Diffusive Shock Acceleration Theory Research Articles

Analyses of ∼0.05–2 MeV Ions Associated with the 2022 February 16 Energetic Storm Particle Event Observed by Parker Solar Probe

We present analyses of 0.05–2 MeV ions from the 2022 February 16 energetic storm particle event observed by Parker Solar Probe's (PSP) IS⊙IS/EPI-Lo instrument at 0.35 au from the Sun. This event was characterized by an enhancement in ion fluxes from a quiet background, increasing gradually with time with a nearly flat spectrum, rising sharply near the arrival of the coronal mass ejection (CME)–driven shock, becoming nearly a power-law spectrum, then decaying exponentially afterward, with a rate that was independent of energy. From the observed fluxes, we determine diffusion coefficients, finding that far upstream of the shock the diffusion coefficients are nearly independent of energy, with a value of 1020 cm2 s−1. Near the shock, the diffusion coefficients are more than 1 order of magnitude smaller and increase nearly linearly with energy. We also determine the source of energetic particles, by comparing ratios of the intensities at the shock to estimates of the quiet-time intensity to predictions from diffusive shock acceleration theory. We conclude that the source of energetic ions is mostly the solar wind for this event. We also present potential interpretations of the near-exponential decay of the intensity behind the shock. One possibility we suggest is that the shock was overexpanding when it crossed PSP and the energetic particle intensity decreased behind the shock to fill the expanding volume. Overexpanding CMEs could well be more common closer to the Sun, and this is an example of such a case.

Uskorenie solnechnykh kosmicheskikh luchey udarnoy volnoy v nizhney korone Solntsa 7 maya 1978 g.

Based on the theory of diffusive shock acceleration of charged particles, we have theoretically studied the spectra of protons produced by a shock wave driven by a coronal mass ejection (CME) in the lower solar corona with known solar plasma parameters for the solar cosmic ray (SCR) event recorded near the Earth’s orbit on May 7, 1978 (ground level enhancement no. 31, GLE31) using numerical methods. The proton flux data from the CPME instrument installed on the IMP-8 spacecraft and the worldwide network of ground-based neutron monitors combined with the particle measurements by the telescopes on the IMP-7 and IMP-8 satellites have been used to analyze the event. In our calculations, to satisfactorily reproduce the proton spectrum measured at the Earth’s orbit, the CME velocity was assumed to be constant and equal to 600 km s–1. The shock acceleration of SCRs proceeded up to 1.4 R⊙ solar radii for 256 s.

Electron acceleration in supernova remnants

Supernova remnants (SNRs) are believed to produce the majority of galactic cosmic rays (CRs). SNRs harbor non-relativistic collisionless shocks responsible for the acceleration of CRs via diffusive shock acceleration (DSA), in which particles gain their energy via repeated interactions with the shock front. Since the DSA theory involves pre-existing mildly energetic particles, a means of pre-acceleration is required, especially for electrons. Electron injection remains one of the most troublesome and still unresolved issues and our physical understanding of it is essential to fully comprehend the physics of SNRs. To study any electron-scale phenomena responsible for pre-acceleration, we require a method capable of resolving these small kinetic scales and particle-in-cell simulations that fulfill this criterion. Here, I report on the latest achievements made by utilizing kinetic simulations of non-relativistic high Mach number shocks. I discuss how the physics of SNR shocks depends on the shock parameters (e.g. the shock obliquity, Mach number, the ion-to-electron mass ratio) as well as the processes responsible for the electron heating and acceleration.

