Non-relativistic Shocks Research Articles

Electron Acceleration at Quasi-parallel Nonrelativistic Shocks: A 1D Kinetic Survey

We present a survey of 1D kinetic particle-in-cell simulations of quasi-parallel nonrelativistic shocks to identify the environments favorable for electron acceleration. We explore an unprecedented range of shock speeds v sh ≈ 0.067–0.267c, Alfvén Mach numbers MA=5–40 , sonic Mach numbers Ms=5–160 , as well as the proton-to-electron mass ratios m i/m e = 16–1836. We find that high Alfvén Mach number shocks can channel a large fraction of their kinetic energy into nonthermal particles, self-sustaining magnetic turbulence and acceleration to larger and larger energies. The fraction of injected particles is ≲0.5% for electrons and ≈1% for protons, and the corresponding energy efficiencies are ≲2% and ≈10%, respectively. The extent of the nonthermal tail is sensitive to the Alfvén Mach number; when MA≲10 , the nonthermal electron distribution exhibits minimal growth beyond the average momentum of the downstream thermal protons, independently of the proton-to-electron mass ratio. Acceleration is slow for shocks with low sonic Mach numbers, yet nonthermal electrons still achieve momenta exceeding the downstream thermal proton momentum when the shock Alfvén Mach number is large enough. We provide simulation-based parameterizations of the transition from thermal to nonthermal distribution in the downstream (found at a momentum around pi,e/mivsh≈3mi,e/mi ), as well as the ratio of nonthermal electron to proton number density. The results are applicable to many different environments and are important for modeling shock-powered nonthermal radiation.

Nonthermal Acceleration of Electrons, Positrons, and Protons at a Nonrelativistic Quasi-parallel Collisionless Shock

Energetic positrons have been observed in the interstellar medium, and high-energy positrons with relativistic energies up to approximately 1 TeV have been detected in Galactic cosmic rays. We conducted a study on the acceleration of particles, specifically positrons, in a nonrelativistic quasi-parallel collisionless shock induced by a plasma consisting of protons, electrons, and positrons. The positron-to-proton number density ratio in the plasma is 0.1. We focused on a representative shock with a sonic Mach number of 17.1 and an Alfvénic Mach number of 16.8 in the rest frame of the shock. To investigate the acceleration mechanisms of particles including positrons in the shock, we utilized 1D particle-in-cell simulations. It was found that all three species of particles in the shock can be accelerated and exhibit power-law spectra. At the shock front, a significant portion of incoming upstream particles are reflected and undergo significant energy increases, and these reflected particles can be efficiently injected into the process of diffusive shock acceleration (DSA). Moveover, the reflected positrons can be further accelerated by an electric field parallel to the magnetic field when they move along the magnetic field upstream of the shock. As a result, positrons can be preferentially accelerated to be injected in the DSA process compared to electrons.

On the Width of a Collisionless Shock and the Index of the Cosmic Rays It Accelerates

Despite being studied for many years, the structure of collisionless shocks is still not fully determined. Such shocks are known to be accelerators of cosmic rays (CRs), which, in turn, modify the shock structure. The shock width λ is known to be connected to the CR spectral index a. Here, we use an instability analysis to derive the shock width in the presence of CRs. We obtain an analytical expression connecting the shock width to the CR index and to the fraction of upstream particles that are accelerated. We find that when this fraction becomes larger than ∼30%, a new instability becomes dominant. The shock undergoes a transition where its width increases by a factor ∼8–10, and the CR acceleration effectively ends. Our analysis is valid for strong, nonrelativistic, and unmagnetized shocks. We discuss the implication of these results on the expected range of CR spectra and flux observed and on the structure of nonrelativistic collisionless shocks.

