Diffusive shock acceleration (DSA) by relativistic shocks is thought to generate the $dN/dE\propto E^{-p}$ spectra of charged particles in various astronomical relativistic flows. We show that for test particles in one dimension (1D), $p^{-1}=1-\ln\left[γ_d(1+β_d)\right]/\ln\left[γ_u(1+β_u)\right]$, where $β_u$ ($β_d)$ is the upstream (downstream) normalized velocity, and $γ$ is the respective Lorentz factor. This analytically captures the main properties of relativistic DSA in higher dimensions, with no assumptions on the diffusion mechanism. Unlike 2D and 3D, here the spectrum is sensitive to the equation of state even in the ultra-relativistic limit, and (for a J{ü}ttner-Synge equation of state) noticeably hardens with increasing $1<γ_u<57$, before logarithmically converging back to $p(γ_u\to\infty)=2$. The 1D spectrum is sensitive to drifts, but only in the downstream, and not in the ultra-relativistic limit.
A new numerical model of the nonlinear diffusive shock acceleration is presented. It is used for modeling of particle acceleration in supernova remnants. The model contains coupled spherically symmetric hydrodynamic equations and the transport equations for energetic protons, ions and electrons. The forward and reverse shocks are included in the consideration. The spectra of cosmic rays released into interstellar medium from a supernova remnant are determined. The role of the reverse shock in the production of CR ions and positrons is discussed.
We present a three-dimensional hybrid simulation of a collisionless perpendicular shock in a partially ionized plasma for the first time. In this simulation, the shock velocity and the upstream ionization fraction are Vsh ~ 1333 km/s and fi ~ 0.5, that are typical values for isolated young supernova remnants in the interstellar medium. We confirm previous two-dimensional simulation results that downstream hydrogen atoms leak into the upstream region, they are accelerated by the pickup process in the upstream region, and large magnetic field fluctuations are generated both in the upstream and d…
We provide a both qualitative and quantitative comparison among different approaches aimed to solve the problem of non-linear diffusive acceleration of particles at shocks. In particular, we show that state-of-the-art models (numerical, Monte Carlo and semi-analytical), even if based on different physical assumptions and implementations, for typical environmental parameters lead to very consistent results in terms of shock hydrodynamics, cosmic ray spectrum and also escaping flux spectrum and anisotropy. Strong points and limits of each approach are also discussed, as a function of the problem…
We present a more accurate numerical scheme for the calculation of diffusive shock acceleration of cosmic rays using Stochastic Differential Equations. The accuracy of this scheme is demonstrated using a simple analytical flow profile that contains a shock of finite width and a varying diffusivity of the cosmic rays, where the diffusivity decreases across the shock. We compare the results for the slope of the momentum distribution with those obtained from a perturbation analysis valid for finite but small shock width. These calculations show that this scheme, although computationally more expe…
The present work considers a plane shock front propagating along a cylindrical jet. Electrons experience the diffusive shock acceleration around the shock front, and subsequently drift away into the downstream flow in which they emit most of their energy. Assuming a proper boundary condition at the interface between the shock zone and the downstream zone, we solve the transport equation for the electrons in the downstream flow zone, where the combined effects of escape, synchrotron and IC cooling in the Thomson regime are taken into account. Using the electron spectrum obtained in this manner …
The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, $M\lesssim 3$, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diffusive shock acceleration (DSA). We examine the effects of DSA by multiple shocks on the spectrum of accelerated CRs by including {\it in situ} injection/acceleration at each shock, followed by repeated re-acceleration at successive shocks in the test-particle regime. For simplicity, the accelerat…
The problem of accelerating cosmic rays is one of fundamental importance, particularly given the uncertainty in the conditions inside the acceleration sites. Here we examine Diffusive Shock Acceleration in arbitrary turbulent magnetic fields, constructing a new model that is capable of bridging the gap between the very weak ($δB/B_0 \ll 1$) and the strong turbulence regimes. To describe the diffusion we provide quantitative analytical description of the "Bohm exponent" in each regime. We show that our results converge to the well known quasi-linear theory in the weak turbulence regime. In the …