Nonthermal electron acceleration at collisionless quasi-perpendicular shocks

Shock waves propagating in collisionless heliospheric and astrophysical plasmas have been studied extensively over the decades. One prime motivation is to understand the nonthermal particle acceleration at shocks. Although the theory of diffusive shock acceleration (DSA) has long been the standard for cosmic-ray acceleration at shocks, plasma physical understanding of particle acceleration remains elusive. In this review, we discuss nonthermal electron acceleration mechanisms at quasi-perpendicular shocks, for which substantial progress has been made in recent years. The discussion presented in this review is restricted to the following three specific topics: The first is stochastic shock drift acceleration (SSDA), which is a relatively new mechanism for electron injection into DSA. The basic mechanism, related in-situ observations and kinetic simulations results, and how it is connected with DSA will be discussed. Second, we discuss shock surfing acceleration (SSA) at very high Mach number shocks relevant to young supernova remnants (SNRs). While the original proposal under the one-dimensional assumption is unrealistic, SSA has now been proven efficient by a fully three-dimensional kinetic simulation. We discuss the multidimensional nature of SSA and its role in electron injection. Finally, we discuss the current understanding of the magnetized Weibel-dominated shock. It is essentially a magnetized shock in which the reflected-gyrating ions dominate the formation of the shock structure but with a substantial magnetic field amplification by the ion-Weibel instability. Spontaneous magnetic reconnection of self-generated current sheets within the shock structure is an interesting consequence of Weibel-generated strong magnetic turbulence. Although the exact condition for active magnetic reconnection has not been clarified, we argue that high Mach number shocks with both Alfvén and sound Mach numbers exceeding 20–40 will likely behave as a Weibel-dominated shock. Despite a number of interesting recent findings, the relative roles of SSDA, SSA, and magnetic reconnection for electron acceleration at collisionless shocks and how the dominant particle acceleration mechanisms change depending on shock parameters remain to be answered.

Interpreting the Observed Positive Correlation between the Event-integrated Fluence and the Rollover Energy of Solar Energetic Particle Events by the PAMELA Mission with Coupled Hydromagnetic Wave Excitation and Proton Acceleration at Shocks in the Low Corona

An analytical model for diffusive shock acceleration (DSA) at one-dimensional stationary planar shocks in the lower corona is presented. The model introduces an upstream escape boundary through which a constant flux of protons streaming upstream out of the system is allowed. The nonvanishing flux of streaming protons out of the system limits the maximum attainable energy of DSA and produces a rollover in the high-energy spectra of the shock-accelerated protons. The condition for the rollover energy derived from the model can account for the approximately linear relation between the natural logarithm of event-integrated fluences and the natural logarithm of rollover energies as demonstrated in Bruno et al. Solar energetic particle (SEP) events with higher integrated fluences in principle exhibit higher rollover energies since proton-excited hydromagnetic waves in the turbulent sheath reduce the proton diffusion coefficient and throttle the upstream streaming of protons. The consistency between the observation and the theory of DSA at shocks in the lower corona serves as evidence for the shock origin of protons of the highest energies in large SEP events.

Bulk Properties of Pickup Ions Derived from the Ulysses Solar Wind Ion Composition Spectrometer Data

Nonthermal, pickup ions (PUIs) represent an energetic component of the solar wind (SW). While a number of theoretical models have been proposed to describe the PUI flow, of major importance are in situ measurements providing us with the vital source of model validation. The Solar Wind Ion Composition Spectrometer (SWICS) instrument on board the Ulysses spacecraft was specifically designed for this purpose. Zhang et al. proposed a new, accurate method for the derivation of ion velocity distribution function in the SW frame on the basis of count rates collected by SWICS. We calculate the moments of these distribution functions for protons (H+) and He+ ions along the Ulysses trajectory for a period of 2 months including the Halloween 2003 solar storm. This gives us the time distributions of PUI density and temperature. We compare these with the results obtained earlier for the same interval of time, in which the ion spectra are converted to the SW frame using the narrow-beam approximation. Substantial differences are identified, which are of importance for the interpretation of PUI distributions in the 3D, time-dependent heliosphere. We also choose one of the shocks crossed by Ulysses during this time interval and analyze the distribution functions and PUI bulk properties in front of and behind it. The results are compared with the test-particle calculations and diffusive shock acceleration theory.