Individual particle approach to diffusive shock acceleration. Effect of the non-uniform flow velocity downstream of the shock

The momentum distribution of particles accelerated at strong non-relativistic shocks may be influenced by the spatial distribution of the flow speed around the shock. This phenomenon becomes evident in the cosmic-ray-modified shock, where the particle spectrum itself determines the profile of the flow velocity upstream. However, the effects of a non-uniform flow speed downstream are unclear. Hydrodynamics indicates that the spatial variation of flow speed over the length scales involved in the acceleration of particles in supernova remnants (SNRs) could be noticeable. In the present paper, we address this question. We initially followed Bell's approach to particle acceleration and then also solved the kinetic equation. We obtained an analytical solution for the momentum distribution of particles accelerated at the cosmic-ray-modified shock with spatially variable flow speed downstream. We parameterised the downstream speed profile to illustrate its effect on two model cases: the test particle and non-linear acceleration. The resulting particle spectrum is generally softer in Sedov SNRs because the distribution of the flow speed reduces the overall shock compression accessible to particles with higher momenta. On the other hand, the flow structure in young SNRs could lead to harder spectra. The diffusive properties of particles play a crucial role as they determine the distance the particles can return from to the shock, and, as a consequence, the flow speed that they encounter downstream. We discuss a possibility that the plasma velocity gradient could be (at least partially) responsible for the evolution of the radio index and for the high-energy break visible in gamma rays from some SNRs. We expect the effects of the gradient of the flow velocity downstream to be prominent in regions of SNRs with higher diffusion coefficients and lower magnetic field, that is, where acceleration of particles is not very efficient.

Saturation Level of Ion Weibel Instability and Isotropization Length Scale in Electron-Ion Weibel-Mediated Shocks

Abstract Ion Weibel instability is considered to be the dominant physics for the dissipation in high-Mach number astrophysical shocks such as supernova remnant shocks and gamma-ray burst shocks. We study the instability dependence on various parameters using theory and particle-in-cell simulations. We demonstrate that electron physics determines the saturation level of the Weibel-generated magnetic field, even though the instability is driven by the ions. We discuss the application to astrophysical and laboratory laser experiment environments to clarify the roles of the ion Weibel instability. We develop a model for the isotropization length scale in Weibel-mediated shocks and compare its value to other characteristic length scales of each system. We find that electron heating to near equipartition is crucial for the formation of ultra-relativistic Weibel-mediated shocks. On the other hand, our results imply that non-relativistic shocks in typical interstellar medium are not purely mediated by the Weibel instability.

Drivers of Magnetic Field Amplification at Oblique Shocks: In Situ Observations

Collisionless shocks are ubiquitous structures throughout the Universe. Shock waves in space and astrophysical plasmas convert the energy of a fast-flowing plasma to other forms of energy, including thermal and magnetic energies. Plasma turbulence and high-amplitude electric and magnetic fluctuations are necessary for effective energy conversion and particle acceleration. We survey and characterize in situ observations of reflected ions and magnetic field amplification rates at quasiperpendicular shocks under a wide range of upstream conditions. We report magnetic amplification rates as high as 25 in our current data set. Reflected ions interacting with the incoming plasma create magnetic perturbations that cause magnetic amplification in upstream and downstream regions of quasiperpendicular shocks. Our observations show that, in general, magnetic amplification increases with the fraction of reflected ions, which itself increases with Mach number. Both parameters plateau once full reflection is reached. Magnetic amplification continuously increases with the inverse of the magnetization parameter of the upstream plasma. We find that the extended foot region upstream of shocks and nonlinear processes within that region are key factors for intense magnetic amplification. Our observations at nonrelativistic shocks provide the first experimental evidence that below a certain magnetization threshold, the magnetic amplification efficiency at quasiperpendicular shocks becomes comparable to that at the quasiparallel shocks.

Kinetic Simulations of Nonrelativistic High-mach-number Perpendicular Shocks Propagating in a Turbulent Medium

Strong nonrelativistic shocks are known to accelerate particles up to relativistic energies. However, for diffusive shock acceleration, electrons must have a highly suprathermal energy, implying the need for very efficient preacceleration. Most published studies consider shocks propagating through homogeneous plasma, which is an unrealistic assumption for astrophysical environments. Using 2D3V particle-in-cell simulations, we investigate electron acceleration and heating processes at nonrelativistic high-Mach-number shocks in electron-ion plasma with a turbulent upstream medium. For this purpose, slabs of plasma with compressive turbulence are simulated separately and then inserted into shock simulations, which require matching of the plasma slabs at the interface. Using a novel procedure of matching electromagnetic fields and currents, we perform simulations of perpendicular shocks setting different intensities of density fluctuations (≲10%) in the upstream region. The new simulation technique provides a framework for studying shocks propagating in turbulent media. We explore the impact of the fluctuations on electron heating, the dynamics of upstream electrons, and the driving of plasma instabilities. Our results indicate that while the presence of turbulence enhances variations in the upstream magnetic field, their levels remain too low to significantly influence the behavior of electrons at perpendicular shocks.

