a < 0.05 then A C are relatively independent of a, while if a > 0.05 then A is an increasing function of a C is a decreasing function of a. In the nonrelativistic case (both center right panels) for all values of s considered, A is weakly dependent on a while C is a decreasing function of a. Comparing the center right panels, we see that differences in the profiles of both A C caused by adopting the simplified dispersion relation only become apparent for larger values of a with s 0.5. In both center right panels, differences between the relativistic corresponding nonrelativistic profiles become more significant for larger values of a, as expected (see Figure 3). [33] In Figure 8 we compare the limiting profiles (55) for the perpendicular electron differential flux with the electron flux measured by the CRRES Medium Electrons A (MEA) experiment [Vampola et al., 199] during the 9 October 1990 geomagnetic storm. Particle wave data for this storm were also analyzed by Meredith et al. [00] Summers et al. [00]. The measured electron flux in Figure 8 is taken during CRRES orbit 19, close to the end of the several-day recovery period of the storm, at L = 5 L =6. Electron energization due to gyroresonant interaction with chorus waves during this recovery period [Summers et al., 00] results in the formation of the high-energy tail in the electron spectrum shown in Figure 8. At L = 5 L =6we adopt the respective spectral indices l = 1.4 l =.1 since these values approximately match the slopes of the measured 11 of 1

can be sensitive to the value of a. The measured flux the limiting profiles (55) are in close agreement for certain values of s a, e.g., see the bottom right panel. However, conclusive comparison between the limiting flux (55) the experimental data is difficult not least because of the range of uncertainty of the parameter a. For comparison purposes, we also show in Figure 8 corresponding limiting spectral profiles (5) due to Schulz Davidson [1988] for which we set ln(1/r) = 3. These limiting solutions, which are valid for large energy E, are independent of the value of a. [34] In Figures 9 10 we compare the limiting solutions (56) with the measured electron flux profiles at Earth as given by the AE-8 model [Vette, 1991]. In Figure 9 we plot the integral fluxes I 4p (E > E 0 ) for E 0 = 100, 150, 00 50 kev, over the range 3 L 8. For the model solutions we assume a nominal value for the spectral index l = 1.6, for the pitch angle index we choose the values Figure 9. (bottom) AE-8 model [Vette, 1991] electron integral fluxes I 4p (E > E 0 ) at Earth, for the specified energies E 0 as a function of L shell, together with corresponding limiting profiles (56) for the indicated values of s, l. (top) Corresponding to the bottom panel, minimum resonant energy for gyroresonant interaction with a whistler mode wave with positive growth rate, as a function of L shell. flux profiles. In the left panels we vary the pitch angle index s we fix values for the cold plasma parameter a. For the latter we use electron density values given by the Sheeley et al. [001] model; we find a = 0.038 at L =5, a = 0.06 at L = 6. We see in the bottom left panel that for s 0.3 the measured differential flux well exceeds the limiting solutions (55). In the top left panel, the measured flux well exceeds the limiting solutions if s 1, slightly exceeds (over most energies) the limiting solution if s = 0.5, is exceeded by the limiting solution if s 0.3. In the right panel we fix s = 0.6, we select values of the parameter a in the range given by Meredith et al. [00]. It is evident from the right panel that the limiting solutions (55) Figure 10. For Earth, at L = 5 L = 6, comparison of the AE-8 model [Vette, 1991] electron integral spectrum I 4p (E > E 0 ) with corresponding limiting profiles (56) for the indicated values of s, l. 1 of 1

9:33 109 I 4p ðe > E 0 Þ ¼ L 3 Dl; ð s; a; E 0 Þ cm sec 1 ; ð67þ where A, C D are given by equations (57), (58) (59). [36] Results (64), (65) (67) are valid in the nonrelativistic approximation in which case the coefficients A, C D are given by equations (60), (61) (6).The nonrelativistic form of result (66) is J? ðpþ ¼ 1:63 106 m e c l L 3 Mx ð m ; y m Þ cm sec 1 sr 1 kev 1 ; E p > p : ð68þ Figure 11. Electron number density N 0 at Jupiter [from Divine Garrett, 1983] as a function of L shell, together with the corresponding profile of the cold plasma parameter a = jw e j /w pe. s = 0.3 s = 1.0. We adopt the Sheeley et al. [001] model for the electron number density. We note the relatively close agreement between the limiting solutions the AE-8 model profiles in the region 6 L 8, for s = 0.3. The concept of self-limitation of stably trapped flux is not expected to apply to the region 3 < L < 5 where parasitic scattering processes lead to the formation of the slot. In Figure 10 we compare the Vette [1991] AE-8 model solutions at L = 5 L = 6 with the limiting spectra (56). We again adopt the Sheeley et al. [001] density model. We choose values for the spectral index that approximately match the data, namely, l = 1.9 at L = 5, l =. at L = 6. In each case we then vary the pitch angle index s. At both L = 5 L = 6 it is evident that the limiting solution (56) is relatively close to the experimental profile for appropriately specified (small) values of s. For moderate or large values of s (say, s 0.5) the measured profiles greatly exceed the limiting solutions. 4.. Jupiter [35] For Jupiter, we set B 0 (L) =B J /L 3 H P = sr J, with B J = 3.9 gauss, R J =11R E s = 1. The limiting results (6), (6), (5), (44) (47) then take the form I 4p ðe > Þ¼ 9:33 109 L 3 Al; ð s; aþ cm sec 1 ; ð64þ J? ðp Þ¼ 1:63 106 L 3 Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; ð65þ J? ðpþ ¼ 1:63 106 L 3 E 0 1 l E = ðm e c l Þ þ @ A ðe= ðm e c ÞÞþ Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; p > p ; ð66þ Figure 1. Electron integral fluxes I 4p (E > E 0 ) at Jupiter measured by the Galileo Energetic Particle Detector (EPD) [Jun et al., 005] for E 0 = 1.5, 11 MeV, as a function of L shell, together with corresponding limiting profiles (67) for the indicated values of s, l. 13 of 1

