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 To address factors dictating similarities and differences between solar system radiation belts, we present comparisons between relativistic electron radiation belt spectra of all five strongly magnetized planets: Earth, Jupiter, Saturn, Uranus, and Neptune. We choose observed electron spectra with the highest intensities near ∼1 MeV and compare them against expectations based on the so-called Kennel-Petschek limit (KP). For evaluating the KP limit, we begin with a recently published relativistic formulation and then add several refinements of our own. Specifically, we utilized a more flexible analytic spectral shape that allows us to accurately fit observed radiation belt spectra, and we examine the differential characteristics of the KP limit. We demonstrate that the previous finding that KP-limited spectra take on an E−1 shape in the nonrelativistic formulation is also roughly preserved with the relativistic formulation; this shape is observed at several of the planets studied. We also conclude that three factors limit the highest relativistic electron radiation belt intensities within solar system planetary magnetospheres: (1) plasma whistler mode interactions that limit differential spectral intensities to a differential Kennel-Petschek limit (Earth, Jupiter, and Uranus), (2) the absence of robust acceleration processes associated with injection dynamics (Neptune), and (3) material interactions between the radiation electrons and clouds of gas and dust (Saturn).
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 All of the strongly magnetized planets of the solar system (Earth, Jupiter, Saturn, Uranus, and Neptune) have robust electron radiation belts at relativistic energies (Figure 1). Acceleration of electrons to relativistic energies within strongly magnetized space environments is clearly a universal process and not one that is peculiar to the special conditions that prevail at Earth or any other specific planet. It is of substantial interest for generalizing space environment acceleration processes to determine the similarities and differences between the radiation belts of these five accessible environments. Are the intensities and characteristics of these environments governed by the same processes and in a predictable and scalable fashion?
 To address this question, we compare the most intense electron spectra, specifically at ∼1 MeV, measured within each of the target magnetospheres. Some of the spectra show remarkable similarities. We address the reason for the similarities by comparing the spectra against a reformulated version of the so-called Kennel-Petschek limit [Kennel and Petschek, 1966]. Kennel and Petschek  noted that electron distributions within magnetic bottle configurations, allowing some losses for particles with velocities aligned with the magnetic field, are intrinsically unstable to the generation of whistler mode plasma waves. The authors postulate that whistler mode waves, propagating roughly parallel to the magnetic field lines, are generated near the magnetic equator of the Earth's inner magnetosphere and propagate to the ends of the field lines anchored in the Earth's ionosphere. At the ionosphere, a small fraction of the wave energy is reflected back into the system. If the fraction of reflected energy is large enough (percent level) then there can be runaway growth of the waves and a resulting strong, nonlinear suppression of the charge particle intensities. The level of the integral intensities above a specified energy value is called the “Kennel-Petschek (KP) limit.” This theory predicts that in the absence of extremely fast acceleration processes, acting faster than the runaway wave growth can act to suppress the particle intensities, integral intensities at energies above a defined minimum energy will reside below the KP limit. There are times when the source rate of energetic electrons is indeed faster than the maximum loss rate stimulated by the Kennel-Petschek processes, and we address that situation in section 10.
Kennel and Petschek  showed measurements of integral intensities (>40 keV) within Earth's magnetosphere and demonstrated that the intensities were generally below their KP limit (derived for the nonrelativistic regime). Other authors have also demonstrated the usefulness of this KP formulation for explaining Earth magnetospheric particle intensities [e.g. Baker et al., 1979, and references therein]. It is significant that the agent of the particle losses that purportedly works to establish the KP limit, whistler mode waves, has been observed at all of the planetary magnetospheres addressed here, but with interesting differences in intensity [Kurth and Gurnett, 1991].
 Since 1966 there have been several refinements to the Kennel-Petschek theory. Schulz and Davidson  derived a differential version of the KP limit. Specifically, they predicted that for integral intensities that are near the KP limit, the spectra will take on an E−1 spectral shape at intermediate energies (up to the energy where the acceleration processes can no longer populate the limiting profile). Evidence was provided in a companion paper [Davidson et al., 1988] that the E−1 shape is a characteristic of the most intense of Earth's electron radiation belt spectra observed well inside of synchronous altitudes (most convincingly near L = 5.4, near the innermost observation position).
 More recently, Summers et al.  created a relativistic version of the Kennel-Petschek theory. They show substantial, although not profound, differences between the predictions based on the relativistic and nonrelativistic versions of the theory. These authors find that the electron radiation belt spectra of Earth, Jupiter, and Uranus have integral intensities consistent with expectations based on their new relativistic formulation of the KP limit. For Earth [e.g., Kennel and Petschek, 1966; Davidson et al., 1988], Jupiter [Barbosa and Coroniti, 1976], and Uranus [Mauk et al., 1987], these findings extend previous similar findings, mostly based on nonrelativistic or, in the case of Barbosa and Coroniti , ultra relativistic formulations. Summers et al.  also challenge the assumption that feedback resulting from ionospheric reflection of the waves plays a role in establishing the KP limit. Their arguments are based on claims about the Poynting flux directionality of measured whistler waves within Earth's magnetosphere [e.g., Burton and Holzer, 1974; LeDocq et al., 1998; Santolík et al., 2003]. However, it is unclear from the literature whether reflected wave power levels (Poynting flux levels) of only 0.25% of that which propagates away from the equator (corresponding to 5% amplitude reflection) has actually been characterized given measurement sensitivities. Note in particular that relatively weak equatorward-propagating whistler waves are observed at frequencies above and below the main band of whistler chorus emission [Santolík et al., 2010]. This finding begs a question about the possible existence of weak equatorward-propagating waves hidden within the main, overwhelming band of Earthward-propagating waves. Very little reflected wave power is needed to sustain the Kennel-Petschek feedback chain. In any case, Summers et al.  adopt the numerical wave growth factors used by Kennel and Petschek  to establish the KP upper limit, but they justify the factors on other grounds.
 Whistler mode waves are not the only waves that can cause scattering and losses to magnetically trapped electron populations. Recent studies have shown that Electromagnetic Ion Cyclotron (EMIC) waves can cause strong losses to electrons, particularly those with energies >1 MeV (recent works include Sandanger et al. , Summers et al. , Miyoshi et al. , Shprits et al. , and Ukhorskiy et al. ). At lower energies, Electron Cyclotron Waves (ECH) have been identified as an agent for electron losses, particularly for the range of relatively low-energy electrons (from the perspective of this paper) that can cause diffuse auroral emissions (selected works include Horne et al.  and Meredith et al. ). These and other mechanisms certainly can play a role in sculpting electron spectral shapes. However, they have not been identified in the literature as defining an upper limit to electron intensities on the basis of a nonlinear feedback mechanism. For example, the most competitive of the possible mechanisms, EMIC waves [see Summers et al., 2007], are thought to be generated by hot ions and not hot electrons. And so, an increase in electron intensities does not in and of itself stimulate a suppression of those same intensities. With this reminder regarding the possible role of other wave modes, we concentrate exclusively on the whistler mode mechanism for the rest of this work.
