Parker Transport Equation Research Articles

Fast and accurate evaluation of deep-space galactic cosmic ray fluxes with HelMod-4/CUDA

The accurate knowledge of cosmic ion fluxes is essential for fundamental physics, deep space missions, and exploration activities in the solar system. In the HelMod-4 model the Parker transport equation is solved using a Monte Carlo approach to evaluate the solar modulation effect on local interstellar spectra of Galactic Cosmic Rays (GCRs). This work presents the latest updates to the HelMod-4 model parameters, focusing on the descending phase of solar cycle 24. The updates are motivated by the latest high-precision measurements from the AMS-02 detector which revealed for the first time with high accuracy the features of GCR fluxes’ evolution during a period of positive interplanetary magnetic field polarity. Furthermore, we present HelMod-4/CUDA, a GPU-accelerated approach for solving the Parker equation in the heliosphere using a stochastic differential equation method. The code is an evolution of the HelMod-4 code, porting the algorithm to GPU architecture using the CUDA programming language. This approach achieves significant speedup compared to a CPU implementation. The HelMod-4/CUDA code has been validated by comparing its results with the most precise and updated experimental GCR spectra observed during high and low solar activity periods, both in the inner and outer heliosphere, at the Earth location, and outside the ecliptic plane. The comparison shows that HelMod-4 and HelMod-4/CUDA can be equivalently used to provide solar-modulated spectra with a similar degree of accuracy in reproducing observed data.

Diffusion - Convection Equation of Galactic Cosmic Rays (GCR) in the Atmosphere and Its Analytical, Numerical Solutions by Using Finite Elements Method Using Parker Transport Equation

This study depicts that to find the Analytical and numerical solutions of Diffusion – Convections Equations of Galactic Cosmic Rays (GCRs) by Finite Element method (FEM) and also to find the Energy Equation of GCRs by using a part of Parker’s transport equation. This considers moreover centres on the exactness and acknowledgment of the FEM strategy by utilizing dissemination blunder, scattering mistake and add up to blunder investigation. The comes about are depicted both graphically and in a unthinkable frame, which essentially guarantees the method’s legitimacy and the algorithm’s proficiency to maintain the exactness, effortlessness, and nonlinear Convection-Diffusion Equation conditions. The proposed method may be connected for tackling any nonlinear convection diffusion equation. We concentrate on assaying the confluence and stability of the nonlinear parabolic partial differential equation. This study focuses on the delicacy and acceptance of FEM method by exercising dispersion error dissipation error, and total error analysis.

Astronaut Radiation Dose Calculation With a New Galactic Cosmic Ray Model and the AMS‐02 Data

AbstractWe present a new calculation of the astronaut dose rate from the galactic cosmic rays in free space at 1 AU. We use the unshielded isotropic fluence‐to‐dose conversion coefficients given in the International Commission on Radiological Protection publication 123. A new 3D and time‐dependent solar modulation model based on Parker's transport equation as originally developed in Song et al. (2021, https://doi.org/10.3847/1538-4365/ac281c) is used to calculate the galactic cosmic ray spectra at 1 AU. This model uses the recent local interstellar spectra of Corti et al. (2019, https://doi.org/10.3847/1538-4357/aafac4), M. J. Boschini et al. (2020, https://doi.org/10.3847/1538-4365/aba901, 2021a, https://doi.org/10.3847/1538-4357/abf11c) to reproduce the PAMELA and AMS‐02 observations between 2006 and 2019. The radiation dose calculated from our model and from the AMS‐02 spectra in the same rigidity region agrees better than 1% for proton and helium in a time‐dependent way, and at 2% level for six most contributing cosmic ray elements averaged over 7 or 8.5 years. The time‐dependent dose rate analysis over 13 years shows an effective dose equivalent rate of 55–58 cSv/yr at solar minimum (January 2010) and 26 cSv/yr at solar maximum (February 2014).

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.

Comprehensive modulation potential for the solar modulation of Galactic cosmic rays

The concept of the modulation potential being utilized to describe the solar modulation of galactic cosmic rays is calculated based on the full three-dimensional Parker transport equation. This is in sharp contrast to the force-field approach, which is based on a series of approximations of this equation and where traditionally all the physical processes are condensed into a single parameter without knowing their relative contribution. In our comprehensive approach to the modulation potential, the contribution of all the different physical processes is given explicitly as derived directly from the parameters of Parker's transport equation. We use the stochastic differential equations approach to study the effects of each of these different physical modulation processes thoroughly and then also determine the rigidity and mass-to-charge-ratio ($A/Z$) of these processes for the first time in the context of a modulation potential. It is found that: (1) This comprehensive modulation potential for galactic cosmic particles with a given rigidity at the Earth is a random variable, which can be fit by an inverse Gaussian distribution; (2) the rigidity and $A/Z$ dependence of this modulation potential can be divided into three categories based on the index value of power law diffusion; (3) the modulation caused by convection becomes weaker with increased rigidity and eventually dissipates, leading to a larger modulation potential for particles with a larger $A/Z$; (4) particle drift significantly reduces this modulation potential and its variation with changes in the tilt angle of the heliospheric current sheet has a distinct pattern for each of the two solar magnetic field polarities.