Xiao et al. [003] who simulated the growth of whistler mode waves in the Io torus during an interchange injection event using data from the Plasma Wave Subsystem (PWS) EPD instruments on Galileo. The measured values of s ranged from about s = 0.4 at the start of the injection event to about s = 0.15 at the time of the most intense waves. [39] In Figure 14 we compare the electron integral spectrum I(E > E 0 ) at Jupiter (L = 7.69) measured by the Low- Energy Charged Particle (LECP) experiment on Voyager 1 [Armstrong et al., 1981] with corresponding limiting solutions (67). We show the limiting spectra in the top panel for l = 1.5 various s values, in the bottom panel for s = 0. various l values. Figure 14 further demonstrates that relatively close agreement between measured limiting flux profiles is possible for a suitably small value of the pitch angle index s in the case that the spectral index l models the slope of the measured spectrum. Figure 13. As in Figure 1, except that the limiting profiles (67) are shown for different sets of values of the parameters s l. [37] In Figure 11 we show the electron number density profile at Jupiter, for 3 L 30, as given by Divine Garrett [1983], together with the corresponding profile of the cold plasma parameter a = jw e j /w pe. [38] In Figures 1 13 we plot electron integral fluxes I 4p (E > E 0 ) at Jupiter for 6 L 15, measured by the Energetic Particle Detector (EPD) on Galileo [Jun et al., 005] for E 0 = 1.5 MeV E 0 = 11 MeV. In Figure 1 we select values of the spectral index l that approximately match the slopes of the data profiles, namely l = 1.5 for E 0 = 1.5 MeV l = 1.6 for E 0 = 11 MeV, we plot the limiting flux profiles (67) for the pitch angle indices s = 0.1 s = 0.5. Similarly, in Figure 13, at each energy, we fix the parameter s plot the limiting profiles for different values of l. Relatively close agreement between the limiting profiles the observed fluxes, at least over a limited L shell range, is possible by appropriately selecting values for l s, namely, for E 0 = 1.5 MeV, l 1. to 1.5 s 0.1 to 0.3; for E 0 = 11 MeV, l 1.6 s 0.1. Typical values for the parameter s at Jupiter have been reported by Figure 14. Electron integral spectrum I(E > E 0 ) at Jupiter (L = 7.69) measured by the Voyager 1 Low-Energy Charged Particle (LECP) experiment [Armstrong et al., 1981] together with corresponding limiting spectra derived from equation (67) for the indicated values of s, l. 14 of 1

Figure 15. (right) Electron integral fluxes I 4p (E > E 0 ) at Jupiter measured by the Galileo Energetic Particle Detector (EPD) [Sorensen et al., 005; Jun et al., 005] for the specified energies E 0,asa function of L shell, together with corresponding limiting profiles (67) for s = 0.3, l = 1.5. (left) Corresponding to the right panel, minimum resonant energy for gyroresonant interaction with whistler mode waves with a positive growth rate. [40] In Figure 15 we show electron integral fluxes I 4p (E > E 0 ) at Jupiter, for E 0 = MeV E 0 = 11 MeV, over the range 6 L 30, measured by the Galileo EPD instrument, reported by Sorensen et al. [005] Jun et al. [005]. We choose here to present a simple comparison between the measured fluxes the corresponding limiting profiles (67) for a fixed pair of nominal values for l s, namely s = 0.3 l = 1.5. Even such a rough comparison reveals that over certain ranges of L shell, dependent on energy, the observed fluxes may be close to the limiting values (67). More detailed comparisons require empirical data on the variation of the spectral index l with L shell. 4.3. Uranus [41] For Uranus, we set B 0 (L) =B U /L 3 H P = LR U /, with B U = 0.3 gauss, R U =4R E. The limiting results (6), (5), (44) (47) then become J? 3:0 10 9 I 4p E > ¼ 5:3 10 5 J? p ¼ ðpþ ¼ L 4 5:3 105 L 4 E L 4 Al; ð s; aþ cm sec 1 ; ð69þ Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; ð70þ 0 1 l E = ðm e c l Þ þ @ A ðe= ðm e c ÞÞþ Cl; ð s; aþ cm sec 1 sr 1 kev 1 ; p > p ; ð71þ 3:0 109 I 4p ðe > E 0 Þ ¼ L 4 Dl; ð s; a; E 0 Þ cm sec 1 ; ð7þ where A, C D are given by equations (57), (58) (59). [4] Results (69), (70) (7) are valid in the nonrelativistic approximation in which case the coefficients are given by equations (60), (61) (6). [43] The nonrelativistic form of result (71) is J? ðpþ ¼ 5:3 105 m e c l L 4 Mx ð m ; y m Þ cm sec 1 sr 1 kev 1 ; E p > p : ð73þ [44] In Figure 16 we show electron differential spectra at Uranus, at the given L shells, measured by the Low Energy Charged Particle (LECP) experiment [Mauk et al., 1987] on Voyager. For comparison we show the limiting solutions (71) for the perpendicular differential spectra, for the specified sets of s values. At each L shell, we choose the spectral index l so that it roughly matches the slope of the data profile over the given energy range; the value of the parameter a = jw e j /w pe is calculated using the electron density observations at Uranus reported by McNutt et al. [1987] Sittler et al. [1987]. As expected, comparisons between the measured spectra the limiting solutions depend somewhat sensitively on the values of the pitch angle index s. In addition, we caution that in Figure 16 15 of 1