Fast Particle Acceleration in 3D Hybrid Simulations of Quasiperpendicular Shocks.

Understanding the conditions conducive to particle acceleration at collisionless, nonrelativistic shocks is important for the origin of cosmic rays. We use hybrid (kinetic ions-fluid electrons) kinetic simulations to investigate particle acceleration and magnetic field amplification at nonrelativistic, weakly magnetized, quasiperpendicular shocks. So far, no self-consistent kinetic simulation has reported nonthermal tails at quasiperpendicular shocks. Unlike 2D simulations, 3D runs show that protons develop a nonthermal tail spontaneously (i.e., from the thermal bath and without preexisting magnetic turbulence). They are rapidly accelerated via shock drift acceleration up to a maximum energy determined by their escape upstream. We discuss the implications of our results for the phenomenology of heliospheric shocks, supernova remnants, and radio supernovae.

The magnetohydrodynamic-particle-in-cell module in athena++: implementation and code tests

ABSTRACT We present a new magnetohydrodynamic-particle-in-cell (MHD-PIC) code integrated into the athena++ framework. It treats energetic particles as in conventional PIC codes, while the rest of thermal plasmas are treated as background fluid described by MHD, thus primarily targeting at multiscale astrophysical problems involving the kinetic physics of the cosmic rays (CRs). The code is optimized towards efficient vectorization in interpolation and particle deposits, with excellent parallel scaling. The code is also compatible with static/adaptive mesh refinement, with dynamic load balancing to further enhance multiscale simulations. In addition, we have implemented a compressing/expanding box framework that allows adiabatic driving of CR pressure anisotropy, as well as the δf method that can dramatically reduce Poisson noise in problems where distribution function f is only expected to slightly deviate from the background. The code performance is demonstrated over a series of benchmark test problems, including particle acceleration in non-relativistic parallel shocks. In particular, we reproduce the linear growth of the CR gyro-resonant (streaming and pressure anisotropy) instabilities, under both the periodic and expanding/compressing box settings. We anticipate the code to open up the avenue for a wide range of astrophysical and plasma physics applications.

Shock Corrugation to the Rescue of the Internal Shock Model in Microquasars: The Single-scale Magnetohydrodynamic View

Questions regarding the energy dissipation in astrophysical jets remain open to date, despite numerous attempts to limit the diversity of the models. Some of the most popular models assume that energy is transferred to particles via internal shocks, which develop as a consequence of the nonuniform velocity of the jet matter. In this context, we study the structure and energy deposition of colliding plasma shells, focusing our attention on the case of initially inhomogeneous shells. This leads to the formation of distorted (corrugated) shock fronts—a setup that has recently been shown to revive particle acceleration in relativistic magnetized perpendicular shocks. Our study shows that the radiative power of the far downstream of nonrelativistic magnetized perpendicular shocks is moderately enhanced with respect to the flat-shock cases. Based on the decay rate of the downstream magnetic field, we make predictions for multiwavelength polarization properties.

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.

Probing particle acceleration at trans-relativistic shocks with off-axis gamma-ray burst afterglows

ABSTRACT Particle acceleration is expected to be different between relativistic and non-relativistic collisionless shocks. We show that electromagnetic counterparts to gravitational waves (GWs), gamma-ray burst (GRB) afterglows, are ideal targets for observing trans-relativistic evolution of accelerated electron distribution because the GWs spot nearby GRBs with off-axis jets, otherwise missed in gamma-ray observations. We find that the relativistic spectral slope begins to change steeply near the peak time of the light curve and approaches the non-relativistic limit in about 10 times the peak time. The trans-relativistic evolution of the afterglow synchrotron spectrum is consistent with GRB 170817A observations within errors, and will be measurable in similar but more distant events at a GW horizon ∼200 Mpc in a denser environment. We roughly estimate that such events represent a fraction of 10–50 per cent of the GRB 170817A-like off-axis short GRBs. We also find that the spectral evolution does not depend on the jet structure if their light curves are similar to each other.

