Outer Radiation Belt Electron Lifetime Model Based on Combined Van Allen Probes and Cluster VLF Measurements

The flux of energetic electrons in the outer radiation belt shows a high variability. The interactions of electrons with very low frequency (VLF) chorus waves play a significant role in controlling the flux variation of these particles. Quantifying the effects of these interactions is crucially important for accurately modeling the global dynamics of the outer radiation belt and to provide a comprehensive description of electron flux variations over a wide energy range (from the source population of 30 keV electrons up to the relativistic core population of the outer radiation belt). Here, we use a synthetic chorus wave model based on a combined database compiled from the Van Allen Probes and Cluster spacecraft VLF measurements to develop a comprehensive parametric model of electron lifetimes as a function of L‐shell, electron energy, and geomagnetic activity. The wave model takes into account the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. We provide general analytical formulas to estimate electron lifetimes as a function of L‐shell (for L = 3.0 to L = 6.5), electron energy (from 30 keV to 2 MeV), and geomagnetic activity parameterized by the AE index. The present model lifetimes are compared to previous studies and analytical results and also show a good agreement with measured lifetimes of 30 to 300 keV electrons at geosynchronous orbit.


Introduction
Energetic electron fluxes in the outer radiation belt show high variability, especially during geomagnetically disturbed conditions where the fluxes of electrons can vary by several orders of magnitude (Reeves et al., 2003) over a period of an hour or less. Such extreme variations in relativistic electron fluxes can cause significant malfunctions and unexpected failures of spacecraft electronics (Horne, 2007). Wave-particle interactions with very low frequency (VLF) chorus waves play an important role in controlling the flux variation of these particles (Bortnik & Meredith, 2008;Bortnik, Thorne, Meredith, & Santolik 2007;Millan & Thorne, 2007;. Chorus waves significantly contribute to the acceleration and scattering of energetic electrons into the loss cone. These waves are right-hand-polarized, intense electromagnetic whistler mode emissions with short, relatively coherent and repetitive rising or falling tones. The most intense emissions are observed during geomagnetically active conditions and are located from premidnight to postmidday Aryan et al., 2014Aryan et al., , 2016Aryan et al., , 2017Boynton et al., 2018;Burtis & Helliwell, 1976;Li et al., 2011;Meredith et al., 2001;Santolík et al., 2004). They are observed in two distinct frequency bands: the lower band chorus 0.1-0.5f ce and the upper band chorus 0.5-0.8f ce with a power gap separating the bands at 0.5f ce (Sazhin & Hayakawa, 1992;Tsurutani & Smith, 1974), where f ce is the equatorial electron gyrofrequency. It is known that the two bands interact with different electron energy populations. The upper band chorus waves interact mainly with the lower energy electrons (<50 keV) , whereas the lower band parallel and oblique chorus waves can affect electrons on a much wider energy range from 0.1 keV to multi-MeVs  and contribute strongly to the quickly evolving dynamics of radiation belt electron fluxes Horne et al., 2005;Li, Ma et al., 2016;Mourenas et al., 2014;Mourenas et al., 2016;Thorne et al., 2013;Tu et al., 2014).
It is crucial to quantify the effects of these interactions for accurately modeling and forecasting the global dynamics of the outer radiation belt and for providing a comprehensive description of electron flux variations over a wide energy range, from the source population of ∼10to 50 keV electrons up to the relativistic core population of the outer radiation belt. This requires complete spatiotemporal coverage of the inner magnetosphere to provide comprehensive and accurate statistics of chorus waves encompassing the entire parameter space. However, it is unlikely to achieve such comprehensive coverage with a given spacecraft mission. Therefore, in this study, we use a synthetic empirical wave model, developed by Agapitov et al. (2018) based on combined VLF measurements from the Van Allen Probes and Cluster spacecraft, to develop a comprehensive parametric model of electron lifetimes in the outer radiation belt as a function of geomagnetic activity (AE), L-shell (L), electron energy (E), and magnetic local time (MLT). The wave model takes into account the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. The resulting synthetic statistical model of chorus wave amplitude, obliquity, and frequency is presented in the polynomial form of geomagnetic activity level in three MLT sectors, L-shell, and for a range of electron energies. In the next section we describe the synthetic chorus wave model in detail and how we use the model to calculate electron lifetimes in the outer radiation belt. We then describe and discuss the results and compare them with analytical results and measured data.

