#### 2.1. Particle Distribution Function

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

with *p*^{2} = *p*_{∥}^{2} + *p*_{⊥}^{2}, where *p*_{∥} = *γm*_{e}*v*_{∥} and *p*_{⊥} = *γm*_{e}*v*_{⊥} are the components of relativistic momentum *p* = *γm*_{e}*v*, *m*_{e} 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 − *v*^{2}/*c*^{2})^{−1/2} = (1 + *p*^{2}/(*m*_{e}*c*)^{2})^{1/2}, with *v*^{2} = *v*_{∥}^{2} + *v*_{⊥}^{2}, 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

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*/(*m*_{e}*c*^{2}) = *γ* − 1, which satisfies the relation

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

is given by

where

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* = *m*_{e}*v*, *E* = (*m*_{e}*v*^{2})/2, and then substitution of distribution (1) into equation (4) gives

where *E*_{*} = *p*_{*}^{2}/(2*m*_{e}).

[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*.

#### 2.2. Wave Growth Rates

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

where *N*_{0} is the cold electron number density,

is a cold plasma parameter where ∣Ω_{e}∣ is the electron gyrofrequency and *ω*_{pe} is the plasma frequency (each defined in Appendix A), and

where *ω* is the (real) wave frequency and *k* is the (real) wave number. *I*_{1}, *I*_{2} are the integrals given by

and

where

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

[10] In general, the integrals *I*_{1} and *I*_{2} cannot be found in closed analytical form and so must be evaluated numerically. We evaluate *I*_{1} and *I*_{2} 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 Δ_{R} = 1 and *γ*_{R} = 1 in formulae (11) and (12) for the integrals *I*_{1} and *I*_{2}. In this special case, the integrals can be obtained analytically and we find

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

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.

Table 1. For the Relativistic Case, Values of *x*_{*}, *x*_{m}, and *E*_{*}(*keV*) for the Given (*s*, *l*) Values; *x*_{*} = *ω*_{*}/∣Ω_{e}∣ is the Normalized Frequency at Which the Whistler Mode Wave Growth Rate Vanishes; *x*_{m} = *ω*_{m}/∣Ω_{e}∣ 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*\*l* | 1.1 | 1.2 | 1.25 | 1.5 | 2 | 2.5 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|

0.1 | *x*_{*} | 0.051 | 0.052 | 0.053 | 0.056 | 0.060 | 0.062 | 0.064 | 0.066 | 0.067 | 0.067 | 0.068 | 0.068 | 0.068 | 0.069 |

*x*_{m} | 0.024 | 0.027 | 0.028 | 0.032 | 0.039 | 0.044 | 0.048 | 0.053 | 0.056 | 0.059 | 0.060 | 0.061 | 0.062 | 0.063 |

*E*_{*} | 179.4 | 173.6 | 171.1 | 161.5 | 150.6 | 144.6 | 140.9 | 136.7 | 134.3 | 132.8 | 131.8 | 131.0 | 130.4 | 130.0 |

0.2 | *x*_{*} | 0.120 | 0.123 | 0.124 | 0.129 | 0.135 | 0.138 | 0.140 | 0.142 | 0.143 | 0.144 | 0.144 | 0.145 | 0.145 | 0.145 |

*x*_{m} | 0.058 | 0.063 | 0.065 | 0.075 | 0.091 | 0.101 | 0.108 | 0.118 | 0.124 | 0.128 | 0.130 | 0.132 | 0.134 | 0.135 |

*E*_{*} | 66.41 | 64.40 | 63.55 | 60.36 | 56.89 | 55.11 | 54.04 | 52.86 | 52.22 | 51.83 | 51.57 | 51.37 | 51.23 | 51.12 |

0.25 | *x*_{*} | 0.153 | 0.156 | 0.157 | 0.163 | 0.169 | 0.172 | 0.174 | 0.176 | 0.177 | 0.178 | 0.178 | 0.179 | 0.179 | 0.179 |

*x*_{m} | 0.073 | 0.080 | 0.083 | 0.096 | 0.115 | 0.128 | 0.137 | 0.148 | 0.155 | 0.159 | 0.162 | 0.165 | 0.167 | 0.168 |

