Energetic electrons in the outer radiation belt can resonate with intense bursts of whistler-mode chorus emission leading to microburst precipitation into the atmosphere. The timescale for removal of outer zone MeV electrons during the main phase of the October 1998 magnetic storm has been computed by comparing the rate of microburst loss observed on SAMPEX with trapped flux levels observed on Polar. Effective lifetimes are comparable to a day and are relatively independent of L shell. The lifetimes have also been evaluated by theoretical calculations based on quasi-linear scattering by field-aligned waves. Agreement with the observations requires average wide-band wave amplitudes comparable to 100 pT, which is consistent with the intensity of chorus emissions observed under active conditions. MeV electron scattering is most efficient during first-order cyclotron resonance with chorus emissions at geomagnetic latitudes above 30 degrees. Consequently, the zone of MeV microbursts tends to maximize in the prenoon (0400–1200 MLT) sector, since nightside chorus is more strongly confined to the equator.
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 Chorus emissions are intense whistler-mode waves, which are excited in discrete bursts following the injection of plasmasheet electrons into the region outside the plasmapause during enhanced convection. The properties of chorus have been described in previous studies [e.g., Tsurutani and Smith, 1977; Meredith et al., 2003; Santolik et al., 2003]. Chorus emissions are important because of their strong interaction with electrons in the outer radiation belt, which leads to nonadiabatic scattering. For pitch angles in the vicinity of the loss cone, such scattering leads to precipitation into the atmosphere and a net removal of energetic electrons from the outer radiation zone. The inherently bursty nature of chorus [Tsurutani and Smith, 1977; Santolik et al., 2003] causes microbursts of precipitation, which can be measured on low-altitude satellites such as SAMPEX [Lorentzen et al., 2001; O'Brien et al., 2003, 2004]. Comparison of the precipitation flux with the trapped content of the radiation belts has shown that chorus-induced microbursts could substantially deplete the MeV electron outer radiation belt during the 1–2 day duration of a magnetic storm. Microburst precipitation tends to be far more intense during the storm main phase, but it can persist during the storm recovery, in competition with acceleration processes that lead to a net flux enhancement [O'Brien et al., 2004]. Quantification of electron microburst loss is therefore important before an accurate assessment can be made of relevant acceleration processes.
 The principal purpose of this study is to quantify the rate of microburst loss for relativistic electrons and to compare the estimated lifetimes with theoretical calculations based on quasi-linear scattering. Specifically, we wish to identify the preferred spatial location for scattering and the L shell dependence of computed lifetimes and to determine the required power spectral intensity of waves to account for this loss. Data from SAMPEX and Polar are combined to obtain an estimate of lifetimes during the main phase of the October 1998 storm (section 2). A simplified formulation of pitch angle diffusion is given in section 3, and numerical results are presented in section 4. We conclude with a discussion of the relative importance of microburst precipitation compared with other loss processes in different phases of a storm.
2. Calculation of MeV Electron Lifetimes From Microbursts
 The calculation of MeV electron lifetimes due to microbursts requires data from two orbiting satellites. First, the number of trapped electrons per unit L is calculated from fluxes observed at Polar. Then, the number of electrons being lost per unit time over a small range of L is calculated from microburst fluxes observed at low altitude on SAMPEX. The trapped content in that small L range is then compared with the loss rate to determine an effective lifetime for microburst loss.
2.1. Relativistic Trapped Electron Content
 Assuming gyrotropy, we model the equatorial perpendicular trapped flux as j⊥(eq)(E, L) = Polar data is used to obtain aL and dL, assuming a dipole field and a pitch angle distribution. The pitch angle index nL at each L shell (given in Table 1) is taken from earlier measurements by Vampola . The relativistic electron content at each L shell can then be obtained from the integral
where the electron energy E is measured in rest mass units (mc2) and the L shell volume for a dipole field is (corrected from Lyons and Williams )
The dipole approximation (2) is accurate at low L but will tend to underestimate V near geostationary orbit, possibly by as much as 40%.