Synthetic Chorus Wave Model
The synthetic model describing chorus waves amplitudes and the wave normal angle distribution (Agapitov et al., , 2018 has been previously derived from the combined statistics of the Van Allen Probes and Cluster satellites VLF measurements. The combined chorus data set includes more than 5 years of Van Allen Probes and 10 years of Cluster VLF measurements. The Van Allen Probes provide excellent coverage of relatively low L-shells and low latitudes, but the coverage becomes more sparse at high latitudes and terminates at L > 6. In contrast, the Cluster spacecraft provide good coverage of high-latitude (up to λ = 45°) and high-L (L > 6) regions, but limited coverage of the low latitudes, especially in the night sector. Hence, the combination of the two data sets provides complementary coverage of geomagnetic latitudes up to 45°i n the chorus frequency range 0.1f ce to 0.8f ce . The database of VLF wave measurements includes the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. The resulting synthetic model of chorus waves properties includes fits for the wave amplitude, wave frequency, and wave normal angle distribution as a function of magnetic latitude (λ), MLT, and L above the plasmapause. The synthetic chorus model provides the wave amplitude distribution from the equator up to λ = 45°for Kp = 0-6 and L ≈ 4-7. For the purpose of this study, we have modified the model to consider three ranges of AE index. The AE index is one of the most influential parameters that define chorus wave activity (Aryan et al., 2014;Boynton et al., 2018;Zhang et al., 2018). The average amplitude of chorus waves can be well parameterized by AE, which provides a measure of injections and convection of energetic electrons that generate such waves (Li et al., 2011;Meredith et al., 2001;Shprits et al., 2007). Therefore, the present study paves the way for developing a future multiparameter wave model that will also include solar wind parameters (Aryan et al., 2014(Aryan et al., , 2017. An approximately dipolar geomagnetic field and McIlwain L-shells are considered in the chorus model (Agapitov et al., , 2018. This dipolar field model can become inaccurate at L > 5 during magnetic storms or strong substorms with AE > 200-300 nT. Therefore, to mitigate the potential impact of errors in L-shells on the chorus model, two relatively wide L-shell bins are considered, extending from the plasmapause (L ∼ 3-4) to L = 5, and from L = 5 to L = 7, respectively (Agapitov et al., , 2018. In addition, there is presently no magnetic field model that accurately incorporates substorm effects through the AE index, and all existing disturbed field models differ significantly from each other (Huang et al., 2008;Lu et al., 1999). The chorus wave model dependence of wave amplitude Bw on λ is presented in Figure 1, and the corresponding tables of coefficients of the model fits can be found in Agapitov et al. (2018). The wave normal angular distribution of lower band chorus is presented as a combination of field-aligned wave normal angle population with Gaussian distribution around the background magnetic field direction and the population of oblique waves with Gaussian distribution in the wave normal range between the local Gendrin angle and the resonance cone as presented in Equation 1 (Agapitov et al., , 2018: where the factor Q depends on L, λ, MLT, and geomagnetic activity with the approximation from Agapitov et al. (2018) (presented in Figure 1), θ 1 and θ 2 , δθ 1 ≈ δθ 2 ≈ 8°were determined by Agapitov et al. (2015) and can be approximated by polynomials on λ:θ 1 = 11.5 + 14.3λ/10 − 8.1(λ/10) 2 + 1.2(λ/10) 3 and θ 2 = 66 + 0.1(λ/10). The model takes into account the significant population of very oblique waves recorded in the night/morning sector by the Van Allen Probes at low latitudes during disturbed periods Li et al., 2016). The chorus mean frequency depends on λ (as reported earlier by Breuillard et al., 2012, and Bunch et al., 2013 that is represented in Agapitov et al. (2018) by the linear dependence of the mean frequency f m : f m /f ce = 0.35 − 0.0125λ with a variance of 0.07. This frequency dependence decreases the latitude of cyclotron resonances and (as Bw goes down with λ) increases 2-5 times the effective wave amplitude (Agapitov et al., 2018).