Diffusive shock acceleration (DSA) at relativistic shocks is expected to be an important acceleration mechanism in a variety of astrophysical objects including extragalactic jets in active galactic nuclei and gamma ray bursts. These sources remain good candidate sites for the generation of ultra-high energy cosmic rays. In this paper, key predictions of DSA at relativistic shocks that are germane to production of relativistic electrons and ions are outlined. The technique employed to identify these characteristics is a Monte Carlo simulation of such diffusive acceleration in test-particle, rel…
Non-detection of $γ$-ray emission from galaxy clusters has challenged diffusive shock acceleration (DSA) of cosmic-ray (CR) protons at weak collisionless shocks that are expected to form in the intracluster medium. As an effort to address this problem, we here explore possible roles of Alfvén waves self-excited via resonant streaming instability during the CR acceleration at parallel shocks. The mean drift of Alfvén waves may either increase or decrease the scattering center compression ratio, depending on the postshock cross-helicity, leading to either flatter or steeper CR spectra. We first …
We present a grid of nonequilibrium ionization models for the X-ray spectra from supernova remnants undergoing efficient diffusive shock acceleration. The calculation follows the hydrodynamics of the blast wave as well as the time-dependent ionization of the plasma behind the shock. The ionization state is passed to a plasma emissivity code to compute the thermal X-ray emission, which is combined with the emission from nonthermal synchrotron emission to produce a self-consistent model for the thermal and nonthermal emission from cosmic-ray dominated shocks. We show how plasma diagnostics such …
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 non-linea…
The processes leading to the exceptionally hard radio spectra of pulsar-wind nebulae (PWNe) are not yet understood. Radio photon spectral indices among $29$ PWNe from the literature show an approximately normal, $α=0.2\pm0.2$ distribution. We present $\sim 3σ$ evidence for a distinct sub-population of PWNe, with a hard spectrum $α=0.01\pm0.06$ near the termination shock and significantly softer elsewhere, possibly due to a recent evacuation of the shock surroundings. Such spectra, especially in the hard sub-population, suggest a Fermi process, such as diffusive shock acceleration, at its extre…
We present a nonlinear Monte Carlo model of efficient diffusive shock acceleration (DSA) where the magnetic turbulence responsible for particle diffusion is calculated self-consistently from the resonant cosmic-ray (CR) streaming instability, together with non-resonant short- and long-wavelength CR-current-driven instabilities. We include the backpressure from CRs interacting with the strongly amplified magnetic turbulence which decelerates and heats the super-alfvenic flow in the extended shock precursor. Uniquely, in our plane-parallel, steady-state, multi-scale model, the full range of part…
Collisionless shocks are often studied in two spatial dimensions (2D), to gain insights into the 3D case. We analyze diffusive shock acceleration for an arbitrary number $N\in\mathbb{N}$ of dimensions. For a non-relativistic shock of compression ratio $\mathcal{R}$, the spectral index of the accelerated particles is $s_E=1+N/(\mathcal{R}-1)$; this curiously yields, for any $N$, the familiar $s_E=2$ (i.e., equal energy per logarithmic particle energy bin) for a strong shock in a mono-atomic gas. A precise relation between $s_E$ and the anisotropy along an arbitrary relativistic shock is derived…
We present results of semi-analytic calculations which show clear evidence for changes in the non-equilibrium ionization behind a supernova remnant forward shock undergoing efficient diffusive shock acceleration (DSA). The efficient acceleration of particles (i.e., cosmic rays) lowers the shock temperature and raises the density of the shocked gas, thus altering the ionization state of the plasma in comparison to the test particle approximation where cosmic rays gain an insignificant fraction of the shock energy. The differences between the test particle and efficient acceleration cases are su…
