### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[1] The dynamics of inner radiation belt electrons are governed by competing source, loss, and transport processes. However, during the recent extended solar minimum period the source was inactive and electron intensity was characterized by steady decay. This provided an opportunity to determine contributions to the decay rate of losses by precipitation into the atmosphere and of diffusive radial transport. To this end, a stochastic simulation of inner radiation belt electron transport is compared to data taken by the IDP instrument on the DEMETER satellite during 2009. For quasi-trapped, 200 keV electrons at*L*= 1.3, observed in the drift loss cone (DLC), results are consistent with electron precipitation losses by atmospheric scattering alone, provided account is taken of non-diffusive wide-angle scattering. Such scattering is included in the stochastic simulation using a Markov jump process. Diffusive small-angle atmospheric scattering, while causing most of the precipitation losses, is too slow relative to azimuthal drift to contribute significantly to DLC intensity. Similarly there is no contribution from scattering by VLF plasma waves. Energy loss, energy diffusion, and azimuthal drift are also included in the model. Even so, observed decay rates of stably-trapped electrons with*L* < 1.5 are slower than predicted by scattering losses alone, requiring radial diffusion with coefficient *D*_{LL} ∼ 3 × 10^{−10} s^{−1} to replenish electrons lost to the atmosphere at low *L* values.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[2] The intensity of inner radiation belt electrons is characterized by occasional rapid increases, or injections, followed by periods of steady decay [*Rosen and Sanders*, 1971]. Natural injections, more frequent at lower energies and during geomagnetically active times, are not well understood. Artificial injection by high-altitude nuclear detonations, particularly by the Starfish test in 1962, provided an opportunity to study the decay process from high intensity levels. It was found to be caused by precipitation losses from electron scattering in the residual upper atmosphere [*Walt*, 1964] combined with radial diffusion [*Newkirk and Walt*, 1968; *Farley*, 1969].

[3] In the decades since nuclear testing other radiation belt processes have been studied, the principal one for inner belt electrons being scattering by whistler mode plasma waves. Such waves originating from ground-based VLF transmitters were thought to cause significant precipitation losses [*Abel and Thorne*, 1998]. (Further out, scattering by plasmaspheric hiss and lightning-generated waves cause losses that form the slot region, separating the inner and outer belts [*Lyons and Thorne*, 1973; *Meredith et al.*, 2007].) Precipitation caused by the naval communication station in western Australia (call sign NWC) has been clearly observed [*Sauvaud et al.*, 2008; *Gamble et al.*, 2008], and this station was not operational during the decay of Starfish electrons. However, recent satellite measurements of VLF wave intensity showed a significant deficit relative to earlier models [*Starks et al.*, 2008], raising doubts as to whether wave-induced precipitation is an important inner-belt process.

[4] The recent solar minimum was a particularly inactive period for the radiation belt [*Russell et al.*, 2010]. During 2009 there were no significant inner-belt injections. It was therefore a time suited to studying the decay of the inner radiation belt with modern instrumentation. Contributions of candidate loss mechanisms can be isolated from radial diffusion, and other source processes, by measuring the intensity of quasi-trapped electrons in the drift loss cone [*Selesnick*, 2006] (the DLC is the region of phase space where eastward drifting electrons are destined to enter the dense atmosphere to the west of the South Atlantic Anomaly, or SAA). Our goal in this work is to determine whether the inner radiation belt DLC intensity, measured by the IDP instrument on the DEMETER satellite [*Sauvaud et al.*, 2006], is consistent with electron precipitation caused by atmospheric scattering alone.

[5] Scattered electrons in the DLC must be replenished in less than a drift period. We will see that accurate description of this process requires inclusion of non-diffusive, wide-angle scattering. This is achieved by Monte Carlo simulation of electron paths through phase space using methods of stochastic calculus.

[6] We begin below by introducing the stochastic model equations as they apply generally to pitch angle diffusion, energy diffusion and loss, radial diffusion, and azimuthal drift (section 2). Following a description of the necessary model inputs (section 3), we review formulae for diffusion and energy transport from atmospheric scattering, including effects of free and bound electrons, plasma ions, and atomic nuclei (section 4). Then we discuss an alternate formulation of the nuclear contribution based on a jump process that includes wide-angle scattering (section 5). Sample stochastic paths illustrate both the diffusive and jump processes (section 6), after which respective model simulations are compared to data as a function of drift phase. Finally, radial diffusion is included to match observed inner belt electron decay rates (section 7).

