Electron radiation belts of the solar system

Electron radiation belts of the solar system

B. H. Mauk1 and N. J. Fox1

Received 11 May 2010; revised 16 July 2010; accepted 16 August 2010; published 9 December 2010.

[1] To address factors dictating similarities and differences between solar system radiationbelts, we present comparisons between relativistic electron radiation belt spectra of allfive strongly magnetized planets: Earth, Jupiter, Saturn, Uranus, and Neptune. We chooseobserved electron spectra with the highest intensities near ∼1MeV and compare them againstexpectations based on the so‐called Kennel‐Petschek limit (KP). For evaluating the KPlimit, we begin with a recently published relativistic formulation and then add severalrefinements of our own. Specifically, we utilized a more flexible analytic spectral shape thatallows us to accurately fit observed radiation belt spectra, and we examine the differentialcharacteristics of the KP limit. We demonstrate that the previous finding that KP‐limitedspectra take on an E−1 shape in the nonrelativistic formulation is also roughly preserved withthe relativistic formulation; this shape is observed at several of the planets studied. We alsoconclude that three factors limit the highest relativistic electron radiation belt intensitieswithin solar system planetary magnetospheres: (1) plasma whistler mode interactions thatlimit differential spectral intensities to a differential Kennel‐Petschek limit (Earth, Jupiter,and Uranus), (2) the absence of robust acceleration processes associated with injectiondynamics (Neptune), and (3) material interactions between the radiation electrons and cloudsof gas and dust (Saturn).

Citation: Mauk, B. H., and N. J. Fox (2010), Electron radiation belts of the solar system, J. Geophys. Res., 115, A12220,doi:10.1029/2010JA015660.

1. Introduction

[2] All of the strongly magnetized planets of the solarsystem (Earth, Jupiter, Saturn, Uranus, and Neptune) haverobust electron radiation belts at relativistic energies(Figure 1). Acceleration of electrons to relativistic energieswithin strongly magnetized space environments is clearly auniversal process and not one that is peculiar to the specialconditions that prevail at Earth or any other specific planet. Itis of substantial interest for generalizing space environmentacceleration processes to determine the similarities and dif-ferences between the radiation belts of these five accessibleenvironments. Are the intensities and characteristics of theseenvironments governed by the same processes and in a pre-dictable and scalable fashion?[3] To address this question, we compare the most intense

electron spectra, specifically at ∼1 MeV, measured withineach of the target magnetospheres. Some of the spectra showremarkable similarities. We address the reason for the simi-larities by comparing the spectra against a reformulated ver-sion of the so‐called Kennel‐Petschek limit [Kennel andPetschek, 1966]. Kennel and Petschek [1966] noted thatelectron distributions within magnetic bottle configurations,allowing some losses for particles with velocities alignedwith

the magnetic field, are intrinsically unstable to the generationof whistler mode plasma waves. The authors postulate thatwhistler mode waves, propagating roughly parallel to themagnetic field lines, are generated near the magnetic equatorof the Earth’s inner magnetosphere and propagate to the endsof the field lines anchored in the Earth’s ionosphere. At theionosphere, a small fraction of the wave energy is reflectedback into the system. If the fraction of reflected energy is largeenough (percent level) then there can be runaway growth ofthe waves and a resulting strong, nonlinear suppression of thecharge particle intensities. The level of the integral intensitiesabove a specified energy value is called the “Kennel‐Petschek(KP) limit.” This theory predicts that in the absence ofextremely fast acceleration processes, acting faster than therunaway wave growth can act to suppress the particle inten-sities, integral intensities at energies above a defined mini-mum energy will reside below the KP limit. There are timeswhen the source rate of energetic electrons is indeed fasterthan themaximum loss rate stimulated by the Kennel‐Petschekprocesses, and we address that situation in section 10.[4] Kennel and Petschek [1966] showed measurements of

integral intensities (>40 keV) within Earth’s magnetosphereand demonstrated that the intensities were generally belowtheir KP limit (derived for the nonrelativistic regime). Otherauthors have also demonstrated the usefulness of this KPformulation for explaining Earth magnetospheric particleintensities [e.g. Baker et al., 1979, and references therein]. Itis significant that the agent of the particle losses that pur-portedly works to establish the KP limit, whistler mode

1Johns Hopkins University Applied Physics Laboratory, Laurel,Maryland, USA.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2010JA015660

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A12220, doi:10.1029/2010JA015660, 2010

A12220 1 of 19

http://dx.doi.org/10.1029/2010JA015660

waves, has been observed at all of the planetary magneto-spheres addressed here, but with interesting differences inintensity [Kurth and Gurnett, 1991].[5] Since 1966 there have been several refinements to the

Kennel‐Petschek theory. Schulz and Davidson [1988]derived a differential version of the KP limit. Specifically,they predicted that for integral intensities that are near the KPlimit, the spectra will take on an E−1 spectral shape at inter-

mediate energies (up to the energy where the accelerationprocesses can no longer populate the limiting profile). Evi-dence was provided in a companion paper [Davidson et al.,1988] that the E−1 shape is a characteristic of the mostintense of Earth’s electron radiation belt spectra observedwell inside of synchronous altitudes (most convincingly nearL = 5.4, near the innermost observation position).[6] More recently, Summers et al. [2009] created a rela-

tivistic version of the Kennel‐Petschek theory. They showsubstantial, although not profound, differences between thepredictions based on the relativistic and nonrelativistic ver-sions of the theory. These authors find that the electronradiation belt spectra of Earth, Jupiter, and Uranus haveintegral intensities consistent with expectations based on theirnew 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 Barbosaand Coroniti [1976], ultra relativistic formulations. Summerset al. [2009] also challenge the assumption that feedbackresulting from ionospheric reflection of the waves plays a rolein establishing the KP limit. Their arguments are based onclaims about the Poynting flux directionality of measuredwhistler waves within Earth’s magnetosphere [e.g., Burtonand Holzer, 1974; LeDocq et al., 1998; Santolík et al.,2003]. However, it is unclear from the literature whetherreflected wave power levels (Poynting flux levels) of only0.25% of that which propagates away from the equator(corresponding to 5% amplitude reflection) has actually beencharacterized given measurement sensitivities. Note in par-ticular that relatively weak equatorward‐propagating whistlerwaves are observed at frequencies above and below the mainband of whistler chorus emission [Santolík et al., 2010]. Thisfinding begs a question about the possible existence ofweak equatorward‐propagating waves hidden within themain, overwhelming band of Earthward‐propagating waves.Very little reflected wave power is needed to sustain theKennel‐Petschek feedback chain. In any case, Summers et al.[2009] adopt the numerical wave growth factors used byKennel and Petschek [1966] to establish the KP upper limit,but they justify the factors on other grounds.[7] Whistler mode waves are not the only waves that can

cause scattering and losses to magnetically trapped electronpopulations. Recent studies have shown that ElectromagneticIon Cyclotron (EMIC) waves can cause strong losses toelectrons, particularly those with energies >1 MeV (recentworks include Sandanger et al. [2007], Summers et al.[2007], Miyoshi et al. [2008], Shprits et al. [2009], andUkhorskiy et al. [2010]). At lower energies, Electron Cyclo-tron Waves (ECH) have been identified as an agent forelectron losses, particularly for the range of relatively low‐energy electrons (from the perspective of this paper) that cancause diffuse auroral emissions (selected works includeHorne et al. [2003] and Meredith et al. [2009]). These andother mechanisms certainly can play a role in sculptingelectron spectral shapes. However, they have not been iden-tified in the literature as defining an upper limit to electronintensities on the basis of a nonlinear feedback mechanism.For example, the most competitive of the possible mechan-isms, EMIC waves [see Summers et al., 2007], are thought tobe generated by hot ions and not hot electrons. And so, an

Figure 1. Spectrograms of energetic electron intensities(color scale) versus energy (vertical scale) and time (horizon-tal scale). Figure 1 is taken from Mauk et al. [1995]. TheEarth data were obtained by the International Sun EarthExplorer (ISEE) mission, and the other data were obtainedby the Voyager encounters of the various outer planet mag-netospheres. The time scales encompass both inbound andoutbound magnetopauses (“M” character above each plot).Energy spans roughly 20 keV to ∼1 MeV. The intensity scaleis fixed for all of the panels (intensities between planets canbe compared). Copyright by the University of Arizona Press(1995).

MAUK AND FOX: ELECTRON RADIATION BELTS OF THE SOLAR SYSTEM A12220A12220

2 of 19

increase in electron intensities does not in and of itself stim-ulate a suppression of those same intensities. With thisreminder regarding the possible role of other wave modes, weconcentrate exclusively on the whistler mode mechanism forthe rest of this work.[8] In the present paper, we extend the relativistic Kennel‐

Petschek whistler mode analysis initiated by Summers et al.[2009] in several ways. All of the previous authors citedhere who have addressed this mechanism have utilized theassumption of a power law spectral shape [Kennel andPetschek, 1966; Schulz and Davidson, 1988; Summerset al., 2009]. The power law representation, as we will

show here, is a good representative of measured spectra onlyover limited ranges of energy. The procedures develop herecan be used for any spectral shape, and we have chosen ananalytic spectral form that is flexible enough to be accuratelyfit to most of the observed electron spectra utilized here. Wealso examine the differential characteristics of the relativistictheory. Our procedure is a relatively simple extension of theprocedures outlined by Summers et al. [2009], but we arguethat the insight gained from this extension is substantial.Finally, we add the planets Neptune and Saturn to the pan-theon of planets examined with respect to the KP theory.Significantly, these planets do not follow the pattern estab-lished by the examination of the other strongly magnetizedplanets, and we derive significant information about the fac-tors that limit radiation belt intensities within planetarymagnetospheres.

2. Motivations From Earth

[9] Recent work by the present authors has motivated amore careful examination of the Kennel‐Petschek limit. Foxet al. [2006] plotted near‐equatorial electron spectra for var-ious L values, as sampled by two instruments (MEA andHEEF, see Acknowledgments) on the Combined Release andRadiation 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 ofthe Fox et al. [2006] study is that the observed intensities atenergies of > 1 MeV are greater than expectations based onadiabatic transport from a source population within the near‐Earth, tail side plasma sheet (those expectations are shown assmooth curved spectra in Figure 2, middle and bottom). Here“adiabatic” is defined to mean processes that preserve the firsttwo adiabatic invariants, those associated with gyration andbounce. For our purposes here, there are other features ofsignificant note in Figure 2. For the lower energies, somewhatless than ∼1 MeV, the observed intensities fall far below theadiabatic transport expectations, suggesting that particle los-ses play a significant role is sculpting the character of thespectra. Significantly, these <1 MeV components display aroughly 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 character-istics 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 generallylimiting integral intensities, but in strongly sculpting thecharacter of the most intense radiation belt electron spectra.The specific spectral features highlighted in Figure 2 aresimilar to the characteristics of the most intense spectrumreported byDavidson et al. [1988] specifically at L = 5.4 (seetheir Figure 5, bottom left) observed by a different mission(SCATHA) and with different instrumentation.