Quantum simulations of hydrodynamics via the Madelung transformation

Developing numerical methods to simulate efficiently nonlinear fluid dynamics on universal quantum computers is a challenging problem. In this paper, a generalization of the Madelung transform is defined to solve quantum relativistic charged fluid equations interacting with external electromagnetic forces via the Dirac equation. The Dirac equation is discretized into discrete-time quantum walks which can be efficiently implemented on universal quantum computers. A variant of this algorithm is proposed to implement simulations using current noisy intermediate scale quantum (NISQ) devices in the case of homogeneous external forces. High resolution (up to $N={2}^{17}$ grid points) numerical simulations of relativistic and nonrelativistic hydrodynamical shocks on current IBM NISQs are performed with this algorithm. This paper demonstrates that fluid dynamics can be simulated on NISQs, and opens the door to simulating other fluids, including plasmas, with more general quantum walks and quantum automata.

The Mechanism of Efficient Electron Acceleration at Parallel Nonrelativistic Shocks

Thermal electrons cannot directly participate in the process of diffusive acceleration at electron–ion shocks because their Larmor radii are smaller than the shock transition width: this is the well-known electron injection problem of diffusive shock acceleration. Instead, an efficient pre-acceleration process must exist that scatters electrons off of electromagnetic fluctuations on scales much shorter than the ion gyroradius. The recently found intermediate-scale instability provides a natural way to produce such fluctuations in parallel shocks. The instability drives comoving (with the upstream plasma) ion–cyclotron waves at the shock front and only operates when the drift speed is smaller than half of the electron Alfvén speed. Here we perform particle-in-cell simulations with the SHARP code to study the impact of this instability on electron acceleration at parallel nonrelativistic, electron–ion shocks. To this end, we compare a shock simulation in which the intermediate-scale instability is expected to grow to simulations where it is suppressed. In particular, the simulation with an Alfvénic Mach number large enough to quench the intermediate instability shows a great reduction (by two orders of magnitude) of the electron acceleration efficiency. Moreover, the simulation with a reduced ion-to-electron mass ratio (where the intermediate instability is also suppressed) not only artificially precludes electron acceleration but also results in erroneous electron and ion heating in the downstream and shock transition regions. This finding opens up a promising route for a plasma physical understanding of diffusive shock acceleration of electrons, which necessarily requires realistic mass ratios in simulations of collisionless electron–ion shocks.

Density jump as a function of magnetic field for switch-on collisionless shocks in pair plasmas

The properties of collisionless shocks, like the density jump, are usually derived from magnetohydrodynamics (MHD), where isotropic pressures are assumed. Yet, in a collisionless plasma, an external magnetic field can sustain a stable anisotropy. We have already devised a model for the kinetic history of the plasma through the shock front (J. Plasma Phys., vol. 84, issue 6, 2018, 905840604), allowing to self-consistently compute the downstream anisotropy, and hence the density jump, in terms of the upstream parameters. This model deals with the case of a parallel shock, where the magnetic field is normal to the front both in the upstream and the downstream. Yet, MHD also allows for shock solutions, the so-called switch-on solutions, where the field is normal to the front only in the upstream. This article consists in applying our model to these switch-on shocks. While MHD offers only one switch-on solution within a limited range of Alfvén Mach numbers, our model offers two kinds of solutions within a slightly different range of Alfvén Mach numbers. These two solutions are most likely the outcome of the intermediate and fast MHD shocks under our model. While the intermediate and fast shocks merge in MHD for the parallel case, they do not within our model. For simplicity, the formalism is restricted to non-relativistic shocks in pair plasmas where the upstream is cold.

The electron foreshock at high-Mach-number non-relativistic oblique shocks

In the Universe, matter outside of stars and compact objects is mostly composed of collisionless plasma. The interaction of a supersonic plasma flow with an obstacle results in collisionless shocks that are often associated with intense nonthermal radiation and the production of cosmic ray particles. Motivated by simulations of non-relativistic high-Mach-number shocks in supernova remnants, we investigate the instabilities excited by relativistic electron beams in the extended foreshock of oblique shocks. The phase-space distributions in the inner and outer foreshock regions are derived with a particle-in-cell simulation of the shock and used as initial conditions for simulations with periodic boundary conditions to study their relaxation toward equilibrium. We find that the observed electron-beam instabilities agree very well with the predictions of a linear dispersion analysis: the electrostatic electron-acoustic instability dominates in the outer region of the foreshock, while the denser electron beams in the inner foreshock drive the gyroresonant oblique-whistler instability.