 In the present paper, we extend the relativistic Kennel-Petschek whistler mode analysis initiated by Summers et al.  in several ways. All of the previous authors cited here who have addressed this mechanism have utilized the assumption of a power law spectral shape [Kennel and Petschek, 1966; Schulz and Davidson, 1988; Summers et al., 2009]. The power law representation, as we will show here, is a good representative of measured spectra only over limited ranges of energy. The procedures develop here can be used for any spectral shape, and we have chosen an analytic spectral form that is flexible enough to be accurately fit to most of the observed electron spectra utilized here. We also examine the differential characteristics of the relativistic theory. Our procedure is a relatively simple extension of the procedures outlined by Summers et al. , but we argue that the insight gained from this extension is substantial. Finally, we add the planets Neptune and Saturn to the pantheon of planets examined with respect to the KP theory. Significantly, these planets do not follow the pattern established by the examination of the other strongly magnetized planets, and we derive significant information about the factors that limit radiation belt intensities within planetary magnetospheres.
2. Motivations From Earth
 Recent work by the present authors has motivated a more careful examination of the Kennel-Petschek limit. Fox et al.  plotted near-equatorial electron spectra for various L values, as sampled by two instruments (MEA and HEEF, see Acknowledgments) on the Combined Release and Radiation Effects Satellite (CRRES) during magnetic storms, and during what these authors describe as a super storm (Figure 2, middle and bottom). Storm and super storm spectra (Figure 2) reveal striking characteristics. The main finding of the Fox et al.  study is that the observed intensities at energies of > 1 MeV are greater than expectations based on adiabatic transport from a source population within the near-Earth, tail side plasma sheet (those expectations are shown as smooth curved spectra in Figure 2, middle and bottom). Here “adiabatic” is defined to mean processes that preserve the first two adiabatic invariants, those associated with gyration and bounce. For our purposes here, there are other features of significant note in Figure 2. For the lower energies, somewhat less than ∼1 MeV, the observed intensities fall far below the adiabatic transport expectations, suggesting that particle losses play a significant role is sculpting the character of the spectra. Significantly, these <1 MeV components display a roughly E−1 shape, just as predicted by Schulz and Davidson [1988, Figure 2, top] on the basis of nonrelativistic KP theory. Finally, the spectra exhibit a sharp transition between the E−1 spectral shape and a much steeper shape at high energies. Again, this feature was anticipated by Schulz and Davidson [1988, Figure 2, top]. These qualitative spectral characteristics significantly amplify previous indications [e.g., Davidson et al., 1988; Summers et al., 2009] that Kennel-Petschek theory has a significant role not only in generally limiting integral intensities, but in strongly sculpting the character of the most intense radiation belt electron spectra. The specific spectral features highlighted in Figure 2 are similar to the characteristics of the most intense spectrum reported by Davidson et al.  specifically at L = 5.4 (see their Figure 5, bottom left) observed by a different mission (SCATHA) and with different instrumentation.
3. Kennel-Petschek Theory
 Because of the complexity of calculations of the growth rate of whistler mode waves in complex geometries, many in the space physics community judge the Kennel-Petschek limit theory to be difficult. However, it is intrinsically a very simple concept. A rough estimate of the net gain G of whistler wave amplitudes as the waves pass through the magnetic equator of the Earth's magnetosphere is
where G is the ratio (Af/Ai) of “final” to “initial” wave amplitudes as the waves propagate all the way through the equatorial unstable region, γ is temporal growth rate (1/s) (not to be confused with the relativistic Lorentz factor used in Appendix A), Vg is wave group speed (cm/s), Rp is planetary radius (cm), dl is differential distance along the magnetic field B in units of planetary radii, and D is the total distance along the magnetic field line (roughly symmetric around the magnetic equator) over which the wave growth remains significant, in units of planetary radii. Traditionally, the value of D is estimated as roughly L, the magnetospheric distance parameter, although at Jupiter, because of the centrifugal confinement of plasmas close to the magnetic equator, a lower value is suggested by Summers et al. .
where here we show the temporal growth rate γ explicitly as a function of the integral omnidirectional particle flux (F(>E), in units of cm−2 s−1) of the electrons above a specified energy (E*). E* is the energy of electrons that are in gyrocyclotron resonance with propagating whistler waves with wave frequency ω*r, but is also the specific energy where the wave growth rate transitions from negative values below that energy (wave frequencies above the corresponding wave frequency ω*r) to positive values above that energy (wave frequencies below the corresponding wave frequency ω*r).
 The Kennel-Petschek limit is defined by the condition that GR = 1, where R is the ionospheric reflection coefficient for wave amplitudes (not wave power). If we assume that R = 0.05, or 5%, then Ln[G] = Ln[1/R] ∼ 3. This level of reflection corresponds to just 0.25% of the wave power (0.05 × 0.05). Fundamental to the usefulness of this entire approach is the fact that the KP limit varies only as the logarithm of the wave gain value G. Therefore, a useful upper limit can be estimated even when there is substantial uncertainty in the value of R. Again, Summers et al.  formulate the problem without relying on ionospheric reflection, but nonetheless, continue to use Ln[G] ∼ 3, partly for the purpose of continuity with previous work. The logarithmic sensitivity allows for a substantial amount of flexibility here.