The effect and properties of drifts in the heliosphere

We investigate the properties of drifts and their effect on cosmic ray modulation in the heliosphere using a numerical modulation model based on the solution of a set of stochastic differential equations that was derived from the Parker transport equation. The illustrative capabilities of the numerical model are exploited to yield a better understanding of the physical modulation processes involved. Various studies have indicated that drifts need to be scaled down towards solar maximum conditions and the present study looks at how this can be achieved. Drifts are scaled down directly by multiplying the drift coefficient by a factor of less than unity as well as indirectly through the drift–diffusion relation, that is, by modifying the diffusion coefficient so as to cause a change in the drift effects through altered gradients in particle intensity. Contour plots of particle exit positions and exit energies are presented for both of these cases, and it is illustrated that drifts in the model lead to larger energy losses. This is explained with the aid of figures indicating the relative amount of time spent by pseudo-particles in different regions of the heliosphere during the modulation process. These figures also indicate that an increase in diffusion leads to a suppression or reduction of drift effects. Finally, the figures also show that drift effects are reduced as a function of increasing particle energy; even though the drift coefficient increases with particle energy, the total drift effect, taking into account the contribution from the increased diffusion associated with larger energies, causes drift effects to be reduced with an increase in energy.

On the numerical modelling of astrospheres, including those of luminous blue variable stars, and the modulation of cosmic rays within

ABSTRACT Luminous blue variable-type stars are massive O-type stars that show variable outflow parameters. These stars can also undergo eruptions where a large amount of mass is ejected in a very short duration. The astrospheres of these stars are modelled using a magnetohydrodynamic model including the effect of radiative cooling. The variation in outflow parameters is demonstrated to lead to a variation in number density up to the termination shock, after which there is no effect in the post-shocked regions. On the other hand, eruptions increase the size of the astrosphere and cause the termination shock to oscillate between the inner boundary and the astropause. Galactic cosmic ray modulation within these types of astrospheres is also studied using a 1D stochastic differential equation model. It is shown that, even though these astrospheres are considerably larger than the heliosphere, calculated cosmic ray differential intensities remain larger than what is observed in the heliosphere at Earth, implying lower levels of modulation related to larger cosmic ray diffusion coefficients. During eruptive events, modelled cosmic ray intensities fluctuate due to the oscillating termination shock, and varying plasma parameters. It is also shown that such behaviour, can in principle, lead to the acceleration of these particles, but the results of this study, due to the 1D approach taken to solving the Parker transport equation, are argued to represent an upper limit to this phenomenon.

A Model of Double Coronal Hard X-Ray Sources in Solar Flares

A number of double coronal X-ray sources have been observed during solar flares by RHESSI, where the two sources reside at different sides of the inferred reconnection site. However, where and how these X-ray-emitting electrons are accelerated remains unclear. Here we present the first model of the double coronal hard X-ray (HXR) sources, where electrons are accelerated by a pair of termination shocks driven by bidirectional fast reconnection outflows. We model the acceleration and transport of electrons in the flare region by numerically solving the Parker transport equation using velocity and magnetic fields from the macroscopic magnetohydrodynamic simulation of a flux rope eruption. We show that electrons can be efficiently accelerated by the termination shocks and high-energy electrons mainly concentrate around the two shocks. The synthetic HXR emission images display two distinct sources extending to >100 keV below and above the reconnection region, with the upper source much fainter than the lower one. The HXR energy spectra of the two coronal sources show similar spectral slopes, consistent with the observations. Our simulation results suggest that the flare termination shock can be a promising particle acceleration mechanism in explaining the double-source nonthermal emissions in solar flares.