*E*_{*} | 47.42 | 46.00 | 45.42 | 43.22 | 40.87 | 39.69 | 38.99 | 38.23 | 37.83 | 37.58 | 37.41 | 37.29 | 37.20 | 37.13 |

0.5 | *x*_{*} | 0.291 | 0.294 | 0.296 | 0.302 | 0.308 | 0.311 | 0.313 | 0.315 | 0.315 | 0.316 | 0.316 | 0.317 | 0.317 | 0.317 |

*x*_{m} | 0.141 | 0.154 | 0.160 | 0.187 | 0.223 | 0.245 | 0.259 | 0.276 | 0.285 | 0.291 | 0.296 | 0.299 | 0.301 | 0.303 |

*E*_{*} | 15.19 | 14.76 | 14.58 | 13.95 | 13.32 | 13.03 | 12.87 | 12.70 | 12.61 | 12.56 | 12.52 | 12.50 | 12.48 | 12.46 |

1.0 | *x*_{*} | 0.470 | 0.473 | 0.475 | 0.480 | 0.484 | 0.486 | 0.487 | 0.488 | 0.489 | 0.489 | 0.489 | 0.489 | 0.490 | 0.490 |

*x*_{m} | 0.240 | 0.264 | 0.276 | 0.324 | 0.382 | 0.412 | 0.429 | 0.448 | 0.458 | 0.464 | 0.469 | 0.472 | 0.474 | 0.476 |

*E*_{*} | 3.996 | 3.891 | 3.849 | 3.706 | 3.577 | 3.523 | 3.493 | 3.464 | 3.449 | 3.440 | 3.434 | 3.429 | 3.426 | 3.424 |

2.0 | *x*_{*} | 0.651 | 0.653 | 0.654 | 0.657 | 0.659 | 0.660 | 0.660 | 0.661 | 0.661 | 0.661 | 0.661 | 0.662 | 0.662 | 0.662 |

*x*_{m} | 0.361 | 0.403 | 0.423 | 0.495 | 0.566 | 0.596 | 0.612 | 0.629 | 0.637 | 0.642 | 0.645 | 0.648 | 0.649 | 0.651 |

*E*_{*} | 0.835 | 0.817 | 0.809 | 0.786 | 0.767 | 0.760 | 0.756 | 0.752 | 0.750 | 0.749 | 0.748 | 0.748 | 0.747 | 0.747 |

3.0 | *x*_{*} | 0.740 | 0.742 | 0.742 | 0.744 | 0.746 | 0.746 | 0.746 | 0.747 | 0.747 | 0.747 | 0.747 | 0.747 | 0.747 | 0.747 |

*x*_{m} | 0.436 | 0.489 | 0.513 | 0.597 | 0.667 | 0.694 | 0.707 | 0.721 | 0.728 | 0.731 | 0.734 | 0.736 | 0.737 | 0.739 |

*E*_{*} | 0.303 | 0.297 | 0.295 | 0.288 | 0.283 | 0.281 | 0.280 | 0.279 | 0.276 | 0.278 | 0.278 | 0.278 | 0.278 | 0.278 |

4.0 | *x*_{*} | 0.793 | 0.795 | 0.795 | 0.796 | 0.797 | 0.797 | 0.798 | 0.798 | 0.798 | 0.798 | 0.798 | 0.798 | 0.798 | 0.798 |

*x*_{m} | 0.487 | 0.549 | 0.575 | 0.665 | 0.731 | 0.754 | 0.765 | 0.776 | 0.782 | 0.785 | 0.787 | 0.789 | 0.790 | 0.791 |

*E*_{*} | 0.143 | 0.140 | 0.139 | 0.137 | 0.135 | 0.134 | 0.133 | 0.133 | 0.133 | 0.133 | 0.133 | 0.133 | 0.133 | 0.133 |

5.0 | *x*_{*} | 0.829 | 0.829 | 0.830 | 0.831 | 0.831 | 0.832 | 0.832 | 0.832 | 0.832 | 0.832 | 0.832 | 0.832 | 0.832 | 0.832 |

*x*_{m} | 0.526 | 0.593 | 0.622 | 0.713 | 0.774 | 0.794 | 0.804 | 0.814 | 0.818 | 0.821 | 0.823 | 0.824 | 0.825 | 0.826 |