### 2. Stochastic Model Description

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[7] The radiation belt electron distribution function *f* is generally a function of the three adiabatic invariants *M*, *J*, and Φ, and of the corresponding phases for gyration, bounce motion, and azimuthal drift [*Schulz and Lanzerotti*, 1974]. The phase dependences are often neglected, but we retain a dependence on drift phase *ϕ* for comparison with the DLC data. Evolution of *f* is usually described by a diffusion equation, perhaps with additional terms depending on the relevant transport processes. This is equivalent to a set of stochastic differential equations describing the motion of individual electrons. The prototype SDE for coordinate *q* as a function of time *t* is *dq* = *μdt* + *σdW* where, in a Hamiltonian system, −*μ* = ∂*D*/∂*q*is the Fokker-Planck drag coefficient, for diffusion coefficient *D*, the Brownian motion (or Wiener process) , and *G*_{0,1} is a Gaussian deviate of zero mean and unit variance. These equations have recently found application in space physics and radiation belt dynamics [*Zhang*, 1999; *Tao et al.*, 2008]. Derivations, and descriptions of stochastic calculus in general, are readily available [e.g., *Gardiner*, 1985; *Klebaner*, 2005].

[8] To describe specific transport processes we change variables to *ϕ*, kinetic energy *E*, *x* = cos *α*_{0} where *α*_{0} is equatorial pitch angle at a reference *ϕ* = *ϕ*_{0}, and *L* = 2*πk*/(*a*Φ) where *k* is the geomagnetic dipole moment and *a* Earth's radius. The *x* coordinate is more convenient than the equivalent mirror magnetic field, *B*_{m}, which is conserved during azimuthal drift while *α*_{0} generally is not. For diffusion in the new coordinates, evaluation of drag coefficients, −*μ* = (∂*DG*/∂*q*)/*G*, involves a Jacobian *G* of the transformation from *M*, *J*, and Φ to *E*, *x*, and *L*, which (neglecting drift shell splitting) is proportional to *γpxS*_{b}/*L*^{2}(generalized for a non-dipolar magnetic field from*Schulz and Lanzerotti* [1974, p. 56], with help from *Roederer* [1970, equation 2.39]). Here *p* = *γmv* is momentum, *γ* is the relativistic Lorentz factor for electron rest mass *m* and speed *v*, and *S*_{b} = *vτ*_{b}/2 for bounce period *τ*_{b}.

[9] Beginning with radial transport, the stochastic differential equation, or SDE, for *L* with diffusion coefficient *D*_{LL} is

Radial diffusion is assumed to conserve *M* and *J* [*Birmingham et al.*, 1967] and so is accompanied by adiabatic changes in the equatorial pitch angle cosine, *x*, and the kinetic energy, *E.* They can be related to *B*_{m} and , which are determined only by the magnetic field model [*Roederer*, 1970]. By setting the total derivative *dK* = 0, the adiabatic change in *x* associated with *dL* is

Conservation of *M* = *p*^{2}/(2*mB*_{m}) results in the adiabatic kinetic energy change,

where

The partial derivatives for equations (2) and (4) are evaluated numerically from the field model (section 3).

[10] The SDE for *x* with diffusion coefficient *D*_{xx} is

It includes *dx*_{a} and a Markov jump process *dA* [*Klebaner*, 2005] that will be included for wide-angle scattering (seesection 5).

[11] The SDE for *E* includes *dE*_{a}, a diffusion coefficient *D*_{EE}, and a non-stochastic energy loss rate*dE*/*dt*:

The drag coefficient, (1/*γp*)∂(*γpD*_{EE})/∂*E*, has been neglected relative to *dE*/*dt.*

[12] Radial diffusion is generally accompanied by *ϕ* diffusion [*Birmingham et al.*, 1967], but we neglect this relative to the azimuthal drift rate *ω*_{d}, so the equation for *ϕ*is non-stochastic:

[13] The set of SDEs (1), (5), (6), and (7) describe possible electron stochastic paths. Because of the friction term in equation (6), density along any path is not conserved, rather *df* = −*C f dt* where

The solution for *f*is the Feynman-Kac formula [*Klebaner*, 2005],

where represents an expectation (or average) over all possible stochastic paths, starting from any location at the initial time *t*_{0} but ending at the current location and time *t.* It is amenable to numerical evaluation by Monte Carlo simulation of a representative set of paths.