Electron Lifetimes
Electron decay time constants (i.e., lifetime) estimates are very useful for radiation belt modeling, for instance, to accurately incorporate electron loss in radial diffusion models with a loss term (Schulz & Lanzerotti, 1974). In this study, the synthetic chorus wave model described in the previous section is used to calculate the local pitch angle diffusion coefficients (D αα ) for electrons over a wide energy range (1 keV to 2 MeV), as a function of geomagnetic activity, L, and MLT. Figure 2 shows the local bounce-averaged pitch angle diffusion coefficients (D αα ) at L = 4 for quiet (left: AE < 100 nT), moderate (middle: 100 ≤ AE ≤ 300 nT), and active (right: AE > 300 nT) geomagnetic conditions as a function of electron energy and equatorial pitch angle (α ∘ eq ). The D αα values are shown for night (row 1: MLT > 21 and MLT ≤ 03), dawn (row 2: 03 < MLT ≤ 09), day (row 3: 09 < MLT ≤ 15), and averaged MLT. Overall, the electron scattering rates are low during quiet conditions but significantly increase during moderate and active geomagnetic conditions, especially for low-energy electrons (<100 keV). For moderate and high geomagnetic activities, a significant scattering is observed for electrons with low energies (less than ∼100 keV) and small pitch angles, which are rapidly precipitated into the atmosphere. However, for quiet geomagnetic conditions, the scattering rates appear relatively weak, and the highest electron scattering rates are observed for electrons with energies in the range of approximately 10-1,000 keV and are enhanced during moderate and high geomagnetic activities. Also, the scattering rates vary noticeably across MLT. Dawn and night sector scattering is generally stronger than dayside scattering, especially for low-energy <100 keV electrons or for high energy >100 keV electrons with high pitch angles, due to the more intense chorus wave activity near the equator (where these electrons reach cyclotron resonance with the waves) on the dawn/night sector compared to day sector. The MLT-averaged diffusion rates have similar features of intense scattering as on the dawn side.
We studied the local pitch angle diffusion coefficients for a range of L-shell from L = 3.0 to L = 6.5. Figure 3 shows the MLT-averaged pitch angle diffusion coefficients (D αα ) at L = 3.0-6.5 (top to bottom) for quiet (left), moderate (middle), and active (right) geomagnetic conditions as a function of electron energy and equatorial pitch angle ( α ∘ eq ). In general, the rate of electron scattering depicts a similar trend across all L-shells (weak scattering during quiet conditions, but intense scattering during moderate and active geomagnetic conditions for low-energy electrons with energies of approximately less than 100 keV). However, there is an increase in the rate of electron scattering at higher L-shells, for all geomagnetic conditions, where chorus wave activity is more intense. The scattering rates can vary by up to an order of a magnitude in MLT, especially, for electrons with energies of ∼10-1,000 keV during quiet conditions. Nevertheless, as we are interested in calculating electron lifetimes over hours to days, and since >30 keV electrons drift azimuthally around the Earth in less than 1 hr at L > 4, it is justified to consider the MLT-averaged diffusion rates actually experienced by such electrons. Such MLT-averaged diffusion rates have been checked in past works to accurately provide equilibrium electron loss rates following storms ).
The lifetimes of electrons are calculated by integrating 1/[4 * tan(α eq ) * D αα ] from small α eq = α LC (at the loss cone) up to α eq ≈ 60°and for the MLT-averaged D αα larger than 10 −8 ) (Albert & Shprits, 2009;Artemyev et al., 2013;Mourenas et al., 2012). Figure 4 shows electron lifetimes as a function of L and electron energy for quiet (left), moderate (middle), and active (right) geomagnetic conditions. During quiet conditions, electron lifetimes are relatively long, >10 days, for a wide range of electron energies. However, the lifetimes become shorter during moderate and active geomagnetic conditions, especially, for low-energy electrons (<100 keV) that can be quickly (in less than 1-3 hr) precipitated into the atmosphere by chorus waves. This strong reduction of low-energy electron lifetimes is mainly due to the increase of MLT-averaged magnetic chorus wave power with AE at low latitudes <20°(see Figure 2), where such electrons reach cyclotron resonance with quasi-parallel waves. In fact, the ∼10to 50 keV electron population provides the free energy source for chorus waves generation (see Figure 1 from Agapitov et al., 2018).
