Pitch Angle Diffusion Research Articles

Modeling the transport and anisotropy of energetic electrons in solar flares

Transport of energetic electrons in the flare loop is important to understanding nonthermal emissions in solar flares. In this work, we model the propagation of electrons by numerically solving the particle transport equation which includes the physics of magnetic mirroring and turbulent pitch-angle diffusion. We find that both the fractions of electrons trapped in the looptop and precipitating into the solar surface display a non-monotonic behavior with increasing scattering rate. In the moderate diffusion regime, the precipitation fraction is highest and we expect intense nonthermal HXR and microwave emissions at the footpoints. With no or weak pitch-angle scattering, the velocity space distribution can be highly anisotropic both in the looptop and loopleg regions. Different patterns of stripes with positive gradients in the perpendicular direction can drive the electron cyclotron maser instability with higher efficiency than the classical loss-cone distribution, facilitating the excitation of coherent solar radio bursts. Our simulation results highlight the effects of turbulent pitch-angle scattering on electron trap/precipitation and anisotropic distribution in solar flares, which may help us understand the precipitation of magnetospheric electrons accounting for the aurora as well.

A Cosmic Ray Acceleration Mechanism Based on Background Flow Velocity Inhomogeneities Yielding Power-law Spectra

In astrophysics, one significant challenge lies in understanding the acceleration of cosmic rays, which leads to the occurrence of a power law. In this article, momentum transport generated by the combined effects of pitch-angle diffusion and background flow velocity inhomogeneities is proposed to obtain a cosmic rays acceleration mechanism, starting from the well-known focused transport equation describing particle diffusion and acceleration. The inhomogeneities of background flow velocity are ubiquitous in the astrophysical environment. The equation for the isotropic part of the distribution function of charged energetic particles is derived, and its solution is obtained, demonstrating the form of momentum power laws of cosmic rays. In addition, if it is assumed that cosmic rays penetrate compressive MHD waves or turbulence, for quasi-steady states, the spectral index δ of the momentum power law spectrum of cosmic rays is found to be in the range [−5, −3], which includes the observed power law indices of galactic cosmic rays. The results obtained in this article demonstrate that the mechanism proposed in this article, along with shock acceleration, may also contribute to the acceleration of galactic cosmic rays. Furthermore, when momentum convection effect and higher-order momentum derivative terms are considered, the indices of power laws should be smaller than −5. This may explain the power laws of solar energetic particle events.

An Evaluation of Different Numerical Methods to Calculate the Pitch-angle Diffusion Coefficient from Full-orbit Simulations: Disentangling a Rope of Sand

The pitch-angle diffusion coefficient (PADC) quantifies the effect of pitch-angle scattering on charged particles propagating through turbulent magnetic fields and is a key ingredient in understanding the diffusion of these particles along the background magnetic field. Despite its significance, only a limited number of studies have calculated the PADC from test-particle simulations in synthetic magnetic turbulence, employing various, often quite different, techniques for this purpose. In this study, we undertake a comparative analysis of nine different methods for calculating the PADC from full-orbit simulations. Our objective is to find the strengths and limitations of each method and to determine the most reliable approach. Although all nine methods should theoretically yield comparable results, certain methods may be ill-suited for numerical investigations, while others may not be applicable under conditions of strong turbulence. Through this investigation, we aim to provide recommendations for best practices when employing these methods in future numerical studies of pitch-angle scattering.

Impact of EMIC Waves on Electron Flux Dropouts Measured by GPS Spacecraft: Insights From ELFIN