We examine the test-particle solution for diffusive shock acceleration, based on simple models for thermal leakage injection and Alfv'enic drift. The critical injection rate, ξ_c, above which the cosmic ray (CR) pressure becomes dynamically significant, depends mainly on the sonic shock Mach number, M, and preshock gas temperature, T_1. In the hot-phase interstellar medium (ISM) and intracluster medium, ξ_c < 10^{-3} for shocks with M < 5, while ξ_c ~ 10^{-4}(T_1/10^6 K)^{1/2} for shocks with M > 10. For T_1=10^6 K, for example, the test-particle solution would be valid if the injection moment…
We point out that particles accelerated in a non-relativistic shock of compression ratio $r$ attain the standard, $p=(r+2)/(r-1)$ spectral index only under certain conditions. Previous derivations of the spectrum, based on the approximations of spatial diffusion or negligible anisotropy, are shown to be unjustified for a general scattering function. We explain and demonstrate numerically that in contrast to previous claims, $p$ can substantially deviate from the standard result for anisotropic scattering. We prove analytically that the standard approach is nevertheless valid in the limit of an…
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 ($Q_\parallel$) shocks, previously suggested by \citet{kang2010}. In the model …
This research was motivated by the recent observations indicating very strong magnetic fields at some supernova remnant shocks, which suggests in-situ generation of magnetic turbulence. The dissertation presents a numerical model of collisionless shocks with strong amplification of stochastic magnetic fields, self-consistently coupled to efficient shock acceleration of charged particles. Based on a Monte Carlo simulation of particle transport and acceleration in nonlinear shocks, the model describes magnetic field amplification using the state-of-the-art analytic models of instabilities in mag…
We explore electron pre-acceleration at high Mach-number nonrelativistic perpendicular shocks at, e.g., young supernova remnants, which are a prerequisite of further acceleration to very high energies via diffusive shock acceleration. Using fully kinetic particle-in-cell simulations of shocks and electron dynamics in them, we investigate the influence of shock-surfing acceleration at the shock foot on the nonthermal population of electrons downstream of the shock. The shock-surfing acceleration is followed by further energization at the shock ramp where the Weibel instability spawns a type of …
Estimating the cosmic-ray acceleration efficiency $ ε$ in supernova remnants (SNRs) through observations is a challenging task in general. Based on the Rankine-Hugoniot shock conditions, we find an anticorrelation between $ ε$ and the power-law spectral index $ α$ of relativistic particle distribution produced via diffusive particle acceleration by nonrelativistic shocks, implying more efficient acceleration in older SNRs with harder radio spectra. Then $ ε$ may be estimated from some hard radio spectral index measurements. Assuming the particle distribution in downstream of strong shocks to b…
Using test particle simulations we study electron acceleration at collisionless shocks with a two-component model turbulent magnetic field with slab component including dissipation range. We investigate the importance of shock normal angle $θ_{Bn}$, magnetic turbulence level $\left(b/B_0\right)^2$, and shock thickness on the acceleration efficiency of electrons. It is shown that at perpendicular shocks the electron acceleration efficiency is enhanced with the decreasing of $\left(b/B_0\right)^2$, and at $\left(b/B_0\right)^2=0.01$ the acceleration becomes significant due to strong drift electr…
We consider a phenomenological model for the thermal leakage injection in the diffusive shock acceleration (DSA) process, in which suprathermal protons and electrons near the shock transition zone are assumed to have the so-called $κ$-distributions produced by interactions of background thermal particles with pre-existing and/or self-excited plasma/MHD waves or turbulence. The $κ$-distribution has a power-law tail, instead of an exponential cutoff, well above the thermal peak momentum. So there are a larger number of potential seed particles with momentum, above that required for participation…
We investigate shock structure and particle acceleration in relativistic magnetized collisionless pair shocks by means of 2.5D and 3D particle-in-cell simulations. We explore a range of inclination angles between the pre-shock magnetic field and the shock normal. We find that only magnetic inclinations corresponding to "subluminal" shocks, where relativistic particles following the magnetic field can escape ahead of the shock, lead to particle acceleration. The downstream spectrum in such shocks consists of a relativistic Maxwellian and a high-energy power-law tail with exponential cutoff. For…
