Diffusive Shock Acceleration Model Research Articles

Tempestuous life beyond R500: X-ray view on the Coma cluster with SRG/eROSITA

This is the second paper in a series of studies of the Coma cluster using the SRG/eROSITA X-ray data obtained during the calibration and performance verification phase of the mission. Here, we focus on the region adjacent to the radio source 1253+275 (radio relic, RR, hereafter). We show that the X-ray surface brightness exhibits its steepest gradient at ∼79′ (∼2.2 Mpc ≈ R200c), which is almost co-spatial to the outer edge of the RR. As in the case of several other relics, the Mach number of the shock derived from the X-ray surface brightness profile (MX ≈ 1.9) appears to be lower than needed to explain the slope of the integrated radio spectrum in the diffusive shock acceleration model (MR ≈ 3.5) if the magnetic field is uniform and the radiative losses are fast. However, the shock geometry is plausibly much more complicated than a spherical wedge centered on the cluster, given the non-trivial correlation between radio, X-ray, and SZ images. While the complicated shock geometry alone might cause a negative bias in MX, we speculate on a few other possibilities that may affect the MX − MR relation, including the shock substructure that might be modified by the presence of non-thermal filaments stretching across the shock and the propagation of relativistic electrons along the non-thermal filaments with a strong magnetic field. We also discuss the “history” of the radio galaxy NGC 4789, which is located ahead of the relic in the context of the Coma-NGC 4839 merger scenario.

Ultra High Energy Cosmic Ray Source Models: Successes, Challenges and General Predictions

Understanding the acceleration of Ultra High Energy Cosmic Rays is one of the great challenges of contemporary astrophysics. In this short review, we summarize the general observational constraints on their composition, spectrum and isotropy which indicate that nuclei heavier than single protons dominate their spectra above ˜ 5 EeV, that they are strongly suppressed above energies ˜ 50 EeV, and that the only significant departure from isotropy is a dipole. Constraints based upon photopion and photodisintegration losses allow their ranges and luminosity density to be estimated. Three general classes of source model are discussed - magnetospheric models (including neutron stars and black holes), jet models (including Gamma Ray Bursts, Active Galactic Nuclei and Tidal Disruption Events) and Diffusive Shock Acceleration models (involving large accretion shocks around rich clusters of galaxies). The value of constructing larger and more capable arrays to measure individual masses at the highest energies and probably identifying their sources is emphasized.

Diffuse radio source candidate in CIZA J1358.9−4750

Abstract We report on results of our upgraded Giant Metrewave Radio Telescope (uGMRT) observations for an early-stage merging galaxy cluster, CIZA J1358.9−4750 (CIZA1359), in Band-3 (300–500 MHz). We achieved the image dynamic range of ∼38000 using the direction dependent calibration and found a candidate of diffuse radio emission at 4σrms significance. The flux density of the candidate at 400 MHz, 24.04 ± 2.48 mJy, is significantly positive compared to noise, where its radio power, 2.40 × 1024 W Hz−1, is consistent with those of typical diffuse radio sources of galaxy clusters. The candidate is associated with a part of the X-ray shock front at which the Mach number reaches its maximum value of $\mathcal {M}\sim 1.7$. The spectral index (Fν ∝ να) of the candidate, α = −1.22 ± 0.33, is in agreement with an expected value derived from the standard diffusive shock acceleration (DSA) model. However, such a low Mach number with a short acceleration time would require seed cosmic rays supplied from active galactic nucleus (AGN) activities of member galaxies, as suggested in some other clusters. Indeed, we found seven AGN candidates inside the diffuse source candidate. Assuming the energy equipartition between magnetic fields and cosmic rays, the magnetic field strength of the candidate was estimated to be 2.1 μG. We also find head–tail galaxies and radio phoenixes or fossils near CIZA1359.