AbstractAlthough the effects of electromagnetic ion cyclotron (EMIC) waves on the dynamics of the Earth's outer radiation belt have been a topic of intense research for more than 20 years, their influence on rapid dropouts of electron flux has not yet been fully assessed. Here, we make use of contemporaneous measurements on the same ‐shell of trapped electron fluxes at 20,000 km altitude by Global Positioning System (GPS) spacecraft and of trapped and precipitating electron fluxes at 450 km altitude by Electron Losses and Fields Investigation (ELFIN) CubeSats in 2020–2022, to investigate the impact of EMIC wave‐driven electron precipitation on the dynamics of the outer radiation belt below the last closed drift shell of trapped electrons. During six of the seven selected events, the strong 1–2 MeV electron precipitation measured at ELFIN, likely driven by EMIC waves, occurs within 1–2 hr from a dropout of relativistic electron flux at GPS spacecraft. Using quasi‐linear diffusion theory, EMIC wave‐driven pitch angle diffusion rates are inferred from ELFIN measurements, allowing us to quantitatively estimate the corresponding flux drop based on typical spatial and temporal extents of EMIC waves. We find that EMIC wave‐driven electron precipitation alone can account for the observed dropout magnitude at 1.5–3 MeV during all events and that, when dropouts extend down to 0.5 MeV, a fraction of electron loss may sometimes be due to EMIC waves. This suggests that EMIC wave‐driven electron precipitation could modulate dropout magnitude above 1 MeV in the heart of the outer radiation belt.

Effects of Superthermal Plasmas on Hiss Wave‐Driven Scattering Loss of Radiation Belt Electrons

AbstractPlasmaspheric hiss plays an important role in the loss of radiation belt electrons via cyclotron resonant interactions. The cold plasma approximation is widely used in the evaluation of hiss‐driven electron losses, which however can break down during disturbed periods of geomagnetic storms and substorms. The kappa particle velocity distribution, characterized by a pronounced high‐energy tail, is well‐established to model the profile of superthermal plasma under disturbed geomagnetic conditions. In the present study, by calculating the electron bounce‐averaged pitch angle diffusion coefficients with kappa plasma dispersion relations, we investigate the sensitivity of hiss‐induced cyclotron‐resonant electron scattering loss to the spectral index κ under a variety of superthermal plasma conditions. Our results demonstrate that, with increasing κ, the diffusion coefficients of ∼20–100 keV radiation belt electrons significantly decrease at lower pitch angles and increase at higher pitch angles. In contrast, for electrons at higher energies, the diffusion coefficients tend to increase at lower pitch angles and decrease at relatively higher pitch angles. We also find that decrease of L‐shell and increase of α* and temperature anisotropy tend to weaken the hiss‐driven pitch angle scattering efficiency of electrons at energies from tens to hundreds of keV with a dip at ∼30–50 keV, while the scattering of higher energy electrons can be enhanced. This study confirms the important role of superthermal plasmas in the hiss‐driven electron loss processes and should be carefully incorporated in future modeling of radiation belt electron dynamics.

Pitch-Angle Diffusion of Radiation Belt Electrons and Precipitating Particle Fluxes: Dependence on VLF Wavefield Parameters

The dependence of the pitch-angle diffusion efficiency of energetic electrons in the Earth’s magnetosphere on the distribution of the whistler wave field along the geomagnetic flux tube is quantitatively studied for parameters corresponding to the location of the Sura and HAARP HF heating facilities. The expansion of the precipitation energy range with the increase of the region of geomagnetic latitudes occupied by the waves is shown. Using the calculated pitch-angle diffusion coefficient for a given spectrum of waves and their distribution along the flux tube, the ratio of the fluxes of precipitating and trapped particles at low altitude is determined. It is shown that at typical wave intensities corresponding to chorus VLF waves and plasmaspheric hiss, the fluxes of precipitating and trapped electrons can be comparable to each other. At the same time, for the wave amplitudes observed as a result of the action of heating facilities, the flux of precipitating electrons is negligible.