Kennel and Petschek  approximate the relationship between γ and the F(>E*) with their equation (2.27) combined with (4.17), which allows them to invert equation (2). After replacing Ln[G] with 3, we represent that result conceptually with
Note that the inversion of γ[F(>E*)], using the equations cited above in the original KP paper, represents only a very rough approximation. Accurate inversion requires numerical calculations and corresponding factors that depend on the exact shape of the electron spectrum [Schulz and Davidson, 1988]. In the present paper, the complexity of the electron spectral shapes makes it much more difficult to continue to specify the KP limit in the form of a limit on integral flux, and we abandon that approach here. Our starting point is the uninverted formulation given in equation (2), again with Ln[G] ∼ 3, yielding
Equation (4) was also the starting point for Summers et al. . The significant new feature that those authors introduced was to use the relativistic formulation of the linear growth rate γ derived by Xiao et al. [1998; an earlier formulation of relativistic wave growth rates is provided by Liemohn, 1967]. The Summers et al.  formulation of the Xiao et al.  results is reproduced in Appendix A. We will reference formulas from Appendix A during later developments. In evaluating equation (4), in either the nonrelativistic or relativistic formulation, there is a choice that must be made. At what frequency or, equivalently, at what gyroresonant energy is equation (4) to be evaluated? The choice made by Summers et al.  and other authors is to evaluate it at the wave frequency ω = ωm corresponding to where the temporal growth rate is maximum. Specifically,
One might alternatively maximize γ/Vg rather than just γ itself. In the present analysis, we have chosen to examine the relativistic KP limit as a function of the minimum resonant energy, which we call the “strictly” parallel resonant energy Er. The strictly parallel resonant energy is the parallel resonant energy for the condition that the perpendicular momentum = 0. We distinguish the general resonant parallel momentum pR or the equivalent parallel energy ER, which prevails for the condition that perpendicular momentum can take on any value, and the strictly parallel resonant energy because the general resonant parallel momentum changes as a function of the perpendicular momentum (Appendix A). There is a unique one-to-one relationship between the strictly parallel energy Er and the resonant wave frequency ωr, whereas there is not such a unique relationship between ER and ωr. The equation that we solve is
which allows us to address the differential characteristics of the KP limit, that is, the KP limit as a function of minimum resonant energy. Equation (A5) provides the relationship between resonant frequency and the strictly parallel momentum by setting = 0. The strictly parallel momentum can be converted to strictly parallel energy with the standard formula (section 4).
4. Electron Distributions
 We have developed an approach to evaluating the KP limit implications that allows for the use of an arbitrary continuous analytic spectral form for the differential intensity I[E, α], where E is energy and α is pitch angle, provided it can be converted to an analytic and continuous function of the parallel and perpendicular momentum. The form that we have chosen to use here is
where C, kT, γ1, E0, and γ2 are fitting parameters for the energy distribution, and “S” represents the so-called anisotropy parameter. The numerator of the energy distribution is the so-called kappa distribution (representing a Maxwellian core with a power law tail), and the denominator adds an additional power law tail contribution at the highest energies. Note that, in the denominator, the power γ2 acts on the energy ratio E/E0 rather than on the entire 1 + E/E0 (the later form was used, for example, by Baker and Van Allen ), because it accommodates the sharp transitions that are often apparent in the observed spectra. This energy distribution shape in equation (7) has been shown to accurately represent a broad range of measured spectral shapes at both Jupiter and Earth [Mauk et al., 2004; Fox et al., 2006].
 It is, of course, convenient to assume, as we have done with equation (7), that the angular variations are fully independent of the energy variations, as was also assumed by Schulz and Davidson , Summers et al. , and others. While this approach is unlikely to be very accurate in representing particle distributions generally measured in space environments, we will argue in section 5 that this approach is in fact the appropriate one for evaluating a spectrum against the KP limit. It is not just a convenience to take this approach.
 The five energy distribution parameters of equation (7) are determined by fitting that equation to a measured spectrum of interest through an error minimization process. We then prepare that optimized spectrum for insertion into the equations of Appendix A for evaluating the relativistic whistler mode growth rate (γ). To do so, we perform the following actions. (1) We transform all “keV” values in equation (7) to ergs using the standard factor. (2) We solve the following standard relativistic equation,
for energy E and substitute the positive solution into equation (7). (3) We substitute (/p)2S for sin2S(α), where is perpendicular momentum and p is total momentum. (4) We convert to phase space density f(p) by forming I/p2. (5) And finally, we substitute all values of p using the equation p2 = 2 + 2. We end up with f(, ), which is just what we need to insert into the expressions in Appendix A. It is important to note that the manipulations specified here can only be carried out in practice using software that accommodates sophisticated symbolic manipulations, like Mathematica®. Long before one reaches the point of performing the final numerical integrations specified in Appendix A, the individual analytic expressions occupy many pages of text.
5. Anisotropy Factor
 After fitting the energy distribution in equation (7), the remaining parameter that must be set is the anisotropy parameter “S.” Kennel and Petschek  analyzed a simple pitch angle diffusion equation, imagining a source of particles at equatorial pitch angles of π/2, imposing various other assumptions and derived the following estimate for the anisotropy factor A (equivalent to our S for the nonrelativistic formulation): A ∼ 1/(2 ln[(1/α0) + O(1)]), where α0 is equatorial loss cone pitch angle and “O(1) is” of order 1). For L = 6, this parameter is roughly 1/6 or 0.17. This is the value used in the authors' final simple expression. There are other simple approaches to theoretically estimating an anisotropy parameter. For example, the asymptotic shape of a distribution governed by pitch angle diffusion coefficients that are independent of pitch angle has the shape: J0[2.40·cos(α)/cos(α0)], where J0 is the Bessel function, α is pitch angle, and α0 is the loss cone pitch angle [Schulz and Lanzerotti, 1974]. For small loss cone pitch angles, this shape can be very accurately fitted with a sin2S(α) shape, with an S value of order 1.3. And so, on simple theoretical grounds, there is a broad range of possible anisotropy values that might be used (0.17 to 1.3 for the two examples given here).
 There are two reasons why we do not just use the pitch angle distributions that are simply observed along with the energy distributions. First, for the broad range of planetary environments addressed here, the needed pitch angle distributions are just not available. The instrumentation, the complexity of the measurement geometries, and the difficulty of the measurement environments (dealing with penetrating backgrounds, etc.) often did not allow for the determination of even rough estimates of the anisotropy parameters in many cases. There is a more central issue, however. We believe that it only makes sense to define the KP limit with a minimal anisotropy parameter. As rapid acceleration and transport pushes an electron distribution up toward the KP limit, the angular distributions will flatten as a part of the processes that define the limit. For example, if an energy distribution is somewhat below the KP limit and the anisotropy parameter is large, it makes no sense to define the KP limit for that particular distribution on the basis of the observed anisotropy parameter. What makes sense is to define the limiting anisotropy parameter on the basis of what the parameter will become as the acceleration processes try to push the distribution to and beyond the KP limit. In essence, we believe that the anisotropy parameter that defines the KP limit is in some sense a universal parameter (representing more or less the approach that Kennel and Petschek  took originally). The trick is to determine what that parameter is. The assumption of a universal anisotropy parameter associated with distributions as they approach and exceed the KP limit justifies the assumption, implicit in equation (7), that the energy distribution is separable from the angle distribution.