Figure 16. Electron differential spectra J at Uranus, at the indicated L shells, measured by the Voyager Low-Energy Charged Particle (LECP) experiment [Mauk et al., 1987], together with corresponding limiting spectra J? (given by equation (71)) for the specified values of s, l. comparisons between the measured spectra the limiting solutions can only be approximate since the measured spectra here are given over a limited energy range. Corresponding to the extreme left panel of Figure 16, we present in Figure 17 the measured electron differential spectrum extended to all energies, determined by Mauk et al. [1987], who used electron data from the LECP experiment [Krimigis et al., 1986] the Cosmic Ray System (CRS) experiment [Stone et al., 1986] on Voyager. The measured differential spectrum in Figure 17 is described by a broken power law, namely, J / 1/E 1.1, for E 1. MeV; J / 1/E 6.8, for E > 1. MeV. Since the particle distribution (1) assumed in the present investigation is characterized by a single power law over all energy, then comparison between our limiting solutions a broken power law spectrum are problematical. Nevertheless, in Figure 17, for illustration, we provide a set of limiting solutions for the perpendicular differential spectrum (71) for a range of values of s l. For comparison, limiting spectra (5) due to Schulz Davidson [1988] are also shown for various s values. [45] Using the measured differential spectra illustrated in Figure 16, we calculated an approximate profile for the electron omnidirectional integral flux I 4p (E > 0.1 MeV) at Uranus as a function of L shell. The result is shown in Figure 18. Limiting flux profiles, given by equation (7), for s = 0.5, 1.0,.0 l values given in Figure 16, are shown for comparison. Representative empirical values for the pitch angle index s at Uranus do not appear to be available. 5. Summary Conclusions [46] We have reexamined the concept of self-limitation of radiation belt particle fluxes originally introduced by Kennel Petschek [1966]. Our study is the first fully relativistic analysis of the self-limiting flux concept. We assume a particle distribution, given by equation (1), in which the energy spectrum, characterized by the spectral index l (>1), the pitch angle distribution, characterized by the pitch angle index s (>0), are specified a priori. We calculate the temporal growth rate of field-aligned whistler mode waves in a relativistic plasma modeled by the assumed distribution, then apply the condition that waves generated at the magnetic equator acquire a specified power gain over a given distance along a field line. Formulae are thereby obtained for the limiting values of the electron integral omnidirectional flux the electron differential flux which have an explicit dependence on l, s the cold plasma parameter a = jw e j /w pe. We apply these formulae to Earth, Jupiter Uranus, test the results against observed electron fluxes at these planets. Our conclusions are as follows: [47] 1. For a given value of a, the limiting solutions can be either moderately or sensitively dependent on the parameters l s, dependent on the region of (l, s) parameter space considered. Likewise, for given values of l s, the limiting solutions can be sensitive to the parameter a. [48]. The relativistic limiting solutions can differ significantly from the corresponding nonrelativistic solutions, for example, in the cases of large values of a, say a 0.1 (for given values of l, s), or small values of s, say s 0.1 (for any given l value moderate values of a). [49] 3. In the comparison of the limiting solutions with observations of electron fluxes at Earth, Jupiter Uranus, we found that the observed fluxes were close to the limiting value in a number of cases. However, it is difficult to make definitive or broad conclusions regarding our comparisons of the limiting solutions with data. The main reason for this is the overall dependence of the limiting solutions on the parameters l, s a, the associated lack of sufficiently accurate empirical knowledge of all three parameters in most cases considered. Our recommendation here is that further comparisons with observed data are needed in order 16 of 1