3. Kennel‐Petschek Theory

[10] Because of the complexity of calculations of thegrowth rate of whistler mode waves in complex geometries,

Figure 2. (top) Prediction from Schulz andDavidson [1988]that medium energy electrons in a spectrum that challengesthe nonrelativisitic Kennel‐Petschek limit will takes on a 1/Espectral shape up to a high energy break, beyond which thespectral shape is dictated by the acceleration process. (middleand bottom) Spectra from Fox et al. [2006] sampled during astrong storm in Earth’s magnetosphere at (middle) L = 6 and(bottom) L = 5 showing the 1/E spectral shape and thediscontinuity at higher energies predicted by Schulz andDavidson [1988]. The curved lines are expectations basedon adiabatic transport from the plasma sheet.


3 of 19

many in the space physics community judge the Kennel‐Petschek limit theory to be difficult. However, it is intrinsi-cally a very simple concept. A rough estimate of the net gainG of whistler wave amplitudes as the waves pass through themagnetic equator of the Earth’s magnetosphere is

G � exp

Z D=2

�D=2

�

VgRpdl

" #� exp

�

VgRpD

� �; ð1Þ

where G is the ratio (Af /Ai) of “final” to “initial” wave am-plitudes as the waves propagate all the way through theequatorial unstable region, g is temporal growth rate (1/s) (notto be confused with the relativistic Lorentz factor used inAppendix A), Vg is wave group speed (cm/s), Rp is planetaryradius (cm), dl is differential distance along themagnetic fieldB in units of planetary radii, and D is the total distance alongthe magnetic field line (roughly symmetric around the mag-netic equator) over which the wave growth remains signifi-cant, in units of planetary radii. Traditionally, the value ofD isestimated as roughly L, the magnetospheric distance param-eter, although at Jupiter, because of the centrifugal confine-ment of plasmas close to the magnetic equator, a lower valueis suggested by Summers et al. [2009].[11] Equation (1) may be rewritten as

DRp�½Fð> E*Þ� � Ln½G�Vg; ð2Þ

where here we show the temporal growth rate g explicitly as afunction 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 gyrocyclotronresonance with propagating whistler waves with wave fre-quency w*r , but is also the specific energy where the wavegrowth rate transitions from negative values below thatenergy (wave frequencies above the corresponding wavefrequency w*r) to positive values above that energy (wavefrequencies below the corresponding wave frequency w*r).[12] The Kennel‐Petschek limit is defined by the condition

that GR = 1, where R is the ionospheric reflection coefficientfor wave amplitudes (not wave power). If we assume that R =0.05, or 5%, then Ln[G] = Ln[1/R] ∼ 3. This level of reflectioncorresponds to just 0.25% of the wave power (0.05 × 0.05).Fundamental to the usefulness of this entire approach is thefact that the KP limit varies only as the logarithm of the wavegain value G. Therefore, a useful upper limit can be estimatedeven when there is substantial uncertainty in the value of R.Again, Summers et al. [2009] formulate the problem withoutrelying on ionospheric reflection, but nonetheless, continue touse Ln[G] ∼ 3, partly for the purpose of continuity withprevious work. The logarithmic sensitivity allows for a sub-stantial amount of flexibility here.[13] Kennel and Petschek [1966] approximate the rela-

tionship between g and the F(>E*) with their equation (2.27)combined with (4.17), which allows them to invertequation (2). After replacing Ln[G] with 3, we represent thatresult conceptually with

Fð> E*Þ � ��1 3Vg

DRp

� �: ð3Þ

Note that the inversion of g[F(>E*)], using the equations citedabove in the original KP paper, represents only a very rough

approximation. Accurate inversion requires numerical cal-culations and corresponding factors that depend on the exactshape of the electron spectrum [Schulz and Davidson, 1988].In the present paper, the complexity of the electron spectralshapes makes it much more difficult to continue to specify theKP limit in the form of a limit on integral flux, and we aban-don that approach here. Our starting point is the uninvertedformulation given in equation (2), againwithLn[G]∼3, yielding

DRp� ¼ 3Vg: ð4Þ

[14] Equation (4) was also the starting point for Summerset al. [2009]. The significant new feature that those authorsintroduced was to use the relativistic formulation of the lineargrowth rate g derived by Xiao et al. [1998; an earlier for-mulation of relativistic wave growth rates is provided byLiemohn, 1967]. The Summers et al. [2009] formulation ofthe Xiao et al. [1998] results is reproduced in Appendix A.We will reference formulas from Appendix A during laterdevelopments. In evaluating equation (4), in either the non-relativistic or relativistic formulation, there is a choice thatmust be made. At what frequency or, equivalently, at whatgyroresonant energy is equation (4) to be evaluated? Thechoice made by Summers et al. [2009] and other authors is toevaluate it at the wave frequency w = wm corresponding towhere the temporal growth rate is maximum. Specifically,

DRp�ð!mÞ ¼ 3Vgð!mÞ: ð5Þ

One might alternatively maximize g/Vg rather than just gitself. In the present analysis, we have chosen to examine therelativistic KP limit as a function of the minimum resonantenergy, which we call the “strictly” parallel resonant energyEr. The strictly parallel resonant energy is the parallel resonantenergy for the condition that the perpendicular momentump? = 0. We distinguish the general resonant parallelmomentum pR or the equivalent parallel energy ER, whichprevails for the condition that perpendicular momentump? can take on any value, and the strictly parallel resonantenergy because the general resonant parallel momentumchanges as a function of the perpendicular momentum(Appendix A). There is a unique one‐to‐one relationshipbetween the strictly parallel energy Er and the resonant wavefrequency wr, whereas there is not such a unique relationshipbetween ER and wr. The equation that we solve is

DRp�½!rðErÞ� ¼ 3Vg½!rðErÞ�; ð6Þ

which allows us to address the differential characteristics ofthe KP limit, that is, the KP limit as a function of minimumresonant energy. Equation (A5) provides the relationshipbetween resonant frequency and the strictly parallel momen-tum by setting p? = 0. The strictly parallel momentum can beconverted to strictly parallel energy with the standard formula(section 4).

4. Electron Distributions

[15] We have developed an approach to evaluating the KPlimit implications that allows for the use of an arbitrarycontinuous analytic spectral form for the differential intensityI[E,a], where E is energy anda is pitch angle, provided it canbe converted to an analytic and continuous function of the


4 of 19

parallel and perpendicular momentum. The form that we havechosen to use here is

I1

ðcm2 s sr keVÞ� �

¼ CEkeVðkTð�1 þ 1Þ þ EkeVÞ�ð�1þ1Þ

1þ EkeVE0

� ��2 sin2Sð�Þ;

ð7Þ

where C, kT, g1, E0, and g2 are fitting parameters for theenergy distribution, and “S” represents the so‐called anisot-ropy parameter. The numerator of the energy distribution isthe so‐called kappa distribution (representing a Maxwelliancore with a power law tail), and the denominator adds anadditional power law tail contribution at the highest energies.Note that, in the denominator, the power g2 acts on the energyratioE/E0 rather than on the entire 1 +E/E0 (the later formwasused, for example, byBaker and Van Allen [1976]), because itaccommodates the sharp transitions that are often apparentin the observed spectra. This energy distribution shape inequation (7) has been shown to accurately represent a broadrange of measured spectral shapes at both Jupiter and Earth[Mauk et al., 2004; Fox et al., 2006].[16] It is, of course, convenient to assume, as we have done

with equation (7), that the angular variations are fully inde-pendent of the energy variations, aswas also assumed by SchulzandDavidson [1988], Summers et al. [2009], and others.Whilethis approach is unlikely to be very accurate in representingparticle distributions generally measured in space environ-ments, wewill argue in section 5 that this approach is in fact theappropriate one for evaluating a spectrum against the KPlimit. It is not just a convenience to take this approach.[17] The five energy distribution parameters of equation (7)

are determined by fitting that equation to a measured spec-trum of interest through an error minimization process. Wethen prepare that optimized spectrum for insertion intothe equations of Appendix A for evaluating the relativisticwhistler mode growth rate (g). To do so, we perform thefollowing actions. (1) We transform all “keV” values inequation (7) to ergs using the standard factor. (2)We solve thefollowing standard relativistic equation,

ðp=ðmecÞÞ2 ¼ ðE=ðmec2ÞÞ½ðE=ðmec

2ÞÞ þ 2�; ð8Þ

for energy E and substitute the positive solution intoequation (7). (3) We substitute (p? /p)

2S for sin2S(a), wherep? 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 equationp2 = p?

2 + pk2. We end up with f (p?, pk), which is just what we

need to insert into the expressions in Appendix A. It is impor-tant to note that the manipulations specified here can only becarried out in practice using software that accommodatessophisticated symbolic manipulations, like Mathematica®.Long before one reaches the point of performing the finalnumerical integrations specified in Appendix A, the indi-vidual analytic expressions occupy many pages of text.

5. Anisotropy Factor

[18] After fitting the energy distribution in equation (7), theremaining parameter that must be set is the anisotropyparameter “S.”Kennel and Petschek [1966] analyzed a simple

pitch angle diffusion equation, imagining a source of particlesat equatorial pitch angles of p/2, imposing various other as-sumptions and derived the following estimate for the anisot-ropy factor A (equivalent to our S for the nonrelativisticformulation): A ∼ 1/(2 ln[(1/a0) + O(1)]), where a0 is equa-torial 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 inthe authors’ final simple expression. There are other simpleapproaches to theoretically estimating an anisotropy param-eter. For example, the asymptotic shape of a distributiongoverned by pitch angle diffusion coefficients that are inde-pendent of pitch angle has the shape: J0[2.40·cos(a)/cos(a0)],where J0 is the Bessel function, a is pitch angle, and a0 is theloss cone pitch angle [Schulz and Lanzerotti, 1974]. For smallloss cone pitch angles, this shape can be very accurately fittedwith a sin2S(a) shape, with an S value of order 1.3. And so, onsimple theoretical grounds, there is a broad range of possibleanisotropy values that might be used (0.17 to 1.3 for the twoexamples given here).[19] There are two reasons why we do not just use the pitch

angle distributions that are simply observed along with theenergy distributions. First, for the broad range of planetaryenvironments addressed here, the needed pitch angle dis-tributions are just not available. The instrumentation, thecomplexity of the measurement geometries, and the difficultyof the measurement environments (dealing with penetratingbackgrounds, etc.) often did not allow for the determination ofeven rough estimates of the anisotropy parameters in manycases. There is a more central issue, however. We believe thatit only makes sense to define the KP limit with a minimalanisotropy parameter. As rapid acceleration and transportpushes an electron distribution up toward the KP limit, theangular distributions will flatten as a part of the processes thatdefine the limit. For example, if an energy distribution issomewhat below the KP limit and the anisotropy parameter islarge, it makes no sense to define the KP limit for that par-ticular distribution on the basis of the observed anisotropyparameter. What makes sense is to define the limitinganisotropy parameter on the basis of what the parameter willbecome as the acceleration processes try to push the distri-bution to and beyond the KP limit. In essence, we believe thatthe anisotropy parameter that defines the KP limit is in somesense a universal parameter (representing more or less theapproach that Kennel and Petschek [1966] took originally).The trick is to determine what that parameter is. Theassumption of a universal anisotropy parameter associatedwith distributions as they approach and exceed the KP limitjustifies the assumption, implicit in equation (7), that theenergy distribution is separable from the angle distribution.[20] Our approach is to use observations during strong

storms within the Earth’s magnetosphere to guide our choicefor the anisotropy parameter to be used in all of the en-vironments studied here. We have fitted pitch angle dis-tributions during various phases of activity and find that thesinn(a) shape does a reasonably good job (Figure 3; note thatn = 2S). The inserted table in Figure 3 shows the result of astudy of the pitch angle parameter n as a function of energyand magnetospheric disturbance level, all for L = 6. Note thatfor storms and superstorms and at lower energies, where onFigure 2 the E−1 spectral shape prevails, the anisotropy is lowand relatively insensitive to energy. At the highest energy,above the break in the spectrum in Figure 2, and therefore