 Our approach is to use observations during strong storms within the Earth's magnetosphere to guide our choice for the anisotropy parameter to be used in all of the environments studied here. We have fitted pitch angle distributions during various phases of activity and find that the sinn(α) shape does a reasonably good job (Figure 3; note that n = 2S). The inserted table in Figure 3 shows the result of a study of the pitch angle parameter n as a function of energy and magnetospheric disturbance level, all for L = 6. Note that for storms and superstorms and at lower energies, where on Figure 2 the E−1 spectral shape prevails, the anisotropy is low and relatively insensitive to energy. At the highest energy, above the break in the spectrum in Figure 2, and therefore above the energy where the KP limit has a strong influence, the anisotropy parameter is larger. The anisotropy parameter (S = n/2) that we would choose for super storms might be 0.25, and for storms might be 0.4. For all of the work described here, for all of the planets, we have arbitrarily rounded up the super storm value and have chosen to use S = 0.3.
 As demonstrated by previous authors [e.g., Schulz and Davidson, 1988; Summers et al., 2009] the KP limit does depend on the choice of “S,” and there is certainly no guarantee that all of the planetary magnetospheres will have the same anisotropy parameter. We provide some sense of the sensitivity of the KP limit to the anisotropy parameter during our examination below of the spectra measured at Earth. However, the anisotropy parameter is not the only parameter that might vary between the different planets. Other such parameters would include the wave reflection coefficients at the ionosphere and the distribution of plasmas along the magnetic field lines. However, the KP limit is a useful tool mostly to extent that a uniform set of procedures can independently be shown to yield coherent and comparable results despite some measure of sensitivity to the various assumed parameters. We expect our results to be judged based on the extent to which we achieve those coherent and comparable results between planets.
6. Calculating the KP Limit
 While the calculations that we perform to evaluate the KP limit here are quite messy, the procedure is really quite simple. The normalization parameter for our spectral shape in equation (7) is “C” For the calculations described here, we will define two different values of C. Cm is the normalization parameter that comes out of the fitting of the observed spectra with equation (7). CK is the normalization parameter that, given all of the other parameters that come out of fitting of the observed spectra, is needed to satisfy the KP equation (6). We will be reporting our results as a ratio: Cm/CK. When Cm/CK is less than 1, then the observed distribution for a given resonant frequency and correspondingly for a given strictly parallel resonant energy is judged to be less than the KP limit for that frequency and energy. If Cm/CK is greater than 1, then correspondingly the distribution for that frequency and corresponding energy is judged to be above the KP limit.
 Here we spell out our procedure a little more carefully. Choose an observed equatorial electron spectrum that is to be compared against the KP limit. Fit the observed spectrum to equation (7) and set the parameter “S” to 0.3 as justified in section 5. Identify for that spectrum the values of L, equatorial magnetic field strength B, and density N. B and N go into the calculation of the equatorial plasma frequency ωpe and equatorial gyro frequency Ωe used in Appendix A. Process the fitted analytic spectrum in the fashion described in section 4 to generate the phase space density f(, ). Plug f(, ) into the equations in Appendix A, making use of the identified values of L, B, and N. For a systematic array of resonant frequencies (ωr) evaluate the equatorial whistler mode wave growth rate and the corresponding strictly parallel particle energy associated with each resonant frequency, plug each of the wave growths for each resonant frequency into equation (6) along with the group wave velocity for each frequency (again Appendix A; equation (A9)) and determine the value of CK that would be needed to satisfy equation (6). Note that the wave growth rate is linearly proportional to C, and so if the growth rate is, say, a factor of 3, less than is needed to satisfy equation (6), then CK/Cm = 3. At the end of this procedure, we end up with a matrix of numbers where each row is (ωr, Er, Cm/CK). The plots that we show are Cm/CK versus Er. Note that, consistent with previous work, the value of “D” used in equation (6) is generally “L,” but for Jupiter, we use a smaller value as suggested by Summers et al. [2009; see section 7.3]. Our procedure was implemented into a single Mathematica® routine that performs both the extensive symbolic manipulations that are required, as well as the final numerical integrations. It is interesting to examine just how the distribution function is sampled when we perform the analysis described here. Figure 4 shows f(, ) derived by fitting the Figure 2 (bottom) spectrum with equation (7), and using an S value of 1, for the sake of making the angular anisotropy a little more visible on this crude display. The black lines are the resonance curves derived from equation (A5), each corresponding to a single resonant frequency ωr. The lines are labeled with the corresponding strictly parallel resonant energy Er in MeV, which corresponds to the energy of the electrons at the position where the black curves cross the = 0 axis. Clearly integrals (specified in Appendix A) along the black curves will have some relationship with the standard integral intensity I(>E). However, particularly for our very flexible spectral shape represented by equation (7), there is no easy one-to-one relationship. Again, we have abandoned the idea of making the connection between the Kennel-Petschek limit and the integral intensity.
7. Earth, Uranus, and Jupiter
 We will be giving some introductory information about each of the planet's magnetospheres, but assume that the reader is well familiar with Earth's [see Kivelson and Russell, 1995]. Figure 5 shows the results of the Kennel-Petschek procedures outlined in section 6 for the Earth spectrum shown in Figure 2 (bottom). The top shows fitted spectra for L = 4, 5, and 6, while the bottom shows our KP analysis for just L = 5. The spectral parameters (equation (7)) derived for these spectra are provided in Table 1, along with the parameters derived for all of the other spectra analyzed for this work. The bottom displays Cm/CK plotted against the strictly parallel energy Er. The blue bar, centered vertically on the Cm/CK = 1 position, represents the region of the plot that is roughly within a factor of 3 of the KP limit, motivated by the statement by Kennel and Petschek  that the derived KP limit is roughly accurate to within a factor of 3. We assume here that when the calculated Cm/CK profile resides within the shaded region, that the spectrum is at least strongly under the influence of the processes that establish the KP limit.
Table 1. Parameters for the Equation (7) Spectra Used in This Study
 In Figure 5 (bottom), several profiles are shown for several different assumed equatorial plasma densities. The red profile shown here and in other plots is our best guess as to the appropriate density to use. For Earth, we have adopted as our best guess densities the ones used by Summers et al. , who in turn obtained densities from the work of Sheeley et al. . As the density increases the profile reaches an asymptotic condition (labeled “∞”) where further density increases have no influence on the shape of the profile within the energy range of interest. Because the spectrum was measured during a very strong magnetic storm, the magnetic field is substantially suppressed, and for the calculations we have used the storm time magnetic field values provided by Tsyganenko and Sitnov .