Table 1. Electron Pitch Angle Indices and Microburst Occurrence Cmb(L) at Each L Shell
2.2. Estimation of MeV Electron Microburst Loss
 The HILT/SSD instrument on SAMPEX measures electrons above 1 MeV with a 20-ms resolution. The local precipitation flux attributed to microbursts can be obtained by subtracting the spiky portion of the flux time series from a smooth baseline which corresponds to particles in the drift-loss cone and hence not locally precipitating [e.g., O'Brien et al., 2004]. Using this integral unidirectional microburst flux Jmb(L), the loss rate of >1 MeV electrons is
We assume that precipitation appears over Ω = 2π steradians in either hemisphere. Jmb(L) is a 20-ms average over the small latitude range (λ1 to λ2) traversed by SAMPEX during each sample. To estimate the corresponding area A(L) for precipitation loss, we further assume that the microbursts are confined to 6 hours of MLT based on the statistical size of the microburst precipitation region [O'Brien et al., 2004]. Consequently, A(L) = 2∣sin λ2 − sin λ1∣(π/2) R2, where R is the nearly constant distance from SAMPEX to the center of the Earth, and the factor of two comes from assuming the conjugate hemisphere sees similar microburst precipitation. The rate of relativistic electron precipitation loss is
The corresponding change in L shell range ΔL over the 20-ms sample can be obtained from the SAMPEX telemetry.
2.3. Calculation of Effective Lifetime
 The Polar and SAMPEX observations yield an instantaneous microburst lifetime
However, this must be adjusted to account for the occurrence Cmb(L) of microbursts as a function of L in the 6-hour dawn sector (where the microbursts are observed) and also for overcounting Q of backscattered microbursts on the second and subsequent bounces. Note that the HILT instrument has a poor angular resolution (due to its large geometric factor) and cannot resolve the local bounce loss cone [Blake et al., 1996]. The effective lifetime from microburst precipitation into the bounce loss cone is therefore
The microburst occurrence Cmb(L) is based on SAMPEX observations in the dawn sector taken from the Kp = 4–6 statistics given in Table 1. Lower values of Cmb(L) indicate less frequent microbursts and thus longer effective lifetimes. The overcounting Q is obtained by the assumption that a fraction q of the microburst flux survives each bounce into the upper atmosphere. We estimate that q is 0.675 based on the observation that an isolated bouncing microburst [Blake et al., 1996] exhibited a drop to 0.14 of its original value on its third return to SAMPEX. This third return could include two or three visits to the opposite hemisphere. We thus assume that the microburst had visited the atmosphere five times, giving q = 0.141/5 = 0.675. Assuming that each given flux spike, identified as a microburst, is actually an infinite superposition of attenuated returns of previous bursts, the oversampling correction factor Q is then given by
 Results from this analysis of microburst lifetimes are shown in Figure 1 as a function of L shell. There is considerable scatter in the instantaneous lifetimes T obtained from the 20-ms data samples on SAMPEX. Use of the dipole approximation to obtain V could cause an underestimate of lifetimes by as much as 30% at geostationary orbit, but such errors are much less at lower L. Averages over a 0.25 L shell range are better ordered and more representative of average loss times when microbursts are active. The adjusted lifetimes (6) shown by the heavy solid line, which takes into account the occurrence rate of microbursts and overcounting, indicate little systematic change with L, especially over the region where microburst occurrence is high (L > 4). The center points are median lifetimes and the error bars are 95% confidence intervals generated using Monte Carlo resampling with an exponential inflation factor of 3 added afterward to account for the strong serial correlation in the time series. The average lifetime during this storm was about a day. The much lower occurrence rate of microburst at low L < 4 is probably related to changes in the location of the plasmapause during the storm.