Numerical Modeling of Spectral Hardening at a Finite-width Shock

Spectral hardening has been identified in solar flare hard X-ray observations for several decades and remains a puzzle. We examine spectral hardening under the diffusive shock acceleration mechanism using numerical simulations. The hardening is related to the finite width of the shock and is controlled by the shock Péclet number. We implement two different types of Monte Carlo simulations. The first is based on the backward stochastic differential equation method, where the Parker transport equation is solved by casting it to a set of stochastic different equations, and by following the trajectories of individual quasiparticles. In the second approach, we follow real particles and particles are assumed to move freely between scatterings from magnetic turbulence in the plasma. The scattering is modeled as either large-angle hard-sphere elastic collision, or small-angle pitch-angle scattering. We show that the results from these two approaches agree well with each other and agree with analytical results. We also use a Pan-spectrum form to fit the resulting spectra.

Modeling Electron Acceleration and Transport in the Early Impulsive Phase of the 2017 September 10th Solar Flare

The X8.2-class limb flare on 2017 September 10 is among the best studied solar flare events owing to its great similarity to the standard flare model and the broad coverage by multiple spacecraft and ground-based observations. These multiwavelength observations indicate that electron acceleration and transport are efficient in the reconnection and flare looptop regions. However, there lacks a comprehensive model for explaining and interpreting the multi-faceted observations. In this work, we model the electron acceleration and transport in the early impulsive phase of this flare. We solve the Parker transport equation that includes the primary acceleration mechanism during magnetic reconnection in the large-scale flare region modeled by MHD simulations. We find that electrons are accelerated up to several MeV and fill a large volume of the reconnection region, similar to the observations shown in microwaves. The electron spatial distribution and spectral shape in the looptop region agree well with those derived from the microwave and hard X-ray emissions before magnetic islands grow large and dominate the acceleration. Future emission modelings using the electron maps will enable direct comparison with microwave and hard X-ray observations. These results shed new light on the electron acceleration and transport in a broad region of solar flares within a data-constrained realistic flare geometry.

Model of galactic cosmic ray spectrum above the Earth’s atmosphere

Abstract Galactic cosmic ray transport in the heliosphere is described by the well-known Parker transport equation. In 1969, Fisk and Axford [1] presented approximate analytical solutions to the cosmic ray transport equation. One of their solutions was later used to construct useful semi-empirical models describing the energy spectra of charged particles during the solar cycle. In this paper, a simplified model of the galactic cosmic ray spectrum is presented. The model can be used to describe the galactic spectra of protons and helium nuclei during the 11-year solar cycle in interplanetary space at about 1 AU.

The transport of galactic cosmic rays in heliosphere: The HelMod model compared with other commonly employed solar modulation models

HelMod model allows one to describe how solar modulation affects the propagation of galactic cosmic rays (GCRs) through the heliosphere by solving the Parker transport equation. HelMod provides modulated spectra for protons and nuclei during high and low solar activity periods, in the inner and outer heliosphere, at Earth location and outside the ecliptic plane, as well as at different distances from Sun in the inner and outer heliosphere. The GCRs mainly constitute the high-energy population of the so-called space radiation environment. Modulated cosmic-ray spectra of protons and nuclei, obtained from HelMod and other solar modulation codes were compared to the AMS–02 spectra ranging from GeV up to TeV energies. Concerning other commonly used solar modulation models, HelMod modulated spectra are available over the entire rigidity range and observation period of AMS–02 mission. A quantitative comparison among models was carried out using ΔΦ, i.e., the relative difference between the experimental and simulated fluences, and ηRMS, i.e., the error-weighted root mean square of the relative difference between AMS–02 experimental data and simulated differential intensities. HelMod was found to achieve a good agreement over the full set of experimental data with, typically, ΔΦ within ±2.5% and ηRMS within 5%. Usually larger or, in a few cases, much larger values for ΔΦ and ηRMS were found for the other models.

Double-power-law Feature of Energetic Particles Accelerated at Coronal Shocks

Abstract Recent observations have shown that in many large solar energetic particle (SEP) events the event-integrated differential spectra resemble double power laws. We perform numerical modeling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation, including protons and heavier ions. We find that for all ion species the energy spectra integrated over the simulation domain can be described by a double power law, and the break energy depends on the ion charge-to-mass ratio as E B ∼ (Q/A) α , with α varying from 0.16 to 1.2 by considering different turbulence spectral indices. We suggest that the double-power-law distribution may emerge as a result of the superposition of energetic particles from different source regions where the acceleration rates differ significantly due to particle diffusion. The diffusion and mixing of energetic particles could also provide an explanation for the increase of Fe/O at high energies as observed in some SEP events. Although further mixing processes may occur, our simulations indicate that either a power-law break or rollover can occur near the Sun and predict that the spectral forms vary significantly along the shock front, which may be examined by upcoming near-Sun SEP measurements from the Parker Solar Probe and Solar Orbiter.