The extensive observations of the supernova SN 1993J at radio wavelengths make this object a unique target for the study of particle acceleration in a supernova shock. To describe the radio synchrotron emission we use a model that couples a semianalytic description of nonlinear diffusive shock acceleration with self-similar solutions for the hydrodynamics of the supernova expansion. The synchrotron emission, which is assumed to be produced by relativistic electrons propagating in the postshock plasma, is worked out from radiative transfer calculations that include the process of synchrotron se…
The spectral index $s$ of particles diffusively accelerated in a relativistic shock depends on the unknown angular diffusion function $\mathcal{D}$, which itself depends on the particle distribution function $f$ if acceleration is efficient. We develop a relaxation code to compute $s$ and $f$ for an arbitrary functional $\mathcal{D}$ that depends on $f$. A local $\mathcal{D}(f)$ dependence is motivated and shown, when rising (falling) upstream, to soften (harden) $s$ with respect to the isotropic case, shift the angular distribution towards upstream (downstream) directions, and strengthen (wea…
Radio relics could be generated by multiple shocks induced in the turbulent intracluster medium during galaxy mergers. Kang (2021) demonstrated that the re-acceleration of cosmic ray (CR) protons via diffusive shock acceleration (DSA) by multiple shocks could enhance the acceleration efficiency and flatten the CR spectrum, compared to a single episode of DSA. Here we examine the CR electron acceleration through multiple re-acceleration by considering energy losses and decompression of the particle distribution and magnetic fields in the postshock region between consecutive shock passages. We f…
Electron acceleration to non-thermal energies in low Mach number (M<5) shocks is revealed by radio and X-ray observations of galaxy clusters and solar flares, but the electron acceleration mechanism remains poorly understood. Diffusive shock acceleration, also known as first-order Fermi acceleration, cannot be directly invoked to explain the acceleration of electrons. Rather, an additional mechanism is required to pre-accelerate the electrons from thermal to supra-thermal energies, so they can then participate in the Fermi process. In this work, we use two- and three-dimensional particle-in-ce…
We present new models for the forward and reverse shock thermal X-ray emission from core-collapse and Type Ia supernova remnants (SNRs) which include the efficient production of cosmic rays via non-linear diffusive shock acceleration (DSA). Our CR-hydro-NEI code takes into account non-equilibrium ionization (NEI), hydrodynamic effects of efficient CR production on the SNR evolution, and collisional temperature equilibration among heavy ions and electrons in both the shocked supernova (SN) ejecta and the shocked circumstellar material. While X-ray emission is emphasized here, our code self-cons…
We explore how wave-particle interactions affect diffusive shock acceleration (DSA) at astrophysical shocks by performing time-dependent kinetic simulations, in which phenomenological models for magnetic field amplification (MFA), Alfvenic drift, thermal leakage injection, Bohm-like diffusion, and a free escape boundary are implemented. If the injection fraction of cosmic-ray (CR) particles is greater than 2x10^{-4}, for the shock parameters relevant for young supernova remnants, DSA is efficient enough to develop a significant shock precursor due to CR feedback, and magnetic field can be ampl…
We investigate effects of upstream density fluctuations on the diffusive shock acceleration by Monte Carlo simulations. The simulations show that particles are reaccelerated in the shock downstream region by a sound wave generated at the shock front. The time scale of turbulent acceleration by the sound wave is estimated. We propose a new back reaction of particles accelerated around the shock front. The accelerated particles generate the upstream density fluctuations by the Drury instability, which are converted to the downstream sound waves by the shock. The downstream sound waves modify the…
Recent gamma ray observations show that middle aged supernova remnants interacting with molecular clouds can be sources of both GeV and TeV emission. Models involving re-acceleration of pre-existing cosmic rays in the ambient medium and direct interaction between supernova remnant and molecular clouds have been proposed to explain the observed gamma ray emission. For the re-acceleration process, standard DSA theory in the test particle limit produces a steady state particle spectrum that is too flat compared to observations, which suggests that the high energy part of the observed spectrum has…