A Multi-Event Study of Early-Stage SEP Acceleration by CME-Driven Shocks—Sun to 1 AU

The solar corona between below 10 solar radii is an important region for early acceleration and transport of solar energetic particles (SEPs) by coronal mass ejection-driven shock waves. There, these waves propagate into a highly variable dynamic medium with steep gradients and rapidly expanding coronal magnetic fields, which modulates the particle acceleration near the shock/wave surfaces, and the way SEPs spread into the heliosphere. We present a study modeling the acceleration of SEPs in global coronal shock events in the corona, as well as their transport to 1 au, based on telescopic observations coupled with dynamic physical models. As part of the project Solar Particle Radiation Environment Analysis and Forecasting—Acceleration and Scattering Transport (SPREAdFAST), we model the interaction of observed off-limb coronal bright fronts (CBF) with the coronal plasma from synoptic magnetohydrodynamic (MHD) simulations. We then simulate the SEP acceleration in analytical diffusive shock acceleration (DSA) model. The simulated fluxes are used as time-dependent inner boundary conditions for modeling the particle transport to 1 au. Resulting flux time series are compared with 1 au observations for validation. We summarize our findings and present implications for nowcasting SEP acceleration and heliospheric connectivity.

Steep Cosmic-Ray Spectra with Revised Diffusive Shock Acceleration

Abstract Galactic cosmic rays (CRs) are accelerated at the forward shocks of supernova remnants (SNRs) via diffusive shock acceleration (DSA), an efficient acceleration mechanism that predicts power-law energy distributions of CRs. However, observations of nonthermal SNR emission imply CR energy distributions that are generally steeper than E −2, the standard DSA prediction. Recent results from kinetic hybrid simulations suggest that such steep spectra may arise from the drift of magnetic structures with respect to the thermal plasma downstream of the shock. Using a semi-analytic model of nonlinear DSA, we investigate the implications that these results have on the phenomenology of a wide range of SNRs. By accounting for the motion of magnetic structures in the downstream, we produce CR energy distributions that are substantially steeper than E −2 and consistent with observations. Our formalism reproduces both modestly steep spectra of Galactic SNRs (∝E −2.2) and the very steep spectra of young radio supernovae (∝E −3).

X-Ray Observations of 1ES 1959+650 in Its High-activity State in 2016–2017 with AstroSat and Swift

Abstract We present a comprehensive multifrequency study of the high-frequency-peaked BL Lac object 1ES 1959+650 using data from various facilities during the period 2016–2017, including X-ray data from AstroSat and Swift during the historically high X-ray flux state of the source observed until 2021 February. The unprecedented quality of X-ray data from high-cadence monitoring with AstroSat during 2016–2017 enables us to establish a detailed description of X-ray flares in 1ES 1959+650. The synchrotron peak shifts significantly between different flux states, in a manner consistent with a geometric (changing Doppler factor) interpretation. A time-dependent leptonic diffusive shock acceleration and radiation transfer model is used to reproduce the spectral energy distributions and X-ray light curves, to provide insight into the particle acceleration during the major activity periods observed in 2016 and 2017. The extensive data of Swift-XRT from 2015 December to 2021 February (exp. = 411.3 ks) reveals a positive correlation between flux and peak position.

Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radio relics

Context. Radio relics in galaxy clusters are known to be good laboratories for verification of the applicability of the diffusive shock acceleration (DSA) model in its canonical version. The need for such verification stems from the inconsistencies in the shock properties resulting from radio observations compared to X-ray observations. Aims. In this article we aim to explore how the presence of a second shock in the vicinity of a relic modifies the spectrum of accelerated electrons and decipher which of the involved parameters can have a significant impact on their shape. Methods. We analytically studied DSA of cosmic rays in two stationary shocks aiming to investigate the change of the distribution function. The latter eventually leads to spectrum slope deviations visible in different observations and simulations that do not appear to be explained by the case wherein cosmic rays interact with a single shock wave. Results. We obtain a complex distribution function f(x, p) depending on many parameters (distance between two shocks, compression ratios, spatial diffusion coefficients, injection value, etc.). This function reveals modifications that occur because of the coupled acceleration in both shocks. Apparently, deviations in the particle spectrum from the pure power law depend on a few parameters such as Q1/Q2, κ1/κ2, r1/r2, and L. Although we do not verify this idea by taking a particular cluster as an example, we demonstrate a potential cause of spectral disturbances in radio relics. In general terms, our findings appear to correlate with results from the literature when the distance between the shocks is of the order of the width of a radio relic and κ1/κ2 ∝ 3.