3. Pitch Angle Diffusion Rate
 Resonant wave-particle scattering can occur whenever the wave frequency ω is Doppler-shifted to some multiple of the relativistic electron gyrofrequency:
Diffusion codes have recently been developed [Horne et al., 2003a, 2005; Albert, 2005; Glauert and Horne, 2005] to evaluate the bounce-averaged rate of pitch angle scattering. Such codes evaluate the scattering at the Landau (n = 0) resonance and each cyclotron harmonic for a prescribed distribution of wave power. Other key inputs to the codes are models for the distribution of magnetic field and plasma density. For an arbitrary distribution of wave energy, scattering by many harmonic resonances needs to be carefully evaluated and, when bounce-averaging is performed, this tends to require a large amount of CPU time [Horne et al., 2005]. However, the propagation vectors of chorus emissions observed on Cluster are predominantly field-aligned in the equatorial source region [Santolik et al., 2003]. For nearly field-aligned waves, first-order cyclotron resonant scattering dominates over other resonances. We take advantage of this property to construct a simple analytic expression for the local rate of pitch angle scattering to evaluate diffusion rates near the loss cone. By direct comparison with results from the exact PADIE (Pitch Angle and Energy Diffusion of Ions and Electrons) diffusion code [Glauert and Horne, 2005], we show that this simple analytic treatment gives results accurate to within a factor of two near the loss cone. This is acceptable for the determination of lifetimes, since there is larger uncertainty in the measured lifetimes and in the properties of the waves. One obvious advantage of using the analytic solution is the ability to rapidly test the sensitivity of computed lifetimes to the adopted model parameters, such as the plasma density distribution and the latitudinal distribution of the waves.
 To evaluate the rate of pitch angle scattering by chorus emissions, we assume that the wave energy is distributed over a Gaussian frequency distribution
where B2(ω) is the power spectral density of the wave magnetic field (in nT2 Hz−1), ωm and δω are the frequency of maximum wave power and bandwidth, respectively (in rads s−1), and
Bw is in units of nT. The wave spectrum is also bounded by upper ωuc and lower ωlc frequency limits and is set to zero outside these limits.
 We also assume that the chorus waves are field-aligned so that scattering is entirely due to the first harmonic cyclotron resonance [e.g., Lyons et al., 1971]. With the additional approximation that the wave frequency can be ignored compared to the electron gyrofrequency in both the wave dispersion relationship so that
and in the resonance condition (8), the local rate of pitch angle scattering Dαα can be written as a simple analytic expression
where E is again measured in units of mc2, B0(λ) is the local magnetic field strength, ν = 1.764 assuming cut-off for the wave power at ωlc = ωm − 2δω and ωuc = ωm + 2δω, and the resonant wave frequency
Here α☆ = (Ωe/ωpe)2 is an important dimensionless parameter which controls the resonance condition.
 It is usually most convenient to classify trapped particles by their energy and equatorial pitch angle αeq. As electrons move along their quasi-adiabatic bounce orbits, both their pitch angle and the local cyclotron resonant frequency (13) will change. The bounce-averaged rates of diffusion [e.g., Lyons et al., 1972] are given by
λm is the mirror latitude of the electrons, v is the electron velocity, and τb(αeq) is the bounce period in a dipole magnetic field.
4. Numerical Results
 We use our analytical expression (12) for Dαα to evaluate the local rate of pitch angle scattering at different latitudes along the quasi-adiabatic bounce orbits of electrons with a prescribed energy and equatorial pitch angle. These results are then bounce-averaged using (14) to obtain the effective scattering rates near the edge of the loss cone. Owing to scattering by chorus waves, trapped electron flux will be subject to exponential decay over a lifetime comparable to the inverse of the diffusion rate near the edge of the loss cone [e.g., Thorne et al., 2005]. Consequently, we estimate that τmb ∼ 〉; the factor of 4 comes from drift averaging over the 6 hours of MLT where microburst scattering is most effective. We adopt realistic parameters for the stormtime properties of lower-band equatorial chorus (ωm = 0.35 Ωe, δω = 0.15 Ωe, Bw = 100 pT), based on the statistical study of Meredith et al. . The trough plasma density model of Sheeley et al. , N = 124 (3/L)4 cm−3, is used to simulate equatorial conditions outside the plasmapause. Density models involving two latitudinal variations are used, the first having constant density with latitude and the second with N(λ) ∼ B0(λ). These represent extreme bounds, with the former being more realistic.
