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Keywords:

  • radiation belts;
  • planetary magnetospheres;
  • trapped energetic particles

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] We reexamine the Kennel-Petschek concept of self-limitation of stably trapped particle fluxes in a planetary magnetosphere. In contrast to the original Kennel-Petschek formulation, we carry out a fully relativistic analysis. In addition, we replace the wave reflection criterion in the Kennel-Petschek theory by the condition that the limit on the stably trapped particle flux is attained in the steady state condition of marginal stability when electromagnetic waves generated at the magnetic equator acquire a specified gain over a given convective growth length. We derive relativistic formulae for the limiting electron integral and differential fluxes for a general planetary radiation belt at a given L shell. The formulae depend explicitly on the spectral index and pitch angle index of the assumed particle distribution and on the ratio of the electron gyrofrequency to the electron plasma frequency. We compare the theoretical limits on the trapped flux with observed energetic electron fluxes at Earth, Jupiter, and Uranus.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] Since the discovery in 1958 of the Earth's radiation belts by James Van Allen and colleagues using Explorer 1 [Van Allen et al., 1958], radiation belt physics has been an active area of theoretical and experimental research. Extensive progress has been made in understanding the basic physical processes occurring in the radiation belts of Earth and the other magnetized planets [e.g., Roederer, 1970; Schulz and Lanzerotti, 1974; Lyons and Williams, 1984; Schulz, 1991; Lemaire et al., 1996; Santos-Costa et al., 2003; Bolton et al., 2004], though many unsolved problems remain. For Earth, current problems include quantifying the variability of the energetic electron flux in the outer radiation belt and understanding the physical processes that control the transport, acceleration and loss of radiation belt electrons [Summers et al., 2007a, 2007b; Li et al., 2007; Baker and Kanekal, 2008; Hudson et al., 2008].

[3] It has been recognized since the very early studies [Dungey, 1963; Cornwall, 1964; Kennel and Petschek, 1966; Kennel, 1969; Roberts, 1969] that electromagnetic waves can scatter magnetospheric electrons into the loss cone by pitch angle diffusion and hence cause electron precipitation into the atmosphere. Dungey [1963] and Cornwall [1964] considered precipitation of radiation belt electrons by externally generated whistler mode waves, for instance by lightning-generated whistlers. Kennel and Petschek [1966] showed that a trapped anisotropic electron population, if sufficiently anisotropic, will itself generate whistler mode waves which will scatter electrons and lead to precipitation. This process in which the pitch angle diffusion is self-excited is inherently nonlinear since the trapped particles act back on themselves by means of the waves. Nevertheless, Kennel and Petschek [1966] assumed that the whistler mode turbulence is weak and were able to use a linear perturbation technique, and, in a seminal paper, obtained the widely quoted result that there exists an upper limit on the integral omnidirectional flux of particles that can be stably trapped in a dipole field; if the limiting flux is exceeded then increased particle precipitation results. Kennel and Petschek [1966] argued that the limiting flux is attained in the steady state situation of marginal stability when the wave energy lost due to internal reflection is balanced by the increase in wave energy due to convective wave growth. We provide a brief technical derivation of the Kennel-Petschek flux limit at the end of our paper in Appendix B. The Kennel-Petschek limit is not an actual physical limit on the flux of particles that can be stably trapped in a planetary magnetosphere, and, further, the limit applies only to the case of weak diffusion wherein the loss cone is partially empty. While a sufficiently intense particle source will lead to an increase in trapped flux beyond the Kennel-Petschek limit and to an increased rate of precipitation, the precipitation rate itself cannot exceed the strong diffusion rate [Kennel, 1969]. Therefore, in principle, if the particle source population continues to increase beyond that required to produce strong diffusion, then the trapped particle flux could increase indefinitely. In general, a trapped particle distribution, in addition to being subject to the self-induced or nonparasitic pitch angle scattering postulated in the Kennel and Petschek [1966] scenario, could also be subject to parasitic scattering by externally generated waves. For instance, energetic electrons in the slot region can be scattered by plasmaspheric hiss [Lyons et al., 1972] while those in the Earth's outer zone can be subject to intense precipitation due to scattering by electromagnetic ion cyclotron waves, under appropriate conditions [Thorne and Kennel, 1971; Lyons and Thorne, 1972; Summers and Thorne, 2003]. The integral flux of stably trapped particles is expected to be much less than the Kennel-Petschek limit if significant precipitation losses occur due to parasitic pitch angle scattering. Other particle loss processes not accounted for in the Kennel and Petschek [1966] formulation include loss due to sweeping by planetary satellites or to absorption by dust clouds each of which occurs, for example, at Saturn [e.g., Santos-Costa et al., 2003]. Jovian radiation belt particles likewise suffer losses due to satellites and rings [e.g., Santos-Costa and Bourdarie, 2001; Sicard and Bourdarie, 2004].

[4] The self-limiting flux concept has been addressed in various studies. Schulz [1974] incorporated the Kennel-Petschek limit on the trapped flux in a nonlinear phenomenological model of particle saturation in the Earth's outer radiation belt. Schulz and Davidson [1988] extended the Kennel-Petschek theory to derive a limiting form of the particle differential flux, which at large energy E varies asymptotically as 1/E. Comparisons of the Kennel-Petschek integral flux limit with observed particle fluxes in the Earth's outer zone have been made by Baker et al. [1979], Davidson et al. [1988], and other authors, as well as by Kennel and Petschek [1966]. Observed values of the electron flux in the region 5 ≤ L ≤ 10, in particular at geosynchronous orbit L = 6.6, were found to be generally below, and often near, the Kennel-Petschek limit. Typically, observed values were within a factor of 2 or 3 of the Kennel-Petschek limit, though some of the data suggest that electron fluxes exhibit greater temporal variability than can be accounted for by the steady state Kennel-Petschek theory. Baker et al. [1979] present data at Earth's geosynchronous orbit that provide evidence for an experimental upper limit to the electron integral flux that well exceeds the Kennel-Petschek limit. Barbosa and Coroniti [1976], assuming an ultrarelativistic electron population, calculated the Kennel-Petschek flux limit for Jupiter's inner magnetosphere (4 ≤ L ≤ 12), and concluded that within theoretical and experimental uncertainties observed fluxes were near the stably trapped limit. Using data from the LECP instrument on the Voyager 2 spacecraft, Mauk et al. [1987] found that measured electron integral fluxes in the inner magnetosphere of Uranus near L = 4.73 were in excess of the Kennel-Petschek limit by an order of magnitude. At the same time, intense whistler mode wave activity was observed by the Voyager 2 plasma wave experiment [Gurnett et al., 1986].