Electron Acceleration by Moderate-Mach-number Low-β Shocks: Particle-in-Cell Simulations

Particle acceleration is ubiquitous at shock waves, occurring on scales ranging from supernova remnants in the universe to coronal-mass-ejection-driven shocks and planetary bow shocks in the heliosphere. The most promising mechanism responsible for the almost universally observed power-law spectra is diffusive shock acceleration (DSA). However, how electrons are preaccelerated by different shocks to the energy required by the DSA theory is still unclear. In this paper, we perform two-dimensional particle-in-cell plasma simulations to investigate how the magnetic field orientations, with respect to simulation planes, affect electron preacceleration in moderate-Mach-number low- β shocks. Simulation results show that instabilities can be different as the simulation planes capture different trajectories of particles. For magnetic fields perpendicular to the simulation plane, electron cyclotron drift instability dominates in the foot. Electrons can be trapped by the electrostatic wave and undergo shock-surfing acceleration. For magnetic fields lying in the simulation plane, whistler waves produced by modified two-stream instability dominate in the foot and scatter the electrons. In both cases, electrons undergo multistage acceleration in the foot, shock surface, and immediate downstream, during which process shock-surfing acceleration takes place as part of the preacceleration mechanism in moderate-Mach-number quasi-perpendicular shocks.

Particle acceleration in winds of star clusters

ABSTRACT The origin of cosmic rays in our Galaxy remains a subject of active debate. While supernova remnant (SNR) shocks are often invoked as the sites of acceleration, it is now widely accepted that the difficulties of such sources in reaching PeV energies are daunting and it seems likely that only a subclass of rare remnants can satisfy the necessary conditions. Moreover, the spectra of cosmic rays escaping the remnants have a complex shape that is not obviously the same as the spectra observed at the Earth. Here, we investigate the process of particle acceleration at the termination shock that develops in the bubble excavated by star clusters’ winds in the interstellar medium. While the main limitation to the maximum energy in SNRs comes from the need for effective wave excitation upstream so as to confine particles in the near-shock region and speed up the acceleration process, at the termination shock of star clusters the confinement of particles upstream is guaranteed by the geometry of the problem. We develop a theory of diffusive shock acceleration at such shock and we find that the maximum energy may reach the PeV region for powerful clusters in the high end of the luminosity tail for these sources. A crucial role in this problem is played by the dissipation of energy in the wind to magnetic perturbations. Under reasonable conditions, the spectrum of the accelerated particles has a power-law shape with a slope 4/4.3, in agreement with what is required based upon standard models of cosmic ray transport in the Galaxy.

Mapping the spectral index of Cassiopeia A: evidence for flattening from radio to infrared

ABSTRACT Synchrotron radiation from supernova remnants is caused by electrons accelerated through diffusive shock acceleration (DSA). The standard DSA theory predicts an electron spectral index of p = 2, corresponding to a radio spectral index of α = −0.5. An extension of DSA theory predicts that the accelerated particles change the shock structure, resulting in a spectrum that is steeper than p > 2 (α < −0.5) at low energies and flattens with energy. For Cassiopeia A, a synchrotron spectral flattening was previously reported for a small part of the remnant in the mid-infrared regime. Here, we present new measurements for spectral flattening using archival radio (4.72 GHz) and mid-infrared (3.6 μm) data, and we produce a complete spectral index map to investigate the spatial variations within the remnant. We compare this to measurements of the radio spectral index from L-band (1.285 GHz) and C-band (4.64 GHz) maps. Our result shows overall spectral flattening across the remnant (αR-IR ∼ −0.5 to −0.7), to be compared with the radio spectral index of αR = −0.77. The flattest values coincide with the locations of most recent particle acceleration. In addition to overall flattening, we detect a relatively steeper region in the south-east of the remnant (αR-IR ∼ −0.67). We explore whether these locally steeper spectra could be the result of synchrotron cooling, which provides constraints on the local magnetic field strengths and the age of the plasma, suggesting B ≲ 2 mG for an age of 100 yr, and even B ≲ 1 mG using the age of Cas A, in agreement with other estimates.