### 3. Model Inputs

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[14] A magnetic field model is required and we use the IGRF-11 [*Finlay et al.*, 2010] evaluated for the year 2010. The drift phase *ϕ* (or dipole longitude) is defined relative to the model dipole axis with zero in the prime meridian. The dipole moment of 2010 is used for computing *L.* The azimuthal drift rate *ω*_{d} = (**v**_{d} · **e**_{ϕ})/*r*_{0} is computed from the model field with **v**_{d} = 2*p*/(*qτ*_{b}*B*)∇*I* × **b**, where *r*_{0} is equatorial radius, **e**_{ϕ} is a unit vector in the azimuthal direction, , and the gradient is evaluated numerically at constant *B*_{m} [*Roederer*, 1970, equation 2.41b]. It is shown versus *ϕ* for sample *E*, *L*, and *x* values in Figure 1. Similarly *S*_{b} is shown in Figure 2.

[15] Evaluation of the transport coefficients also requires a model atmosphere. For the neutral atmosphere we use the NRLMSISE-00 model [*Picone et al.*, 2002] which provides number densities of neutral H, He, N, N_{2}, O, O_{2}, and Ar. For plasma we use the GCPM [*Gallagher et al.*, 2000] which provides number densities of H^{+}, He^{+}, O^{+} ions, and free electron density as their sum. For each we use a simplified parameterization by altitude, latitude, and solar radio flux *F*_{10.7} [*Selesnick et al.*, 2007]. For plasma temperature we use *T* = 0.0138*h*^{0.35} eV, where *h* is altitude in km, which is a rough fit to day and night model profiles [*Pierrard and Voiculescu*, 2011].

[16] An initial condition *f*(*t*_{0}) is required for the model solution (9). This is obtained from stably-trapped electron intensities measured by DEMETER/IDP on 1 January 2009. Sample values are shown as a function of*E* and *L* in Figure 3. The DEMETER altitude ∼700 km means that stable trapping is observed only in the SAA region, where *x* values are low enough. Data were selected when the IDP axis attained its lowest *x* value for a given *L* during the day.

### 4. Atmospheric Scattering

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[17] Transport coefficients *D*_{xx} and *dE*/*dt* for scattering of radiation belt electrons by free electrons, protons, and neutral atoms have been derived previously [*MacDonald and Walt*, 1961; *Walt and Farley*, 1976]. Generalizing to include plasma ions, the *x* diffusion coefficient is

where *r*_{e} = *e*^{2}/(4*πϵ*_{0}*mc*^{2}) is the classical electron radius, *y*^{2} = 1 − *x*^{2} = sin^{2}*α*_{0}, *B* and *B*_{0} are respectively the local and equatorial magnetic field magnitude, and *n*_{e} is the free electron number density. The first summation includes all plasma ion species with density *n*_{i} and charge state *Q*_{i}; the second summation includes all species of nuclei, inside neutral atoms and ions, with density *n*_{j} and atomic number *Z*_{j}. The angular brackets indicate a bounce average of the enclosed quantity:

where the integral is along the magnetic field direction between mirror points with length element *ds* and *α* is local pitch angle.

[18] Coulomb logarithms of the form *λ* = ln(1/*θ*_{m}) are defined by minimum scattering angles *θ*_{m} related to the distance from the scattering center at which the central charge is screened out. For plasma electrons and ions, with temperature *T* and density *n*_{e}, this is the Debye length, *λ*_{D} = (a factor accounts for ions and electrons with the same *T*), and

where the reduced mass *m*_{r} is *m*/2 for *λ*_{e} and *m* for *λ*_{p}. For collisions with nuclei, screening is by bound electrons and

where *α*_{f} = 1/137 is the fine structure constant. Diffusion coefficients computed with equation (10), for sample *x*, *E*, *L*, and *F*_{10.7} values, are shown versus longitude in Figure 4.

[19] Angular scattering also results in radial displacements of the electron guiding center by ∼*ρ*, the gyroradius, causing radial diffusion at constant *E* and *x.* However, the diffusion coefficient *D*_{LL} ∼ (*ρ*/*a*)^{2}*D*_{xx} is negligible relative to that derived below (section 7) and is not included in the model.