5 of 19

above the energywhere the KP limit has a strong influence, theanisotropy parameter is larger. The anisotropy parameter (S =n/2) that wewould choose for super stormsmight be 0.25, andfor stormsmight be 0.4. For all of the work described here, forall of the planets, we have arbitrarily rounded up the superstorm value and have chosen to use S = 0.3.[21] As demonstrated by previous authors [e.g., Schulz and

Davidson, 1988; Summers et al., 2009] the KP limit doesdepend on the choice of “S,” and there is certainly no guar-antee that all of the planetary magnetospheres will have thesame anisotropy parameter. We provide some sense of thesensitivity of the KP limit to the anisotropy parameter duringour examination below of the spectra measured at Earth.However, the anisotropy parameter is not the only parameterthat might vary between the different planets. Other suchparameters would include the wave reflection coefficients atthe ionosphere and the distribution of plasmas along themagnetic field lines. However, the KP limit is a useful toolmostly to extent that a uniform set of procedures can inde-pendently be shown to yield coherent and comparable resultsdespite some measure of sensitivity to the various assumedparameters. We expect our results to be judged based on theextent to which we achieve those coherent and comparableresults between planets.

6. Calculating the KP Limit

[22] While the calculations that we perform to evaluate theKP limit here are quite messy, the procedure is really quitesimple. The normalization parameter for our spectral shape inequation (7) is “C” For the calculations described here, wewill define two different values of C. Cm is the normalizationparameter that comes out of the fitting of the observed spectrawith 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). Wewill be reporting our results as a ratio: Cm/CK. When Cm/CK

is less than 1, then the observed distribution for a given res-onant frequency and correspondingly for a given strictlyparallel resonant energy is judged to be less than the KP limitfor that frequency and energy. IfCm/CK is greater than 1, thencorrespondingly the distribution for that frequency and cor-responding energy is judged to be above the KP limit.[23] Here we spell out our procedure a little more carefully.

Choose an observed equatorial electron spectrum that is to becompared against the KP limit. Fit the observed spectrum toequation (7) and set the parameter “S” to 0.3 as justified insection 5. Identify for that spectrum the values of L, equatorialmagnetic field strength B, and density N. B and N go into thecalculation of the equatorial plasma frequency wpe andequatorial gyro frequencyWe used inAppendixA. Process thefitted analytic spectrum in the fashion described in section 4to generate the phase space density f (p?, pk). Plug f (p?, pk)into the equations in AppendixA,making use of the identifiedvalues of L, B, and N. For a systematic array of resonantfrequencies (wr) evaluate the equatorial whistler mode wavegrowth rate and the corresponding strictly parallel particleenergy associated with each resonant frequency, plug eachof the wave growths for each resonant frequency intoequation (6) along with the group wave velocity for eachfrequency (again Appendix A; equation (A9)) and determinethe 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 isneeded to satisfy equation (6), then CK/Cm = 3. At the end ofthis procedure, we end up with a matrix of numbers whereeach row is (wr, Er, Cm/CK). The plots that we show are Cm/CK versus Er. Note that, consistent with previous work, thevalue of “D” used in equation (6) is generally “L,” but forJupiter, we use a smaller value as suggested by Summers et al.[2009; see section 7.3]. Our procedure was implemented intoa single Mathematica® routine that performs both the exten-sive symbolic manipulations that are required, as well as thefinal numerical integrations. It is interesting to examine justhow the distribution function is sampled whenwe perform theanalysis described here. Figure 4 shows f (p?, pk) derived byfitting the Figure 2 (bottom) spectrum with equation (7), andusing an S value of 1, for the sake of making the angularanisotropy a little more visible on this crude display. Theblack lines are the resonance curves derived fromequation (A5), each corresponding to a single resonant fre-quency wr. The lines are labeled with the correspondingstrictly parallel resonant energy Er in MeV, which corre-sponds to the energy of the electrons at the position where theblack curves cross the p? = 0 axis. Clearly integrals (specifiedin Appendix A) along the black curves will have some rela-tionship with the standard integral intensity I(>E). However,particularly for our very flexible spectral shape represented byequation (7), there is no easy one‐to‐one relationship. Again,we have abandoned the idea of making the connectionbetween the Kennel‐Petschek limit and the integral intensity.

7. Earth, Uranus, and Jupiter

7.1. Earth

[24] We will be giving some introductory informationabout each of the planet’s magnetospheres, but assume that

Figure 3. Updated figure from Fox et al. [2006] showingmeasured (solid lines near top) pitch angle distributions fromEarth’s magnetosphere at L = 6 for 1.2 MeV electrons duringa “Super Storm,” during a Storm, and during Quiet periods asobtained by CRRSS [Fox et al., 2006]. Overlaying the mea-surements near the top are fits using the expression I =Const ×sinn[PA]. The table provides n values for the sinn[PA] fits tothe pitch angle distributions as a function of energy andmagnetospheric activity level.


6 of 19

the reader is well familiar with Earth’s [see Kivelson andRussell, 1995]. Figure 5 shows the results of the Kennel‐Petschek procedures outlined in section 6 for the Earthspectrum shown in Figure 2 (bottom). The top shows fittedspectra for L = 4, 5, and 6, while the bottom shows our KPanalysis for just L = 5. The spectral parameters (equation (7))derived for these spectra are provided in Table 1, along withthe parameters derived for all of the other spectra analyzed forthis work. The bottom displays Cm/CK plotted against thestrictly parallel energy Er. The blue bar, centered vertically ontheCm/CK = 1 position, represents the region of the plot that isroughly within a factor of 3 of the KP limit, motivated by thestatement by Kennel and Petschek [1966] that the derived KPlimit is roughly accurate to within a factor of 3. We assumehere that when the calculatedCm/CK profile resides within theshaded region, that the spectrum is at least strongly under theinfluence of the processes that establish the KP limit.[25] 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 theappropriate density to use. For Earth, we have adopted as ourbest guess densities the ones used by Summers et al. [2009],who in turn obtained densities from the work of Sheeley et al.[2001]. As the density increases the profile reaches an

Figure 4. Two‐dimensional display of the phase space den-sity f (pk, p?) for the Earth L = 5 spectrum [Fox et al., 2006]and for an anisotropy factor of S = 1 (larger than the S = 0.3factor used in this report in order to make the anisotropy morevisible on this crude display). The solid black lines are thewhistler resonance curves (equation (A5)) for magneticfield = 149 nT andN = 1. The lines are labeledwith the strictlyparallel resonant energy (MeV) at the position where theblack lines cross the p? = 0 axis. The whistler wave growthcalculation in Appendix A for a specific wave frequency, orequivalently for a specific strictly parallel resonant energy Er,involves performing integrals along the black resonant curvelines.

Figure 5. (top) Fits (using equation (7)) of the Earth strongstorm spectra presented by Fox et al. [2006] for radial posi-tions of L = 4, 5, 6; the L = 6 and 5 spectra are shown inFigure 2 (middle and bottom). For this top plot the energyscale corresponds simply to the energy at which eachdifferential spectrum is evaluated. (bottom) Our Kennel‐Petschek analysis for just the L = 5 spectrum. Cm/CK is theratio of two determinations of the normalization parameter“C” in equation (7). Cm is the value that comes out of thefitting of the data (“m” for measured) and CK is the valueneeded to satisfy the Kennel‐Petschek limit as expressed inequation (6). For this plot the energy scale is the strictlyparallel resonant energy, representing the low‐energy start-ing point (p? = 0) for the resonance curves illustrated inFigure 4 over which one must integrate. A single storm timeequatorial magnetic field value is used (149 nT = BDipole

minus the storm perturbations provided by Tsyganenko andSitnov [2007]). When the purely dipole field value is used,the Cm/CK values are roughly 50% lower. The plot is gen-erated for various equatorial densities (N in units of 1/cm3),and the red curve represents the density used by Summerset al. [2009] from the work of Sheeley et al. [2001]. Thehorizontal blue shaded region brackets the Cm/CK = 1 line bya factor of 3, based on the Kennel and Petschek [1966]estimate that these evaluations are accurate to that sort oflevel. For the portion of the Cm/CK analysis that goes tolower energies than does the top panel spectrum itself, thespectrum has been extrapolated in the analysis.