Kinetic simulation of electron, proton and helium acceleration in a non-relativistic quasi-parallel shock

ABSTRACT In addition to accelerating electrons and protons, non-relativistic quasi-parallel shocks are expected to possess the ability to accelerate heavy ions. The shocks in supernova remnants are generally supposed to be accelerators of Galactic cosmic rays, which consist of many species of particles. We investigate the diffusive shock acceleration of electrons, protons and helium ions in a non-relativistic quasi-parallel shock through a 1D particle-in-cell simulation with a helium-to-proton number density ratio of 0.1, which is relevant for Galactic cosmic rays. The simulation indicates that waves can be excited by the flow of energetic protons and helium ions upstream of a non-relativistic quasi-parallel shock with a sonic Mach number of 14 and an Alfvén Mach number of 19.5 in the shock rest frame, and that the charged particles are scattered by the self-generated waves and accelerated gradually. Moreover, the spectra of the charged particles downstream of the shock are thermal with a non-thermal tail, and the acceleration is efficient, with about $7{{\ \rm per\ cent}}$ and $5.4{{\ \rm per\ cent}}$ of the bulk kinetic energy transferred into the non-thermal protons and helium ions, respectively, in the near downstream region by the end of the simulation.

Thermal Electrons in Mildly Relativistic Synchrotron Blast Waves

Abstract Numerical models of collisionless shocks robustly predict an electron distribution composed of both thermal and nonthermal electrons. Here, we explore in detail the effect of thermal electrons on the emergent synchrotron emission from subrelativistic shocks. We present a complete “thermal + nonthermal” synchrotron model and derive properties of the resulting spectrum and light curves. Using these results, we delineate the relative importance of thermal and nonthermal electrons for subrelativistic shock-powered synchrotron transients. We find that thermal electrons are naturally expected to contribute significantly to the peak emission if the shock velocity is ≳0.2c, but would be mostly undetectable in nonrelativistic shocks. This helps explain the dichotomy between typical radio supernovae and the emerging class of “AT2018cow-like” events. The signpost of thermal electron synchrotron emission is a steep optically-thin spectral index and a ν 2 optically-thick spectrum. These spectral features are also predicted to correlate with a steep postpeak light-curve decline rate, broadly consistent with observed AT2018cow-like events. We expect that thermal electrons may be observable in other contexts where mildly relativistic shocks are present and briefly estimate this effect for gamma-ray burst afterglows and binary–neutron-star mergers. Our model can be used to fit spectra and light curves of events and accounts for both thermal and nonthermal electron populations with no additional physical degrees of freedom.

Lepton-driven Nonresonant Streaming Instability

Abstract A strong super-Alfvénic drift of energetic particles (or cosmic rays) in a magnetized plasma can amplify the magnetic field significantly through nonresonant streaming instability (NRSI). While the traditional analysis is done for an ion current, here we use kinetic particle-in-cell simulations to study how the NRSI behaves when it is driven by electrons or by a mixture of electrons and positrons. In particular, we characterize the growth rate, spectrum, and helicity of the unstable modes, as well the level of the magnetic field at saturation. Our results are potentially relevant for several space/astrophysical environments (e.g., electron strahl in the solar wind, at oblique nonrelativistic shocks, around pulsar wind nebulae), and also in laboratory experiments.

Time-dependent Treatment of Cosmic-ray Spectral Steepening Due to Turbulence Driving

Abstract Cosmic-ray acceleration at non-relativistic shocks relies on scattering by turbulence that the cosmic rays drive upstream of the shock. We explore the rate of energy transfer from cosmic rays to non-resonant Bell modes and the spectral softening it implies. Accounting for the finite time available for turbulence driving at supernova-remnant shocks yields a smaller spectral impact than found earlier with steady-state considerations. Generally, for diffusion scaling with the Bohm rate by a factor η, the change in spectral index is at most η divided by the Alfvénic Mach number of the thermal sub-shock. For M A ≲ 50 it is well below this limit. Only for very fast shocks and very efficient cosmic-ray acceleration can the change in spectral index reach 0.1. For standard SNR parameters, it is negligible. Independent confirmation is derived by considering the synchrotron energy losses of electrons: if intense nonthermal multi-keV emission is produced, the energy loss, and hence the spectral steepening, is very small for hadronic cosmic rays that produce TeV-band gamma-ray emission.