Modeling of Cosmic-Ray Production and Transport and Estimation of Gamma-Ray and Neutrino Emissions in Starburst Galaxies

Abstract Starburst galaxies (SBGs) with copious massive stars and supernova (SN) explosions are the sites of active cosmic-ray production. Based on the predictions of nonlinear diffusive shock acceleration theory, we model the cosmic-ray proton (CRP) production by both pre-SN stellar winds (SWs) and supernova remnants (SNRs) from core-collapse SNe inside the starburst nucleus. Adopting different models for the transport of CRPs, we estimate the γ-ray and neutrino emissions due to pp collisions from nearby SBGs such as M82, NGC 253, and Arp 220. We find that with the current γ-ray observations by Fermi-LAT, Veritas, and H.E.S.S., it would be difficult to constrain CRP production and transport models. Yet, the observations are better reproduced with (1) the combination of a single power-law (PL) momentum distribution for SNR-produced CRPs and a diffusion model in which the CRP diffusion is mediated by the strong Kolmogorov-type turbulence of δ B / B ∼ 1 , and (2) the combination of a double-PL model for SNR-produced CRPs and a diffusion model in which the scattering of CRPs is controlled mostly by self-excited waves rather than the preexisting turbulence. The contribution of SW-produced CRPs could be substantial in Arp 220, where the star formation rate is higher and the slope of the initial mass function would be flatter. We suggest that M82 and NGC 253 might be detectable as point sources of high-energy neutrinos in the upcoming KM3NET and IceCube-Gen2, when optimistic models are applied. Future observations of neutrinos, as well as γ-rays, would provide constraints for the production and diffusion of CRPs in SBGs.

A perfect power-law spectrum even at the highest frequencies: The Toothbrush relic

Radio relics trace shock fronts generated in the intracluster medium (ICM) during cluster mergers. The particle acceleration mechanism at the shock fronts is not yet completely understood. We observed the Toothbrush relic with the Effelsberg and Sardinia Radio Telescope at 14.25 GHz and 18.6 GHz, respectively. Unlike previously claimed, the integrated spectrum of the relic closely follows a power law over almost three orders of magnitude in frequency, with a spectral index ofα58 MHz18.6 GHz= −1.16 ± 0.03. Our finding is consistent with a power-law injection spectrum, as predicted by diffusive shock acceleration theory. The result suggests that there is only little magnetic field strength evolution downstream of the shock. From the lack of spectral steepening, we find that either the Sunyaev–Zeldovich decrement produced by the pressure jump is less extended than ∼600 kpc along the line of sight or, conversely, that the relic is located far behind in the cluster. For the first time, we detect linearly polarized emission from the “brush” at 18.6 GHz. Compared to 8.3 GHz, the degree of polarization across the brush increases at 18.6 GHz, suggesting a strong Faraday depolarization toward lower frequencies. The observed depolarization is consistent with an intervening magnetized screen that arises from the dense ICM containing turbulent magnetic fields. The depolarization, corresponding to a standard deviation of the rotation measures as high asσRM = 212 ± 23 rad m−2, suggests that the brush is located in or behind the ICM. Our findings indicate that the Toothbrush relic can be consistently explained by the standard scenario for relic formation.

Solar Cosmic Ray Acceleration at the Front of a Fast Shock in the Lower Solar Corona

Based on the theory of diffusive shock acceleration of charged particles, we have numerically studied the spectra of protons produced by a fast shock with a speed of 5000 km s–1 in the lower solar corona with known solar plasma parameters. We show that protons with energies $$ \gtrsim $$ 105 MeV can be produced at a distance up to 3 $${{R}_{ \odot }}$$ ( $${{R}_{ \odot }}$$ is the solar radius) within 274 s.