[20] The energy loss rate to free and bound electrons was also derived previously [*Walt and Farley*, 1976]. It is

where *β* = *v*/*c* and the *I*_{j} are mean ionization potentials (we use values from *Nakamura et al.* [2010, Figure 27.5]).

[21] Energy diffusion results from range straggling by fluctuations in the energy loss per collision. Previously it has been neglected relative to *dE*/*dt*, but we include it as follows: After many scattering events the energy distribution of an initially mono-energetic beam approximates a Gaussian with variance*σ*_{E}^{2} = *ξE*_{max}(1 − *β*^{2}/2), where *E*_{max} is the maximum energy loss per collision and *ξ* = 2*πr*_{e}^{2}*mc*^{2}*s* ∑ *n*_{j}*Z*_{j}/*β*^{2} is the Vavilov parameter for path length *s* [*Seltzer and Berger*, 1964]. Then the energy diffusion coefficient is *vσ*_{E}^{2}/(2*s*). For electrons with *E*_{max} = *E*, and after bounce averaging,

Equations (14) and (15) strictly apply only to neutral atoms, but we include ions in the summations with *Z*_{j} − *Q*_{j} replacing *Z*_{j} (thus excluding H^{+} with no bound electrons).

### 5. Jump Model

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[22] Instead of using *D*_{xx}, it is possible to calculate discrete *x* changes from individual collisions as a jump process. It is also a Markov process [*Klebaner*, 2005] because the jumps are independent of one another.

[23] In each collision the scattering angle *θ* is related geometrically to a trapped electron's initial pitch angle *α*, final pitch angle *α*′, and change in gyrophase angle *ζ* by

The probability density per unit time of scattering through *θ* is *v* ∑ *n*_{i}(*dσ*_{i}/*d*Ω), where *n*_{i} is the density of scatterers as above. The differential cross section for relativistic electron scattering from an unshielded nucleus (or ion) is [*Mott and Massey*, 1965, equation IX.47]

Changing variables and bounce averaging, the probability density per unit time of scattering from *x* to *x*′, irrespective of gyrophase, is

where, by *M* conservation at *ϕ* = *ϕ*_{0}, *x*^{2} = 1 − (*B*_{0}/*B*) sin^{2} *α* and similarly . The upper limit *ζ*_{m} accounts for shielding and is obtained from equation (16) with *θ* = *θ*_{m}, the minimum scattering angle defined above. If there is no *ζ* for which *x* can change to *x*′ by scattering through *θ*_{m}, that is equation (16) gives |cos *ζ*_{m}| > 1, then *ζ*_{m} = *π.* The integral over *ζ* can be done analytically, resulting in a complicated expression, but the bounce average must be carried out numerically. Equation (18) applies to *x* and *x*′ in the range −1 to 1 but, because of bounce averaging, the model range is 0 to 1. Therefore, *P*(−*x*, *x*′) + *P*(*x*, *x*′) should be included and henceforth this sum is implied by *P*(*x*, *x*′).

[24] The jump process *dA* = *x*′ − *x* for equation (5) with time step *dt* is obtained by random sampling from *P*(*x*, *x*′)*dt.* Because the Monte Carlo simulations go backward in time, *x*′ is set to the current value of *x* and a new value obtained for the value of *x* prior to scattering. The Markov condition requires the time step be short enough that most samples give *dA* = 0, that is no scattering event occurs within *dt*, with only occasional nonzero *dA.*

[25] A jump process could entirely replace the diffusion and drag terms in equation (5), but it is impractical to calculate jumps for ion or electron scattering because, with Debye shielding, the minimum scattering angles are so small that *P*(*x*, *x*′) approximates a *δ*-function. Instead we calculate jumps only for nuclear scattering, which has relatively large*θ*_{m} due to the small atomic size, dropping the third term in the square brackets of equation (10) in favor of *dA* but retaining the first two.

[26] Scattering functions *P*(*x*, *x*′) are shown versus *x* for sample values of *x*′, longitude, *E*, and *L* in Figure 5. They show that small-angle scattering is always dominant, as expected, but wide-angle scattering is more frequent than would be predicted by an equivalent diffusion process, for which the scattering functions are Gaussian. The equivalent diffusion coefficient is*D*_{xx} = ∫ *P*(*x*, *x*′)(*x* − *x*′)^{2}*dx*′/2. This gives similar values to equation (10), but not identical because there is no small-angle assumption in its derivation.