7 of 19

asymptotic condition (labeled “∞”) where further densityincreases have no influence on the shape of the profile withinthe energy range of interest. Because the spectrum wasmeasured during a very strong magnetic storm, the magneticfield is substantially suppressed, and for the calculations wehave used the storm time magnetic field values provided byTsyganenko and Sitnov [2007].[26] For our best guess density, the observed spectrum for

Earth L = 5 resides fairly close to the KP limit (within theshaded region) from ∼30 keV to almost ∼1 MeV. The peakvalue of Cm/CK is 0.60, very close to 1.0 given the roughassumptions that go into the generation of the KP limit. Forvarious assumed anisotropy parameters, that peak value is0.72, 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 qualitativeconclusions about the influence of the KP limit on the spectralshape would be roughly the same. It the anisotropy parameterwas indeed 0.1 at Earth, contrary to Figure 3, we might havequestioned the quantitative usefulness of the KP analysis forthis particular spectrum.[27] It is of interest to consider the shape of the asymptotic

Cm/CK profile, specifically the fact that it is flat and nearlyhorizontal at the lower energies. We note that for energiesbelow about 0.1 MeV, the KP analysis is performed using anextrapolation of the measured spectrum, and so the discussionhere should be considered theoretical in nature. As accelera-tion processes drive the distribution closer and closer to theKP limit, it is reasonable to assume that the processes thatlimit the intensities start to act first on those portions of thedistribution (those collections of electrons that reside along alimited set of the resonant curves plotted in Figure 4) that riseabove the differential KP limit. And so as the process pro-ceeds, one expects the Cm/CK profile to flatten itself againstthe Cm/CK = 1 line. Significantly, this condition is qualita-tively achieved for a spectrum that has an E−1 shape at thelower energies, just as predicted by Schulz and Davidson[1988]. To test this conclusion more accurately, we havemodified our Earth spectrum to a perfect power law, with theshape 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 theone for g = 1.0, as we expected from the nonrelativistic pre-diction of Schulz and Davidson [1988]. More detailed anal-ysis shows that g ≈ 0.9 provides themost horizontal profile forthe Earth situation represented in Figure 6. For the relativisticcalculations performed here, then we roughly confirm thepredictions of Schulz and Davidson [1988] using the non-relativistic formulation that the saturation shape of the KP‐limited spectrum is approximately E−1.[28] The KP limit calculations for three different Earth

L values (4, 5, and 6) are shown in Figure 7, but in each case

Table 1. Parameters for the Equation (7) Spectra Used in This Study

Planet Position (L) Energy Range (MeV) C kT g1 E0 g2 Source

Earth 4 0.1–5 2.34E7 ∼0 1.290 1324 6.7145 0.1–5 2.38E6 ∼0 0.978 1748 7.036 Fox et al. [2006]6 0.1–5 7.96E6 ∼0 1.324 1715 5.676

Earth 6.67 0.03–0.3 4.0E16 9.45 5.05 a a Baker et al. [1979]Uranus 4.7 0.03–5 6.5E6 ∼0 1.101 1189 5.604 Mauk et al. [1987]Jupiter 8.25 0.05–30 1.0E6 ∼0 0.8367 2546 1.741 Garrett et al. [2003]

3 0.2–30 2.52E4 ∼0 0.3269 6119 1.841 Baker and Van Allen [1976]Neptune 7.4 0.03–5 5.49E6 5.2 1.560 625 2.745 Mauk et al. [1995]Cassini Saturn 3.25 0.03–9 2.17E3 ∼0 0.5122 2000 7.0 This workVoyager Saturnb 2.8 0.03–5 7.58E2 1007 −0.21 680 3.28

3.35E7 0 2.22 500c 10c

1.9E4d 101d 1234d

3.6 0.03–5 3.26E5 4027 3.94 630 3.94 Krimigis et al. [1983]1.61E7 0 2.26 500c 10c Van Allen [1984]2E3d 109d 1133d

4.4 0.03–5 5.41E2 1837 −0.14 750 71.82E7 0 2.46 500c 10c

1.0E3d 117d 1155d

aThe denominator of equation (7) is not used.bThe complex spectra of Voyager Saturn require the addition of three components.cThese parameters keep low‐energy components from interfering at high energies.dThe third component is Gaussian: I3 = C exp[−(E − E0)2/(2 kT2)].

Figure 6. Test of the proposition that Kennel‐Petschek lim-ited spectra take on the E−1 spectral shape for the relativisticKP analysis presented in this paper as predicted with the non-relativistic formulation by Schulz and Davidson [1988]. Herewe repeat the analysis for the bottom panel of Figure 5 butalso (a) use a strictly power law energy distribution, (b)assume a very large density (otherwise the curves roll downon the left side), and (c) assume a somewhat higher value ofCm just so that we can center the lines on a plot grid line. Wefind that the spectral index near 1 yields the most horizontalline (our metric is how horizontal the line is; see text). Furtherrefinements yield that the most horizontal line comes with aspectral shape of ∼E−0.9.


8 of 19

just for our best guess as to the equatorial plasma density.Two points are of particular interest. The spectral shape of theL = 6 spectrum (top) is steeper at the lower energies than isthe L = 5 spectrum (E−1.32 versus E−0.98; Table 1), and indeedthe Cm/CK profile (bottom) is flatter for the spectrum that iscloser to E−1 than it is for the steeper spectrum. Second, theL = 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 thanare the other profiles because of the differing magnetosphericparameters. Onemight wonder why the L = 4 spectrum retainsthe qualitative characteristics that are observed within theother two spectra. Our guess is that the L = 5 spectrum isultimately the source of the L = 4 spectrum. The generalfinding here that the KP limit has an important role in sculpt-ing spectral intensities at Earth, of course just confirms thefindings of a number of previous authors [e.g., Baker et al.,1979; Davidson et al., 1988; Summers et al., 2009].

7.2. Uranus

[29] A novel feature of Uranus is that its spin axis, at thetime of the Voyager 2 encounter, pointed roughly in thedirection of the Sun (see the book Bergstralh et al. [1991], foradditional information about all aspects of Uranus). Thatconfiguration, combined with the large ∼58° tilt of the mag-netic axis with respect to the spin axis, caused a decoupling ofthe rotational and solar wind‐driven electric fields within themagnetosphere [Selesnick and Richardson, 1986; Vasyliunas,1986; Selesnick andMcNutt, 1987] allowing the solar wind todrive dynamical processes within the system [Mauk et al.,

1987, 1994; Sittler et al., 1987; Belcher et al., 1991]. TheUranian magnetosphere has a subsolar magnetopause loca-tion of roughly 18 RU.[30] Our Uranus spectrum (Figure 8), substantially more

intense at ∼1MeV than any spectrum sampled elsewhere, wastaken at the most planetward penetration of the Voyagerspacecraft, at L = 4.73 (the spectrum is from Mauk et al.[1987], and it combines measurements from two instruments;see Krimigis et al. [1986], Stone et al. [1986], and Selesnickand Stone [1991]). The exact spectral index or shape of thespectrum above the breakpoint energy near ∼1 MeV is poorlyconstrained at this L = 4.73 position, but for our purposes hereit is sufficient to know that the spectrum drops precipitouslyabove that energy [Stone et al., 1986]. The Uranus spectrumis remarkably similar to the Earth spectra (compared insection 7.4), and clearly the Cm/CK levels (bottom) proclaimthat the KP limit must play a role in sculpting this spectrum.This finding confirms the findings of previous authors that theKP limit is matched or exceeded in this environment (Mauket al. [1987], for the nonrelativistic analysis and Summerset al. [2009], for the relativistic analysis). The big uncertaintyhere is the equatorial density. The largest density observedat the innermost L excursion of Voyager was N = 2.2 cm−3

[Belcher et al., 1991; Kurth et al., 1991] but measured welloff the magnetic equator (18°) and with a sensor that mea-

Figure 7. Repeat of the analysis given in Figure 5, but for allthree of the spectra in the top plot. Here a single density N isused for each of the L value spectra, the one recommended bySummers et al. [2009] from the work of Sheeley et al. [2001].

Figure 8. Same analysis as described in Figure 5 but for themost intense (at 1 MeV) spectrum sampled in the magneto-sphere of Uranus [Mauk et al., 1987]. Here the most likelydensity is uncertain because of the high magnetic latitudesampling (text) and so two values of the profiles are shown inred. On each of these plots the curve on the bottom labeled“∞” corresponds to density above which no change occurs inthe profile. The magnetic field is the dipole value.


9 of 19

sures only energies >10 eV. We have arbitrarily adopted N =5 as our most likely equatorial density when we comparethese profiles with other planets.

7.3. Jupiter

[31] Jupiter’s hugemagnetosphere is thought to be poweredby the rapid planetary rotation, which energizes and trans-ports the plasmas that are continuously generated in the innermagnetosphere (∼5.9RJ) by the volcanoes of theMoon Io [seeDessler, 1983; Bagenal et al., 2004]. The outward transportof the energized plasmas occurs episodically in the form ofsmall scale injections in the inner regions (<10 RJ) [Boltonet al., 1997; Kivelson et al., 1997; Thorne et al., 1997] andlarge‐scale injections in the middle regions (>9 RJ) [Mauket al., 1999]. A very novel feature of this magnetosphere isthe formation of a magnetodisc at distances >15–20 RJ, with aneutral sheet configuration that wraps all the way around theplanet, and not just on the tail side as occurs at Earth. Anothernovelty is that Jupiter has the only radiation belt that can be

sensed and imaged remotely with radio wave synchrotronradiation, resulting from a combination of high electronintensities and strong magnetic fields very close to the planet(<3RJ; reviewed by Bolton et al. [2004]). Jupiter’s subsolarmagnetopause occurs at roughly 100 RJ from the planet.[32] Two Jupiter spectra are shown in Figure 9, one (L =

8.3) obtained from an internal NASA JPL report [Garrettet al., 2003] and the other (L = 3) representing the mostintense spectra presented by Baker and Van Allen [1976] forthe inner magnetosphere as sampled by Pioneer. The L =3 directional intensity spectrum is estimated from the pub-lished omnidirectional intensity spectrum by dividing by 4pand represents electron populations that contribute to thesynchrotron radiation sensed from Earth [Bolton et al., 2004].Because the Garrett et al. [2003] information is not readilyavailable, we provide additional information in Appendix B.A significant difference between the Jupiter spectra and thespectra of Earth and Uranus is that the Jupiter spectra extendto much higher energies without a sharp spectral break(section 7.4). Again, as with the Earth and Uranus spectra, theKP 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. [2009] andbased on the observed scale height of the rotationally con-fined plasmas [Mei et al., 1995]. The Cm/CK values would besubstantially higher than shown (by a factor of ∼8/3) had weused D = L.[33] The KP analysis of the Jupiter L = 3 spectrum is not

shown here. The peak Cm/CK for the L = 3 asymptotic profileis at minimum a factor 20 below the KP limit, and thus itappears that the Jupiter L = 3 spectrum, and more specificallythe spectra of those electrons thought to participate in thegeneration of the remarkable synchrotron radiation from Ju-piter’s inner magnetosphere, are not limited by KP processes.Aswith our discussionof the EarthL= 4 spectrum (section 7.1),we hypothesize that the electron populations energized fur-ther out, possibly in the vicinity of the Jupiter L = 8 region,provides the source of the populations in the more interiorregions. This scenario was suggested by Horne et al. [2008].