[5] The Kennel-Petschek concept of self-limitation of radiation belt particle fluxes remains important in magnetospheric physics, though the conditions relating to its validity are imperfectly understood and warrant further investigation. Almost all previous investigations of flux limiting, including the original work by Kennel and Petschek [1966], have used the nonrelativistic approximation. Here for the first time, we present an exact, fully relativistic analysis of the Kennel-Petschek trapping limit. When relativistic effects are included, wave growth properties can be significantly different from those derived from a nonrelativistic treatment, for electron thermal energies above 100 keV, especially in regions of low plasma density [e.g., Xiao et al., 1998]. We present in section 2 our derivation of the limiting electron integral flux in a fully relativistic regime. To achieve this, we derive the temporal growth rate for field-aligned whistler mode waves in a relativistic plasma modeled by our adopted distribution function (equation (1)), and we apply the condition that waves generated at the magnetic equator acquire a specified convective gain over a given growth region. It is important to note that we apply the latter condition to replace the condition used by Kennel and Petschek [1966] involving wave reflection. The idea of Kennel and Petschek [1966] that wave packets generated near the equator are subsequently reflected back from higher latitudes has not been borne out by observations. In section 3 we briefly examine in the context of the present study the aforementioned limiting energy spectrum derived by Schulz and Davidson [1988]. In section 4, we apply our trapping limits to Earth, Jupiter and Uranus, and compare our results with observed energetic electron fluxes. Finally, in section 5 we summarize the results and state our conclusions.

2. Stably Trapped Flux Limit

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

2.1. Particle Distribution Function

[6] To model the (equatorial) electron phase-space density we use the distribution function

  • equation image

with p2 = p2 + p2, where p = γmev and p = γmev are the components of relativistic momentum p = γmev, me is the electron rest mass, v is the electron velocity with components v and v respectively parallel and perpendicular to the ambient magnetic field, and γ = (1 − v2/c2)−1/2 = (1 + p2/(mec)2)1/2, with v2 = v2 + v2, and c is the speed of light; α = tan−1(p/p) is the electron pitch angle; l is the spectral index and s is the pitch angle index; p* is a minimum value of the momentum to be specified below. Since the electron differential number flux J is given by

  • equation image

then J(p*) in equation (1) is the perpendicular (α = π/2) differential number flux at p = p*. We introduce the electron kinetic energy E, given by E/(mec2) = γ − 1, which satisfies the relation

  • equation image

Then, by using equations (1) and (2), the electron integral omnidirectional flux,

  • equation image

is given by

  • equation image

where

  • equation image

and Γ is the gamma function.

[7] Barbosa and Coroniti [1976] used a particle distribution of the form (1) in their study of relativistic electrons and whistlers in Jupiter's magnetosphere, and Schulz and Lanzerotti [1974] used a nonrelativistic form of equation (1) in their determination of the limit on trapped particle flux in the Earth's magnetosphere. In the nonrelativistic case, γ = 1, p = mev, E = (mev2)/2, and then substitution of distribution (1) into equation (4) gives

  • equation image

where E* = p*2/(2me).

[8] Taking into consideration the form of the distribution function (1) and relation (7), we restrict our attention to the physically feasible parameter values l > 1 and s > 0. In the top panel of Figure 1, for l = 2 and s = 0.5, we show typical contours of constant phase-space density for distribution (1); in the bottom panel we illustrate the pitch angle variation of the distribution for different values of s.

image

Figure 1. (top) Contours of constant phase space density f = constant, where f is given by equation (1), for s = 0.5, l = 2; the contours are labeled by values of λ where [(p/(mec))2 + (p/(mec))2]l+s+1 = λ[p/(mec)]2s. (bottom) The pitch angle variation (at a fixed energy) for distribution (1), for the indicated values of the pitch angle index s.

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2.2. Wave Growth Rates

[9] In Appendix A we derive the temporal growth rate ωi for electromagnetic R-mode waves propagating parallel to a uniform magnetic field, corresponding to the anisotropic electron distribution (1) in the presence of a cold background electron population. In the fully relativistic case, we find

  • equation image

where N0 is the cold electron number density,

  • equation image

is a cold plasma parameter where ∣Ωe∣ is the electron gyrofrequency and ωpe is the plasma frequency (each defined in Appendix A), and

  • equation image

where ω is the (real) wave frequency and k is the (real) wave number. I1, I2 are the integrals given by

  • equation image

and

  • equation image

where

  • equation image
  • equation image
  • equation image

and x, y satisfy the cold plasma R-mode dispersion relation,

  • equation image

[10] In general, the integrals I1 and I2 cannot be found in closed analytical form and so must be evaluated numerically. We evaluate I1 and I2 by employing the MATLAB function quadgk which uses an adaptive Gauss-Kronrod quadrature method [Shampine, 2008]. In addition, it is efficient to evaluate the integral occurring in equation (5), and also the similar integral in equation (46) below, by using quadgk even though these integrals can be evaluated analytically for certain values of the parameter l.

[11] The nonrelativistic approximation to the wave growth rate can be recovered by setting ΔR = 1 and γR = 1 in formulae (11) and (12) for the integrals I1 and I2. In this special case, the integrals can be obtained analytically and we find

  • equation image

Substitution of results (17) into equation (8) yields the nonrelativistic growth rate,

  • equation image

which can be shown to agree with the growth rate formula (2.65) given by Schulz and Lanzerotti [1974]. We find it useful to provide, in Appendix A, an equivalent derivation of expression (18) in which the parameter s is specifically identified as the electron pitch angle anisotropy in the nonrelativistic formulation (see equation (A15)).

[12] In Figure 2 we select typical cases to illustrate the dependence of the relativistic and nonrelativistic profiles of the whistler mode wave growth rate on the values of the parameters s and l; in the top panel we keep s fixed and vary l, and in the bottom panel we keep l fixed and vary s. The top panel indicates that a change in l value does not significantly change the frequency band for wave growth but does have some effect on the frequency at which the wave growth maximizes. The bottom panel shows that the value of s can significantly influence the frequency band over which wave growth occurs, and consequently also the value of the frequency at which the wave growth maximizes. The dependence on s and l of the frequencies at which the wave growth rate vanishes and maximizes is illustrated in detail for the relativistic case in Table 1, discussed below.

image

Figure 2. Relativistic and nonrelativistic whistler mode wave growth rates (given by equations (8) and (18)) for the case a = ∣Ωe2/ωpe2 = 0.05 and the indicated values of the pitch angle index s and the spectral index l.