 We first consider the accuracy of using the simple analytic expression (12) by comparing (see Figure 2) our approximate results, with exact solutions from the BAS diffusion code PADIE [Horne et al., 2005]. The computations are performed for 1 MeV electrons at L = 4, assuming a constant density along the field line. Two exact solutions are shown. The first (dash-dot) has a wave normal distribution confined within 1° of the ambient magnetic field, and only scattering from the dominant (n = 1) first-order cylotron harmonic is included. In the second (dashed), the waves are distributed over 30° and all resonances up to the fifth harmonic are included. In our approximate results (solid line) the neglect of ω in the resonance condition (8) and also the approximation on the whistler dispersion relation (11) lead to a misplacement of the local resonant scattering peaks, particularly for equatorial pitch angles near 90°. However, our bounce-averaged scattering rates near the edge of the loss cone are remarkably similar to both exact solutions, indicating that we can estimate electron lifetimes to within a factor of two. Because of the computational efficiency, we will use the approximate solutions to test the sensitivity of lifetimes to key parameters.
 First, consider the contribution to MeV microburst loss from local scattering at different latitudes. The lower part of Figure 3 shows local scattering rates at L = 4 for a constant density model. Wave frequencies are fixed to the equatorial values listed above, and the waves are also assumed to have amplitudes independent of latitude. Near the equator, resonance with MeV electrons is confined to a very narrow range of pitch angles near 80°. At higher latitude the peak scattering rate diminishes and moves to lower equatorial pitch angles. Significant scattering near the loss cone (αeq ∼ 5°) only begins to occur at latitudes above 30°. High-latitude chorus is therefore required to produce MeV microbursts. When bounce-averaging is performed, the net rate of scattering is greatly reduced near the loss cone but still gives an effective lifetime comparable to 2 days.
 The dependence of scattering lifetimes on L shell and on the adopted latitudinal distribution of density have also been investigated. The top part of Figure 4 indicates a very modest decrease in lifetime over the L range from 3 to 7 for the case of N(λ) constant, assuming that wave amplitude remains at 100 pT for all L. For the density model with N(λ) ∼ B0 (λ), lifetimes become longer than a year, which is clearly unrealistic. The reason for this great difference in scattering rates near the loss cone is the sensitivity of the resonance condition to the parameter α* = (Ωe/ωpe)2, which scales as B02(λ) for a constant density model and as B0(λ) when N(λ) ∼ B0(λ). The more rapid increase in α* for the constant density model allows MeV electrons to resonate effectively with chorus at smaller latitude, leading to larger bounce-averaged diffusion rates near the loss cone.
 During different phases of a storm, plasma density at a given L shell outside the plasmapause can be expected to exhibit substantial variability with MLT [Sheeley et al., 2001]. Lowest densities are expected on the nightside and during the onset of the main phase, as the plasmapause reforms at lower L, and enhanced convection transports plasma of ionospheric origin to the dayside magnetopause. In the storm recovery, ionospheric outflow causes an increase in plasma density, particularly on the dayside. We consequently evaluate the effect of the large variability in trough plasma densities reported by Sheeley et al.  on MeV scattering rates. The results shown in Figure 5 indicate that scattering lifetimes are extremely sensitive to changes in density. Lifetimes at L = 4 could drop below a day for the lower densities (10/cc) expected during the main phase of a storm and subsequently increase to over 3 days as densities build up to 50/cc during the storm recovery, without any change in the properties of the scattering waves. This conclusion is consistent with the results of O'Brien et al. , who showed that microburst loss is far more effective during the storm main phase.
 A statistical survey of CRRES chorus emissions [Meredith et al., 2003] indicates that nightside emissions are generally confined to within 15° of the equator, whereas dayside chorus becomes more intense at higher latitudes. We explore the effect of this difference on the precipitation loss of MeV electrons in Figure 6. For this simulation, the amplitude of chorus is held constant at 100 pT, up to a maximum latitude λm, and then set to zero at higher latitudes. For the case of waves confined within 15° of the equator, there is no scattering near the loss cone (∼5°). We therefore conclude that nightside chorus should not produce MeV microbursts, although scattering can occur at much lower energies. Waves must extend to at least 30° latitude for significant MeV microburst loss. Such intense high-latitude chorus emissions are found on the dayside, particularly in the prenoon sector [Horne et al., 2005] where microbusts are mainly observed [O'Brien et al., 2004].