7.4. Earth, Uranus, and Jupiter Compared

[34] The more intense spectra from Earth, Uranus, andJupiter have remarkable similarities (Figure 10). Theirintensities at ∼1 MeV are essentially identical. They all havespectral shapes below ∼1 MeV close to E−1. All three haverelatively flat and horizontal Cm/CK profiles between 0.1 and1MeV, with values relatively close toCm/CK = 1.We believethat the peak intensities and the shapes of these three spectraare determined and sculpted by the processes that establishthe differential KP limit.[35] Both the Earth and Uranus spectra have sharp spectral

breaks somewhat above the 1 MeV level, but the Jupiterspectrum extends to much higher energies. The energizationprocesses at Jupiter must be more extreme than they are in theother environments. But even at the higher energies theintensities are limited by the differential KP limit. By con-firming that the E−1 spectral shape is still theoretically antici-pated with the relativistic theory and showing that these threeradiation belts roughly satisfy that expectation with intensitylevels close to the KP limit, we have amplified the findings ofprevious authors, most recently Summers et al. [2009],

Figure 9. Same analysis as that performed in Figure 5 butfor a Jupiter spectrum generated by Garrett et al. [2003] (seeAppendixB) for L= 8.3RJ. The L3 spectrum in the top is fromthe work of Baker and Van Allen [1976]. Magnetic field is thedipole value, and the density used for the red curve is from thework ofMei et al. [1995]. The one difference about the Jupiteranalysis from all of the others shown in this paper is the choicefor the value of D = 3 for equation (6). Generally we use thevalue of L. For this plot we use D = 3, corresponding to thefull‐width scale height for the centrifugally confined coldplasmas as quantified by Mei et al. [1995]. The choice ofusing a narrower value of D was recommended by Summerset al. [2009]. Had L been used, the Cm/CK profiles wouldhave been roughly a factor of 8/3 higher.


10 of 19

concerning the influence that the KP processes have on theradiation belts of Earth, Uranus, and Jupiter.

8. Neptune

[36] Neptune’s magnetosphere is notable in a number ofways, including the strong tilt to its magnetic dipole (47°) andthe presence of the Moon Triton (14 RN from Neptune) thathas a retrograde orbit and is one of the two Moons in thesolar system with a robustly collisional atmosphere [seeCruikshank, 1995]. For the present discussion the mostnotable attribute of Neptune’s magnetosphere is that it is thequietest magnetosphere of the strongly magnetized planets inthe solar system. Temporal injection processes are common atEarth [e.g., DeForest and McIlwain, 1971], Jupiter [Boltonet al., 1997; Mauk et al., 1999], Saturn [Burch et al., 2005;Hill et al., 2005; Mauk et al., 2005], and Uranus [Belcheret al., 1991; Mauk et al., 1987, 1994; Sittler et al., 1987].However, no injection features were observed within Nep-ture’s magnetosphere during the one encounter by Voyager 2in 1989 [Mauk et al., 1995]. Neptune’s radiation belts are alsoremarkably symmetric (Figure 1), an attribute that was notfound within the other magnetospheres. Injections appear tobe driven by solar wind interactions at Earth and Uranus, andby strong internally generated plasma energized by rapidplanetary rotations in the cases of Jupiter and Saturn. Neptunehas neither a strong interaction with the solar wind nor does ithave a strong internal source of plasma (Triton is in the outermagnetoshere; Neptune’ subsolar magnetopause is roughly at

∼18 RN). It is of interest whether that absence of injections hasan influence on the radiation belt of Neptune. Horne et al.[2003], for example, suggested that injections are key toseeding the dramatic acceleration of Jovian radiation belts.[37] 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 tocreate this spectrum, seeKrimigis et al. [1989] and Stone et al.[1989]). Inside L = 7.4, the intensities decreased as thespacecraft plunged very close to Neptune’s cloud tops. TheKP analysis of this Neptune spectrum (Figure 11) shows thatwhile the spectrum is KP‐limited near 0.1 MeV, the Cm/CK

profile resides a factor of 30 below the KP limit near ∼1MeV.This result is distinctly different than our finding, near∼1 MeV, for the other planets presented up to this point, as ismade clear in Figure 12. Figure 12 shows that not only is theNeptune spectral intensity nearly 2 orders of magnitudebelow those of the other planets near 1 MeV, but a factor of10 separates the Neptune Cm/CK profile at ∼1 MeV fromthose of the other planets. We speculate that the differencesmight be explained by the absence of injection dynamics,unique to the Voyager 2 visit to Neptune.

9. Saturn

9.1. Saturn During the Cassini Epoch

[38] We separate our analysis of Saturn’s radiation belt intotwo epochs: the present Cassini epoch (2004 to present) andFigure 10. Here we compare spectra and KP analysis for

Earth, Uranus, and Jupiter, as extracted from Figures 5, 8,and 9. The densities used in the bottom are for the red profilesin the referenced figures.

Figure 11. Spectrum and KP analysis for Neptune in thefashion performed for Earth in Figure 5. The spectrum isfrom Mauk et al. [1995]. The most likely density is fromRichardson et al. [1995].


11 of 19

the Voyager epoch (1981). Saturn is a rotationally dominatedmagnetosphere whose plasma populations are mostly pro-vided by geyser‐like plumes emanating from the southernpole of the Moon Enceladus at a radial position of roughly4 RS from Saturn (see Dougherty et al. [2009], for reviewinformation about Saturn as diagnosed by the Cassini space-craft). This source of materials generates relatively denseclouds of gas and dust; the dust cloud was called the “E‐ring”long before the Enceladus plumes were discovered. Anothernotable feature of Saturn’s magnetosphere is that it sup-ports 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 radialposition >22 RS.[39] 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 radialL values down even to regions inside of the visible rings andsearched for the highest rates measured by electron channelsmeasuring energies near 1 MeV. Factors of ±3 variabilitywere observed in the orbit‐to‐orbit measurement in thehighest per orbit ∼1 MeV intensity. The channel rates, thespectral fit to those rates, and the resulting spectrum for theregion and time with the highest ∼1MeV intensity is shown inFigure 13. Our KP analysis of that spectrum, sampled at L =3.25, is provided in Figure 14. In this case the entire spectrumis more than a factor of 20 below the KP limit. We have

investigated the possibility that the spectral intensities near1 MeV remain relatively high with increasing L values, andthat the corresponding Cm/CK values climb upward towardthe KP limit. This investigation was motivated by the findingsat both Earth and Jupiter where high intensity spectra at L = 3and 4, respectively, resided well below the KP limit, whilesimilar intensities at higher L values (5 and 8, respectively)challenged the KP limit. Saturn during the Cassini epochturns out to be different. As one moves outward in L from theL ∼ 3.25 position, the Cm/CK values of the lower energies(0.1 MeV) do climb significantly upward, but both theintensities and the Cm/CK values at relativistic energies, andspecifically near ∼1MeV, drop precipitously, unlike the casesof Earth and Jupiter.[40] The findings regarding Saturn’s Cm/CK values, along

with qualitative features of the electron spectrogram inFigure 15 (specifically the diminution of the lower energyintensities near the center of the display) suggest that the∼1 MeV intensities are limited not by the KP limit but bylosses associated with the broad E‐ring gases and dust. The

Figure 12. Addition of our Neptune spectrum and KP anal-ysis (Figure 11) to Figure 10, which now compares four of ourfive planets.

Figure 13. (top) Rates (1/s) of Cassini MIMI instrumentelectron channels [Krimigis et al., 2004] for the measuredspectrum that has the highest rate near 1 MeV and for thoseelectron channels for which the MIMI team has the greatestconfidence (D. G. Mitchell, MIMI Instrument Scientist, pri-vate communication, 2010). Both measured rates and themodel rates for an optimized analytic spectral shape(equation (7)) are shown. The optimization routine integratesover the latest (D. G. Mitchell, private communication, 2010)channel energy bandwidths [Krimigis et al., 2004]. (bottom)The bottom shows an optimized spectrum. There may existcontributions from penetrating background, and for thepresent study, the spectrum shown should be viewed as anupper boundary.


12 of 19

pre‐Cassini modeling of Saturn’s radiation belt by Santos‐Costa et al. [2003] anticipated strong losses in the vicinityof the E‐ring. The intensities that we find here (specifically atthe ∼0.3 MeV energy highlighted by the authors) are an orderof magnitude lower than those authors anticipated, but as wediscuss in section 9.3, those intensities may be quite variable.The effects of gas and dust generally on energetic protons andelectrons have been addressed during the Cassini epoch byParanicas et al. [2007, 2008].

9.2. Five Planets Compared

[41] The uniqueness of the Saturn findings relative to theother planets is emphasized in Figure 16. At 1 MeV thespectral intensities and the Cm/CK values for Neptune andSaturn are similar, but the shapes of these entities as a functionof energy are dramatically different. That is a key reason whywe speculate that the root causes for the diminution of the

Figure 14. Kennel‐Petschek analysis of the Saturn spec-trum in Figure 13 in the fashion that was performed forEarth in Figure 5. The most likely density (red curve) isextrapolated from Persoon et al. [2005].

Figure 15. Energy‐time‐intensity spectrogram of Cassini electron intensities from the Cassini MIMIinstrument [Krimigis et al., 2004] at Saturn’s magnetic equator from (left and right) about 14 RS in to(middle) about 4 RS.

Figure 16. KP analysis of the radiation belt electron spec-tra from all five of the planets considered here: E, Earth;U, Uranus; J, Jupiter; N, Neptune and; S, Saturn. Resultsare concatenated from Figures 5, 8, 9, 11, and 14.


13 of 19

radiation belts of Neptune and Saturn are very different(dynamics versus material interactions).

9.3. Saturn During the Voyager Epoch

[42] The electron radiation belt at Saturn measured by theVoyager spacecraft has important differences with that mea-sured by the Cassini spacecraft (Figure 17). The Voyagerspectrum (1981) is highly structured and substantially moreintense than that observe by Cassini in 2009. The Voyagerpeak near ∼1 MeV was interpreted as drift resonance with theMoons in the inner magnetosphere (e.g., the MoonsEnceladus andMimas) [Krimigis et al., 1983;Van Allen et al.,1984]. Specifically, for electrons with magnetic drift speedsthat nearly match the orbital speed of a Moon, the Moon‐electron absorption probability is drastically reduced. Theshift in the peak of the spectrum with radial distance roughlymatches expectations from the resonance theory [Van Allenet al., 1984]. The KP analysis of the Voyager spectra(Figure 18; each spectrum is represented by adding togetherthree different components, including a Gaussian componentcentered near ∼1MeV; Table 1) shows that the high degree ofstructure within the Voyager spectra is allowed by the KPtheory, since the KP limit is only challenged by the spectraover a limited energy range.[43] 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 havebeen quantitatively different between the two time periods.The emission of materials from Enceladus is known to bevariable [Esposito et al., 2005; Melin et al., 2009]. It is alsopossible that there occurred a dramatic temporal event, forexample, an interplanetary shock striking and passingthrough the magnetosphere, just prior to the Voyagerencounter. There are uncertainties in understanding of theresponses of the channels of both the Cassini and Voyagerinstruments, but significant discussions within the Cassini/

Voyager instrument teams, particularly with T. P. Armstrong,a member of both teams with substantial experience inunderstanding channel responses (private communication,2010), resulted in our conclusion that very large differencesbetween the spectra observed by both Voyager and Cassinicannot be explained by uncertainties in the responses of thetwo detector systems.