Shock acceleration efficiency in radio relics

Context. Radio relics in galaxy clusters are giant diffuse synchrotron sources powered in cluster outskirts by merger shocks. Although the relic–shock connection has been consolidated in recent years by a number of observations, the details of the mechanisms leading to the formation of relativistic particles in this environment are still not well understood. Aims. The diffusive shock acceleration (DSA) theory is a commonly adopted scenario to explain the origin of cosmic rays at astrophysical shocks, including those in radio relics in galaxy clusters. However, in a few specific cases it has been shown that the energy dissipated by cluster shocks is not enough to reproduce the luminosity of the relics via DSA of thermal particles. Studies based on samples of radio relics are required to further address this limitation of the mechanism. Methods. In this paper, we focus on ten well-studied radio relics with underlying shocks observed in the X-rays and calculate the electron acceleration efficiency of these shocks that is necessary to reproduce the observed radio luminosity of the relics. Results. We find that in general the standard DSA cannot explain the origin of the relics if electrons are accelerated from the thermal pool with an efficiency significantly smaller than 10%. Our results show that other mechanisms, such as shock re-acceleration of supra-thermal seed electrons or a modification of standard DSA, are required to explain the formation of radio relics.

DHybridR: A Hybrid Particle-in-cell Code Including Relativistic Ion Dynamics

Abstract We present the first plasma simulations obtained with the code dHybridR, a hybrid particle-in-cell code with fluid electrons and both thermal and energetic ions that retain relativistic dynamics. dHybridR is constructed to study astrophysical and space-physics problems where a few energetic nonthermal particles (i.e., cosmic rays, CRs) affect the overall dynamics of a nonrelativistic plasma, such as CR-driven instabilities, collisionless shocks, magnetic reconnection, turbulence, etc. In this method paper we provide some applications to linear (resonant/nonresonant CR streaming instability) and strongly nonlinear (parallel shocks) problems that show the capabilities of the code. In particular, we provide the first self-consistent hybrid runs that show the acceleration of relativistic ions at nonrelativistic shocks; CRs develop a power law in momentum, which translates into a broken power law in energy that exhibits a steepening around the ion rest mass, as predicted by the theory of diffusive shock acceleration. We present examples of 2D dHybridR runs relevant for fast shocks in radio supernovae, whose evolution can be followed in real time, and 3D runs of low-Mach-number heliospheric shocks, which can be compared with in situ spacecraft observations.

Steepening of Cosmic-Ray Spectra in Shocks with Varying Magnetic Field Direction

Abstract Cosmic-ray (CR) spectra, both measured upon their arrival at the Earth’s atmosphere and inferred from the emission in supernova remnants (SNRs), appear to be significantly steeper than the “standard” diffusive shock acceleration (DSA) theory predicts. Although the reconstruction of the primary spectra introduces an additional steepening due to propagation effects, there is a growing consensus in the CR community that these corrections fall short to explain the newest high-precision data. Using 2D hybrid simulations, we investigate a new mechanism that may steepen the spectrum during the acceleration in SNR shocks. Most of the DSA treatments are limited to homogeneous shock environments. To investigate whether inhomogeneity effects can produce the necessary extra steepening, we assume that the magnetic field changes its angle along the shock front. The rationale behind this approach is the strong dependence of the DSA efficiency upon the field angle, θ Bn. Our results show that the variation of shock obliquity along its face results in a noticeable steepening of the DSA spectrum. Compared to simulations of quasi-parallel shocks, we observe an increase of the spectral index by Δq = 0.1–0.15. Possible extrapolation of the limited simulation results to more realistic SNR conditions are briefly considered.

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.

Solar Cosmic Ray Acceleration by a Shock Wave in the Lower Solar Corona on November 22, 1977

Based on the theory of diffusive shock acceleration of charged particles, we have investigated the spectra of protons recorded in the solar cosmic ray event near the Earth’s orbit on November 22, 1977 (ground level enhancement no. 30, GLE30). The proton flux data from the CPME instrument installed on the IMP-8 spacecraft and the worldwide network of neutron monitors have been used to analyze the event. Using GLE30 as an example, we have shown for the first time that solar cosmic rays of relativistic energies can be produced by a shock wave with a relatively low speed of 560 km s–1 in the lower solar corona at a distance up to 1.6 $${{R}_{ \odot }}$$ ( $${{R}_{ \odot }}$$ is the solar radius) within 615 s. The calculated proton spectra satisfactorily reproduce the measurements in the Earth’s orbit.