Various studies have noted the importance of electron loss due to resonant chorus wave-particle interactions (Albert & Shprits, 2009;Thorne, 2010). The present results are comparable with previous studies, in particular with the results from Orlova and Shprits (2014) Figure 1 here; see also Mourenas et al., 2014;Agapitov et al., 2018). Here, we shall consider for simplicity the expression derived for quasi-parallel waves , valid for E[keV] > 30(6.6/L) 2 (for cyclotron resonance to be available), assuming a plasma density variation given by the statistical model of Sheeley et al. (2001).
At L = 6.6 (i.e., Geosynchronous Earth Orbit), we use MLT-averaged and latitude-averaged (over 10-30°) chorus wave amplitudes Bw = 6 and 12 pT during quiet and moderately disturbed geomagnetic conditions, respectively, in rough agreement with Figures 1a-1c results. During active periods, however, the wave amplitude decreases significantly at higher latitudes. In addition, we should consider the wave amplitude Bw at the latitude of cyclotron resonance, that is, near 13°for 30 keV, 18°for 100 keV, and 30°for 1 MeV (e.g., see Figure 2 from Agapitov et al., 2018). Note that the decrease of the wave mean frequency toward

10.1029/2020JA028018
Journal of Geophysical Research: Space Physics higher latitudes in our chorus model leads to a decrease of the latitude of electron cyclotron resonance with the waves, as compared with the case of a constant mean frequency ∼ 0.35f ce , and it gives also a slightly faster decrease of this latitude of cyclotron resonance from high to low electron energy (Agapitov et al., 2018). All this leads us to take approximately Bw = (30 pT) * 1/(1+E[keV]/511) 1/2 during active periods. Figure 5a shows electron lifetimes as a function of electron energy, allowing us to compare the results of the present full numerical model (solid curves) with the above analytical lifetime estimates (dashed curves) valid only for E > 30(6.6/L) 2 keV-that is for electron diffusion through cyclotron resonance with the waves (Mourenas et al., 2012). The present model results are in rough agreement with the analytical estimates over many decades. In particular, the model lifetimes follow approximately the same analytical scaling with energy given by Equation 2, except at low energies <70 keV during periods of elevated AE activity. This discrepancy at low energies for high AE is due to the steep peak of wave power present at low latitudes (where cyclotron resonance occurs at low energies) in the night/dawn sector when AE > 300 nT (see Figure 1d), which is not fully taken into account in the above analytical estimates. For a constant chorus wave amplitude Bw (that would not decrease with latitude), both the analytical and full numerical lifetimes would increase less rapidly with energy in Figure 5a, especially during moderate to  high AE periods, characterized by a faster increase of lifetimes with energy due to the faster decrease of Bw with latitude. Figure 5b allows us to compare the model results with the measured average electron lifetimes at L = 6.6 , together with 10th and 90th percentiles. The model lifetimes, corresponding to quiet periods, agree well with measured lifetimes for moderate energy (<300 keV) electrons. At higher energies (>300 keV), the measured lifetimes become smaller than model lifetimes during low AE activity (green solid line), falling in closer agreement with model lifetimes for moderate activity (100 < AE < 300 nT). This behavior is probably partly due to the upper limit of <20 days imposed on measured lifetimes by the method of empirical lifetime determination, in the presence of low measured fluxes at high E . Possible physical causes of discrepancies between estimated and measured lifetimes will be discussed in the next section.