IMF Bx, By, and Bz dependence of energetic particle precipitation in the magnetosphere

The orientation of Interplanetary Magnetic Field (IMF B) components plays a major role in magnetospheric Energetic Particles Precipitation (EPP). This is because the IMF B components in the Geocentric Solar Magnetospheric (GSM) coordinate system control magnetic reconnection at the subsolar magnetopause. This study intends to unravel the dependence of the preceding conditions of the orientations of IMF B components that drove geomagnetic storms on the EPPs produced in the Van Allen radiation belt. We classified the geomagnetic storms based on their drivers and IMF Bz orientations prior to the storms’ onset. Energetic particle fluxes, IMF B, and solar wind data were used. The study showed that electrons are more reactive to solar wind speed than protons. Additionally, storms with southward IMF Bz orientation caused more EPP enhancement than with northward orientation. Our results also showed that protons are more influenced by magnetospheric compression than electrons at the northward orientation of IMF Bz. In fact, at this point, the proton’s motion violates adiabatic invariants. As particles interact with electric fields, a significant population of EPPs is produced by pitch angle diffusion. Generally, IMF Bx and IMF By are often anticorrelated. The preceding conditions of orientations of the IMF Bx indicated weaker dependence on auroral indices, and invariably on energetic particles precipitation than the IMF By.

Field Line Curvature (FLC) Scattering in the Dayside Off‐Equatorial Minima Regions

AbstractMagnetic field line curvature (FLC) scattering is an effective mechanism for collisionless particle scattering. In the terrestrial magnetosphere, the FLC scattering plays an essential role in shaping the outer boundary of protons radiation belt, the rapid decay of ring current, and the formation of proton isotropic boundary (IB). However, previous studies have yet to adequately investigate the influence of FLC scattering on charged particles in the Earth's dayside magnetosphere, particularly in the off‐equatorial magnetic minima regions. This study employs T89 magnetic field model to investigate the impacts of FLC scattering on ring current protons in the dayside magnetosphere, with a specific focus on the off‐equatorial minimum regions. We analyze the spatial distributions of single and dual magnetic minima regions, adiabatic parameter, and pitch angle diffusion coefficients due to FLC scattering as functions of Kp. The results show that the effects of FLC scattering are significant not only on the dusk and dawn sides but also in the off‐equatorial minima regions on the noon. Additionally, we investigate the role of dipole tilt angle in the hemispheric asymmetry of FLC scattering effects. The dipole tilt angle controls the overall displacement of the dayside magnetosphere, resulting in different FLC scattering effects in the two hemispheres. Our study holds significance for understanding the FLC scattering effects in the off‐equatorial region of Earth's dayside magnetosphere and for constructing a more accurate dynamic model of particles.

Statistical Properties of Exohiss Waves and Associated Scattering Losses of Radiation Belt Electrons

AbstractExohiss waves are a type of structureless whistler‐mode waves that exist in the low density plasmatrough outside the plasmapause and may potentially perturb the motions of electrons in the radiation belt and ring current. Using data from Van Allen Probe A, we analyze the distribution of magnetic power spectral density (PSD) of exohiss waves in different magnetic local time (MLT) and L‐shell regions near the geomagnetic equator. The results reveal that the peak magnetic PSD of exohiss waves is the weakest at MLT = 0–6 and the strongest at MLT = 12–18. The magnetic PSDs of exohiss waves are much lower than those of chorus and hiss waves except for the MLT = 12–18 sector. In addition, we calculated the quasi‐linear bounce‐averaged pitch angle and momentum diffusion coefficients (〈Dαα〉 and 〈Dpp〉) of electrons caused by exohiss waves. The diffusion coefficients are then compared with those caused by chorus and hiss waves. The peak 〈Dαα〉 of electrons driven by exohiss waves becomes stronger as L‐shell increases at all MLTs and is the greatest on the dayside, especially in the sector of MLT = 12–18. Exohiss waves have more significant effect on the loss of radiation belt electrons with specific energy levels related to MLT and L‐shell region compared to chorus and hiss waves. On the other hand, 〈Dpp〉 of electrons caused by exohiss waves is very small, which illustrates that exohiss waves have almost no acceleration effect on electrons.