 For our best guess density, the observed spectrum for Earth L = 5 resides fairly close to the KP limit (within the shaded region) from ∼30 keV to almost ∼1 MeV. The peak value of Cm/CK is 0.60, very close to 1.0 given the rough assumptions that go into the generation of the KP limit. For various assumed anisotropy parameters, that peak value is 0.72, 0.60, 0.42, and 0.18 for the anisotropy parameters 0.4, 0.3, 0.2, and 0.1. For the first three values our qualitative conclusions about the influence of the KP limit on the spectral shape would be roughly the same. It the anisotropy parameter was indeed 0.1 at Earth, contrary to Figure 3, we might have questioned the quantitative usefulness of the KP analysis for this particular spectrum.
 It is of interest to consider the shape of the asymptotic Cm/CK profile, specifically the fact that it is flat and nearly horizontal at the lower energies. We note that for energies below about 0.1 MeV, the KP analysis is performed using an extrapolation of the measured spectrum, and so the discussion here should be considered theoretical in nature. As acceleration processes drive the distribution closer and closer to the KP limit, it is reasonable to assume that the processes that limit the intensities start to act first on those portions of the distribution (those collections of electrons that reside along a limited set of the resonant curves plotted in Figure 4) that rise above the differential KP limit. And so as the process proceeds, one expects the Cm/CK profile to flatten itself against the Cm/CK = 1 line. Significantly, this condition is qualitatively achieved for a spectrum that has an E−1 shape at the lower energies, just as predicted by Schulz and Davidson . To test this conclusion more accurately, we have modified our Earth spectrum to a perfect power law, with the shape E−g. In Figure 6, we show the Cm/CK profiles for g = 0.7, 1.0, and 1.3. The most horizontal of these profiles is the one for g = 1.0, as we expected from the nonrelativistic prediction of Schulz and Davidson . More detailed analysis shows that g ≈ 0.9 provides the most horizontal profile for the Earth situation represented in Figure 6. For the relativistic calculations performed here, then we roughly confirm the predictions of Schulz and Davidson  using the nonrelativistic formulation that the saturation shape of the KP-limited spectrum is approximately E−1.
 The KP limit calculations for three different Earth L values (4, 5, and 6) are shown in Figure 7, but in each case just for our best guess as to the equatorial plasma density. Two points are of particular interest. The spectral shape of the L = 6 spectrum (top) is steeper at the lower energies than is the L = 5 spectrum (E−1.32 versus E−0.98; Table 1), and indeed the Cm/CK profile (bottom) is flatter for the spectrum that is closer to E−1 than it is for the steeper spectrum. Second, the L = 4 spectrum is roughly as intense as is the L = 5 spectrum, but the L = 4 Cm/CK profile is further below the KP limit than are the other profiles because of the differing magnetospheric parameters. One might wonder why the L = 4 spectrum retains the qualitative characteristics that are observed within the other two spectra. Our guess is that the L = 5 spectrum is ultimately the source of the L = 4 spectrum. The general finding here that the KP limit has an important role in sculpting spectral intensities at Earth, of course just confirms the findings of a number of previous authors [e.g., Baker et al., 1979; Davidson et al., 1988; Summers et al., 2009].
 Our Uranus spectrum (Figure 8), substantially more intense at ∼1 MeV than any spectrum sampled elsewhere, was taken at the most planetward penetration of the Voyager spacecraft, at L = 4.73 (the spectrum is from Mauk et al. , and it combines measurements from two instruments; see Krimigis et al. , Stone et al. , and Selesnick and Stone ). The exact spectral index or shape of the spectrum above the breakpoint energy near ∼1 MeV is poorly constrained at this L = 4.73 position, but for our purposes here it is sufficient to know that the spectrum drops precipitously above that energy [Stone et al., 1986]. The Uranus spectrum is remarkably similar to the Earth spectra (compared in section 7.4), and clearly the Cm/CK levels (bottom) proclaim that the KP limit must play a role in sculpting this spectrum. This finding confirms the findings of previous authors that the KP limit is matched or exceeded in this environment (Mauk et al. , for the nonrelativistic analysis and Summers et al. , for the relativistic analysis). The big uncertainty here is the equatorial density. The largest density observed at the innermost L excursion of Voyager was N = 2.2 cm−3 [Belcher et al., 1991; Kurth et al., 1991] but measured well off the magnetic equator (18°) and with a sensor that measures only energies >10 eV. We have arbitrarily adopted N = 5 as our most likely equatorial density when we compare these profiles with other planets.
 Jupiter's huge magnetosphere is thought to be powered by the rapid planetary rotation, which energizes and transports the plasmas that are continuously generated in the inner magnetosphere (∼5.9 RJ) by the volcanoes of the Moon Io [see Dessler, 1983; Bagenal et al., 2004]. The outward transport of the energized plasmas occurs episodically in the form of small scale injections in the inner regions (<10 RJ) [Bolton et al., 1997; Kivelson et al., 1997; Thorne et al., 1997] and large-scale injections in the middle regions (>9 RJ) [Mauk et al., 1999]. A very novel feature of this magnetosphere is the formation of a magnetodisc at distances >15–20 RJ, with a neutral sheet configuration that wraps all the way around the planet, and not just on the tail side as occurs at Earth. Another novelty is that Jupiter has the only radiation belt that can be sensed and imaged remotely with radio wave synchrotron radiation, resulting from a combination of high electron intensities and strong magnetic fields very close to the planet (<3RJ; reviewed by Bolton et al. ). Jupiter's subsolar magnetopause occurs at roughly 100 RJ from the planet.
 Two Jupiter spectra are shown in Figure 9, one (L = 8.3) obtained from an internal NASA JPL report [Garrett et al., 2003] and the other (L = 3) representing the most intense spectra presented by Baker and Van Allen  for the inner magnetosphere as sampled by Pioneer. The L = 3 directional intensity spectrum is estimated from the published omnidirectional intensity spectrum by dividing by 4π and represents electron populations that contribute to the synchrotron radiation sensed from Earth [Bolton et al., 2004]. Because the Garrett et al.  information is not readily available, we provide additional information in Appendix B. A significant difference between the Jupiter spectra and the spectra of Earth and Uranus is that the Jupiter spectra extend to much higher energies without a sharp spectral break (section 7.4). Again, as with the Earth and Uranus spectra, the KP limit is clearly predicted to have a role in sculpting the L = 8.3 spectrum. For our analysis of the Jupiter L = 8.3 spectrum, the “D” value (equation (6)) that we have used is D = 3, motivated by the suggestion from Summers et al.  and based on the observed scale height of the rotationally confined plasmas [Mei et al., 1995]. The Cm/CK values would be substantially higher than shown (by a factor of ∼8/3) had we used D = L.