 Intense chorus emissions which are excited during enhanced convection events are able to resonate with relativistic electrons at high latitudes leading to MeV microburst precipitation. Analysis of microbursts observed on SAMPEX during the main phase of the October 1998 magnetic storm indicates an effective lifetime of about a day throughout the entire outer radiation zone exterior to the plasmapause. Our theoretical study of bounce-averaged diffusion rates yields similar lifetimes for average wave amplitudes comparable to 100 pT, provided that such waves are present at latitudes above 30°. This confines the spatial region of MeV microbursts to the dawn to noon sector, where high-latitude chorus is usually observed [Meredith et al., 2003]. Our theoretical analysis also indicates that the rate of microburst loss is enhanced when magnetospheric plasma densities are lowest, as expected during the main phase of a storm. Microburst loss rates are expected to decrease during the storm recovery, but the persistence of chorus emissions could lead to local acceleration on timescales comparable to a few days [Horne et al., 2005]. Currently, there is little published information on the angular distribution of high-latitude dayside chorus. It is possible that the angular distribution of chorus could change with both latitude (as predicted by ray tracing [e.g., Horne and Thorne, 2003]) and MLT. The effect of any change in the wave angular distribution will need to be investigated when better information becomes available.
 Interestingly, our determination of the effective lifetime of relativistic electrons by microburst precipitation during the main phase of a storm (∼day) is comparable to the recent calculations for the combined scattering by stormtime plasmaspheric hiss and EMIC waves [Albert, 2003; Summers and Thorne, 2003], suggesting that all three waves play an important role in stormtime electron dynamics. Outer zone MeV electron lifetimes of less than a day (for storm main phase conditions) have also been reported from radial diffusion modeling [Shprits and Thorne, 2004]. Such rapid net rate of loss will compete with any potential acceleration process and needs to be carefully taken into account. It appears that the more rapid loss during the main phase of a storm generally exceeds any acceleration source and thus leads to a net depletion of the relativistic outer zone population. A net increase in MeV electron flux can occur during the recovery of certain storms [Reeves et al., 2003], presumably when source processes dominate over weakened loss mechanisms.
 Enhanced radial diffusion [e.g., Elkington et al., 2003] and local stochastic acceleration during resonant interactions with chorus [e.g., Horne et al., 2005] are leading candidates to explain the observed relativistic electron flux enhancements during the storm recovery phase. Both processes are expected to be important due to the observed stormtime enhancement in the power spectral density of ULF and VLF waves [O'Brien et al., 2003]. It is intriguing that scattering by chorus can act simultaneously as a loss mechanism (owing to pitch angle scattering near the loss cone) and as a source of local acceleration. This is possible because the rate of energy diffusion maximizes for pitch angles well away from the loss cone [e.g., Horne et al., 2005], while the bounce-averaged pitch angle diffusion rates minimize near the loss cone (Figure 2). Theoretical modeling [Thorne et al., 2005] has shown that the combined effect of pitch angle and energy diffusion leads to the observed development of flat-topped pitch angle distributions (with steep gradients near the loss cone) [Horne et al., 2003b] and peaks in phase space density [Green and Kivelson, 2004]. The effect of radial diffusion needs to be included in future studies by an in-depth analysis with three-dimensional diffusion codes in order to quantify the relative role of each class of wave on stormtime electron dynamics.
 This research was funded in part by NASA grants NAG5-11922 and NNG04GN44G. The work was also supported by NSF GEM grant ATM-0402615. We thank R. Selesnick and M. Looper for assistance with the Polar and SAMPEX data. D. S. acknowledges support from the Natural Sciences and Engineering Research Council of Canada under grant A-0621.
 Arthur Richmond thanks Jay Albert and Reiner Friedel for their assistance in evaluating this paper.