Particle Scattering in the Presence of Moderate to Strong Adiabatic Focusing

AbstractStarting from the Vlasov‐Boltzmann equation, we derive corrections to the quasi linear Fokker‐Planck pitch angle scattering coefficient caused by a non‐uniformity of the background magnetic field. Integration of the perturbed particle distribution function is performed along a modified unperturbed trajectory that takes into account the effects of focusing and mirroring on pitch angle and gyrophase. We calculate the changes to the quasi‐linear scattering rate owing to a variations in the field strength and the gyrophase drift that affects the resonance between particles and magnetic fluctuations. It is shown that the scattering coefficient has an oscillating behavior in pitch angle when the focusing lengthscale is comparable to the correlation length of the turbulence and the particle's Larmor radius. The applicability of analytic results are demonstrated with a simple model of charged particles trapped in a magnetic field of a planet. It is suggested that the pitch angle diffusion coefficient could become negative for the case of very strong focusing.

The Relationship Between Electron Precipitation and the Population of Trapped Electrons in LEO: New Evidence Supporting a Natural Limit to the Flux of Energetic Electrons

AbstractEnergetic particle precipitation into the atmosphere has been identified as a key loss process for electrons in the Earth's outer radiation belt region. However, direct measurements of the electron flux precipitating into the atmosphere are challenging from high altitude low inclination spacecraft, such as the Van Allen Probes, due to the small angular size of the loss cone along the orbital path of these high‐altitude spacecraft. Here we use data from the Polar Orbiting Environmental Satellites (POES)/Space Environment Monitor in low‐Earth orbit to assess the relationship between the trapped and precipitating electron flux in integral energy channels >30, >100, and >300 keV. Our results highlight that there is a strong non‐linear relationship between the flux of trapped and precipitating electrons, with the ratio of precipitating to trapped flux only becoming significantly enhanced once the trapped flux reaches a critical level. This transition from low to high levels of precipitation is consistent with the theory proposed by Kennel and Petschek (K‐P) (1966) https://doi.org/10.1029/JZ071i001p00001, whereby intense chorus waves are excited and trigger pitch angle diffusion, resulting in energetic particle precipitation and limiting the trapped flux. Using electron flux data from POES and chorus wave data from the Van Allen Probes, we further test the observations against predictions from the K‐P theory. A particle tracing model is also utilized to illustrate a direct link between the drift paths of injected electrons, the occurrence of chorus waves and the spatial distribution of strong electron precipitation into the atmosphere at different energies.

Bounce Resonance Between Energetic Electrons and Magnetosonic Waves: A Parametric Study

AbstractMagnetosonic (MS) waves are electromagnetic emissions from a few to 100 Hz primarily confined near the magnetic equator both inside and outside the plasmasphere. Previous studies proved that MS waves can transport equatorially mirroring electrons from an equatorial pitch angle of 90° down to lower values by bounce resonance. However, the dependence of the bounce resonance effect on wave or background plasma parameters is still unclear. Here, we applied a test particle simulation to investigate electron transport coefficients, including diffusion and advection coefficients in energy and pitch angle, due to bounce resonance with MS waves. We investigate five wave field parameters, including wave frequency width, wave center frequency, latitudinal distribution width, wave normal angle root‐mean‐square of wave magnetic amplitude, and two background parameters, L‐shell value and plasma density. We find different transport coefficient peaks resulting from different bounce resonance harmonics. Higher‐order harmonic resonances exist, but the effect of fundamental resonance is much stronger. As the wave center frequency increases, higher‐order harmonics start to dominate. With wave frequency width increasing, the energy range of effective bounce resonance broadens, but the effect itself weakens. The bounce resonance effect will increase when we decrease the wave normal angle, or increase the wave amplitude, latitudinal distribution width, L‐shell value, and plasma density. The parametric study will advance our understanding of the favorable conditions of bounce resonance.

Nonlinear electron scattering by electrostatic waves in collisionless shocks

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$ . A quasi-linear scaling ( $\nu \approx 2$ ) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$ , while the diffusion is nonlinear ( $1<\nu <2$ ) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$ , while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$ . We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.