Solar Energetic Particle Events Observed by the PAMELA Mission

Abstract Despite the significant progress achieved in recent years, the physical mechanisms underlying the origin of solar energetic particles (SEPs) are still a matter of debate. The complex nature of both particle acceleration and transport poses challenges to developing a universal picture of SEP events that encompasses both the low-energy (from tens of keV to a few hundreds of MeV) observations made by space-based instruments and the GeV particles detected by the worldwide network of neutron monitors in ground-level enhancements (GLEs). The high-precision data collected by the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) satellite experiment offer a unique opportunity to study the SEP fluxes between ∼80 MeV and a few GeV, significantly improving the characterization of the most energetic events. In particular, PAMELA can measure for the first time with good accuracy the spectral features at moderate and high energies, providing important constraints for current SEP models. In addition, the PAMELA observations allow the relationship between low- and high-energy particles to be investigated, enabling a clearer view of the SEP origin. No qualitative distinction between the spectral shapes of GLE, sub-GLE and non-GLE events is observed, suggesting that GLEs are not a separate class, but are the subset of a continuous distribution of SEP events that are more intense at high energies. While the spectral forms found are to be consistent with diffusive shock acceleration theory, which predicts spectral rollovers at high energies that are attributed to particles escaping the shock region during acceleration, further work is required to explore the relative influences of acceleration and transport processes on SEP spectra.

Diffusive shock acceleration and turbulent reconnection

Diffusive Shock Acceleration (DSA) cannot efficiently accelerate particles without the presence of self-consistently generated or pre-existing strong turbulence ($ \delta B/B \sim 1 $) in the vicinity of the shock. The problem we address in this article is: if large amplitude magnetic disturbances are present upstream and downstream of a shock then Turbulent Reconnection (TR) will set in and will participate not only in the elastic scattering of particles but also in their heating and acceleration. We demonstrate that large amplitude magnetic disturbances and Unstable Current Sheets (UCS), spontaneously formed in the strong turbulence in the vicinity of a shock, can accelerate particles as efficiently as DSA in {\bf large scale systems and on long time scales}. We start our analysis with "elastic" scatterers upstream and downstream and estimate the energy distribution of particles escaping from the shock, recovering the well known results from the DSA theory. Next we analyze the additional interaction of the particles with active scatterers (magnetic disturbances and UCS) upstream and downstream of the shock. We show that the asymptotic energy distribution of the particles accelerated by DSA/TR has very similar characteristics with the one due to DSA alone, but the synergy of DSA with TR is much more efficient: The acceleration time is an order of magnitude shorter and the maximum energy reached two orders of magnitude higher. We claim that DSA is the dominant acceleration mechanism in a short period before TR is established, and then strong turbulence will dominate the heating and acceleration of the particles. In other words, the shock serves as the mechanism to set up a strongly turbulent environment, in which the acceleration mechanism will ultimately be the synergy of DSA and TR.

Particle Acceleration and Alfv’en Wave Generation at the April 4, 2001 Interplanetary Shock

Based on the theory of diffusive shock acceleration of charged particles, we investigate the formation of proton spectra and the generation of an Alfv’en turbulence spectrum by accelerated (storm) particles at the interplanetary shock front in the event occurred at 14:22 UT on April 4, 2001. We formulate a scenario whereby a satisfactory theoretical description of the observed proton and Alfv’en wave spectra obtained from measurements becomes possible: the propagation of solar cosmic rays in interplanetary space before the interplanetary shock arrival at the Earth’s orbit creates an enhanced level of Alfv’en turbulence in the solar wind, which reduces the diffusion coefficient of particles and increases the efficiency of their acceleration.