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Table 1. For the Relativistic Case, Values of x*, xm, and E*(keV) for the Given (s, l) Values; x* = ω*/∣Ωe∣ is the Normalized Frequency at Which the Whistler Mode Wave Growth Rate Vanishes; xm = ωm/∣Ωe∣ is the Normalized Frequency at Which the Growth Rate Maximizes; and E*(keV) is the Minimum Electron Resonant Energy for Interaction With Waves of Frequency ω*
s\l1.11.21.251.522.5345678910
0.1x*0.0510.0520.0530.0560.0600.0620.0640.0660.0670.0670.0680.0680.0680.069
xm0.0240.0270.0280.0320.0390.0440.0480.0530.0560.0590.0600.0610.0620.063
E*179.4173.6171.1161.5150.6144.6140.9136.7134.3132.8131.8131.0130.4130.0
0.2x*0.1200.1230.1240.1290.1350.1380.1400.1420.1430.1440.1440.1450.1450.145
xm0.0580.0630.0650.0750.0910.1010.1080.1180.1240.1280.1300.1320.1340.135
E*66.4164.4063.5560.3656.8955.1154.0452.8652.2251.8351.5751.3751.2351.12
0.25x*0.1530.1560.1570.1630.1690.1720.1740.1760.1770.1780.1780.1790.1790.179
xm0.0730.0800.0830.0960.1150.1280.1370.1480.1550.1590.1620.1650.1670.168
E*47.4246.0045.4243.2240.8739.6938.9938.2337.8337.5837.4137.2937.2037.13
0.5x*0.2910.2940.2960.3020.3080.3110.3130.3150.3150.3160.3160.3170.3170.317
xm0.1410.1540.1600.1870.2230.2450.2590.2760.2850.2910.2960.2990.3010.303
E*15.1914.7614.5813.9513.3213.0312.8712.7012.6112.5612.5212.5012.4812.46
1.0x*0.4700.4730.4750.4800.4840.4860.4870.4880.4890.4890.4890.4890.4900.490
xm0.2400.2640.2760.3240.3820.4120.4290.4480.4580.4640.4690.4720.4740.476
E*3.9963.8913.8493.7063.5773.5233.4933.4643.4493.4403.4343.4293.4263.424
2.0x*0.6510.6530.6540.6570.6590.6600.6600.6610.6610.6610.6610.6620.6620.662
xm0.3610.4030.4230.4950.5660.5960.6120.6290.6370.6420.6450.6480.6490.651
E*0.8350.8170.8090.7860.7670.7600.7560.7520.7500.7490.7480.7480.7470.747
3.0x*0.7400.7420.7420.7440.7460.7460.7460.7470.7470.7470.7470.7470.7470.747
xm0.4360.4890.5130.5970.6670.6940.7070.7210.7280.7310.7340.7360.7370.739
E*0.3030.2970.2950.2880.2830.2810.2800.2790.2760.2780.2780.2780.2780.278
4.0x*0.7930.7950.7950.7960.7970.7970.7980.7980.7980.7980.7980.7980.7980.798
xm0.4870.5490.5750.6650.7310.7540.7650.7760.7820.7850.7870.7890.7900.791
E*0.1430.1400.1390.1370.1350.1340.1330.1330.1330.1330.1330.1330.1330.133
5.0x*0.8290.8290.8300.8310.8310.8320.8320.8320.8320.8320.8320.8320.8320.832
xm0.5260.5930.6220.7130.7740.7940.8040.8140.8180.8210.8230.8240.8250.826
E*0.0780.0770.0770.0750.0740.0740.0740.0740.0740.0740.0730.0730.0730.073
10.0x*0.9080.9080.9080.9080.9080.9090.9090.9090.9090.9090.9090.9090.9090.909
xm0.6330.7160.7480.8330.8750.8870.8930.8980.9010.9030.9040.9040.9050.905
E*0.0110.0110.0110.0110.0110.0110.0110.0110.0110.0110.0110.0110.0110.011

2.3. Limiting Electron Flux

[13] We assume that a stably trapped electron flux can be maintained close to its limiting value by whistler mode waves of sufficient power generated at the magnetic equator. The power flow Pw of an electromagnetic wavefield is proportional to the magnitude of the Poynting vector and hence to the square of the wave amplitude Bw [e.g., Swanson, 1989]. We write

  • equation image

where G is the wave gain defined by

  • equation image

The path integral (20) is taken along a flux tube with element ds; ωi is the wave growth rate and vg = ∣/dk∣ is the group speed.

[14] If we apply condition (20) to a region in the vicinity of a planet with a dipole or dipole-like magnetic field, then we shall require a gain of 3 e-foldings in wave power (i.e., G = 3/2) over a distance LRP/2 where RP is the radius of the planet. Then equation (20) takes the approximate form

  • equation image

If we apply equation (20) to a region with a stretched magnetic field or current sheet, then we specify a wave gain G = 3/2 over the distance σRP, where the thickness of the current sheet is 2σRP and σ is a scalar constant independent of L shell. In this case, we replace equation (21) by the condition

  • equation image

Since the power gain in decibels (dB) is

  • equation image

then setting G = 3/2 is equivalent to a power gain of 30 log10e ∼ 13 dB over the specified region of convective wave growth. In both equations (21) and (22) we evaluate ωi and vg at the frequency ω = ωm corresponding to the maximum value of the wave temporal growth rate.

[15] From equation (16) we find the group speed of a whistler mode wave is given by

  • equation image

[16] We emphasize that the wave gain conditions (21) or (22) replace the original condition used by Kennel and Petschek [1966] involving wave reflection. The idea of wave packets bouncing back and forth along a field line used by Kennel and Petschek [1966] has not been validated since observations have established that the Poynting flux of the waves appears to be always directed away from the equatorial source region. As well, it has been recently found that waves are not well guided by the magnetic field lines and so, at least locally, a wave generated on one flux tube would not be fed back along the same flux tube via reflection. This makes the global modeling problem all the more difficult. Regarding the gain required to ensure self-sustaining waves, we have elected to make our formulation equivalent to that of Kennel and Petschek [1966] by setting G = 3/2, corresponding to a net wave gain of 13 dB, as we have stated. In fact, there is some latitude in the specification of G. Analysis of whistler mode wave growth using data from the Combined Release and Radiation Effects Satellite (CRRES) [Li et al., 2008] and the Time History of Events and Macroscale Interactions during Substorms (THEMIS) satellite [Li et al., 2009] suggests that strong waves require a gain of about 50 dB to grow from background noise level. This corresponds to G ∼ 6 and would consequently increase our estimate of the limit on stably trapped flux by a factor of 4. It can be argued that substantial whistler mode wave growth, at least at Earth, requires a gain in the range G ∼ 3 to 6. The comparisons below of our limiting solutions with observed electron fluxes at Earth should be viewed with this caveat.

[17] Application of condition (21) or (22) to result (8) for the growth rate ωi serves to determine the limiting value of J(p*) and hence (by equation (5)) the upper limit on the integral omnidirectional flux I4π(E > E*). We find the limiting values are

  • equation image

and

  • equation image

where

  • equation image
  • equation image

and HP is the convective growth length given by

  • equation image

We identify p* (or E*) as the minimum electron momentum (or energy) for which gyroresonance can occur with a wave having a positive value for the growth rate. From equations (8) and (16), we see that ωi > 0 for 0 < ω < ω* where x* = ω*/∣Ωe∣ and y* = ck*/∣Ωe∣ satisfy

  • equation image

and

  • equation image

Numerical values for x*, y* are determined from the simultaneous solution of equations (30) and (31). By setting ω = ω*, k = k*, v = v*, and v = 0 in the relativistic gyroresonance condition,

  • equation image

we can show that

  • equation image

where

  • equation image

and

  • equation image

[18] Result (26), with E* given by equations (33)(35), gives the limiting omnidirectional electron flux for E > E* in the fully relativistic case, and is a key new result in this paper.