Modeling the Contribution of Precipitation Loss to a Radiation Belt Electron Dropout Observed by Van Allen Probes

AbstractA drift‐diffusion model is used to simulate the low‐altitude electron distribution, accounting for azimuthal drift, pitch angle diffusion, and atmospheric backscattering effects during a rapid electron dropout event on 21st August 2013, at L = 4.5. Additional external loss effects are introduced during times when the low‐altitude electron distribution cannot be reproduced by diffusion alone. The model utilizes low‐altitude electron count rate data from five POES/MetOp satellites to quantify pitch angle diffusion rates. Low‐altitude data provides critical constraint on the model because it includes the drift loss cone region where the electron distribution in longitude is highly dependent on the balance between azimuthal drift and pitch angle diffusion. Furthermore, a newly derived angular response function for the detectors onboard POES/MetOp is employed to accurately incorporate the bounce loss cone measurements, which have been previously contaminated by electrons from outside the nominal field‐of‐view. While constrained by low‐altitude data, the model also shows reasonable agreement with high‐altitude data. Pitch angle diffusion rates during the event are quantified and are faster at lower energies. Precipitation is determined to account for all of the total loss observed for 450 keV electrons, 88% for 600 keV and 38% for 900 keV. Predictions made in the MeV range are deemed unreliable as the integral energy channels E3 and P6 fail to provide the necessary constraint at relativistic energies.

Long‐Term Variations of Energetic Electrons Scattered by Signals From the North West Cape Transmitter

AbstractVery‐low‐frequency (VLF) signals emitted from ground‐based transmitters for submarine communication can penetrate the ionosphere and leak into the magnetosphere, leading to electron precipitation via wave‐particle interaction and thereby providing a potential means for radiation belt remediation. In this study, we systematically analyze the dependence of quasi‐trapped electron fluxes scattered by signals from the North West Cape (NWC) transmitter on electron energy, L‐shell, and geomagnetic activity (i.e., the Dst index) using long‐term measurements from the DEMETER satellite. Considering potentially changed theoretical cyclotron resonant condition, we find that the variations of wave normal angle (WNA) of NWC transmitter signals or of the background electron density can explain the variated “wisp” positions in energy versus L plane. The long‐term data analyzation suggests that the energy‐dependences increases can help to distinguish the different source mechanisms of quasi‐trapped electrons. The enhancement of quasi‐trapped electron fluxes induced by NWC transmitter signals is more obvious at L = 1.8 than L = 1.6 due to higher trapped flux levels and strong pitch angle diffusion induced by transmitter signals.

Chorus wave power at the strong diffusion limit overcomes electron losses due to strong diffusion

Earth’s radiation belts consist of high-energy charged particles trapped by Earth’s magnetic field. Strong pitch angle diffusion of electrons caused by wave-particle interaction in Earth’s radiation belts has primarily been considered as a loss process, as trapped electrons are rapidly diffused into the loss cone and lost to the atmosphere. However, the wave power necessary to produce strong diffusion should also produce rapid energy diffusion, and has not been considered in this context. Here we provide evidence of strong diffusion using satellite data. We use two-dimensional Fokker-Planck simulations of electron diffusion in pitch angle and energy to show that scaling up chorus wave power to the strong diffusion limit produces rapid acceleration of electrons, sufficient to outweigh the losses due to strong diffusion. The rate of losses saturates at the strong diffusion limit, whilst the rate of acceleration does not. This leads to the surprising result of an increase, not a decrease in the trapped electron population during strong diffusion due to chorus waves as expected when treating strong diffusion as a loss process. Our results suggest there is a tipping point in chorus wave power between net loss and net acceleration that global radiation belt models need to capture to better forecast hazardous radiation levels that damage satellites.