As shown in Figure 5a the model lifetimes at GEO follow approximately the same analytical scaling with energy given by Equation 2. Here we refine and generalize Equation 2, deriving analytical lifetime fits to the full model lifetimes as a function of energy and L-shell in the range of 30 ≤ E ≤ 2,000 keV and 3 ≤ L ≤ 6.5, respectively. We use the analytical formula in Equation 2 to find numerically the polynomial function Bw(L, E) that provides the best agreement at all (E, L) with model lifetimes. Table 1 shows the average Bw that provides the best agreement between analytical and model lifetimes as a function of E and L for quiet (AE < 100 nT), moderate (100 ≤ AE ≤ 300 nT), and active (AE > 300 nT) geomagnetic conditions. In general, for quiet and moderate geomagnetic conditions the average Bw does not change significantly with E. Therefore, for quiet and moderate geomagnetic conditions we simply drive polynomial fits for Bw as a function of L given by Equations 3 and 4, respectively. The polynomial functions presented in Equations 3-5 are substituted into Equation 2 to provide general analytical formulas given by Equations 6-8: ð−0:5L þ 9:1ÞðL=6:6Þ 3=4 2 (6) ð−2:2L þ 28:9ÞðL=6:6Þ 3=4 2 (7)

Journal of Geophysical Research: Space Physics
The generalized analytical lifetime formulas (Equations 6-8) can be used to estimate electron lifetimes for different geomagnetic conditions as a function of energy and L-shell in the range of 30 ≤ E ≤ 2,000 keV and 3 ≤ L ≤ 6.5, respectively. Figure 6 shows electron lifetimes as a function of L and electron energy for quiet (left), moderate (middle), and active (right) geomagnetic conditions. The solid curves indicate numerical model lifetimes at L = 0.5 intervals for 3 ≤ L ≤ 6.5. The black dashed curves represent the derived best fit analytical lifetime estimates calculated using Equations 6-8 for different geomagnetic conditions. It is clear that the best fit analytical lifetime estimates agree well with full numerical model lifetimes and can be used in general to estimate lifetimes as a function of L and E in the electron energy range of 30≤ E ≤ 2,000 keV and L-shell in the range of 3≤ L ≤ 6.5, although the analytical lifetime energy scaling used in the fits can become inaccurate for E[keV] <30(6.6/L) 2 (for all AE), as noted earlier .

Discussion
Many studies have been devoted to calculate electron lifetimes throughout the radiation belts (Albert & Shprits, 2009;Baker et al., 2013;Claudepierre et al., 2020;Fennell et al., 2013;Meredith et al., 2002). Precise calculations of electron lifetimes are crucial for accurately modeling and forecasting the global dynamics of the outer radiation belt and for providing a comprehensive description of electron flux variations over a wide energy range. Predicting the magnitude and duration of potentially hazardous conditions may help satellite operators to switch off nonessential satellite electronic systems to reduce malfunctions and unexpected failures only during the most dangerous periods (Horne, 2007). In this study a synthetic chorus wave model is used to calculate the local pitch angle diffusion coefficients (D αα ) for electrons over a wide energy range (1 keV to 2 MeV), as a function of geomagnetic activity, L, and MLT. The pitch angle diffusion rates are calculated numerically, assuming that the refractive index of very oblique chorus waves cannot exceed some realistic limits imposed by Landau damping and hot plasma effects (see details in Li, Thorne et al., 2014, andMourenas et al., 2014, in good agreement with observations; Ma et al., 2017). We then used the full simulation results of MLT-averaged pitch angle diffusion rates to calculate electron lifetimes in a range of L-shells (from L = 3.0 to L = 6.5) and for a wide range of electron energies (from 1 keV to 2 MeV) as a function of geomagnetic activity.
The importance of electron loss due to resonant chorus wave-particle interactions have been highlighted by various studies (e.g., Albert & Shprits, 2009;Thorne, 2010). Here the results of the present model of electron lifetimes are compared with analytical lifetime estimates provided by Mourenas et al. (2014) and with actual lifetime measurements at GEO ). The present model results (at L = 6.5) are in rough agreement with analytical estimates (at L = 6.5) over many decades. In particular, the model lifetimes follow approximately the analytical scaling with energy given by Equation 2, except at low energies (<70 keV) and for periods of moderate to high AE activity. This is probably partly due to the steep increase of the wave amplitude at low latitudes during disturbed periods, which is not fully taken into account in the analytical estimates. This discrepancy could also be related to the presence of an additional small population of very oblique chorus waves that may reduce lifetimes by a factor 2 as compared with quasi-parallel waves alone for electron energies 30-100 keV (Li, Thorne et al., 2014;Mourenas et al., 2014).