### 6. Sample Stochastic Paths

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[27] Stochastic paths are obtained by numerically integrating the SDEs backward in time from a final location. Independent Gaussian deviates are required for each Brownian motion *dW* at each time step *dt.* The time steps are adjusted based on the size of the transport coefficients at each location. For example, shorter steps are required at locations with higher atmospheric densities. Sample paths in *x* and *E* are shown in Figures 6–9.

[28] To illustrate the diffusion and jump processes separately, the three *x* paths in Figure 6 were computed with *D*_{xx} from electron and ion scattering only, as described above, and with *dA* = 0; those in Figure 7 were computed with *dA* from nuclear scattering but *D*_{xx} = 0. The *x* and *E* paths in Figures 8 and 9 were computed with contributions from all of the transport coefficients, including *D*_{xx} from ions and electrons and *dA* from nuclei, except that *D*_{LL} = 0. In each case the paths were followed backward from a common final location for a maximum of 30 days. By coincidence, one path in each figure was terminated early because it was scattered into the dense atmosphere where the energy rapidly increased above 1 MeV, the upper limit for the simulation. When this occurs the initial condition *f*(*t*_{0}) = 0 for equation (9). The upper limit was chosen because there is rarely any significant inner-belt electron intensity with*E* > 1 MeV (see also Figure 3).

[29] As shown in Figure 9, energy diffusion sometimes causes *E* to increase with time. This is unphysical for a single electron because its energy can only decrease as a result of atomic collisions. Ideally, energy changes should be computed from a jump process incorporating the energy loss probability, instead of from diffusion combined with steady energy loss. However, for an ensemble of paths the diffusion process should provide an accurate distribution of energies over time.

### 7. Model Simulations and Data Comparisons

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[30] Satellite electron data generally are accumulated over fixed intervals in *t*, *E*, and *α.*Model simulations can account for this simply by choosing final conditions for the stochastic paths by random sampling from within each interval, with appropriate weighting for instrumental response functions. Then all paths do not end at the same point in phase space, but the Feynman-Kac solution(9) still applies by averaging over this new selection of paths. The measured intensity *j* = *p*^{2}*f* is derived from the simulated *f* using the final *p*, while the initial *f* is derived from the initial *j* at the start of each path using the derived starting *p.* Data from the IDP instrument are well resolved in *E* and *t* [*Sauvaud et al.*, 2006], and sampling is required only for an interval in *x.*It is achieved by choosing a random direction from an isotropic distribution within the IDP field-of-view, fixed at 25° half-angle, and converting it to*x* using the IDP pointing direction and the satellite *B*/*B*_{0}. (A field-of-view wider than the nominal 17° [*Sauvaud et al.*, 2006] was required to match data in the SAA region, perhaps due to scattering within the collimator, but this choice does not influence other conclusions of the study.)

[31] Any subset of the data can be chosen for simulation. To compare stably-trapped and DLC intensity, the data shown inFigure 10were chosen from a 3-day interval, 10–12 January 2009, at each crossing of*L* = 1.3 and from the *E* = 197 keV energy channel. Electron intensity *j* is shown as a function of *ϕ.* The maximum intensity near *ϕ*= 300° is from the SAA. Data from similar longitudes in the northern hemisphere are taken inside the bounce loss cone (BLC) and have low or zero intensity. Because the model does not depend on bounce phase and uses bounce-averaged transport coefficients, it cannot accurately represent BLC intensity. Data taken at other longitudes, where northern and southern intensities are similar, are from the DLC. Although loss cone angles cannot be sharply defined at low*L*, regions representing stably-trapped (SAA), quasi-trapped (DLC), and untrapped (BLC) electrons are clearly visible in the data (though electrons observed near*ϕ* = 10° are not easily categorized). While useful for describing the data, these distinctions are unnecessary because the model transport coefficients include the full dependence on atmospheric densities and the model is independent of any loss cone definition.