Non-linear diffusive shock acceleration: A recipe for injection of electrons

Prescriptions for electron injection into the diffusive shock acceleration process are required in many practical considerations of cosmic-ray astrophysics, particularly in modeling of the synchrotron emission of astrophysical sources. In particle-in-cell simulations of quasi-parallel magnetized collisionless shocks, we analyse the evolution of particle spectra. We find that in the later stages of shock evolution, the initially strong suprathermal part in the ion spectra fades, thus leaving the spectra composed of a Maxwellian and a power law. Once the electron and ion spectra flatten and become parallel, we find that the amounts of cosmic ray ions and electrons become similar. We make the step towards relating the micro and macro-scale physics by applying this injection rule to Blasi’s semi-analytical model of non-linear diffusive shock acceleration, in order to obtain the particle spectra and electron-to-proton ratio Kep at high energies. By using shock jump conditions that include the electron heating, we find Kep as a function of Mach number. For Mach number ∼ 100, our model finely reproduce the typically observed ratio for Galactic cosmic-rays Kep∼ 1:100 in the test particle regime.

The application of the adiabatic compression scenario to the radio relic in the galaxy cluster Abell 3411−3412

ABSTRACT Although radio relics are understood to originate in intracluster shock waves resulting from merger shocks, the most widely used model for describing this (re-)acceleration process at shock fronts, the diffusive shock acceleration (DSA) model, has several challenges, including the fact that it is inefficient at low shock Mach numbers. In light of these challenges, it is worthwhile to consider alternative mechanisms. One possibility is the adiabatic compression by a shock wave of a residual fossil electron population which has been left over from a radio galaxy jet. This paper applies this model to the relic hosted in the merging galaxy cluster Abell 3411−3412, where a radio bridge between the relic and a radio galaxy has been observed, with the aim to reproduce the spatial structure of the spectral index of the relic. Four scenarios are presented, in which different effects are investigated, such as effects behind the shock front and different shock strengths. The results show that the adiabatic compression model can reproduce the observed spectral indices across the relic for a shock Mach number that is lower than the value required by the DSA-type modelling of this relic and is in accordance with the values derived from X-ray observations, if other mechanisms, such as an expansion phase or post-shock turbulence, are effective behind the shock front.

Gamma-Ray and Neutrino Emissions due to Cosmic-Ray Protons Accelerated at Intracluster Shocks in Galaxy Clusters

Abstract We examine the cosmic-ray protons (CRp) accelerated at collisionless shocks in galaxy clusters using cosmological structure formation simulations. We find that in the intracluster medium (ICM) within the virial radius of simulated clusters, only ∼7% of shock kinetic energy flux is dissipated by the shocks that are expected to accelerate CRp—that is, supercritical, quasi-parallel (Q ∥) shocks with sonic Mach number M s ≥ 2.25. The rest is dissipated at subcritical shocks and quasi-perpendicular shocks, both of which may not accelerate CRp. Adopting the diffusive shock acceleration (DSA) model recently presented in Ryu et al., we quantify the DSA of CRp in simulated clusters. The average fraction of the shock kinetic energy transferred to CRp via DSA is assessed at ∼(1–2) × 10−4. We also examine the energization of CRp through reacceleration using a model based on the test-particle solution. Assuming that the ICM plasma passes through shocks three times on average through the history of the universe and that CRp are reaccelerated only at supercritical Q ∥-shocks, the CRp spectrum flattens by ∼0.05–0.1 in slope and the total amount of CRp energy increases by ∼40%–80% from reacceleration. We then estimate diffuse γ-ray and neutrino emissions, resulting from inelastic collisions between CRp and thermal protons. The predicted γ-ray emissions from simulated clusters lie mostly below the upper limits set by Fermi-LAT for observed clusters. The neutrino fluxes toward nearby clusters would be ≲10−4 of the IceCube flux at E ν = 1 PeV and ≲10−6 of the atmospheric neutrino flux in the energy range of E ν ≤ 1 TeV.