The model lifetimes calculated at L = 6.5 have also been compared in Figure 5b with empirical electron lifetimes obtained from 20 years of daily averaged measurements performed by LANL spacecraft at GEO, mainly during low geomagnetic activity. It is worth noting that such GEO measurements actually span a finite range of L-shells. Baker et al. (2019) have notably shown that the geosynchronous spacecraft GOES 15 can span L ∼ 6 to 8 (more often L ∼ 6.4-7.5) depending on local time and geomagnetic conditions. Nevertheless, the considered L bin of the chorus wave model is relatively wide, extending from L = 5 to L = 7, and chorus wave parameters (amplitude and wave normal angle) remain constant within this bin.
Since model lifetimes are approximately proportional to ∼1/(Bw 2 L 3/2 ) (see Equation 2), they should vary by less than 15% about their value at L = 6.5 inside the range 6 ≤ L ≤ 7, limiting potential discrepancies with the measured lifetimes due to differences in L-shells.
The present model results at L = 6.5 are in rough agreement with actual lifetimes measured at GEO mainly during weak geomagnetic activity with average Dst ∼ −20 to −11 nT, often during the late recovery phase of storms . In particular, the model lifetimes for nearly quiet periods (AE < 100 nT) agree well with the measured lifetimes of moderate energy (20-300 keV) electrons. At higher energies (>300 keV), however, the measured lifetimes become smaller than model lifetimes during low AE activity, falling in closer agreement with model lifetimes for moderate activity (100 < AE < 300 nT). This behavior is probably partly due to the upper limit of <20 days imposed on measured lifetimes by the method of lifetime determination used by Boynton et al. (2014). Lifetimes larger than ≃18-20 days, corresponding to less than 18-20% reductions in electron flux over a typical flux decay interval (for lifetime evaluation) of 4 days, are indeed very unlikely to be identified by the considered method at energies of 1 MeV or 2 MeV, where electron fluxes are much smaller than at lower energy and a small fluctuation in electron count rates can easily suppress such a weak decay . Based on the model lifetimes displayed in Figure 5b, such relatively short <20 days measured 1-MeV electron lifetimes can be obtained only during moderately active periods with AE > 100 nT-corresponding to higher average chorus wave amplitudes Bw and shorter lifetimes than when AE < 100 nT (Li et al., 2011;Shprits et al., 2007). Note that MeV electron fluxes generally reach their peak level at the end of prolonged periods of high substorm activity (AE > 200-500 nT, often during early storm recovery), because such electrons need to be accelerated by chorus waves or inward radial diffusion from the bulk of seed <200-to 300-keV electrons directly injected from the plasma sheet (Mourenas et al., 2019;Reeves et al., 2003;. Such peaks of MeV electron flux often start to decay a few days later at GEO Mourenas et al., 2019), during periods that can sometimes remain moderately active, with AE ∼ 100-150 nT. Overall, the present model lifetimes therefore appear in relatively good agreement with observations.