Characterizing Radiation‐Belt Energetic Electron Precipitation Spectra: A Comparison of Quasi‐Linear Diffusion Theory With In Situ Measurements

AbstractHigh energy electron precipitation from the Earth's radiation belts is important for loss from the radiation belts and atmospheric chemistry. We follow up investigations presented in Reidy et al. (2021, https://doi.org/10.1029/2020ja028410) where precipitating flux is calculated inside the field of view of the POES T0 detector using quasi‐linear theory and pitch angle diffusion coefficients (Dαα) from the British Antarctic Survey (BAS). These results showed good agreements at >30 keV for L* >5 on the dawnside but the flux were too low at higher energies. We have investigated the effect of changing parameters in the calculation of the precipitating flux to improve the results for the higher energies using comparisons of in situ flux and cold plasma measurements from GOES‐15 and RBSP. We find that the strength of the diffusion coefficients rather than the shape of the source spectrum has the biggest effect on the calculated precipitation. In particular we find decreasing the cold plasma density used in the calculation of Dαα increases the diffusion and hence the precipitation at the loss cone for the higher energies, improving our results. The method of calculating Dαα is also examined, comparing co‐located rather than averaged RBSP measurements. We find that the method itself has minimal effect but using RBSP derived Dαα improved our results over using Dαα calculated using the entire BAS wave data base; this is potentially due to better measurements of the cold plasma density from RBSP than the other spacecraft included in the BAS wave data base (e.g., THEMIS).

New Chorus Diffusion Coefficients for Radiation Belt Modeling

AbstractWhistler mode chorus is an important magnetospheric wave emission playing a major role in radiation belt dynamics, where it contributes to both the acceleration and loss of relativistic electrons. In this study we compute bounce and drift averaged chorus diffusion coefficients for 3.0 < L* < 6.0, using the TS04 external magnetic field model, taking into account co‐located near‐equatorial measurements of the wave intensity and fpe/fce, by combining the Van Allen probes measurements with data from a multi‐satellite VLF wave database. The variation of chorus wave normal angle (WNA) with spatial location and fpe/fce is also taken into account. We find that chorus propagating at small WNAs has the dominant contribution to the diffusion rates in most MLT sectors. However, in the region 4 ≤ MLT < 11 high WNAs dominate at intermediate pitch angles. In the region 3 < L* < 4, the bounce and drift averaged pitch angle and energy diffusion rates during active conditions are primarily larger than those in our earlier models by up to a factor of 10 depending on energy and pitch angle. Further out, the results are similar. We find that the bounce and drift averaged energy and pitch angle diffusion rates can be significantly larger than the new model in regions of low , where the differences can be up to a factor of 10 depending on energy and pitch angle.

Differentiating Between Simultaneous Loss Drivers in Earth's Outer Radiation Belt: Multi‐Dimensional Phase Space Density Analysis

AbstractWe analyzed the contribution of electromagnetic ion cyclotron (EMIC) wave driven electron loss to a flux dropout event in September 2017. The evolution of electron phase space density (PSD) through the dropout showed the formation of a radially peaked PSD profile as electrons were lost at high L*, resembling distributions created by magnetopause shadowing. By comparing 2D Fokker Planck simulations of pitch angle diffusion to the observed change in PSD, we found that the μ and K of electron loss aligned with maximum scattering rates at dropout onset. We conclude that, during this dropout event, EMIC waves produced substantial electron loss. Because pitch angle diffusion occurred on closed drift paths near the last closed drift shell, no radial PSD minimum was observed. Therefore, the radial PSD gradients resembled solely magnetopause shadowing loss, even though the local pitch angle scattering produced electron losses of several orders of magnitude of the PSD.

Parametrization of coefficients for sub-grid modeling of pitch-angle diffusion in global magnetospheric hybrid-Vlasov simulations

Sub-grid models are key tools to accurately describe the physical processes at play in a system when high-resolution simulations are not feasible. We previously developed a sub-grid model for pitch-angle diffusion in hybrid-Vlasov simulations of Earth's magnetosphere. However, a more precise description of the pitch-angle diffusion coefficient is required to apply this model to global simulations. In this study, we use an existing method to parametrize pitch-angle diffusion coefficients from monotonic distribution functions and adapt it to bi-Maxwellian distributions. We determine these coefficients for various values of the ion temperature anisotropy and plasma β∥. We use these newly parametrized coefficients in our sub-grid model and show that it accurately models reduction of temperature anisotropy in both local simulations and global simulations of the Earth's magnetosphere, while using minimal computational resources.