[19] In Table 1 we present values of x*, xm and E* in the relativistic case for the specified array of (s, l) values, and we set the cold plasma parameter a = ∣Ωe2/ωpe2 = 0.05 which typically represents the Earth's trough region outside the plasmasphere near L = 4 [Sheeley et al., 2001]. We deduce from Table 1 that for any fixed l value, the values of x*, xm and E* depend sensitively on s, with x* and xm increasing and E* decreasing as s increases. For any fixed s value, x* and E* are relatively insensitive to l, while xm increases moderately as l increases.

[20] The nonrelativistic forms of results (25) and (26), which are obtained from equations (18), (21), (22) and (7) (or equations (25), (17) and (7)), are

  • equation image

and

  • equation image

with

  • equation image

where p* and E* assume nonrelativistic values of the minimum momentum and minimum energy; the function M(x, y) is the nonrelativistic limit of the function K(x, y). The nonrelativistic growth rate (18) is positive for frequencies such that 0 < ω < ω*, where ω* is given by

  • equation image

at which, by equation (16), the corresponding wave number is

  • equation image

Then, setting ω = ω*, k = k*, and v = v* in the nonrelativistic gyroresonance condition

  • equation image

leads to v*/c = −(1 − x*)/y*, which, using equations (39) and (40), yields the required nonrelativistic values

  • equation image

and

  • equation image

[21] Result (37), with p* and E* defined by formulae (42) and (43), gives the limiting omnidirectional electron flux in the nonrelativistic case. Kennel and Petschek [1966], Schulz and Lanzerotti [1974] and other authors have given various versions of formula (37). In Appendix B we derive the formula for the stably trapped flux limit originally given by Kennel and Petschek [1966].

[22] Our results (25) for the limiting perpendicular differential flux J(p*) and (26) for the limiting integral omnidirectional flux I4π(E > E*) can be readily extended. For, from relations (1) and (2) it follows that

  • equation image

Result (44) is therefore the upper limit for the perpendicular differential flux, where in the fully relativistic case J(p*) is given by equation (25) and p* by equation (33), and in the nonrelativistic approximation J(p*) is given by equation (36) and p* by equation (42). Further, since the integral omnidirectional flux of electrons exceeding the energy E0 is

  • equation image

where we take E0 > E*, then, using equations (1), (2) and (25), we find

  • equation image

Expression (46) gives the limiting integral omnidirectional electron flux for energies exceeding E0, where E0 > E*. The nonrelativistic approximation to result (46), obtained by substituting the nonrelativistic form of equations (1) and (2) into equation (45), and by using equation (36), is

  • equation image

[23] In Figure 3 we plot x*, xm and E* as functions of the parameter a = ∣Ωe2/ωpe2 in both the relativistic and nonrelativistic cases, for the pitch angle indices s = 0.1, 0.5, 2.0, and the fixed spectral index l = 2. Since aB02/N0, then for a given background magnetic field, a “large” value of a corresponds to a “small” value of the electron density N0. Thus we observe from Figure 3 that, as expected, the differences between the relativistic and nonrelativistic profiles of x*, xm and E* become more pronounced as a becomes “large” or N0 becomes “small”. The sensitive dependence of x*, xm and E* on the pitch angle index s, as is evident from Table 1 (in the relativistic case) and as noted earlier, is also illustrated in Figure 3.

image

Figure 3. (top) Values of the whistler mode wave frequency x* = ω*/∣Ωe∣ at which the growth rate vanishes, as a function of a = ∣Ωe2/ωpe2, in the relativistic and nonrelativistic cases, for the indicated values of s, l. (middle) Corresponding to the top panel, values of the wave frequency xm = ωm/∣Ωe∣ at which the growth rate maximizes. (bottom) Values of the minimum resonant energy E* for whistler mode wave-electron interaction corresponding to the wave frequency x* given in the top panel.

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3. Limiting Energy Spectrum

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[24] In the formulation of Kennel and Petschek [1966], and also in the present work, a limiting form for the trapped particle flux is calculated for a specified particle distribution, namely the energy spectrum and the pitch angle distribution are assumed a priori. Accordingly, for a particle distribution of the form (1), the limiting flux strictly depends on the spectral index l and the pitch angle index s. In a modification of the Kennel-Petschek theory, Schulz and Davidson [1988] avoided an explicit assumption concerning the energy spectrum and took a particle distribution of the form,

  • equation image

The corresponding omnidirectional differential particle flux,

  • equation image

where equation image = p2equation image, is

  • equation image

In a nonrelativistic analysis, Schulz and Davidson [1988] calculated the limiting form of the function g(p), for an energetic particle population of given pitch angle anisotropy s, that corresponds to marginal stability for parallel-propagating electromagnetic cyclotron waves over all frequencies ω such that 0 < ω/∣Ωe∣ < s/(s + 1). The limiting omnidirectional differential flux due to Schulz and Davidson [1988], for large energy E, can be written

  • equation image

which corresponds to the limiting perpendicular particle differential flux,

  • equation image

To derive result (51), in which R(<1) is a coefficient of wave reflection, Schulz and Davidson [1988] employed a wave gain condition similar to that used by Kennel and Petschek [1966]; see condition (B1) below. The condition requires that in the limiting state the loss of wave energy due to imperfect reflection at the ionosphere is balanced by path-integrated wave growth along the field line. Expression (51) is not strictly dependent on the form of wave gain condition actually used by Schulz and Davidson [1988]. If alternatively one specifies a wave gain of ln(1/R) over the convective growth length LRP, where RP is the planetary radius and L denotes shell parameter, then the limiting energy spectrum (51) likewise follows. In the following section, where appropriate, we incorporate the limiting spectrum (52) in comparisons of the limiting solutions obtained in the present paper with observational data.

4. Comparison of Stably Trapped Flux Limits With Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[25] Relativistic expressions (25), (26), (44) and (46) for J(p*), I4π(E > E*), J(p) and I4π(E > E0), respectively, may be evaluated for any given planetary radiation belt. These results depend partly on B0 = B0(L), the equatorial magnetic field strength at a given L shell, and the wave convective growth length HP. The values of the minimum momentum p*, the minimum energy E*, and the function K(xm, ym) each in general depends on the pitch angle index s, the spectral index l, and the cold plasma parameter a = ∣Ωe2/ωpe2B02(L)/N0(L), where N0(L) is the electron number density at a given L shell. In the relativistic formulation, for any particular case, the parameters s, l and a are specified and then x* and xm must be found; x* corresponds to the frequency at which the growth rate (8) vanishes, for which we must solve equations (30) and (31), while xm corresponds to the frequency of maximum wave growth, for which we must maximize the function (8) with respect to x. Once the values of x* and xm are determined (which may take nontrivial computing effort), then all the remaining quantities p*, E*, J(p*), I4π(E > E*), J(p) and I4π(E > E0) are readily found.

[26] In the nonrelativistic formulation, calculations are much more straightforward and require less labor. Here x* = s/(s + 1) is known (result (39)), and p*, E* likewise take simple forms (results (42) and (43)). Determination of xm in the nonrelativistic case requires maximizing the function (18) with respect to x. Once xm is found, the nonrelativistic quantities J(p*), I4π(E > E*), J(p) and I4π(E > E0) easily follow from equations (36), (37), (44), and (47) respectively.