10. Discussion on Time Scales

[44] Our focus here has been on magnetospheric electronspectra that are intense at relativistic energies, specificallynear 1 MeV. The Kennel‐Petschek limit has been used withother populations as well, and it is instructive to revisit thoseapplications. Of specific interest is the finding of Baker et al.[1979] that during very active magnetospheric conditions(magnetic activity index, Kp ∼ 5–6) within the geosynchronousorbit (6.67 RE), integral particle intensities can exceed theclassic KP limit by an order of magnitude. The positedexplanation is that the acceleration mechanism can be fasterthan the times associated with the losses that arise with evenstrong whistler wave scattering. The so‐called strong diffu-sion loss time associated with strong wave scattering can beestimated as the transit time of the electrons along the field

Figure 17. Comparison of the analytic Saturn electron spec-trum from Figure 13 frommeasurements made in 2009 by theCassini MIMI instrument and a spectrum sampled by theVoyager LECP instrument in 1981 [Krimigis et al., 1983].The Voyager data are channel‐average intensities (notintensities at specific energies), and the model (using para-meters in Table 1) is also integrated to represent the infor-mation as channel averages. True model intensity spectra areshown in Figure 18.

Figure 18. KP analysis for two Voyager‐epoch Saturn elec-tron spectra (VS) sampled at L = 3.6 and 2.8 RS and ourCassini Saturn spectra (CS) sampled at 3.3 RS. Because ofthe structure within the Voyager spectra, a simple fitting toequation (7) does not suffice. For each spectrum, we addtwo separate equation (7) fittings (one for low energy and onefor high) plus an additional Gaussian term centered near1 MeV (Table 1). Even for the Voyager epoch, densities fromthe work of Persoon et al. [2005] were used.


14 of 19

line (some fraction of a bounce time) times the ratio of thesolid angle contained within the loss cones and 4p. A moreaccurate calculation results in the following expression for anidealized minimum e‐folding loss time [e.g., Lyons andWilliams, 1984]:

Tm ¼ 4LRP=v

2:2 sin2½�lc�; ð9Þ

whereRP is radius of the planet (cm), v is particle speed (cm/s),andalc is the loss cone angle. For the geosynchronous regionsduring storm time, the idealized minimum e‐folding loss timeis several minutes, and more realistic minimum loss times arelikely several times the idealized value, perhaps 10 min oreven longer. And so if particle transport and accelerationtimes are faster than ∼10 min in the geosynchronous orbit,then indeed one might expect the particle accelerations tooverwhelm even the relatively fast loss times associated withthe Kennel‐Petschek type processes.[45] An evaluation of the Baker et al. [1979] 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 KPanalysis for the Earth L = 5 spectrum presented in Figure 5.

Indeed, for typical geosynchronous densities (N = 3 [Sheeleyet al., 2001]) the Baker et al. spectrum does substantiallyexceed the KP limit at minimum resonant energies <0.2MeV.There is a possibility that the density could have beenunusually low at geosynchronous orbit during this very activeperiod (N ≤ 0.3), therebymaintaining consistencywith the KPlimit. However, most likely, as suggested by Baker et al.[1979], the processes (injections) that populate this regiondo indeed act in a fashion that is faster (minutes) than the lossprocess associated with compliance to the KP limit.[46] And so while it appears that fast acting processes can

overpower the processes acting to establish the KP limit, sucha reversal of time scales did not happen for the high energy,relativistic spectra studied here. It is also true that this reversalof times scales was not observed to happen to either low‐ orhigh‐energy electrons within the environments where therelativistic electron populations were intense.

11. Summary and Conclusions

[47] We compared the most intense (near ∼1 MeV) radia-tion belt electron spectra measured within the magneto-spheres of the five strongly magnetized planets within thesolar system: Earth, Jupiter, Saturn, Uranus, andNeptune.Wecompared the spectra with each other and also with respect toexpectations from the well known Kennel and Petschek[1966] theoretical limit on radiation belt intensities. Toderive the KP expectations, we started with the relativistictheory of Summers et al. [2009] and added several refine-ments. Specifically, we developed an approach that allows usto use more realistic spectral shapes that can be accurately fitto observed spectra. Also, our approach allowed us toexamine the differential characteristics of the KP limit. Weroughly confirmed, for this relativistic theory, the finding ofSchulz and Davidson [1988], for the nonrelativistic theorythat KP‐saturated spectra take on the E−1 spectral shape. Weshow that those measured spectra that reside near the KP limitindeed are observed to have the ∼E−1 spectral shape up to anenergy where the spectrum no long resides at the level of thedifferential KP limit.[48] With our comparison of the five strongly magnetized

planets of the solar system, we find that the radiation belts ofEarth, Jupiter, and Uranus can be categorized together ashaving spectra that appear to be strongly sculpted by Kennel‐Petschek processes, whereas those of Neptune and Saturnmust be differently categorized. It is significant that Kurthand Gurnett [1991] also categorized Neptune and Saturndifferently from Earth, Jupiter, and Saturn with regard to thepresence (E, J, U) or absence (N, S) of whistler modes withsufficient intensity to cause strong particle losses. These au-thors caution that observing geometry could have a role toplay in the observed differences.[49] Our more specific conclusions about the different en-

vironments are provided here: The most intense measuredspectra at Earth, Uranus, and Jupiter reside near to the dif-ferential KP limit for energies between ∼0.1 and ∼1MeV. Thespectral shapes below ∼1 MeV are roughly E−1, as expectedby 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 pro-cesses, specifically by strong scattering by whistler modewaves as the electron intensities become too high. These

Figure 19. KP analysis of an Earth spectra sampled byBaker et al. [1979] at geosynchronous orbit on 18 Septem-ber 1976 at about 0915 UT during strong magnetic stormconditions (magnetic activity index Kp = 5–6). (top) The bluediamonds are differential channels obtained by differencingthe integral channels displayed in the paper. The fit para-meters are provided in Table 1. That spectrum is comparedwith the Fox et al. [2006] L = 5 spectrum shown in Figure 5.(bottom) The KP analysis for both the Baker et al. [1979]spectrum (black) and the Fox et al. [2006] spectrum (red).


15 of 19

findings amplify on related findings of other authors, mostrecently those of Summers et al. [2009].[50] The most intense spectrum measured in Neptune’s

magnetosphere has intensities close to the differential KPlimit near ∼0.1 MeV but has intensities a factor of 30 belowthe KP limit near ∼1 MeV. Magnetospheric processes atNeptune apparently do not robustly energize electrons to the1 MeV level to the extent that the KP limits allow. No sig-nificant dynamical events were observed within Neptune’smagnetosphere, and specifically, no injection‐like eventswere observed. The lack of dynamics may be a consequenceof the fact that Neptune is coupled only weakly to the solarwind, and there is no strong internal source of plasma deepwithin the magnetosphere such as that which occurs at bothJupiter and Saturn. We speculate that such injections areneeded to robustly energize electrons to the higher energies.This speculation is stimulated in part by Horne et al. [2008],who suggested that injection‐like events are critical to ex-plaining the high electron energies and intensities at Jupiter.[51] The most intense Cassini‐epoch electron spectrum at

Saturn is well below the differential KP limit at all measuredenergies. We speculate that the Saturn electron intensities arestrongly suppressed by the gases and dust populationsreleased by the plumes of Enceladus. The difference betweenthe suppression seen at Neptune and that seen at Saturn is thatat Neptune the low‐energy (0.1 MeV) intensities are high andthe high energies (1 MeV) are low. This finding is inconsis-tent with suppression arising from interactions with materialssuch as gas and dust. Gas and dust are expected to suppressthe low energies to a greater extent than they do the higherenergies. The Cassini (2005) spectra of Saturn qualitativelymatch expectations from interactions with such materials.The Voyager (1981) electron spectra are significantly dif-ferent from the Cassini (2005) spectra, showing higherintensities and substantial energy structure. We speculate thatthe gas and dust environment was different during the Voy-ager encounter than it is during the Cassini observations. Theenergy structure is allowed since the Voyager spectra do notchallenge the differential KP limit at most energies. Thepresence of the E‐ring of particulates was known prior toCassini, and modeling has been used to estimate the con-sequences of the E‐ring on radiation belt intensities.[52] The differential Kennel‐Petschek limit of relativistic

electron populations is not observed to be significantly ex-ceeded by the spectral intensities observed within any of thestrongly magnetized planets of the solar system. Fordynamically active magnetospheres, absent absorbing mate-rials, the KP theory is predictive of the shape of the spectra atenergies below a cutoff, specifically near ∼1 MeV for mag-netospheres studied. It remains an open question as towhether the characteristics of whistler mode emissions sup-port the simplicity of the KP theory as utilized here. It is likelythat full closure involves interactions and wave propagationsoccurring within relatively large volumes of space and notjust within the confines of single flux tubes.

Appendix A

[53] Provided here are expressions for the temporal growthrate of whistler waves propagating parallel to the magneticfield and properly taking account of the condition that theelectrons that provide the power for the growth extend in

energy to relativistic values. These expressions are in the formprovided by Summers et al. [2009] (Copyright by theAmerican Geophysical Union, 2009) who modified themfrom the expressions by Xiao et al. [1998]. An earlier treat-ment was published by Liemohn [1967]. The linear e‐foldingtemporal growth rate is

� ¼ �!2pe�rel Arel � Acf g

2!r þ !2pe Wej j=ð!r � Wej jÞ2

h i ; ðA1Þ

where g is the e‐folding temporal growth rate (1/s), wpe isplasma frequency [√(4pNe2/me)], wr is resonant frequency(radians/s), We is the electron gyrofrequency [eB/(mec)], N isdensity (1/cm3), e is electron charge (statcoulombs), me iselectron mass (g), c is speed of light (cm/s), B is equatorialmagnetic field strength (Gauss), and

�rel ¼ �með!r � Wej jÞN0k

�Z10

dp?DR

p2?@f

@p?

� �pk¼pR

; ðA2Þ

defined as the fraction of the energetic electron distributionnear cyclotron resonance,

Arel ¼k

með!r� Wej jÞR10

dp?p2?DR�R

� p? @f@pk

� pk@f@p?

h ipk¼pRR1

0

dp?DR

p2? � @f@p?

h ipk¼pR

; ðA3Þ

the relativistic distribution pitch angle anisotropy factor,

�R ¼�1þ ck

!r

� �ck!r

� �2�1

� �1þ p2?