A Diffusive Shock Acceleration Model for Protons in Weak Quasi-parallel Intracluster Shocks

Abstract Low sonic Mach number shocks form in the intracluster medium (ICM) during the formation of the large-scale structure of the universe. Nonthermal cosmic-ray (CR) protons are expected to be accelerated via diffusive shock acceleration (DSA) in those ICM shocks, although observational evidence for the γ-ray emission of hadronic origin from galaxy clusters has yet to be established. Considering the results obtained from recent plasma simulations, we improve the analytic test-particle DSA model for weak quasi-parallel ( ) shocks, previously suggested by Kang & Ryu. In the model CR spectrum, the transition from the postshock thermal to CR populations occurs at the injection momentum, p inj, above which protons can undergo the full DSA process. As the shock energy is transferred to CR protons, the postshock gas temperature should decrease accordingly and the subshock strength weakens due to the dynamical feed of the CR pressure to the shock structure. This results in the reduction of the injection fraction, although the postshock CR pressure approaches an asymptotic value when the CR spectrum extends to the relativistic regime. Our new DSA model self-consistently accounts for such behaviors and adopts better estimations for p inj. With our model DSA spectrum, the CR acceleration efficiency ranges from η ∼ 10−3–0.01 for supercritical, -shocks with sonic Mach number 2.25 ≲ M s ≲ 5 in the ICM. Based on Ha et al., on the other hand, we argue that proton acceleration would be negligible in subcritical shocks with M s < 2.25.

Spectrum of Electrons Accelerated in Supernova Remnants.

Using a semianalytic model of nonlinear diffusive shock acceleration, we model the total spectrum of cosmic ray (CR) electrons accelerated by supernova remnants (SNRs). Because electrons experience synchrotron losses in the amplified magnetic fields characteristic of SNRs, they exhibit substantially steeper spectra than protons. In particular, we find that the difference between the electron and proton spectral index (power law slope) ranges from 0.1 to 0.4. Our findings must be reckoned with theories of Galactic CR transport, which often assume that electrons and protons are injected with the same slope, and may especially have implications for the observed "positron excess".

Study of Time Evolution of the Bend-over Energy in the Energetic Particle Spectrum at a Parallel Shock

Abstract Shock acceleration is considered one of the most important mechanisms for the acceleration of astrophysical energetic particles. In this work, we calculate the trajectories of a large number of test charged particles accurately in a parallel shock with magnetic turbulence. We investigate the time evolution of the accelerated particle energy spectrum in the downstream of the shock, in order to understand the acceleration mechanism of energetic particles. From simulation results we obtain power-law energy spectra with a bend-over energy, E 0, increasing with time. With the particle mean acceleration time and mean momentum change during each cycle of the shock crossing from the diffusive shock acceleration model (following Drury), a time-dependent differential equation for the maximum energy, E acc, of particles accelerated at the shock can be approximately obtained. We assume the theoretical bend-over energy as E acc. It is found that the bend-over energy from simulations agrees well with the theoretical bend-over energy using the nonlinear diffusion theory, NLGCE-F, in contrast to that using the classic quasi-linear theory.

Spectral hardening of cosmic ray protons and helium nuclei in supernova remnant shocks * * Supported by the National Natural Science Foundation of China (11433004, U1738211, 11563009), the Yunnan Applied Basic Research Projects (2015FB103, 2016FB001, 2018FY001 (-003)), and National Key R&D Program of China (2018YFA0404204)

The observed hardening of the spectra of cosmic ray protons and helium nuclei is studied within the model of nonlinear diffusive shock acceleration of supernova remnants (SNRs). In this model, the injected particles with energies below the spectral " knee” are assumed to be described by two populations with different spectral indexes around 200 GeV. The high-energy population is dominated by the particles with energies above 200 GeV released upstream of the shock of SNR, and the low-energy population is attributed to the particles with energies below 200 GeV released downstream of the shock of SNR. In this scenario, the spectral hardening of cosmic ray protons and helium nuclei observed by PAMELA, AMS-02, and CREAM experiments can be reproduced.