[32] Simulations of the data are also shown in Figure 10. They were derived from the initial condition data (Figure 3) taken 10 days earlier, and from stochastic paths that included transport in *x*, *E*, and *ϕ.* Radial transport was not included (*D*_{LL} = 0). The *x*-transport included diffusion with electron, ion, and nuclear scattering (all of the terms inequation (10)), but no jump process (*dA* = 0). The simulations match the data closely in the SAA region, but nowhere else. In the DLC region, virtually all paths reached *E* > 1 MeV prior to reaching the initial time, and so the simulated intensities were virtually all zero. Points with *j* = 0 are not shown in the figure, either for data or simulations. For each data point in the DLC region, ∼10^{6} separate paths were computed; fewer were required in the SAA region.

[33] The same data are shown in Figure 11 with a different set of simulations. In this case the *x*-transport was by diffusion only for electron and ion scattering, with the jump process*dA* now included for nuclear scattering. Sample paths of this type were shown in Figure 8. Now the simulations closely match the data at nearly all longitudes, allowing for statistical errors. The jump process includes frequent small-angle and occasional wide-angle scattering, while the diffusion model ofFigure 10was based only on small-angle scattering. Therefore, wide-angle scattering is the process by which simulated quasi-trapped electrons were transported from the stably-trapped population.

[34] The simulations described above were for only a short, 3-day interval. Over longer periods, intensities at a fixed location decay with time. The decay is fast initially (assuming a nearly isotropic injection) and then approaches a slower but relatively steady decay. The decay can be exponential in time, depending on the shape of the initial energy spectrum. The fast initial decay is not represented in the data because it occurred following an injection of electrons into to inner belt, which was prior to our initial time of 1 January 2009. The simulations inFigures 10 and 11 were increased by a factor 3 to empirically account for this. The fast initial decay occurs as the *x*-dependence of the initial intensity adjusts. We do not have measurements of the*x*-dependence in the initial-condition data, and chose by trial-and-error*j* ∼ (1 − *x*^{2})^{4} to minimize the fast decay period. The *x*-distributions adjust relatively quickly at low*L* and so the simulation results are otherwise insensitive to the choice of initial *x*-dependence for*j.*

[35] Decay of stably-trapped electrons is illustrated inFigure 12. A single data point was chosen every 10 days for all of 2009, again at *L* = 1.3 and a fixed *E* = 304 keV. The chosen points have the lowest observed *x*during the day, which generally corresponds to the highest stably-trapped intensity from the SAA. There is a general and steady decay throughout the year, but with some variability that is partly due to slight differences in satellite location and detector pointing direction. Four separate simulations of the data are also shown inFigure 12. To illustrate the effect of changing atmospheric density, simulations are shown with *F*_{10.7} values of 70 and 150. The 70 value is typical of solar quiet conditions and was close to the actual value throughout 2009. It was used for all simulations except those shown in Figure 12 with the high 150 value typical of solar active conditions. Faster decay for higher *F*_{10.7} is a result of higher atmospheric density.

[36] For each *F*_{10.7} value, simulations are shown for each of the models described above, with *x* diffusion only and including the jump process, but *D*_{LL}= 0 always. It is seen that the diffusion model has a faster initial decay than the jump model, but after ∼30 days the decay rates are similar for the remainder of the year. Therefore, the decay of the stably-trapped electrons is not significantly influenced by wide-angle scattering and is equally-well described by the diffusion or jump models.

[37] Simulated decay rates in Figure 12 are significantly faster than the observed decay rate, the data being a factor ∼50 more intense than the *F*_{10.7} = 70 simulation after a year. This is illustrated again in Figure 13. The same data are shown for *L* = 1.3 with additional data for *L* = 1.2, 1.25, 1.4, and 1.6, based on similar selection criteria. Two simulations are shown for each *L* value. One uses the same diffusion model described above, with *dA* = *D*_{LL} = 0. At lower *L* values, this simulation shows much faster decay than the data, with smaller differences at higher *L.* The other simulation differs only by the addition to the model of radial transport, with *D*_{LL} = 3 × 10^{−10} s^{−1}. With this addition the simulated decay rates closely match those of the data, which are similar for all *L.* This is a result of *L* diffusion replenishing losses at low *L* caused by *x* diffusion. For simulations with radial transport, initial conditions are used from the range *L* = 1.1 to 3. If a path reaches outside this range then it is terminated and its initial *j* = 0.