ðme2Þ2� �

� !rWej j

� �2þ1

� �1=2ck!r

� �2�1

� �� !r

Wej j� � ;

ðA4Þ

the relativistic Lorentz factor at resonant energies (not to beconfused with the wave linear growth rate g in equation (A1)and in the main body of this paper),

pR ¼ me

kð�R!r � Wej jÞ; ðA5Þ

the parallel momentum for resonant particles,

DR ¼ 1� !rpRmec2k�R

; ðA6Þ

Ac ¼ !r

ð Wej j � !rÞ ; ðA7Þ

the latter being the minimum required anisotropy factor forwave instability, and

c2k2

!2r

¼ 1þ !2pe

!rð Wej j � !rÞ ; ðA8Þ


16 of 19

the wave dispersion relation for parallel propagating whis-tler waves. In these expressions, k is the wave spatial fre-quency (2p/l where l is wavelength in centimeter), pk ismomentum parallel to the magnetic field, and p? ismomentum perpendicular to the magnetic field. The lastexpression (A8) is the wave dispersion relationship, fromwhich the wave group velocity can be derived by differenti-ation (Vg = ∂w/∂k). The wave group velocity, needed to solveequation (6), is

Vg ¼ 2c2k Wej jð1� !r= Wej jÞ2ð!2

pe þ 2 Wej j!rð1� !r= Wej jÞ2 : ðA9Þ

Appendix B

[54] The Jupiter spectrum for L = 8.3 in section 7.3 wastaken from the NASA Jet Propulsion Laboratory publication:Garrett et al. [2003]. Because this report, while available, isnot as readily available as other papers, we provide a fewdetails here. Jupiter spectra between ∼8 and ∼16 RJ weredeveloped by performing radiation ray tracing calculations onthe Galileo Energetic Particle Detector [Williams et al., 1992]to better interpret the channel rate information provided by theinstrument. That information was combined with the higherenergy (31 MeV) channel information from Pioneer [VanAllen et al., 1974, 1975], which sampled Jupiter many yearsbefore Galileo did. The results were captured in the form of ananalytic fit of form similar to that used by Baker and VanAllen [1976],

I1

cm2 s sr keV

� ¼ J0 EMeVð Þ�A 1þ EMeV

E0

� �B

: ðB1Þ

Table B1 provides the fitting parameters for each value of theradial distance L, and Figure B1 shows a plot of all of thespectra generated. The innermost spectrum was sampled atL = 8.25 RJ, and that is the one that is most intense at 1 MeV,and the one used here for our KP analysis.

[55] Acknowledgments. We appreciate substantial discussions withmembers of the Cassini Magnetosphereic Imaging Instrument (MIMI) teamand the Voyager Low Energy Charge Particle (LECP) instrument teamregarding the relative responses of the two instruments to radiation electronsin 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 studyof Fox et al. [2006] and updated here were taken by the Medium Electrons Ainstrument (MEA; A. L. Vampola and E. G. Mullen, Principal Investigators)and the High Energy Electron Fluxmeter instrument (HEEF; E. G. Mullenand D. H. Brautigam, Principal Investigators) on the Combined Release andRadiation Effects Satellite (CRRES) and obtained from the National SpaceScience 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 ofthese data.[56] Masaki Fujimoto thanks the reviewers for their assistance in evalu-

ating this paper.

ReferencesBagenal, F., T. E. Dowling, and W. B. McKinnon (Eds.) (2004), Jupiter —The Planet, Satellites and Magnetosphere, Cambridge Univ. Press, Cam-bridge, UK.

Baker, D., and J. Van Allen (1976), Energetic electrons in the JovianMagnetosphere, J. Geophys. Res., 81(4), 617–632, doi:10.1029/JA081i004p00617.

Baker, D. N., P. Stauning, E. W. Hones Jr., P. R. Higbie, and R. D. Belian(1979), Strong electron pitch angle diffusion observed at geostationaryorbit, Geophys. Res. Lett., 6(3), 205, doi:10.1029/GL006i003p00205.

Barbosa, D. D., and F. V. Coroniti (1976), Relativistic electrons and whis-tlers in Jupiter’s magnetosphere, J. Geophys. Res., 81(25), 4531–4536,doi:10.1029/JA081i025p04531.

Belcher, J. W., R. L. McNutt Jr., J. D. Richardson, R. S. Selesnick, E. C.Sittler Jr., and F. Bagenal (1991), The plasma environment of Uranus, inUranus, edited by J. T. Bergstralh et al., pp. 780–829, Univ. of ArizonaPress, Tucson, Ariz.

Bergstralh, J. T., E. D. Miner, and M. S. Mattheews (1991), Uranus,pp. 410–465, Univ. of Arizona Press, Tucson, Ariz.

Bolton, S. J., R. M. Thorne, D. A. Gurnett, W. S. Kurth, and D. J. Williams(1997), Enhanced whistler mode emissions: Signatures of interchangemotion in the Io torus, Geophys. Res. Lett., 24(17), 2123–2126,doi:10.1029/97GL02020.

Bolton, S. J., R. M. Thorne, S. Bourdarie, I. Depater, and B. Mauk (2004)Jupiter’s inner radiation belts, in Jupiter, edited by F. Bagenal et al., p. 671,Cambridge Planet. Sci. Ser., Cambridge Univ. Press, Cambridge, UK.

Burch, J. L., J. Goldstein, T. W. Hill, D. T. Young, F. J. Crary, A. J. Coates,N. André, W. S. Kurth, and E. C. Sittler (2005), Properties of localplasma injections in Saturn’s magnetosphere, Geophys. Res. Lett., 32,L14S02, doi:10.1029/2005GL022611.

Burton, R. K., and R. E. Holzer (1974), The origin and propagation of chorusin the outer magnetosphere, J. Geophys. Res., 79(7), 1014–1023,doi:10.1029/JA079i007p01014.

Figure B1. Jupiter electron spectra plotted from parameters(Table B1) given by Garrett et al. [2003].

Table B1. Spectral Parameters for Equation (B1) for Jupiter FromGarrett et al. [2003]

L Jo A B E0

8.25 4720 0.753 1.86 2.268.75 3850 0.961 2.24 4.739.25 2970 1.2 2.07 5.999.75 2160 1.42 2.02 9.2510.25 2103 1.39 2.58 11.610.75 2300 1.38 2.27 6.8911.25 2080 1.36 2.13 5.0511.75 1540 1.54 2.11 6.0412.25 2420 1.33 2.33 3.3612.75 1380 1.65 2.51 6.7913.25 550 2.22 10.1 14513.75 465 2.15 5.26 49.914.25 439 2.08 5 37.914.75 469 1.92 2.21 7.5215.25 299 2.15 1.65 7.5115.75 187 2.26 3.68 34.916.25 179 1.98 15 66


17 of 19

Cruikshank, D. P. (1995), Neptune and Triton, University of Arizona Press,Tucson, Arizona.

Davidson, G. T., P. C. Filbert, R. W. Nightingale, W. L. Imhof, J. B. Reagan,and E. C. Whipple (1988), Observations of intense trapped electron fluxesat synchronous altitudes, J. Geophys. Res., 93(A1), 77–95, doi:10.1029/JA093iA01p00077.

DeForest, S., and C. McIlwain (1971), Plasma clouds in the magnetosphere,J. Geophys. Res., 76(16), 3587–3611, doi:10.1029/JA076i016p03587.

Dessler, A. J. (1983), The Physics of the Jovian Magnetosphere, Cam-bridge Planet. Sci. Ser., Cambridge Univ. Press, Cambridge, UK.

Dougherty, M. K., L. W. Esposito, and S. M. Krimigis (2009), SaturnFrom Cassini‐Huygens, Springer, Dordrecht.

Esposito, L. W., et al. (2005), Ultraviolet imaging spectroscopy shows anactive Saturnian system, Science, 307(5713), 1251–1255, doi:10.1126/science.1105606.

Fox, N. J., B. H. Mauk, and J. B. Blake (2006), Role of nonadiabatic pro-cesses in the creation of the outer radiation belts, Geophys. Res. Lett., 33,L18108, doi:10.1029/2006GL026598.

Garrett, H. B., I. Jun, J. M. Ratliff, R. W. Evans, G. A. Clough, and R. W.McEntire (2003), Galileo Interim Radiation Electron Model, JPL Publ.03‐006, 72 pp., Jet Propulsion Lab., California Inst. of Technol., Pasadena,Calif.

Hill, T. W., et al. (2005), Evidence for rotationally driven plasma transportin Saturn’s magnetosphere, Geophys. Res. Lett. , 32 , L14S10,doi:10.1029/2005GL022620.

Horne, R. B., R. M. Throne, N. P. Meredith, and R. R. Anderson (2003),Diffuse auroral electron scattering by electron cyclotron harmonic andwhistler mode waves during an isolated substorm, J. Geophys. Res.,108(A7), 1290, doi:10.1029/2002JA009736.

Horne, R. B., R. M. Thorne, S. A. Glauert, J. D. Menietti, Y. Y. Shprits,and D. A. Gurnett (2008), Gyro‐resonant electron acceleration at Jupiter,Nat. Phys., 4, 301–304, doi:10.1038/nphys897.

Kennel, C. F., and H. E. Petschek (1966), Limit on stably trapped particlefluxes, J. Geophys. Res., 71(1), 1–28, doi:10.1029/JZ071i001p00001.

Kivelson, M. G., and C. T. Russell (1995), Introduction to Space Physics,588 pp., Cambridge Univ. Press, New York.

Kivelson, M. G., K. K. Khurana, C. T. Russell, and R. J. Walker (1997),Intermittent short‐duration magnetic field anomalies in the Io torus: Evi-dence for plasma interchange?, Geophys. Res. Lett., 24(17), 2127–2130,doi:10.1029/97GL02202.

Krimigis, S., J. Carbary, E. Keath, T. Armstrong, L. Lanzerotti, and G.Gloeckler (1983), General characteristics of hot plasma and energeticparticles in the Saturnian magnetosphere: Results from the VoyagerSpacecraft, J. Geophys. Res., 88(A11), 8871–8892, doi:10.1029/JA088iA11p08871.

Krimigis, S. M., T. P. Armstrong, W. I. Axford, A. F. Cheng, G. Gloeckler,D. C. Hamilton, E. P. Keath, L. J. Lanzerotti, and B. H. Mauk (1986),The magnetosphere of Uranus: Hot plasma and radiation environment,Science, 233, 97.

Krimigis, S. M., et al. (1989), Hot plasma and energetic particles in neptune’smagnetosphere, Science, 246(4936), 1483–1489, doi:10.1126/sci-ence.246.4936.1483

Krimigis, S. M., et al. (2004), Magnetospheric Imaging Instrument (MIMI)on the Cassini Mission to Saturn/Titon, Space Sci. Rev., 114, 233–329.

Krimigis, S. M., et al. (2005), Dynamics of Saturn’s magnetosphere fromMIMI during Cassini’s orbital insertion, Science, 307, 1270–1273,doi:10.1126/science.1105978.

Kurth, W. S., and D. A. Gurnett (1991), Plasma waves in planetary magne-tospheres, J. Geophys. Res., 96(suppl.), 18,977–18,991.

Kurth, W. S., D. A. Gurnett, F. L. Scarf, and F. V. Coroniti (1991), Wave‐particle interactions in the magnetosphere of Uranus, in Uranus, editedby J. T. Bergstralh et al., p. 926, Univ. of Arizona Press, Tucson, Ariz.

LeDocq, M. J., D. A. Gurnett, and G. B. Hospodarsky (1998), Chorussource locations from VLF Poynting flux measurements with the Polarspacecraft, Geophys. Res. Lett., 25(21), 4063–4066, doi:10.1029/1998GL900071.