Early-stage Solar Energetic Particle Acceleration by Coronal Mass Ejection-driven Shocks with Realistic Seed Spectra. I. Low Corona

Abstract An outstanding problem in heliospheric physics is understanding the acceleration of solar energetic particles (SEPs) in coronal mass ejections (CMEs) and flares. A fundamental question is whether the acceleration occurs in interplanetary space or near the Sun. Recent work has shown that CME-driven shocks may produce SEPs while still below five solar radii. In this work we explore SEP acceleration during the onset of CMEs and shocks even lower in the corona, using realistic suprathermal spectra, for a selection of events. We have calculated quiet-time, pre-event suprathermal particle spectra from 1 au observations, and scaled them back to the low corona to serve as seed spectra. For each event, AIA observations and the CASHeW framework were used to model the compressive/shock wave kinematics and its interaction with the corona. The proton acceleration was then modeled using an analytic diffusive shock acceleration model as the shock waves propagate between ∼1.05 and ∼1.3 solar radii. We demonstrate the capability of low coronal shock-related extreme ultraviolet waves to accelerate protons to multi-MeV energies in a matter of minutes, in the very early stages of the associated solar eruptions. We find that strong proton energization occurs for high values of the density jump, Alfvén Mach number, and shock speed. In future work the results of this early-stage shock acceleration will be used to model the continued acceleration higher in the corona.

X-ray and Radio Observations of the Radio Relic Galaxy Clusters 1RXS J0603.3+4214 and RXC J1053.7+5453

We study two galaxy clusters with radio relics, 1RXS J0603.3+4214 and RXC J1053.7+5453, through X-ray and radio observations. Radio relics are diffuse non-thermal radio sources found in outskirts of galaxy clusters. Because of their shape and location, they are thought to be related to cluster merger shocks. The galaxy cluster 1RXS J0603.3+4214 has a well-known linear-shape “toothbrush” radio relic. We investigate the temperature structure across the relic to constrain the Mach number of the associated shock. The results are compared with radio spectral results, which suggest that a simple diffusive shock acceleration model does not hold for this relic. The RXC J1053.7+5453 harbors a standard arc-like relic. We also get the Mach number from the temperature profile. In addition, we found an edge-like structure in the X-ray image between the X-ray peak and relic. We investigate the density and temperature profiles across the edge and found that the structure is likely relevant to not a shock but a contact discontinuity.

On the Foundation of Equipartition in Supernova Remnants

Abstract A widely accepted paradigm is that equipartition (eqp) between the energy density of cosmic rays (CRs) and the energy density of the magnetic field cannot be sustained in supernova remnants (SNRs). However, our 3D hydrodynamic supercomputer simulations, coupled with a nonlinear diffusive shock acceleration model, provide evidence that eqp may be established at the end of the Sedov phase of evolution in which most SNRs spend the longest portions of their lives. We introduce the term “constant partition” for any constant ratio between the CR energy density and the energy density of the magnetic field in an SNR, while the term “equipartition” should be reserved for the case of approximately the same values of the energy density (also, it is constant partition in the order of magnitude) of ultra-relativistic electrons only (or CRs in total) and the energy density of the magnetic field. Our simulations suggest that this approximate constant partition exists in all but the youngest SNRs. We speculate that since evolved SNRs at the end of the Sedov phase of evolution can reach eqp between CRs and magnetic fields, they may be responsible for initializing this type of eqp in the interstellar medium. Additionally, we show that eqp between the electron component of CRs and the magnetic field may be used for calculating the magnetic field strength directly from observations of synchrotron emission from SNRs. The values of magnetic field strengths in SNRs given here are approximately 2.5 times lower than values calculated by Arbutina et al.