4.1. Earth

[27] For Earth, we set B0(L) = BE/L3 and HP = LRE/2, with BE = 0.312 gauss and RE = 6.4 × 108 cm. Then the limiting formulae (26), (25), (44) and (47) can be expressed in the form

  • equation image
  • equation image
  • equation image

and

  • equation image

where

  • equation image
  • equation image

and

  • equation image

In Figure 4 we show two-dimensional (l, s) plots of the parameters A and C and the corresponding minimum electron resonant energy E* in the case a = ∣Ωe2/ωpe2 = 0.05. Figure 4 demonstrates that the values of A and C can be either moderately or strongly dependent on the values adopted for l and s, subject to the particular region of (l, s) parameter space considered.

image

Figure 4. Two-dimensional (l, s) plots of the parameters A(l, s, a) and C(l, s, a), and the corresponding electron resonant energy E*, in the relativistic case with a = ∣Ωe2/ωpe2 = 0.05.

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[28] In the nonrelativistic approximation, results (53), (54) and (56) remain valid except that the coefficients A, C and D are then given by

  • equation image
  • equation image

and

  • equation image

The nonrelativistic version of equation (55) is

  • equation image

[29] Complementary to Figure 4, in Figure 5 we show the variation of A and C (given by equations (57) and (58)) as a function of s, for 0.1 ≤ s ≤ 5, in the cases l = 1.1, 2.0 and 4.0, with a = ∣Ωe2/ωpe2 = 0.05. Also, in Figure 5 we show the corresponding nonrelativistic forms of A and C (given by equations (60) and (61)). Differences between the relativistic and nonrelativistic forms of A and C are relatively small if s ≥ 1, though may become more significant if s ≪ 1.

image

Figure 5. Relativistic and nonrelativistic values of the parameters A(l, s, a) and C(l, s, a) as functions of the pitch angle index s, for a = ∣Ωe2/ωpe2 = 0.05 and the indicated values of l.

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[30] A typical range of s values measured in Earth's outer radiation belt is 0 < s < 1.5; see Table 1 in Thorne et al. [2005] which was reproduced from an unpublished report by A. Vampola. Analysis of electron data measured by CRRES [Li et al., 2008] and THEMIS [Li et al., 2009] during nightside electron injection events when intense whistler mode chorus is present yields peak values of s in the range 1 < s < 1.5.

[31] In the left panels of Figure 6, for Earth we show the relativistic and nonrelativistic forms of A and C as functions of L shell for the spectral index l = 2 and the pitch angle indices s = 0.1, 0.5 and 2.0. We assume a dipole magnetic field and the electron number density variation N0 = 124(3/L)4 cm−3 (due to Sheeley et al. [2001]) in the trough region outside the plasmasphere; see Figure 7. In the relativistic case, A is a weakly decreasing function of L if s > 0.5 and a moderately decreasing function of L if s < 0.5 (approximately), while C is an increasing function of L for all s values. In the nonrelativistic case, C is likewise an increasing function of L, while A remains independent of L to a good approximation, for all s values.

image

Figure 6. Relativistic and nonrelativistic values of the parameters A(l, s, a) and C(l, s, a) for l = 2 and the indicated values of s. In the left panels, A and C are plotted as functions of L shell, assuming the electron density model N0 = 124(3/L)4 cm−3 [Sheeley et al., 2001] and a dipole magnetic field in the Earth's trough region outside the plasmasphere. In the center and right panels, A and C are plotted as functions of a = ∣Ωe2/ωpe2, assuming the full whistler mode dispersion equation (16) in the center panels, and the simplified dispersion equation (B2) in the right panels.

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image

Figure 7. Electron number density N0 at Earth (from the Sheeley et al. [2001] trough density model) as a function of L shell, together with the corresponding profile of the cold plasma parameter a = ∣Ωe2/ωpe2.

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[32] In the center and right panels of Figure 6 we plot the relativistic and nonrelativistic forms of A and C as functions of the cold plasma parameter a = ∣Ωe2/ωpe2, for l = 2 and s = 0.1, 0.5, 2.0; we assume the full whistler mode dispersion relation (16) in the center panels and the simplified dispersion relation (B2) used by Kennel and Petschek [1966] in the right panels. In the relativistic case (both center and right panels) for s ≥ 0.5, if a < 0.05 then A and C are relatively independent of a, while if a > 0.05 then A is an increasing function of a and C is a decreasing function of a. In the nonrelativistic case (both center and right panels) for all values of s considered, A is weakly dependent on a while C is a decreasing function of a. Comparing the center and right panels, we see that differences in the profiles of both A and C caused by adopting the simplified dispersion relation only become apparent for larger values of a with s ≥ 0.5. In both center and right panels, differences between the relativistic and corresponding nonrelativistic profiles become more significant for larger values of a, as expected (see Figure 3).

[33] In Figure 8 we compare the limiting profiles (55) for the perpendicular electron differential flux with the electron flux measured by the CRRES Medium Electrons A (MEA) experiment [Vampola et al., 1992] during the 9 October 1990 geomagnetic storm. Particle and wave data for this storm were also analyzed by Meredith et al. [2002] and Summers et al. [2002]. The measured electron flux in Figure 8 is taken during CRRES orbit 192, close to the end of the several-day recovery period of the storm, at L = 5 and L = 6. Electron energization due to gyroresonant interaction with chorus waves during this recovery period [Summers et al., 2002] results in the formation of the high-energy tail in the electron spectrum shown in Figure 8. At L = 5 and L = 6 we adopt the respective spectral indices l = 1.4 and l = 2.1 since these values approximately match the slopes of the measured flux profiles. In the left panels we vary the pitch angle index s and we fix values for the cold plasma parameter a. For the latter we use electron density values given by the Sheeley et al. [2001] model; we find a = 0.038 at L = 5, and a = 0.026 at L = 6. We see in the bottom left panel that for s ≥ 0.3 the measured differential flux well exceeds the limiting solutions (55). In the top left panel, the measured flux well exceeds the limiting solutions if s ≥ 1, slightly exceeds (over most energies) the limiting solution if s = 0.5, and is exceeded by the limiting solution if s ≤ 0.3. In the right panel we fix s = 0.6, and we select values of the parameter a in the range given by Meredith et al. [2002]. It is evident from the right panel that the limiting solutions (55) can be sensitive to the value of a. The measured flux and the limiting profiles (55) are in close agreement for certain values of s and a, e.g., see the bottom right panel. However, conclusive comparison between the limiting flux (55) and the experimental data is difficult not least because of the range of uncertainty of the parameter a. For comparison purposes, we also show in Figure 8 corresponding limiting spectral profiles (52) due to Schulz and Davidson [1988] for which we set ln(1/R) = 3. These limiting solutions, which are valid for “large” energy E, are independent of the value of a.

image

Figure 8. Measured electron differential spectrum J at L = 5 and L = 6 from the MEA experiment on CRRES (orbit 192) during the geomagnetic storm of 9 October 1990, together with corresponding limiting profiles (55) for the indicated values of s, l, and a = ∣Ωe2/ωpe2. Also shown are limiting energy spectra due to Schulz and Davidson [1988].