*E*_{*} | 0.078 | 0.077 | 0.077 | 0.075 | 0.074 | 0.074 | 0.074 | 0.074 | 0.074 | 0.074 | 0.073 | 0.073 | 0.073 | 0.073 |

10.0 | *x*_{*} | 0.908 | 0.908 | 0.908 | 0.908 | 0.908 | 0.909 | 0.909 | 0.909 | 0.909 | 0.909 | 0.909 | 0.909 | 0.909 | 0.909 |

*x*_{m} | 0.633 | 0.716 | 0.748 | 0.833 | 0.875 | 0.887 | 0.893 | 0.898 | 0.901 | 0.903 | 0.904 | 0.904 | 0.905 | 0.905 |

*E*_{*} | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 |

#### 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 *P*_{w} of an electromagnetic wavefield is proportional to the magnitude of the Poynting vector and hence to the square of the wave amplitude *B*_{w} [e.g., *Swanson*, 1989]. We write

where *G* is the wave gain defined by

The path integral (20) is taken along a flux tube with element *ds*; *ω*_{i} is the wave growth rate and *v*_{g} = ∣*dω*/*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 *LR*_{P}/2 where *R*_{P} is the radius of the planet. Then equation (20) takes the approximate form

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 *σ**R*_{P}, where the thickness of the current sheet is 2*σ**R*_{P} and *σ* is a scalar constant independent of *L* shell. In this case, we replace equation (21) by the condition

Since the power gain in decibels (*dB*) is

then setting *G* = 3/2 is equivalent to a power gain of 30 log_{10}*e* ∼ 13 *dB* over the specified region of convective wave growth. In both equations (21) and (22) we evaluate *ω*_{i} and *v*_{g} at the frequency *ω* = *ω*_{m} corresponding to the maximum value of the wave temporal growth rate.

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

[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 *ω*_{i} serves to determine the limiting value of *J*_{⊥}(*p*_{*}) and hence (by equation (5)) the upper limit on the integral omnidirectional flux *I*_{4π}(*E* > *E*_{*}). We find the limiting values are

and

where

and *H*_{P} is the convective growth length given by

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 *ω*_{i} > 0 for 0 < *ω* < *ω*_{*} where *x*_{*} = *ω*_{*}/∣Ω_{e}∣ and *y*_{*} = *ck*_{*}/∣Ω_{e}∣ satisfy

and

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,

we can show that

where

and

[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*_{*}, *x*_{m} and *E*_{*} in the relativistic case for the specified array of (*s*, *l*) values, and we set the cold plasma parameter *a* = ∣Ω_{e}∣^{2}/*ω*_{pe}^{2} = 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*_{*}, *x*_{m} and *E*_{*} depend sensitively on *s*, with *x*_{*} and *x*_{m} increasing and *E*_{*} decreasing as *s* increases. For any fixed *s* value, *x*_{*} and *E*_{*} are relatively insensitive to *l*, while *x*_{m} 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

and

with

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

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

Then, setting *ω* = *ω*_{*}, *k* = *k*_{*}, and *v*_{∥} = *v*_{*} in the nonrelativistic gyroresonance condition

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

and

[22] Our results (25) for the limiting perpendicular differential flux *J*_{⊥}(*p*_{*}) and (26) for the limiting integral omnidirectional flux *I*_{4π}(*E* > *E*_{*}) can be readily extended. For, from relations (1) and (2) it follows that

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 *E*_{0} is

where we take *E*_{0} > *E*_{*}, then, using equations (1), (2) and (25), we find

Expression (46) gives the limiting integral omnidirectional electron flux for energies exceeding *E*_{0}, where *E*_{0} > *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

[23] In Figure 3 we plot *x*_{*}, *x*_{m} and *E*_{*} as functions of the parameter *a* = ∣Ω_{e}∣^{2}/*ω*_{pe}^{2} 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 *a* ∝ *B*_{0}^{2}/*N*_{0}, then for a given background magnetic field, a “large” value of *a* corresponds to a “small” value of the electron density *N*_{0}. Thus we observe from Figure 3 that, as expected, the differences between the relativistic and nonrelativistic profiles of *x*_{*}, *x*_{m} and *E*_{*} become more pronounced as *a* becomes “large” or *N*_{0} becomes “small”. The sensitive dependence of *x*_{*}, *x*_{m} 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.