### 8. Summary and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Stochastic Model Description
- 3. Model Inputs
- 4. Atmospheric Scattering
- 5. Jump Model
- 6. Sample Stochastic Paths
- 7. Model Simulations and Data Comparisons
- 8. Summary and Discussion
- Acknowledgments
- References

[38] A stochastic model of electron transport has been developed to describe the spatial distribution and decay of the inner radiation belt. A Markov jump process is included to account for atmospheric scattering through wide-angles. Other modeled processes are small-angle diffusive scattering, energy loss, energy diffusion, and azimuthal drift. The transport coefficients for these processes were derived from a model magnetic field and from model atmosphere and plasma densities with no free or adjustable parameters. In addition, radial diffusion is included with an adjustable coefficient.

[39] The quasi-trapped electron distribution in the drift loss cone (DLC) must be replenished on the ∼several hour timescale of azimuthal drift. The model simulations (Figures 10 and 11) show that, in January 2009 at *L* = 1.3 and *E*= 200 keV, quasi-trapped electrons derived only from wide-angle, non-diffusive atmospheric scattering of stably-trapped electrons. Although not shown, the same is true at other times and for nearby*L* and *E*values. Small-angle scattering and other scattering processes, such as caused by VLF plasma waves, do not contribute significantly to the DLC electron intensity in this region.

[40] Loss rates of stably-trapped electrons due to atmospheric scattering do not require account of wide-angle scattering (Figure 12) and may be computed from the simpler diffusion process. If VLF wave scattering is fully described by a similar diffusion process, it may also cause significant electron losses without contributing an observable DLC intensity. This would have to be evaluated from the wave intensity, but there is no evidence in the electron data of scattering by waves.

[41] Decay rates from atmospheric scattering alone, including *x* and *E* diffusion and steady *E* loss, are much faster than observed decay rates at low *L.* This conclusion is insensitive to the choice of initial *x*-distribution. However, inclusion of radial diffusion with*D*_{LL} = 3 × 10^{−10} s^{−1}, independent of *L*, leads to decay at a rate similar to those observed at *E* = 300 keV and *L* = 1.2 to 1.6 (Figure 13). This result is consistent with conclusions reached from observed decay rates of Starfish electrons [*Farley*, 1969], although with somewhat different *D*_{LL} values. In each case *D*_{LL} is much higher than would be expected by extrapolation of quiet time values derived from outer belt electron data, which depend on ∼*L*^{10} [*Selesnick et al.*, 1997], perhaps indicating that inner belt radial diffusion results from electric rather than magnetic impulses.

[42] Because of energy loss, decay rates from atmospheric scattering are sensitive to the shape of the initial electron energy spectrum. Naturally occurring spectra (Figure 3) are considerably softer than *β*-decay spectra from nuclear fission fragments and therefore lead to faster decay than was predicted for the Starfish electrons [*Walt*, 1964].

[43] The quasi-trapped, DLC intensity is not directly influenced by radial diffusion. Rather, radial diffusion determines the intensity of stably-trapped electrons. Then the fraction of those electrons that enter the DLC is determined solely by scattering in the atmosphere and plasmasphere. The required radial diffusion coefficient was derived empirically and represents the rate required to overcome precipitation losses. The loss rates were derived from the same scattering model that accurately predicted the DLC intensity, without any empirical adjustment.

[44] For *L*> 1.6 it is known that there are significant electron losses from wave scattering, including possible contributions from ground-based antennae, lightning, and plasmaspheric hiss. If transport coefficients for these processes were available they could be incorporated into the model. Then even higher*D*_{LL}values may be required to match observed decay rates, but the conclusions regarding quasi-trapped electron intensity would remain unchanged.

[45] Improvements to the atmospheric scattering model could be made. The transport coefficients were derived from simplified scattering cross sections. Angular deflections by elastic scattering from nuclei, ions, and free electrons, were considered separately from energy loss to free electrons and ionization of bound electrons. This separation is accurate for heavy atoms or ions, in which elastic scattering from the nucleus dominates that from bound electrons because of a *Z*^{2}dependence in the cross section formula. For neutral hydrogen its suitability is less clear. Ideally, inelastic double-differential cross sections for collisions with atoms and ions should be used to compute angular and energy transport in combination. This may lead to more accurate transport coefficients and derived decay rates for inner radiation belt electrons.