Shock Acceleration Model for the Toothbrush Radio Relic

Abstract Although many of the observed properties of giant radio relics detected in the outskirts of galaxy clusters can be explained by relativistic electrons accelerated at merger-driven shocks, significant puzzles remain. In the case of the so-called Toothbrush relic, the shock Mach number estimated from X-ray observations ( ) is substantially weaker than that inferred from the radio spectral index ( ). Toward understanding such a discrepancy, we here consider the following diffusive shock acceleration (DSA) models: (1) weak-shock models with and a preexisting population of cosmic-ray electrons (CRe) with a flat energy spectrum, and (2) strong-shock models with and either shock-generated suprathermal electrons or preexisting fossil CRe. We calculate the synchrotron emission from the accelerated CRe, following the time evolution of the electron DSA, and the subsequent radiative cooling and postshock turbulent acceleration (TA). We find that both models could reproduce reasonably well the observed integrated radio spectrum of the Toothbrush relic, but the observed broad transverse profile requires the stochastic acceleration by downstream turbulence, which we label “turbulent acceleration” or TA to distinguish it from DSA. Moreover, to account for the almost uniform radio spectral index profile along the length of the relic, the weak-shock models require a preshock region over 400 kpc with a uniform population of preexisting CRe with a high cutoff energy ( ). Due to the short cooling time, it is challenging to explain the origin of such energetic electrons. Therefore, we suggest the strong-shock models with low-energy seed CRe ( ) are preferred for the radio observations of this relic.

Hydrodynamical and radio evolution of young supernova remnant G1.9+0.3 based on the model of diffusive shock acceleration

The radio evolution of, so far the youngest known, Galactic supernova remnant (SNR) G1.9+0.3 is investigated by using three-dimensional (3D) hydrodynamic modelling and non-linear kinetic theory of cosmic ray (CR) acceleration in SNRs. We include consistent numerical treatment of magnetic field amplification (MFA) due to resonant streaming instability. Under the assumption that SNR G1.9+0.3 is the result of a Type Ia supernova explosion located near the Galactic Centre, using widely accepted values for explosion energy 1051 erg and ejecta mass 1.4 M⊙, the non-thermal continuum radio emission is calculated. The main purpose of this paper is to explain radio flux brightening measured over recent decades and also predict its future temporal evolution. We estimate that the SNR is now ∼120 yr old, expanding in an ambient density of 0.02 cm−3, and explain its steep radio spectral index only by means of efficient non-linear diffusive shock acceleration (NLDSA). We also make comparison between simulations and observations of this young SNR, in order to test the models and assumptions suggested. Our model prediction of a radio flux density increase of ∼1.8 per cent yr−1 during the past two decades agrees well with the measured values. We synthesize the synchrotron spectrum from radio to X-ray and it fits well the Very Large Array, Molonglo Observatory Synthesis Telescope, Effelsberg, Chandra and NuSTAR measurements. We also propose a simplified evolutionary model of the SNR in gamma rays and suggest it may be a promising target for gamma-ray observations at TeV energies with the future generation of instruments like Cherenkov Telescope Array. SNR G1.9+0.3 is the only known Galactic SNR with the increasing flux density and we present here the prediction that the flux density will start to decrease approximately 500 yr from now. We conclude that this is a general property of SNRs in the free expansion phase.

COMBINING DIFFUSIVE SHOCK ACCELERATION WITH ACCELERATION BY CONTRACTING AND RECONNECTING SMALL-SCALE FLUX ROPES AT HELIOSPHERIC SHOCKS

ABSTRACT Computational and observational evidence is accruing that heliospheric shocks, as emitters of vorticity, can produce downstream magnetic flux ropes and filaments. This led Zank et al. to investigate a new paradigm whereby energetic particle acceleration near shocks is a combination of diffusive shock acceleration (DSA) with downstream acceleration by many small-scale contracting and reconnecting (merging) flux ropes. Using a model where flux-rope acceleration involves a first-order Fermi mechanism due to the mean compression of numerous contracting flux ropes, Zank et al. provide theoretical support for observations that power-law spectra of energetic particles downstream of heliospheric shocks can be harder than predicted by DSA theory and that energetic particle intensities should peak behind shocks instead of at shocks as predicted by DSA theory. In this paper, a more extended formalism of kinetic transport theory developed by le Roux et al. is used to further explore this paradigm. We describe how second-order Fermi acceleration, related to the variance in the electromagnetic fields produced by downstream small-scale flux-rope dynamics, modifies the standard DSA model. The results show that (i) this approach can qualitatively reproduce observations of particle intensities peaking behind the shock, thus providing further support for the new paradigm, and (ii) stochastic acceleration by compressible flux ropes tends to be more efficient than incompressible flux ropes behind shocks in modifying the DSA spectrum of energetic particles.