Numerical Modeling of Latitudinal Gradients for Galactic Cosmic-Ray Protons during Solar Minima: Comparing with Ulysses Observations

Abstract The latitudinal gradients of galactic cosmic-ray (GCR) protons measured by Ulysses during two successive minima provide a unique opportunity to study the modulation effects in polar regions of the heliosphere. In this work, a GCR modulation model based on numerically solving the Parker transport equation is used to study the latitudinal distribution of GCR protons in the inner heliosphere. Modifications of the standard Parker heliospheric magnetic field, the reduction of particle drifts, the latitudinal-dependent magnetic turbulence characteristics, and the anisotropic perpendicular diffusion coefficient are incorporated in the numerical model to investigate the corresponding modulation effects. It is found that the latitudinal-dependent magnetic turbulence magnitude, which makes the parallel diffusion coefficient decrease with the increasing of latitude, is crucial to obtain the negative latitude gradient in the inner heliosphere during the negative-polarity solar cycle. For the A > 0 period, on the other hand, the latitudinal diffusion coefficient in the inner heliosphere and the reduced drift velocity in the polar region are more important, while the anisotropic perpendicular diffusion coefficient at high latitude might be not essential. Finally, the proton latitudinal gradient and the corresponding differential intensity along the trajectory of Ulysses during its first and third fast latitude scans are computed, and the results show good agreement with the spacecraft observations.

Uncertainties implicit to the use of the force-field solutions to the Parker transport equation in analyses of observed cosmic ray antiproton intensities

Spacecraft observations of cosmic ray antimatter intensities can potentially shed new light on previously unknown physics and hence play an essential role in the search for dark matter. At lower energies, however, these particles experience considerable modulation in the heliosphere. Since the advent of cosmic ray modulation studies, simplified approaches to solving the Parker transport equation have been used to glean insights from observations as to the behavior of these particles. We show, due to the inherent limitations imposed by the assumptions made in simplifying the Parker equation, that the use of these approaches can introduce unknown uncertainties into analyses based on them, and thus that these approaches should be used with caution, especially when novel insights into potentially new physics are sought.

Investigating 1st and 2nd order Fermi acceleration of energetic particles by small-scale magnetic flux ropes at 1AU

A new telegrapher-type Parker transport equation was derived from the existing underlying focused transport equation to model the acceleration of energetic particles by contracting and reconnecting small-scale magnetic flux ropes (SMFRs) in the large-scale solar wind. Time-dependent and steady-state analytical solutions were found that unify all SMFR acceleration mechanisms present in the transport equation, showing that SMFR acceleration by the parallel reconnection electric field in the mixed spatial and momentum derivative transport term is constrained by and requires the presence of 2nd order Fermi SMFR acceleration. We explore the potential of these solutions in reproducing energetic proton flux enhancements and spectral evolution between ∼50 keV-5 MeV in dynamic SMFR regions near large-scale reconnecting current sheets in the solar wind at 1 AU. It is shown that both 2nd order Fermi SMFR acceleration involving the variance in SMFR compression and incompressible parallel shear flow, and 1st order SMFR Fermi acceleration due to mean SMFR compression are both workable options in reproducing observed flux amplification factors when using reasonable SMFR parameters. However, the predicted substantial quantitative differences in the spatial evolution of the accelerated spectra through the SMFR region might provide a diagnostic to distinguish between 1st and 2nd order Fermi SMFR acceleration in observations.

Solar Energetic Particles Produced by a Slow Coronal Mass Ejection at ∼0.25 au

Abstract We present an analysis of Parker Solar Probe (PSP) IS⊙IS observations of ∼30–300 keV n−1 ions on 2018 November 11 when PSP was about 0.25 au from the Sun. Five hours before the onset of a solar energetic particle (SEP) event, a coronal mass ejection (CME) was observed by STEREO-A/COR2, which crossed PSP about a day later. No shock was observed locally at PSP, but the CME may have driven a weak shock earlier. The SEP event was dispersive, with higher energy ions arriving before the lower energy ones. Timing suggests the particles originated at the CME when it was at ∼7.4R ⊙. SEP intensities increased gradually from their onset over a few hours, reaching a peak, and then decreased gradually before the CME arrived at PSP. The event was weak, having a very soft energy spectrum (−4 to −5 spectral index). The earliest arriving particles were anisotropic, moving outward from the Sun, but later, the distribution was observed to be more isotropic. We present numerical solutions of the Parker transport equation for the transport of 30–300 keV n−1 ions assuming a source comoving with the CME. Our model agrees well with the observations. The SEP event is consistent with ion acceleration at a weak shock driven briefly by the CME close to the Sun, which later dissipated before arriving at PSP, followed by the transport of ions in the interplanetary magnetic field.