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[34] In Figures 9 and 10 we compare the limiting solutions (56) with the measured electron flux profiles at Earth as given by the AE-8 model [Vette, 1991]. In Figure 9 we plot the integral fluxes I4π(E > E0) for E0 = 100, 150, 200 and 250 keV, over the range 3 ≤ L ≤ 8. For the model solutions we assume a nominal value for the spectral index l = 1.6, and for the pitch angle index we choose the values s = 0.3 and s = 1.0. We adopt the Sheeley et al. [2001] model for the electron number density. We note the relatively close agreement between the limiting solutions and the AE-8 model profiles in the region 6 ≤ L ≤ 8, for s = 0.3. The concept of self-limitation of stably trapped flux is not expected to apply to the region 3 < L < 5 where parasitic scattering processes lead to the formation of the slot. In Figure 10 we compare the Vette [1991] AE-8 model solutions at L = 5 and L = 6 with the limiting spectra (56). We again adopt the Sheeley et al. [2001] density model. We choose values for the spectral index that approximately match the data, namely, l = 1.9 at L = 5, and l = 2.2 at L = 6. In each case we then vary the pitch angle index s. At both L = 5 and L = 6 it is evident that the limiting solution (56) is relatively close to the experimental profile for appropriately specified (small) values of s. For moderate or large values of s (say, s ≥ 0.5) the measured profiles greatly exceed the limiting solutions.

image

Figure 9. (bottom) AE-8 model [Vette, 1991] electron integral fluxes I4π(E > E0) at Earth, for the specified energies E0 as a function of L shell, together with corresponding limiting profiles (56) for the indicated values of s, l. (top) Corresponding to the bottom panel, minimum resonant energy E* for gyroresonant interaction with a whistler mode wave with positive growth rate, as a function of L shell.

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image

Figure 10. For Earth, at L = 5 and L = 6, comparison of the AE-8 model [Vette, 1991] electron integral spectrum I4π(E > E0) with corresponding limiting profiles (56) for the indicated values of s, l.

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4.2. Jupiter

[35] For Jupiter, we set B0(L) = BJ/L3 and HP = σRJ, with BJ = 3.9 gauss, RJ = 11RE and σ = 1. The limiting results (26), (25), (44) and (47) then take the form

  • equation image
  • equation image
  • equation image

and

  • equation image

where A, C and D are given by equations (57), (58) and (59).

[36] Results (64), (65) and (67) are valid in the nonrelativistic approximation in which case the coefficients A, C and D are given by equations (60), (61) and (62).The nonrelativistic form of result (66) is

  • equation image

[37] In Figure 11 we show the electron number density profile at Jupiter, for 3 ≤ L ≤ 30, as given by Divine and Garrett [1983], together with the corresponding profile of the cold plasma parameter a = ∣Ωe2/ωpe2.

image

Figure 11. Electron number density N0 at Jupiter [from Divine and Garrett, 1983] as a function of L shell, together with the corresponding profile of the cold plasma parameter a = ∣Ωe2/ωpe2.

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[38] In Figures 12 and 13 we plot electron integral fluxes I4π(E > E0) at Jupiter for 6 ≤ L ≤ 15, measured by the Energetic Particle Detector (EPD) on Galileo [Jun et al., 2005] for E0 = 1.5 MeV and E0 = 11 MeV. In Figure 12 we select values of the spectral index l that approximately match the slopes of the data profiles, namely l = 1.5 for E0 = 1.5 MeV and l = 1.6 for E0 = 11 MeV, and we plot the limiting flux profiles (67) for the pitch angle indices s = 0.1 and s = 0.5. Similarly, in Figure 13, at each energy, we fix the parameter s and plot the limiting profiles for different values of l. Relatively close agreement between the limiting profiles and the observed fluxes, at least over a limited L shell range, is possible by appropriately selecting values for l and s, namely, for E0 = 1.5 MeV, l ∼ 1.2 to 1.5 and s ∼ 0.1 to 0.3; and for E0 = 11 MeV, l ∼ 1.6 and s ∼ 0.1. Typical values for the parameter s at Jupiter have been reported by Xiao et al. [2003] who simulated the growth of whistler mode waves in the Io torus during an interchange injection event using data from the Plasma Wave Subsystem (PWS) and EPD instruments on Galileo. The measured values of s ranged from about s = 0.4 at the start of the injection event to about s = 0.15 at the time of the most intense waves.

image

Figure 12. Electron integral fluxes I4π(E > E0) at Jupiter measured by the Galileo Energetic Particle Detector (EPD) [Jun et al., 2005] for E0 = 1.5, 11 MeV, as a function of L shell, together with corresponding limiting profiles (67) for the indicated values of s, l.

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image

Figure 13. As in Figure 12, except that the limiting profiles (67) are shown for different sets of values of the parameters s and l.

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[39] In Figure 14 we compare the electron integral spectrum I(E > E0) at Jupiter (L = 7.69) measured by the Low-Energy Charged Particle (LECP) experiment on Voyager 1 [Armstrong et al., 1981] with corresponding limiting solutions (67). We show the limiting spectra in the top panel for l = 1.5 and various s values, and in the bottom panel for s = 0.2 and various l values. Figure 14 further demonstrates that relatively close agreement between measured and limiting flux profiles is possible for a suitably small value of the pitch angle index s in the case that the spectral index l models the slope of the measured spectrum.

image

Figure 14. Electron integral spectrum I(E > E0) at Jupiter (L = 7.69) measured by the Voyager 1 Low-Energy Charged Particle (LECP) experiment [Armstrong et al., 1981] together with corresponding limiting spectra derived from equation (67) for the indicated values of s, l.

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[40] In Figure 15 we show electron integral fluxes I4π(E > E0) at Jupiter, for E0 = 2 MeV and E0 = 11 MeV, over the range 6 ≤ L ≤ 30, measured by the Galileo EPD instrument, reported by Sorensen et al. [2005] and Jun et al. [2005]. We choose here to present a simple comparison between the measured fluxes and the corresponding limiting profiles (67) for a fixed pair of nominal values for l and s, namely s = 0.3 and l = 1.5. Even such a rough comparison reveals that over certain ranges of L shell, dependent on energy, the observed fluxes may be close to the limiting values (67). More detailed comparisons require empirical data on the variation of the spectral index l with L shell.

image

Figure 15. (right) Electron integral fluxes I4π(E > E0) at Jupiter measured by the Galileo Energetic Particle Detector (EPD) [Sorensen et al., 2005; Jun et al., 2005] for the specified energies E0, as a function of L shell, together with corresponding limiting profiles (67) for s = 0.3, l = 1.5. (left) Corresponding to the right panel, minimum resonant energy E* for gyroresonant interaction with whistler mode waves with a positive growth rate.

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4.3. Uranus

[41] For Uranus, we set B0(L) = BU/L3 and HP = LRU/2, with BU = 0.23 gauss, RU = 4RE. The limiting results (26), (25), (44) and (47) then become

  • equation image
  • equation image
  • equation image

and

  • equation image

where A, C and D are given by equations (57), (58) and (59).

[42] Results (69), (70) and (72) are valid in the nonrelativistic approximation in which case the coefficients are given by equations (60), (61) and (62).