Electron acceleration to non-thermal energies is known to occur in low Mach number (M<5) shocks in galaxy clusters and solar flares, but the electron acceleration mechanism remains poorly understood. Using two-dimensional (2D) particle-in-cell (PIC) plasma simulations, we showed in Paper I that electrons are efficiently accelerated in low Mach number (M=3) quasi-perpendicular shocks via a Fermi-like process. The electrons bounce between the upstream region and the shock front, with each reflection at the shock resulting in energy gain via shock drift acceleration. The upstream scattering is pr…
Nonlinear diffusive shock acceleration with upstream escape reproduces DAMPE observations
State Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所粒子天体物理国家重点实验室）；Key Laboratory of cosmic ray, Ministry of education, Tibet University（西藏大学教育部宇宙射线重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
Relativistic particle acceleration in collisionless shocks of supernova remnants is accompanied by magnetic field amplification from cosmic ray (CR) driven plasma instabilities. Bell's fast CR-current instability is predicted to produce turbulence with a non-zero mean electric field in the shock precursor. We present a Monte Carlo model of Fermi shock acceleration explicitly taking into account an effective mean upstream electric field. Our model is nonlinear and includes the backreaction effects of efficient Fermi acceleration on the shock structure.
The non-thermal particle spectra responsible for the emission from many astrophysical systems are thought to originate from shocks via a first order Fermi process otherwise known as diffusive shock acceleration. The same mechanism is also widely believed to be responsible for the production of high energy cosmic rays. With the growing interest in collisionless shock physics in laser produced plasmas, the possibility of reproducing and detecting shock acceleration in controlled laboratory experiments should be considered. The various experimental constraints that must be satisfied are reviewed.…
The analytical theory of diffusive cosmic ray acceleration at parallel stationary shock waves with magnetostatic turbulence is generalized to arbitrary shock speeds $V_s=β_1c$, including in particular relativistic speeds. This is achieved by applying the diffusion approximation to the relevant Fokker-Planck particle transport equation formulated in the mixed comoving coordinate system. In this coordinate system the particle's momentum coordinates $p$ and $μ=p_{\parallel }/p$ are taken in the rest frame of the streaming plasma, whereas the time and space coordinates are taken in the observer's …
We show that the diffusion approximation breaks down for particle acceleration at oblique shocks with velocities typical of young supernova remnants. Higher order anisotropies flatten the spectral index at quasi-parallel shocks and steepen the spectral index at quasi-perpendicular shocks. We compare the theory with observed spectral indices.
Using large-scale fully-kinetic two-dimensional particle-in-cell simulations, we investigate the effects of shock rippling on electron acceleration at low-Mach-number shocks propagating in high-$β$ plasmas, in application to merger shocks in galaxy clusters. We find that the electron acceleration rate increases considerably when the rippling modes appear. The main acceleration mechanism is stochastic shock-drift acceleration, in which electrons are confined at the shock by pitch-angle scattering off turbulence and gain energy from the motional electric field. The presence of multi-scale magnet…
We examine limits to the energy to which cosmic rays can be accelerated by relativistic shocks, showing that acceleration of light ions as high as 100 EeV is unlikely. The implication of our estimates is that if ultra-high energy cosmic rays are accelerated by shocks, then those shocks are probably not relativistic.
We present the results of large hybrid (kinetic ions - fluid electrons) simulations of particle acceleration at non-relativistic collisionless shocks. Ion acceleration efficiency and magnetic field amplification are investigated in detail as a function of shock inclination and strength, and compared with predictions of diffusive shock acceleration theory, for shocks with Mach number up to 100. Moreover, we discuss the relative importance of resonant and Bell's instability in the shock precursor, and show that diffusion in the self-generated turbulence can be effectively parametrized as Bohm di…
We show that the energy required to turbulently amplify magnetic field during cosmic ray (CR) acceleration by shocks extracts energy from the CR and steepens the CR energy spectrum.