Impact of solar magnetic field amplitude and geometry on cosmic rays diffusion coefficients in the inner heliosphere

Cosmic rays are remarkable tracers of solar events when they are associated with solar flares, but also galactic events such as supernova remnants when they come from outside our solar system. Solar Energetic Particles (SEPs) are correlated with the 11-year solar cycle while Galactic Cosmic Rays (GCRs) are anti-correlated due to their interaction with the heliospheric magnetic field and the solar wind. Our aim is to quantify separately the impact of the amplitude and the geometry of the magnetic field, both evolving during the solar cycle, on the propagation of cosmic rays of various energies in the inner heliosphere (within Earth orbit). We focus especially on the diffusion caused by the magnetic field along and across the field lines. To do so, we use the results of 3D magnetohydrodynamics (MHD) wind simulations running from the lower corona up to 1 AU. This gives us the structure of the wind and the corresponding magnetic field. The wind is modeled using a polytropic approximation, and fits and power laws are used to account for the turbulence. Using these results, we compute the parallel and perpendicular diffusion coefficients of the Parker cosmic ray transport equation, yielding 3D maps of the diffusion of cosmic rays in the inner heliosphere. By varying the amplitude of the magnetic field, we change the amplitude of the diffusion by the same factor, and the radial gradients by changing the spread of the current sheet. By varying the geometry of the magnetic field, we change the latitudinal gradients of diffusion by changing the position of the current sheets. By varying the energy, we show that the distribution of values for SEPs is more peaked than GCRs. For realistic solar configurations, we show that diffusion is highly non-axisymmetric due to the configuration of the current sheets, and that the distribution varies a lot with the distance to the Sun with a drift of the peak value. This study shows that numerical simulations, combined with theory, can help quantify better the influence of the various magnetic field parameters on the propagation of cosmic rays. This study is a first step towards the resolution of the complete Parker transport equation to generate synthetic cosmic rays rates from numerical simulations.

The Acceleration of Energetic Particles at Coronal Shocks and Emergence of a Double Power-law Feature in Particle Energy Spectra

Abstract We present numerical modeling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation with spatial diffusion both along and across the magnetic field. We show that the location on the shock where the high-energy particle intensity is the largest, depends on the energy of the particles and on time. The acceleration of particles to more than 100 MeV mainly occurs in the shock-streamer interaction region, due to perpendicular shock geometry and the trapping effect of closed magnetic fields. A comparison of the particle spectra to that in a radial magnetic field shows that the intensity at 100 MeV (200 MeV) is enhanced by more than one order (two orders) of magnitude. This indicates that the streamer-like magnetic field can be an important factor in producing large solar energetic particle events. We also show that the energy spectrum integrated over the simulation domain consists of two different power laws. Further analysis suggests that it may be a mixture of two distinct populations accelerated in the streamer and open field regions, where the acceleration rate differs substantially. Our calculations also show that the particle spectra are affected considerably by a number of parameters, such as the streamer tilt angle, particle spatial diffusion coefficient, and shock compression ratio. While the low-energy spectra agree well with standard diffusive shock acceleration theory, the break energy ranges from ∼1 MeV to ∼90 MeV and the high-energy spectra can extend to ∼1 GeV with a slope of ∼2–3.

Large-scale Compression Acceleration during Magnetic Reconnection in a Low-β Plasma

Abstract In solar flares and other astrophysical systems, a major challenge for solving the particle acceleration problem associated with magnetic reconnection is the enormous scale separation between kinetic scales and the observed reconnection scale. Because of this, it has been difficult to draw any definite conclusions by just using kinetic simulations. A particle acceleration model that solves the energetic particle transport equation can capture the main acceleration physics found in kinetic simulations and thus provide a practical way to make observable predictions and directly compare model results with observations. Here we study compression particle acceleration in magnetic reconnection by solving the Parker (diffusion–advection) transport equation using velocity and magnetic fields from two-dimensional magnetohydrodynamics (MHD) simulations of a low-β high-Lundquist-number reconnection layer. We show that the compressible reconnection layer can give significant particle acceleration, leading to the formation of power-law particle energy distributions. We analyze the acceleration rate and find that the acceleration in the reconnection layer is a mixture of first- and second-order Fermi processes. When including a guide field, we find that the spectrum becomes steeper and both the power-law cutoff energy and maximum particle energy decrease as plasma becomes less compressible. This model produces a 2D particle distribution that one can use to generate a radiation map and directly compare with solar flare observations. This provides a framework to explain particle acceleration at large-scale astrophysical reconnection sites, such as solar flares.