Non-linear particle acceleration at non-relativistic shock waves in the presence of self-generated turbulence

Particle acceleration at astrophysical shocks may be very efficient if magnetic scattering is self-generated by the same particles. This non-linear process adds to the non-linear modification of the shock due to the dynamical reaction of the accelerated particles on the shock. Building on a previous general solution of the problem of particle acceleration with arbitrary diffusion coefficients, we present here the first semi-analytical calculation of particle acceleration with both effects taken into account at the same time; charged particles are accelerated in the background of Alfvén waves that they generate due to the streaming instability, and modify the dynamics of the plasma in the shock vicinity.     

Particle acceleration by ultrarelativistic shocks: theory and simulations

We consider the acceleration of charged particles near ultrarelativistic shocks, with Lorentz factor     . We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultrarelativistic shocks is a power law,     , with a nearly universal value     for the slope of this power law.
We confirm that the ultrarelativistic equivalent of the Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurrence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle     around the shock normal.
We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount:     . This limits the energy change per shock crossing cycle to     , except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor     for those particles that are scattered back across the shock into the upstream region.     

Cosmic rays in central regions of active galaxies

We consider the acceleration of charged particles by the Fermi mechanism on magnetic field irregularities in active galactic regions. The relativistic particles are shown to be accelerated most efficiently, while the acceleration of nonrelativistic particles by this mechanism is possible only in highly nonuniform galactic nuclei, that is, in nuclei with strong turbulization. The conditions for the acceleration of charged particles in active galactic nuclei at various stages of their evolution are investigated.     

Shock surfing acceleration

Analytical and numerical analysis identify shock surfing acceleration as an ideal pre-energization mechanism for the slow pick-up ions at quasiperpendicular shocks. After gaining sufficient energy by shock surfing, pick-up ions undergo diffusive acceleration to reach their observed energies. Energetic ions upstream of the cometary bow shock, acceleration of solar energetic particles by magnetosonic waves in corona, ion enhancement in interplanetary shocks, generation of anomalous cosmic rays from interstellar pick-up ions at the termination shock are some of the cases where shock surfing acceleration apply. Inclusion of the lower-hybrid wave turbulence into the laminar model of shock surfing can explain the preferential acceleration of heavier particles as observed by Voyager at the termination shock. At relativistic energies, unlimited acceleration of ions is theoretically possible; because for sufficiently strong shocks main limitation of the mechanism, caused by the escape of accelerated particles downstream of the shock during acceleration no longer exists.     

Stochastic Acceleration of Energetic Particles in the Magnetosphere of Saturn

Voyager's plasma probe observations suggest that there are at least three fundamentally different plasma regimes in Saturn: the hot outer magnetosphere, the extended plasma sheet, and the inner plasma torus. At the outer regions of the inner torus some ions have been accelerated to reach energies of the order of 43 keV. We develop a model that calculates the acceleration of charged particles in the Saturn's magnetosphere. We propose that the stochastic electric field associated to the observed magnetic field fluctuations is responsible of such acceleration. A random electric field is derived from the fluctuating magnetic field – via a Monte Carlo simulation – which then is applied to the momentum equation of charged particles seeded in the magnetosphere. Taking different initial conditions, like the source of charged particles and the distribution function of their velocities, we find that particles injected with very low energies ranging from 0.129 eV to 5.659 keV can be strongly accelerated to reach much higher energies ranging from 22.220 eV to 9.711 keV as a result of 125,000 hitting events (the latter are used in the numerical code to produce the particle acceleration over a predetermined distance).     

Generation of magnetic fluctuations near a shock front in a partially ionized medium

We investigate the generation mechanism of long-wavelength Alfvénic disturbances near the front of a collisionless shock that propagates in a partially ionized plasma. The wave generation and dissipation rates are calculated in the linear approximation. The instability is attributable to a current of energetic particles upstream of the shock front. The generation of long-wavelength magnetic fluctuations is most pronounced for strong shocks, but the effect is retained for shocks with a moderate particle acceleration efficiency without any noticeable modification of the shock structure by the pressure of accelerated particles. The mode generation time for supernova remnants in a partially ionized interstellar medium is shown to be shorter than their age. Long-wavelength magnetic disturbances determine the limiting energies of the particles accelerated at a shock by the Fermimechanism. We discuss the application of the mechanism under consideration to explaining the observed properties of the SN 1006 remnant.     