 The KP analysis of the Jupiter L = 3 spectrum is not shown here. The peak Cm/CK for the L = 3 asymptotic profile is at minimum a factor 20 below the KP limit, and thus it appears that the Jupiter L = 3 spectrum, and more specifically the spectra of those electrons thought to participate in the generation of the remarkable synchrotron radiation from Jupiter's inner magnetosphere, are not limited by KP processes. As with our discussion of the Earth L = 4 spectrum (section 7.1), we hypothesize that the electron populations energized further out, possibly in the vicinity of the Jupiter L = 8 region, provides the source of the populations in the more interior regions. This scenario was suggested by Horne et al. .
7.4. Earth, Uranus, and Jupiter Compared
 The more intense spectra from Earth, Uranus, and Jupiter have remarkable similarities (Figure 10). Their intensities at ∼1 MeV are essentially identical. They all have spectral shapes below ∼1 MeV close to E−1. All three have relatively flat and horizontal Cm/CK profiles between 0.1 and 1 MeV, with values relatively close to Cm/CK = 1. We believe that the peak intensities and the shapes of these three spectra are determined and sculpted by the processes that establish the differential KP limit.
 Both the Earth and Uranus spectra have sharp spectral breaks somewhat above the 1 MeV level, but the Jupiter spectrum extends to much higher energies. The energization processes at Jupiter must be more extreme than they are in the other environments. But even at the higher energies the intensities are limited by the differential KP limit. By confirming that the E−1 spectral shape is still theoretically anticipated with the relativistic theory and showing that these three radiation belts roughly satisfy that expectation with intensity levels close to the KP limit, we have amplified the findings of previous authors, most recently Summers et al. , concerning the influence that the KP processes have on the radiation belts of Earth, Uranus, and Jupiter.
 Neptune's magnetosphere is notable in a number of ways, including the strong tilt to its magnetic dipole (47°) and the presence of the Moon Triton (14 RN from Neptune) that has a retrograde orbit and is one of the two Moons in the solar system with a robustly collisional atmosphere [see Cruikshank, 1995]. For the present discussion the most notable attribute of Neptune's magnetosphere is that it is the quietest magnetosphere of the strongly magnetized planets in the solar system. Temporal injection processes are common at Earth [e.g., DeForest and McIlwain, 1971], Jupiter [Bolton et al., 1997; Mauk et al., 1999], Saturn [Burch et al., 2005; Hill et al., 2005; Mauk et al., 2005], and Uranus [Belcher et al., 1991; Mauk et al., 1987, 1994; Sittler et al., 1987]. However, no injection features were observed within Nepture's magnetosphere during the one encounter by Voyager 2 in 1989 [Mauk et al., 1995]. Neptune's radiation belts are also remarkably symmetric (Figure 1), an attribute that was not found within the other magnetospheres. Injections appear to be driven by solar wind interactions at Earth and Uranus, and by strong internally generated plasma energized by rapid planetary rotations in the cases of Jupiter and Saturn. Neptune has neither a strong interaction with the solar wind nor does it have a strong internal source of plasma (Triton is in the outer magnetoshere; Neptune' subsolar magnetopause is roughly at ∼18 RN). It is of interest whether that absence of injections has an influence on the radiation belt of Neptune. Horne et al. , for example, suggested that injections are key to seeding the dramatic acceleration of Jovian radiation belts.
 Our Neptune spectrum (Figure 11) was sampled by Voyager 2 near L = 7.4 at the peak in the ∼1 MeV rate profile [Mauk et al., 1995] (again multiple sensors were used to create this spectrum, see Krimigis et al.  and Stone et al. ). Inside L = 7.4, the intensities decreased as the spacecraft plunged very close to Neptune's cloud tops. The KP analysis of this Neptune spectrum (Figure 11) shows that while the spectrum is KP-limited near 0.1 MeV, the Cm/CK profile resides a factor of 30 below the KP limit near ∼1 MeV. This result is distinctly different than our finding, near ∼1 MeV, for the other planets presented up to this point, as is made clear in Figure 12. Figure 12 shows that not only is the Neptune spectral intensity nearly 2 orders of magnitude below those of the other planets near 1 MeV, but a factor of 10 separates the Neptune Cm/CK profile at ∼1 MeV from those of the other planets. We speculate that the differences might be explained by the absence of injection dynamics, unique to the Voyager 2 visit to Neptune.
9.1. Saturn During the Cassini Epoch
 We separate our analysis of Saturn's radiation belt into two epochs: the present Cassini epoch (2004 to present) and the Voyager epoch (1981). Saturn is a rotationally dominated magnetosphere whose plasma populations are mostly provided by geyser-like plumes emanating from the southern pole of the Moon Enceladus at a radial position of roughly 4 RS from Saturn (see Dougherty et al. , for review information about Saturn as diagnosed by the Cassini spacecraft). This source of materials generates relatively dense clouds of gas and dust; the dust cloud was called the “E-ring” long before the Enceladus plumes were discovered. Another notable feature of Saturn's magnetosphere is that it supports both small-scale and large-scale injection dynamics [reviewed by Mitchell et al., 2009; Mauk et al., 2009]. Saturn's subsolar magnetopause resides roughly at radial position >22 RS.
 We performed analyses on data obtained from the Low Energy Magnetospheric Measurement System (LEMMS) sensor, part of the Magnetospheric Imaging Instrument (MIMI), on the Saturn orbiter, Cassini [Krimigis et al., 2004, 2005]. We scanned many orbits of Cassini, covering all radial L values down even to regions inside of the visible rings and searched for the highest rates measured by electron channels measuring energies near 1 MeV. Factors of ±3 variability were observed in the orbit-to-orbit measurement in the highest per orbit ∼1 MeV intensity. The channel rates, the spectral fit to those rates, and the resulting spectrum for the region and time with the highest ∼1 MeV intensity is shown in Figure 13. Our KP analysis of that spectrum, sampled at L = 3.25, is provided in Figure 14. In this case the entire spectrum is more than a factor of 20 below the KP limit. We have investigated the possibility that the spectral intensities near 1 MeV remain relatively high with increasing L values, and that the corresponding Cm/CK values climb upward toward the KP limit. This investigation was motivated by the findings at both Earth and Jupiter where high intensity spectra at L = 3 and 4, respectively, resided well below the KP limit, while similar intensities at higher L values (5 and 8, respectively) challenged the KP limit. Saturn during the Cassini epoch turns out to be different. As one moves outward in L from the L ∼ 3.25 position, the Cm/CK values of the lower energies (0.1 MeV) do climb significantly upward, but both the intensities and the Cm/CK values at relativistic energies, and specifically near ∼1 MeV, drop precipitously, unlike the cases of Earth and Jupiter.