Method for testing diffusive shock acceleration and diffusion propagation of 1-100 TeV cosmic electrons with multiwavelength observations of the Geminga halo and pulsar wind nebula
Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Tianfu Cosmic Ray Research Center（天府宇宙线研究中心）；Beijing Planetarium（北京天文馆）；Beijing Academy of Science and Technology（北京市科学技术研究院）；The School of Physical Science and Technology, Southwest Jiaotong University（西南交通大学物理科学与技术学院）
Collisionless shocks with low sonic Mach numbers, $M_{\rm s} \lesssim 4$, are expected to accelerate cosmic ray (CR) protons via diffusive shock acceleration (DSA) in the intracluster medium (ICM). However, observational evidence for CR protons in the ICM has yet to be established. Performing particle-in-cell simulations, we study the injection of protons into DSA and the early development of a nonthermal particle population in weak shocks in high $β$ ($\approx 100$) plasmas. Reflection of incident protons, self-excitation of plasma waves via CR-driven instabilities, and multiple cycles of sho…
We study diffusive shock acceleration (DSA) of electrons in non-relativistic quasi-perpendicular shocks using self-consistent one-dimensional particle-in-cell (PIC) simulations. By exploring the parameter space of sonic and Alfvénic Mach numbers we find that high Mach number quasi-perpendicular shocks can efficiently accelerate electrons to power-law downstream spectra with slopes consistent with DSA prediction. Electrons are reflected by magnetic mirroring at the shock and drive non-resonant waves in the upstream. Reflected electrons are trapped between the shock front and upstream waves and …
Diffusive Shock Acceleration (DSA) cannot efficiently accelerate particles without the presence of self-consistently generated or pre-existing strong turbulence ($ δB/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), spontaneo…
We use 2D and 3D hybrid (kinetic ions - fluid electrons) simulations to investigate particle acceleration and magnetic field amplification at non-relativistic astrophysical shocks. We show that diffusive shock acceleration operates for quasi-parallel configurations (i.e., when the background magnetic field is almost aligned with the shock normal) and, for large sonic and Alfvénic Mach numbers, produces universal power-law spectra proportional to p^(-4), where p is the particle momentum. The maximum energy of accelerated ions increases with time, and it is only limited by finite box size and ru…
The current paradigm of cosmic ray (CR) origin states that the most part of galactic CRs is produced by supernova remnants. The interaction of supernova ejecta with the interstellar medium after supernova's explosions results in shocks responsible for CR acceleration via diffusive shock acceleration (DSA). We use particle-in-cell (PIC) simulations and a combined PIC-magnetohydrodynamic (PIC-MHD) technique to investigate whether DSA can occur in oblique high Mach number shocks. Using the PIC method, we follow the formation of the shock and determine the fraction of the particles that gets invol…
The early acceleration of protons and electrons in the nonrelativistic collisionless shocks with three obliquities are investigated through 1D particle-in-cell simulations. In the simulations, the charged particles possessing a velocity of $0.2\, c$ flow towards a reflecting boundary, and the shocks with a sonic Mach number of $13.4$ and a Alfven Mach number of $16.5$ in the downstream shock frame are generated. In these quasi-parallel shocks with the obliquity angles $θ= 15^\circ$, $30^\circ$, and $45^\circ$, some of the protons and the electrons can be injected into the acceleration processe…
The shock waves produce relativistic particles via the diffusive shock acceleration (DSA) mechanism. Among various circumstances, fast acceleration has been expected for perpendicular shocks. We investigate the acceleration time and the energy spectrum of particles accelerated at a perpendicular shock. In our model, the upstream perpendicular magnetic field has no fluctuation, and the downstream region is highly turbulent. Then, the particle motion is the gyration in the upstream region and Bohm-like diffusion downstream. Under this situation, we derive an analytical form of the acceleration t…