[43] The nonrelativistic form of result (71) is

  • equation image

[44] In Figure 16 we show electron differential spectra at Uranus, at the given L shells, measured by the Low Energy Charged Particle (LECP) experiment [Mauk et al., 1987] on Voyager 2. For comparison we show the limiting solutions (71) for the perpendicular differential spectra, for the specified sets of s values. At each L shell, we choose the spectral index l so that it roughly matches the slope of the data profile over the given energy range; the value of the parameter a = ∣Ωe2/ωpe2 is calculated using the electron density observations at Uranus reported by McNutt et al. [1987] and Sittler et al. [1987]. As expected, comparisons between the measured spectra and the limiting solutions depend somewhat sensitively on the values of the pitch angle index s. In addition, we caution that in Figure 16 comparisons between the measured spectra and the limiting solutions can only be approximate since the measured spectra here are given over a limited energy range. Corresponding to the extreme left panel of Figure 16, we present in Figure 17 the measured electron differential spectrum extended to all energies, determined by Mauk et al. [1987], who used electron data from the LECP experiment [Krimigis et al., 1986] and the Cosmic Ray System (CRS) experiment [Stone et al., 1986] on Voyager 2. The measured differential spectrum in Figure 17 is described by a broken power law, namely, J ∝ 1/E1.12, for E ≤ 1.2 MeV; J ∝ 1/E6.8, for E > 1.2 MeV. Since the particle distribution (1) assumed in the present investigation is characterized by a single power law over all energy, then comparison between our limiting solutions and a broken power law spectrum are problematical. Nevertheless, in Figure 17, for illustration, we provide a set of limiting solutions for the perpendicular differential spectrum (71) for a range of values of s and l. For comparison, limiting spectra (52) due to Schulz and Davidson [1988] are also shown for various s values.

image

Figure 16. Electron differential spectra J at Uranus, at the indicated L shells, measured by the Voyager 2 Low-Energy Charged Particle (LECP) experiment [Mauk et al., 1987], together with corresponding limiting spectra J (given by equation (71)) for the specified values of s, l.

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image

Figure 17. Profile of the electron differential spectrum J at Uranus (L = 4.73) reported by Mauk et al. [1987], together with limiting spectra J (given by equation (71)) for the indicated values of s and l. Also shown are limiting spectra due to Schulz and Davidson [1988].

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[45] Using the measured differential spectra illustrated in Figure 16, we calculated an approximate profile for the electron omnidirectional integral flux I4π(E > 0.12 MeV) at Uranus as a function of L shell. The result is shown in Figure 18. Limiting flux profiles, given by equation (72), for s = 0.5, 1.0, 2.0 and l values given in Figure 16, are shown for comparison. Representative empirical values for the pitch angle index s at Uranus do not appear to be available.

image

Figure 18. (bottom) Electron omnidirectional integral flux I4π(E > 0.12 MeV) at Uranus constructed from measurements reported by Mauk et al. [1987] from the Voyager 2 LECP experiment. Also shown are limiting flux profiles (given by equation (72)) for s = 0.5, 1.0, 2.0 and the l values specified in Figure 16. (top) Corresponding minimum resonant energy E* for gyroresonant interaction with whistler mode waves with a positive growth rate.

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5. Summary and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[46] We have reexamined the concept of self-limitation of radiation belt particle fluxes originally introduced by Kennel and Petschek [1966]. Our study is the first fully relativistic analysis of the self-limiting flux concept. We assume a particle distribution, given by equation (1), in which the energy spectrum, characterized by the spectral index l (>1), and the pitch angle distribution, characterized by the pitch angle index s (>0), are specified a priori. We calculate the temporal growth rate of field-aligned whistler mode waves in a relativistic plasma modeled by the assumed distribution, and then apply the condition that waves generated at the magnetic equator acquire a specified power gain over a given distance along a field line. Formulae are thereby obtained for the limiting values of the electron integral omnidirectional flux and the electron differential flux which have an explicit dependence on l, s and the cold plasma parameter a = ∣Ωe2/ωpe2. We apply these formulae to Earth, Jupiter and Uranus, and test the results against observed electron fluxes at these planets. Our conclusions are as follows:

[47] 1. For a given value of a, the limiting solutions can be either moderately or sensitively dependent on the parameters l and s, dependent on the region of (l, s) parameter space considered. Likewise, for given values of l and s, the limiting solutions can be sensitive to the parameter a.

[48] 2. The relativistic limiting solutions can differ significantly from the corresponding nonrelativistic solutions, for example, in the cases of “large” values of a, say a ≥ 0.1 (for given values of l, s), or “small” values of s, say s ≤ 0.1 (for any given l value and moderate values of a).

[49] 3. In the comparison of the limiting solutions with observations of electron fluxes at Earth, Jupiter and Uranus, we found that the observed fluxes were close to the limiting value in a number of cases. However, it is difficult to make definitive or broad conclusions regarding our comparisons of the limiting solutions with data. The main reason for this is the overall dependence of the limiting solutions on the parameters l, s and a, and the associated lack of sufficiently accurate empirical knowledge of all three parameters in most cases considered. Our recommendation here is that further comparisons with observed data are needed in order to determine the natural conditions in which the limiting solutions are strictly applicable.

[50] 4. The Kennel-Petschek concept of self-limitation of radiation belt particle fluxes can be criticized because the flux limit depends on the spectral index l and pitch angle index s of an assumed particle distribution which has been specified in advance (the limiting energy spectrum due to Schulz and Davidson [1988] which depends on a predetermined pitch angle index can be similarly criticized). In fact, the energy spectrum and pitch angle distribution in the limiting case should, in principle, be determined by finding the steady state solution of a self-consistent coupled set of equations for the wave spectrum and particle distribution in the presence of a specified particle source. Such a problem is formidable and may also require the incorporation of radial transport of particles from the source. A more limited seed-particle calculation was carried out by Summers and Ma [2000] who calculated the steady state solution of a model kinetic equation for the energy distribution of radiation belt electrons under magnetic storm conditions. The equation incorporated an energy diffusion coefficient corresponding to a given spectrum of whistler mode waves, a source term representing substorm produced (lower energy) electrons, and a loss term representing electron precipitation due to pitch angle scattering by whistler mode waves and electromagnetic ion cyclotron waves. The model analyzed by Summers and Ma [2000] was based on the mechanism proposed by Summers et al. [1998] to account for the generation of relativistic electrons during the recovery phase of a magnetic storm by combining energy diffusion during cyclotron interaction with whistler mode chorus outside the plasmasphere with electron precipitation due to pitch angle scattering by the aforementioned waves. It is interesting to note that the limiting energetic electron spectra obtained by Summers and Ma [2000] did not follow a simple power law in energy. Indeed, there is no reason to suppose that the limiting energy spectrum in an electron radiation belt should necessarily be of a power law type under natural conditions.