 The findings regarding Saturn's Cm/CK values, along with qualitative features of the electron spectrogram in Figure 15 (specifically the diminution of the lower energy intensities near the center of the display) suggest that the ∼1 MeV intensities are limited not by the KP limit but by losses associated with the broad E-ring gases and dust. The pre-Cassini modeling of Saturn's radiation belt by Santos-Costa et al.  anticipated strong losses in the vicinity of the E-ring. The intensities that we find here (specifically at the ∼0.3 MeV energy highlighted by the authors) are an order of magnitude lower than those authors anticipated, but as we discuss in section 9.3, those intensities may be quite variable. The effects of gas and dust generally on energetic protons and electrons have been addressed during the Cassini epoch by Paranicas et al. [2007, 2008].
9.2. Five Planets Compared
 The uniqueness of the Saturn findings relative to the other planets is emphasized in Figure 16. At 1 MeV the spectral intensities and the Cm/CK values for Neptune and Saturn are similar, but the shapes of these entities as a function of energy are dramatically different. That is a key reason why we speculate that the root causes for the diminution of the radiation belts of Neptune and Saturn are very different (dynamics versus material interactions).
9.3. Saturn During the Voyager Epoch
 The electron radiation belt at Saturn measured by the Voyager spacecraft has important differences with that measured by the Cassini spacecraft (Figure 17). The Voyager spectrum (1981) is highly structured and substantially more intense than that observe by Cassini in 2009. The Voyager peak near ∼1 MeV was interpreted as drift resonance with the Moons in the inner magnetosphere (e.g., the Moons Enceladus and Mimas) [Krimigis et al., 1983; Van Allen et al., 1984]. Specifically, for electrons with magnetic drift speeds that nearly match the orbital speed of a Moon, the Moon-electron absorption probability is drastically reduced. The shift in the peak of the spectrum with radial distance roughly matches expectations from the resonance theory [Van Allen et al., 1984]. The KP analysis of the Voyager spectra (Figure 18; each spectrum is represented by adding together three different components, including a Gaussian component centered near ∼1 MeV; Table 1) shows that the high degree of structure within the Voyager spectra is allowed by the KP theory, since the KP limit is only challenged by the spectra over a limited energy range.
 We do not know the reason for the differences between the Cassini epoch and the Voyager epoch spectra. However, our best guess is that the gas and dust environment may have been quantitatively different between the two time periods. The emission of materials from Enceladus is known to be variable [Esposito et al., 2005; Melin et al., 2009]. It is also possible that there occurred a dramatic temporal event, for example, an interplanetary shock striking and passing through the magnetosphere, just prior to the Voyager encounter. There are uncertainties in understanding of the responses of the channels of both the Cassini and Voyager instruments, but significant discussions within the Cassini/Voyager instrument teams, particularly with T. P. Armstrong, a member of both teams with substantial experience in understanding channel responses (private communication, 2010), resulted in our conclusion that very large differences between the spectra observed by both Voyager and Cassini cannot be explained by uncertainties in the responses of the two detector systems.
10. Discussion on Time Scales
 Our focus here has been on magnetospheric electron spectra that are intense at relativistic energies, specifically near 1 MeV. The Kennel-Petschek limit has been used with other populations as well, and it is instructive to revisit those applications. Of specific interest is the finding of Baker et al.  that during very active magnetospheric conditions (magnetic activity index, Kp ∼ 5–6) within the geosynchronous orbit (6.67 RE), integral particle intensities can exceed the classic KP limit by an order of magnitude. The posited explanation is that the acceleration mechanism can be faster than the times associated with the losses that arise with even strong whistler wave scattering. The so-called strong diffusion loss time associated with strong wave scattering can be estimated as the transit time of the electrons along the field line (some fraction of a bounce time) times the ratio of the solid angle contained within the loss cones and 4π. A more accurate calculation results in the following expression for an idealized minimum e-folding loss time [e.g., Lyons and Williams, 1984]:
where RP is radius of the planet (cm), v is particle speed (cm/s), and αlc is the loss cone angle. For the geosynchronous regions during storm time, the idealized minimum e-folding loss time is several minutes, and more realistic minimum loss times are likely several times the idealized value, perhaps 10 min or even longer. And so if particle transport and acceleration times are faster than ∼10 min in the geosynchronous orbit, then indeed one might expect the particle accelerations to overwhelm even the relatively fast loss times associated with the Kennel-Petschek type processes.
 An evaluation of the Baker et al.  example using the tools developed in the present work is shown in Figure 19. In Figure 19, the results are compared to the spectrum and KP analysis for the Earth L = 5 spectrum presented in Figure 5. Indeed, for typical geosynchronous densities (N = 3 [Sheeley et al., 2001]) the Baker et al. spectrum does substantially exceed the KP limit at minimum resonant energies <0.2 MeV. There is a possibility that the density could have been unusually low at geosynchronous orbit during this very active period (N ≤ 0.3), thereby maintaining consistency with the KP limit. However, most likely, as suggested by Baker et al. , the processes (injections) that populate this region do indeed act in a fashion that is faster (minutes) than the loss process associated with compliance to the KP limit.
 And so while it appears that fast acting processes can overpower the processes acting to establish the KP limit, such a reversal of time scales did not happen for the high energy, relativistic spectra studied here. It is also true that this reversal of times scales was not observed to happen to either low- or high-energy electrons within the environments where the relativistic electron populations were intense.
11. Summary and Conclusions
 We compared the most intense (near ∼1 MeV) radiation belt electron spectra measured within the magnetospheres of the five strongly magnetized planets within the solar system: Earth, Jupiter, Saturn, Uranus, and Neptune. We compared the spectra with each other and also with respect to expectations from the well known Kennel and Petschek  theoretical limit on radiation belt intensities. To derive the KP expectations, we started with the relativistic theory of Summers et al.  and added several refinements. Specifically, we developed an approach that allows us to use more realistic spectral shapes that can be accurately fit to observed spectra. Also, our approach allowed us to examine the differential characteristics of the KP limit. We roughly confirmed, for this relativistic theory, the finding of Schulz and Davidson , for the nonrelativistic theory that KP-saturated spectra take on the E−1 spectral shape. We show that those measured spectra that reside near the KP limit indeed are observed to have the ∼E−1 spectral shape up to an energy where the spectrum no long resides at the level of the differential KP limit.
 With our comparison of the five strongly magnetized planets of the solar system, we find that the radiation belts of Earth, Jupiter, and Uranus can be categorized together as having spectra that appear to be strongly sculpted by Kennel-Petschek processes, whereas those of Neptune and Saturn must be differently categorized. It is significant that Kurth and Gurnett  also categorized Neptune and Saturn differently from Earth, Jupiter, and Saturn with regard to the presence (E, J, U) or absence (N, S) of whistler modes with sufficient intensity to cause strong particle losses. These authors caution that observing geometry could have a role to play in the observed differences.