Liemohn, H. B. (1967), Cyclotron‐resonance amplification of VLF andULF whistlers, J. Geophys. Res. , 72(1), 39–55, doi:10.1029/JZ072i001p00039.

Lyons, L. R., and D. J. Williams (1984), Quantitative Aspects of Magneto-spheric Physics, D. Reidel, Dordrecht, Boston.

Mauk, B., E. Keath, and S. Krimigis (1994), Unusual Satellite‐Electron sig-nature within the uranian magnetosphere and its implications regardingwhistler electron loss processes, J. Geophys. Res., 99(A10), 19,441–19,450, doi:10.1029/94JA01658.

Mauk, B., D. Williams, R. McEntire, K. Khurana, and J. Roederer (1999),Storm‐like dynamics of Jupiter’s inner and middle magnetosphere,J. Geophys. Res., 104(A10), 22,759–22,778, doi:10.1029/1999JA900097.

Mauk, B. H., S. M. Krimigis, E. P. Keath, A. F. Cheng, T. P. Armstrong,L. J. Lanzerotti, G. Gloeckler, and D. C. Hamilton (1987), The hotplasma and radiation environment of the Uranian magnetosphere, J. Geo-phys. Res., 92(A13), 15,283, doi:10.1029/JA092iA13p15283.

Mauk, B. H., S. M. Krimigis, A. F. Cheng, and R. S. Selesnick (1995),Energetic particles and hot plasmas of Neptune, in Neptune and Triton,edited by D. P. Cruikshank, pp. 169–232, Univ. of Arizona Press, Tucson,Ariz.

Mauk, B. H., D. G. Mitchell, R. W. McEntire, C. P. Paranicas, E. C.Roelof, D. J. Williams, S. M. Krimigis, and A. Lagg (2004), Energeticion characteristics and neutral gas interactions in Jupiter’s magneto-sphere, J. Geophys. Res., 109, A09S12, doi:10.1029/2003JA010270.

Mauk, B. H., et al. (2005), Energetic particle injections in Saturn’s magne-tosphere, Geophys. Res. Lett., 32, L14S05, doi:10.1029/2005GL022485.

Mauk, B. H., et al. (2009), Fundamental plasma processes in Saturn’s mag-netosphere, in Saturn From Cassini Huygens, edited by M. Dougherty etal., pp. 281–331, Springer, Dordrecht.

Mei, Y., R. Thorne, and F. Bagenal (1995), Analytical model for the densitydistribution in the Io plasma torus, J. Geophys. Res., 100(A2), 1823–1828,doi:10.1029/94JA02359.

Melin, H., D. E. Shemansky, and X. Liu (2009), The distribution of atomichydrogen and oxygen in the magnetosphere of Saturn, Planet. Space Sci.,57, 1743–1753, doi:10.16/j.pss.2009.04.014.

Meredith, N. P., R. B. Horne, R. M. Thorne, and R. R. Anderson (2009),Survey of upper band chorus and ECH waves: Implications for the dif-fuse aurora, J. Geophys. Res., 114, A07218, doi:10.1029/2009JA014230.

Mitchell, D. G., J. F. Carbary, S. W. H. Cowley, T. W. Hill, and P. Zarka(2009), The dynamics of Saturn’s magnetosphere, in Saturn FromCassini Huygens, edited by M. Dougherty et al., pp. 257–280, Springer,Dordrecht.

Miyoshi, Y., K. Sakaguchi, K. Shiokawa, D. Evans, J. Albert, M. Connors,and V. Jordanova (2008), Precipitation of radiation belt electrons byEMIC waves, observations from ground and space, Geophys. Res. Lett.,35, L23101, doi:10.1029/2008GL035727.

Paranicas, C., et al. (2007), Energetic electrons injected into Saturn’s neutralgas cloud, Geophys. Res. Lett., 34, L02109, doi:10.1029/2006GL028676.

Paranicas, C., et al. (2008), Sources and losses of energetic protons in Sa-turn’s magnetosphere, Icarus, 197, 516–525.

Persoon, A. M., D. A. Gurnett, W. S. Kurth, G. B. Hospodarsky, J. B.Groene, P. Canu, and M. K. Dougherty (2005), Equatorial electron den-sity measurements in Saturn’s inner magnetosphere, Geophys. Res. Lett.,32, L23105, doi:10.1029/2005GL024294.

Richardson, J. D., et al. (1995), The plasma environment of Neptune, inNeptune and Triton, edited by D. P. Cruikshank, p. 279, Univ. of ArizonaPress, Tucson, Ariz.

Sandanger, M., F. Søraas, K. Aarsnes, K. Oksavik, and D. S. Evans (2007),Loss of relativistic electrons: Evidence for pitch angle scattering by elec-tromagnetic ion cyclotron waves excited by unstable ring current protons,J. Geophys. Res., 112, A12213, doi:10.1029/2006JA012138.

Santolík, O., D. A. Gurnett, J. S. Pickett, M. Parrot, and N. Cornilleau‐Wehrlin (2003), Spatio‐temporal structure of storm time chorus, J.Geophys.Res., 108(A7), 1278, doi:10.1029/2002JA009791.

Santolík, O., J. S. Pickett, D. A. Gurnett, J. D. Menietti, B. T. Tsurutani,and O. Verkhoglyodova (2010), Survey of Poynting flux of whistler modechorus in the outer zone, J. Geophys. Res., 115, A00F13, doi:10.1029/2009JA014925.

Santos‐Costa, D., M. Blanc, S. Maurice, and S. J. Bolton (2003), Model-ing the electron and proton radiation belts of Saturn, Geophys. Res.Lett., 30(20), 2059, doi:10.1029/2003GL017972.

Schulz, M., and G. T. Davidson (1988), Limiting energy spectrum of a satu-rated radiation belt, J. Geophys. Res., 93(A1), 59, doi:10.1029/JA093iA01p00059.

Schulz, M., and L. J. Lanzerotti (1974), Particle Diffusion in the RadiationBelts, Springer, Berlin.

Selesnick, R. S., and R. L. McNutt Jr. (1987), Voyager 2 plasma ion obser-vations in the magnetosphere of Uranus, J. Geophys. Res., 92(A13),15,249, doi:10.1029/JA092iA13p15249.

Selesnick, R. S., and J. D. Richardson (1986), Plasmasphere formation inarbitrarily oriented magnetospheres, Geophys. Res. Lett., 13(7), 624–627,doi:10.1029/GL013i007p00624.

Selesnick, R., and E. Stone (1991), Energetic electrons at Uranus: Bimodaldiffusion in a satellite limited radiation belt, J. Geophys. Res., 96(A4),5651–5665, doi:10.1029/90JA02696.

Sheeley, B. W., M. B. Moldwin, H. K. Rassoul, and R. R. Anderson(2001), An empirical plasmasphere and trough density model: CRRESobservations, J. Geophys. Res., 106(A11), 25,631–25,641, doi:10.1029/2000JA000286.

Shprits, Y. Y., L. Chen, and R. M. Thorne (2009), Simulations of pitchangle scattering of relativistic electrons with MLT dependent diffusion


18 of 19

coeff ic ients , J . Geophys . Res. , 114 , A03219, doi :10.1029/2008JA013695.

Sittler, E. C., Jr., K. W. Ogilvie, and R. S. Selesnick (1987), Survey of elec-trons in the Uranian magnetospshere, Voyager 2 observations, J. Geophys.Res., 92(A13), 15,263–15,281, doi:10.1029/JA092iA13p15263

Stone, E. C., J. F. Cooper, A. C. Cummings, F. B. McDonald, J. H. Trainor,N. Lal, R. McGuire, and D. L. Chenette (1986), Energetic charged par-ticles in the Uranian magnetosphere, Science, 233, 93.

Stone, E. C., A. C. Cummings, M. D. Looper, R. S. Selesnick, N. Lai, F. B.McDoonald, J. H. Trinor, and D. L. Chenette (1989), Energetic chargedparticles in the magnetosphere of Neptune, Science, 246, 1489–1494.

Summers, D., B. Ni, and N. P. Meredith (2007), Timescales for radiation beltelectron acceleration and loss due to resonant wave‐particle interactions:2. Evaluation for VLF chorus, ELF hiss, and electromagnetic ion cyclotronwaves, J. Geophys. Res., 112, A04207, doi:10.1029/2006JA011993.

Summers, D., R. Tang, and R. M. Thorne (2009), Limit on stably trappedparticle fluxes in planetary magnetospheres, J. Geophys. Res., 114,A10210, doi:10.1029/2009JA014428.

Thorne, R. M., T. P. Armstrong, S. Stone, D. J. Williams, R. W. McEntire,S. J. Bolton, D. A. Gurnett, and M. G. Kivelson (1997), Galileo evidencefor rapid interchange transport in the Io torus, Geophys. Res. Lett.,24(17), 2131–2134, doi:10.1029/97GL01788.

Tsyganenko, N. A., and M. I. Sitnov (2007), Magnetospheric configura-tions from a high‐resolution data‐based magnetic field model, J. Geo-phys. Res., 112, A06225, doi:10.1029/2007JA012260.

Ukhorskiy, A. Y., Y. Y. Shprits, B. J. Anderson, K. Takahashi, and R. M.Thorne (2010), Rapid scattering of radiation belt electrons by storm timeEMIC waves, Geophys. Res. Lett . , 37 , L09101, doi:10.1029/2010GL042906.

Van Allen, J. A. (1984), Energetic particles in the inner magnetosphere ofSaturn, in Saturn, edited by T. Gehrels et al., pp. 281–317, Univ. of ArizonaPress, Tucson, Ariz.

Van Allen, J. A., D. N. Baker, B. A. Randall, M. F. Thomsen, D. D. Sentman,and H. R. Flindt (1974), Energetic electrons in the magnetosphere of Jupi-ter, Science, 183, 309–311.

Van Allen, J. A., B. A. Randall, D. N. Baker, C. K. Goertz, D. D. Sentman,M. F. Thomsen, and H. R. Flindt (1975), Pioneer 11 observations ofenergetic particles in the jovian magnetosphere, Science, 188, 459–462.

Vasyliunas, V. M. (1986), The convection‐dominated magnetosphere ofUranus, Geophys. Res. Let t . , 13 (7) , 621–623, doi :10.1029/GL013i007p00621.

Williams, D. J., R. W. McEntire, S. Jaskulek, and B. Wilken (1992), TheGalileo energetic particles detector, Space Sci. Rev., 60, 385–412.

Xiao, F., R. M. Thorne, and D. Summers (1998), Instability of electromag-netic R‐mode waves in a relativistic plasma, Phys. Plasmas, 5, 2489.

N. J. Fox and B. H. Mauk, Johns Hopkins University Applied PhysicsLaboratory, 11100 Johns Hopkins Rd., Laurel, MD 20723, USA.([email protected]; [email protected])


19 of 19

Electron radiation belts of the solar system

Documents

Transcript of Electron radiation belts of the solar system