[51] The general concept of limitation of stably trapped particle fluxes remains of great interest to radiation belt physicists [e.g., O'Brien et al., 2007]. The Kennel-Petschek limit on stably trapped particle fluxes, while subject to criticism as an oversimplification, likewise remains of interest. Further progress on the problem will probably require the solution of self-consistent wave-particle interaction equations and the incorporation of oblique waves. The role of nonlinear wave growth and nonlinear wave trapping [Omura and Summers, 2006; Summers and Omura, 2007; Katoh et al., 2008; Omura et al., 2009] in the self-limiting flux problem should also be evaluated.

Appendix A:: Derivation of Wave Growth Rates

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[52] We derive the linear growth rate for R-mode electromagnetic waves propagating parallel to a uniform magnetic field in a spatially homogeneous plasma, for a hot anisotropic relativistic electron population with a distribution function f(p, p) in the presence of a dominant cold electron population; p = γmev and p = γmev are the components of the relativistic momentum p = γmev, where me is the electron rest mass, v is the electron velocity with components v and v respectively parallel and perpendicular to the ambient magnetic field, γ = (1 − v2/c2)−1/2 = (1 + p2/(mec)2)1/2, with v2 = v2 + v2 and p2 = p2 + p2, and c is the speed of light. We assume that ωpe = (4πN0e2/me)1/2 is the plasma frequency, and ∣Ωe∣ = eB0/(mec) is the electron gyrofrequency, where −e is the electron charge, N0 is the cold electron number density, and B0 is the magnitude of the zeroth-order magnetic field. Then, following Xiao et al. [1998] and considering Fourier components of the wavefield (ei(kxωt)) with real wave number k (>0) and complex wave frequency ω = ωr + i, we can write the relativistic growth/damping rate in the form,

  • equation image

where

  • equation image

is the fraction of the relativistic particle distribution near resonance,

  • equation image

is the relativistic pitch angle anisotropy of the resonant particles,

  • equation image

is the resonant value of the Lorentz factor,

  • equation image

is the resonant value of the electron parallel momentum,

  • equation image
  • equation image

is the minimum anisotropy required for wave instability (ωi > 0), and

  • equation image

is the cold plasma R-mode dispersion relation.

[53] Substitution of the particle distribution (1) into equations (A2) and (A3) leads to the results

  • equation image

and

  • equation image

where I1 and I2 are the integrals given by formulae (11) and (12), in which equation imageR = pR/(mec), ΔR = 1 − ωrequation imageR/(ckγR), and γR is given by equation (A4) with p2/(mec)2 replaced by z2. Expression (8) for the growth rate then readily follows by substituting results (A9) and (A10) into (A1). In section 2 we have written the real part of the wave frequency (ωr > 0) as ω.

[54] In order to recover the nonrelativistic approximation, we set γ = 1, γR = 1 and ΔR = 1. Then equation (A1) becomes

  • equation image

where

  • equation image

and

  • equation image

with pR = me(ωr − ∣Ωe∣)/k. Results equivalent to equations (A11)(A13) were originally derived by Kennel and Petschek [1966].

[55] Substitution of the nonrelativistic form of distribution (1) into equations (A12)(A13) leads to the relations

  • equation image

and

  • equation image

The nonrelativistic growth rate formula (18) now immediately follows from equations (A11), (A14) and (A15).

Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[56] For completeness, and for comparison with material presented herein, we give a condensed derivation of the stably trapped flux limit obtained by Kennel and Petschek [1966]. Kennel and Petschek (KP) considered electromagnetic waves generated by an anisotropic distribution of magnetospherically trapped particles in the nonrelativistic regime. They proposed that the integral omnidirectional flux of particles achieves a limiting value when the loss of wave energy due to imperfect reflection at the end of a field line is balanced by sufficient wave growth along the field line. This steady state situation is described by the condition

  • equation image

where R(<1) is the reflection coefficient; LRE is the typical path length between the points of reflection, where RE is the Earth's radius and L denotes magnetic shell; ωi is the wave growth rate, and vg = ∣/dk∣ is the group speed. [KP considered two cases: (1) whistler mode waves interacting with electrons and (2) ion cyclotron waves interacting with protons. Here we consider only case (1)]. We adopt the simplified whistler mode dispersion relation,

  • equation image

(namely, equation (A8) with the “one” on the right-hand side omitted), where ω is the real wave frequency and the other symbols are as defined previously. Then, for a given electron distribution f, the wave growth rate is given by

  • equation image

where equation image (given by equation (A12)) is the fraction of the electron distribution near resonance, equation image (given by equation (A13)) is the pitch angle anisotropy of the resonant particles, and Ac (given by equation (A7)) is the critical anisotropy. Result (B3) follows from equation (A11) after taking account of our use of the simplified dispersion relation (B2), and setting ωr = ω. KP used the particle distribution,

  • equation image

with

  • equation image

where v is the particle speed, α is the pitch angle, α0 is the loss cone angle, and Q is a normalizing constant. The integral omnidirectional flux, for electrons exceeding the minimum resonant energy ER, is

  • equation image

where ER = equation imagemevR2, and vR = (ω − ∣Ωe∣)/k.

[57] Substituting distribution (B4), with relation (B5), into equation (B6), and correspondingly evaluating equation image using equation (A12), we find after some algebraic manipulations that

  • equation image

From equations (B1), (B3), and (B7), we obtain the KP equatorial stably trapped flux limit as

  • equation image

where

  • equation image

Result (B8) applies to a distribution with a given anisotropy equation image and waves of a given frequency ω; equation (B8) can be similarly obtained by adopting the pitch angle variation ϕ(α) = (sin α)2s used in our main text. In their original formulation, KP relate the anisotropy equation image to the size of the loss cone.

[58] Taking the Earth's equatorial magnetic field strength as B0 = BE/L3 where BE = 0.312 gauss, adopting the nominal values ln(1/R) = 3, ω/∣Ωe∣ = 0.25, equation imageAc = 0.2, ∣Ωe2/ωpe2 = 0.1, and using the standard values for the Earth's radius RE = 6.4 × 108 cm, the speed of light c = 3 × 1010 cm sec−1, and the electronic charge e = 4.8 × 10−10esu, we find from equations (B8)(B9) that a typical value for the KP limiting electron omnidirectional flux is

  • equation image

for ER ≈ 40 keV.

[59] As a criterion for specifying the limiting value of the omnidirectional electron flux, we prefer to use (in section 2.3) a condition requiring a specified wave gain over a given region of convective wave growth rather than the concept used by KP relating to the loss of wave energy due to internal wave reflection.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

[60] This work is supported by the Natural Sciences and Engineering Research Council of Canada under grant A-0621. Further support is acknowledged from the WCU grant R31-10016 funded by the Korean Ministry of Education, Science and Technology, and from NASA grant NNX07AL27G.

[61] Amitava Bhattacharjee thanks Brian Kress and another reviewer for their assistance in evaluating this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Stably Trapped Flux Limit
  5. 3. Limiting Energy Spectrum
  6. 4. Comparison of Stably Trapped Flux Limits With Data
  7. 5. Summary and Conclusions
  8. Appendix A:: Derivation of Wave Growth Rates
  9. Appendix B:: Derivation of the Classical Kennel-Petschek Flux Limit
  10. Acknowledgments
  11. References
  12. Supporting Information
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