 Our more specific conclusions about the different environments are provided here: The most intense measured spectra at Earth, Uranus, and Jupiter reside near to the differential KP limit for energies between ∼0.1 and ∼1 MeV. The spectral shapes below ∼1 MeV are roughly E−1, as expected by both the nonrelativistic and now the relativistic KP theory. We conclude that the radiation belt intensities of Earth, Uranus, and Jupiter are limited by Kennel-Petschek processes, specifically by strong scattering by whistler mode waves as the electron intensities become too high. These findings amplify on related findings of other authors, most recently those of Summers et al. .
 The most intense spectrum measured in Neptune's magnetosphere has intensities close to the differential KP limit near ∼0.1 MeV but has intensities a factor of 30 below the KP limit near ∼1 MeV. Magnetospheric processes at Neptune apparently do not robustly energize electrons to the 1 MeV level to the extent that the KP limits allow. No significant dynamical events were observed within Neptune's magnetosphere, and specifically, no injection-like events were observed. The lack of dynamics may be a consequence of the fact that Neptune is coupled only weakly to the solar wind, and there is no strong internal source of plasma deep within the magnetosphere such as that which occurs at both Jupiter and Saturn. We speculate that such injections are needed to robustly energize electrons to the higher energies. This speculation is stimulated in part by Horne et al. , who suggested that injection-like events are critical to explaining the high electron energies and intensities at Jupiter.
 The most intense Cassini-epoch electron spectrum at Saturn is well below the differential KP limit at all measured energies. We speculate that the Saturn electron intensities are strongly suppressed by the gases and dust populations released by the plumes of Enceladus. The difference between the suppression seen at Neptune and that seen at Saturn is that at Neptune the low-energy (0.1 MeV) intensities are high and the high energies (1 MeV) are low. This finding is inconsistent with suppression arising from interactions with materials such as gas and dust. Gas and dust are expected to suppress the low energies to a greater extent than they do the higher energies. The Cassini (2005) spectra of Saturn qualitatively match expectations from interactions with such materials. The Voyager (1981) electron spectra are significantly different from the Cassini (2005) spectra, showing higher intensities and substantial energy structure. We speculate that the gas and dust environment was different during the Voyager encounter than it is during the Cassini observations. The energy structure is allowed since the Voyager spectra do not challenge the differential KP limit at most energies. The presence of the E-ring of particulates was known prior to Cassini, and modeling has been used to estimate the consequences of the E-ring on radiation belt intensities.
 The differential Kennel-Petschek limit of relativistic electron populations is not observed to be significantly exceeded by the spectral intensities observed within any of the strongly magnetized planets of the solar system. For dynamically active magnetospheres, absent absorbing materials, the KP theory is predictive of the shape of the spectra at energies below a cutoff, specifically near ∼1 MeV for magnetospheres studied. It remains an open question as to whether the characteristics of whistler mode emissions support the simplicity of the KP theory as utilized here. It is likely that full closure involves interactions and wave propagations occurring within relatively large volumes of space and not just within the confines of single flux tubes.
 Provided here are expressions for the temporal growth rate of whistler waves propagating parallel to the magnetic field and properly taking account of the condition that the electrons that provide the power for the growth extend in energy to relativistic values. These expressions are in the form provided by Summers et al.  (Copyright by the American Geophysical Union, 2009) who modified them from the expressions by Xiao et al. . An earlier treatment was published by Liemohn . The linear e-folding temporal growth rate is
where γ is the e-folding temporal growth rate (1/s), ωpe is plasma frequency [√(4πNe2/me)], ωr is resonant frequency (radians/s), Ωe is the electron gyrofrequency [eB/(mec)], N is density (1/cm3), e is electron charge (statcoulombs), me is electron mass (g), c is speed of light (cm/s), B is equatorial magnetic field strength (Gauss), and
defined as the fraction of the energetic electron distribution near cyclotron resonance,
the relativistic distribution pitch angle anisotropy factor,
the relativistic Lorentz factor at resonant energies (not to be confused with the wave linear growth rate γ in equation (A1) and in the main body of this paper),
the parallel momentum for resonant particles,
the latter being the minimum required anisotropy factor for wave instability, and
the wave dispersion relation for parallel propagating whistler waves. In these expressions, k is the wave spatial frequency (2π/λ where λ is wavelength in centimeter), is momentum parallel to the magnetic field, and is momentum perpendicular to the magnetic field. The last expression (A8) is the wave dispersion relationship, from which the wave group velocity can be derived by differentiation (Vg = ∂ω/∂k). The wave group velocity, needed to solve equation (6), is
 The Jupiter spectrum for L = 8.3 in section 7.3 was taken from the NASA Jet Propulsion Laboratory publication: Garrett et al. . Because this report, while available, is not as readily available as other papers, we provide a few details here. Jupiter spectra between ∼8 and ∼16 RJ were developed by performing radiation ray tracing calculations on the Galileo Energetic Particle Detector [Williams et al., 1992] to better interpret the channel rate information provided by the instrument. That information was combined with the higher energy (31 MeV) channel information from Pioneer [Van Allen et al., 1974, 1975], which sampled Jupiter many years before Galileo did. The results were captured in the form of an analytic fit of form similar to that used by Baker and Van Allen ,
Table B1 provides the fitting parameters for each value of the radial distance L, and Figure B1 shows a plot of all of the spectra generated. The innermost spectrum was sampled at L = 8.25 RJ, and that is the one that is most intense at 1 MeV, and the one used here for our KP analysis.
 We appreciate substantial discussions with members of the Cassini Magnetosphereic Imaging Instrument (MIMI) team and the Voyager Low Energy Charge Particle (LECP) instrument team regarding the relative responses of the two instruments to radiation electrons in Saturn's environment, particularly T. P. Armstrong and C. P. Paranicas, but also S. M. Krimigis and D. G. Mitchell. The Earth data used in the study of Fox et al.  and updated here were taken by the Medium Electrons A instrument (MEA; A. L. Vampola and E. G. Mullen, Principal Investigators) and the High Energy Electron Fluxmeter instrument (HEEF; E. G. Mullen and D. H. Brautigam, Principal Investigators) on the Combined Release and Radiation Effects Satellite (CRRES) and obtained from the National Space Science Data Center (http://nssdc.gsfc.nasa.gov/database/MasterCatalog?sc=1990-065A&ds=*). We thank members of the CRESS instrument teams, particularly J. Bernard Blake and D. H. Brautigam, for help with the use of these data.
 Masaki Fujimoto thanks the reviewers for their assistance in evaluating this paper.