Besides, MeV electron fluxes at GEO often experience rapid dropouts via magnetopause shadowing, caused by sudden impulses of solar wind dynamic pressure or increased southward interplanetary magnetic field (Onsager et al., 2007). However, such rapid dropouts of electron flux by factors >10 over less than 1 day (Boynton et al., 2016;Onsager et al., 2007) are automatically excluded from consideration by the procedure of lifetime selection used by Boynton et al. (2014), limiting the measured lifetimes to slow, wave-driven decay timescales >0.4 days. Although magnetopause shadowing and the related rapid dropouts cannot be invoked as a direct cause of the relatively short lifetimes found by Boynton et al. (2014) at geosynchronous orbit, they could still be the indirect cause of a somewhat slower loss, by leading to a steepening of the negative outward gradient of MeV electron phase space density, thereby helping outward radial diffusion of MeV electrons toward higher L. Indeed, various studies have shown that there is usually a transition from positive to negative gradients of electron phase space density in the outer radiation belt for a magnetic moment μ ≈ 200 MeV/G, corresponding at L ∼ 6.6 to 0.5-0.2 MeV for pitch angles α 0 ∼ 40-90° (Turner et al., 2012). By preferentially scattering high-energy electrons outward (toward lower phase space density), radial diffusion can make the observed lifetimes at a fixed L = 6.6 appear smaller above 300 keV. Based on an analytical formulation for the electric field radial diffusion rate due to ultralow frequency (ULF) waves (Ozeke et al., 2014), the corresponding loss timescale of 1-MeV electrons due to outward radial diffusion can be very roughly estimated as ∼10 days for Kp ∼ 1 and a typical phase space density gradient scale length of 1 Earth radius at geosynchronous orbit . This effect could therefore explain the shorter mean measured lifetimes <10 days at 1 MeV. Numerical simulations incorporating realistic geomagnetic fields, radial diffusion, chorus-induced loss, and the observed electron phase space density gradients would be necessary to better assess this point.
The general polynomial functions given in Equations 6-8 should be very useful to accurately estimate electron lifetimes needed in radiation belt models, under different geomagnetic conditions and for electrons in the energy range of 30 ≤ E ≤ 2,000 keV, for L-shells in the range of 3 ≤ L ≤ 6.5. Interestingly, Figure 6 shows that the general polynomial fits given in Equations 6-8 are valid over L = 3.0-6.5, thanks to the weak variation of model lifetimes with L for a fixed energy. This is due to the relatively weak variation of the lifetimes ∼1/L 3/2 for constant Bw(L) (see the analytical lifetime estimate in Equation 2) and to the relatively weak variation of the MLT-averaged Bw with L at a given latitude of cyclotron resonance, below ∼30°(e.g., see Figure 1).

Conclusion
In this study, we used a synthetic chorus wave model, developed by Agapitov et al. (2018), based on the combined VLF measurements from the Van Allen Probes and Cluster spacecraft to develop a comprehensive parametric model of electron lifetimes in the outer radiation belts. The model takes into account the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. We used the resulting comprehensive synthetic chorus wave model to calculate the local pitch angle diffusion coefficients (D αα ) for electrons over a wide energy range (1 keV to 2 MeV), as a function of geomagnetic activity level, L, and MLT. We then used the results to estimate electron lifetimes at a range of L-shell (from L = 3.0 to L = 6.5) as a function of electron energy. The results were compared with previous studies, including analytical results and measured data at GEO. We generalized the analytical formula in Equation 2 by deriving numerically the polynomial function Bw(L, E). The resulting generalized analytical formulas, given by Equations 6-8, can be used to estimate electron lifetimes as a function of L-shell (for L = 3.0 to L = 6.5), electron energy (from 30 keV to 2 MeV), and different geomagnetic conditions. Overall, the present model lifetimes appear in relatively good agreement with observations, previous studies and analytical results. The results presented in this study are useful for the scientific community. Precise calculations of electron lifetimes are crucial for accurately modeling and forecasting the global dynamics of the outer radiation belt.

Data Availability Statement
The data of the synthetic chorus wave model are available in the form of tables of coefficients in Agapitov et al. (2018), and the original wave data used to develop this model are freely available in the RBSP/EFW database (http://www.space.umn.edu/missions/rbspefw-home-university-of-minnesota/) and in the Cluster Active Archive (https://caa.esac.esa.int/caa/). Wave data from the Van Allen Probes were obtained by the EMFISIS and EFW instruments (Kletzing et al., 2013;Wygant et al., 2013). The Cluster data are gained by STAFF instrument (Cornilleau-Wehrlin et al., 1997), which is part of the WEC wave consortium controlled by DWP (Woolliscroft et al., 1997). Lifetimes measured at Geostationary Earth Orbit are available in Boynton et al. (2014).