Particle acceleration during the explosive phase of a solar flare

Starting with the quasi-linear equation of the distribution function of particles in a regular electric field, a combined diffusion coefficient in the momentum space conbining the effects of the regular field and a turbulent field is obtained and a combined mechanism of acceleration by the regular and turbulent fields in the neutral sheet of solar proton flares is proposed. It is shown by calculation that conditions in solar proton flares are such that the charged particles can be effectively accelerated to tens of MeV, even ~1 GeV. It is shown that the combined acceleration by a regular electric field and ion-acoustic turbulence pumps the protons and other heavy ions into ranges of energy where they can be accelerated by Langmuir turbulence. By considering the combined acceleration by Langmuir turbulence and the regular electric field, the observed spectrum of energetic protons and the power-law spectrum of energetic electrons can be reproduced.     

On the role of injection in kinetic approaches to non-linear particle acceleration at non-relativistic shock waves

The dynamical reaction of the particles accelerated at a shock front by the first-order Fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated particles. These models predict the appearance of multiple solutions, all physically allowed. We discuss here the role of injection in selecting the real solution, in the framework of a simple phenomenological recipe, which is a variation of what is sometimes referred to as thermal leakage. In this context we show that multiple solutions basically disappear and when they are present they are limited to rather peculiar values of the parameters. We also provide a quantitative calculation of the efficiency of particle acceleration at cosmic ray modified shocks and we identify the fraction of energy which is advected downstream and that of particles escaping the system from upstream infinity at the maximum momentum. The consequences of efficient particle acceleration for shock heating are also discussed.     

The origin of galactic cosmic rays

One century ago Viktor Hess carried out several balloon flights that led him to conclude that the penetrating radiation responsible for the discharge of electroscopes was of extraterrestrial origin. One century from the discovery of this phenomenon seems to be a good time to stop and think about what we have understood about Cosmic Rays. The aim of this review is to illustrate the ideas that have been and are being explored in order to account for the observable quantities related to cosmic rays and to summarize the numerous new pieces of observation that are becoming available. In fact, despite the possible impression that development in this field is somewhat slow, the rate of new discoveries in the last decade or so has been impressive, and mainly driven by beautiful pieces of observation. At the same time scientists in this field have been able to propose new, fascinating ways to investigate particle acceleration inside the sources, making use of multifrequency observations that range from the radio, to the optical, to X-rays and gamma rays. These ideas can now be confronted with data.I will mostly focus on supernova remnants as the most plausible sources of Galactic cosmic rays, and I will review the main aspects of the modern theory of diffusive particle acceleration at supernova remnant shocks, with special attention for the dynamical reaction of accelerated particles on the shock and the phenomenon of magnetic field amplification at the shock. Cosmic-ray escape from the sources is discussed as a necessary step to determine the spectrum of cosmic rays at the Earth. The discussion of these theoretical ideas will always proceed parallel to an account of the data being collected especially in X-ray and gamma-ray astronomy.In the end of this review I will also discuss the phenomenon of cosmic-ray acceleration at shocks propagating in partially ionized media and the implications of this phenomenon in terms of width of the Balmer line emission. This field of research has recently experienced a remarkable growth, in that Hα lines have been found to bear information on the cosmic-ray acceleration efficiency of supernova shocks.     

Comparison of the Fermi and betatron acceleration efficiencies in collapsing magnetic traps

The acceleration of charged particles in the solar corona during flares is investigated in terms of a model in which the electrons and ions preaccelerated in the magnetic reconnection region are injected into a collapsing magnetic trap. Here, the particle energy increases rapidly simultaneously through the Fermi and betatron mechanisms. Comparison of the efficiencies of the two mechanisms shows that the accelerated electrons in such a trap produce more intense hard X-ray (HXR) bursts than those in a trap where only the Fermi acceleration mechanism would be at work. This effect explains the Yohkoh and RHESSI satellite observations in which HXR sources more intense than the HXR emission from the chromosphere were detected in the corona.     

Relativistic blast waves and synchrotron emission

Relativistic shocks can accelerate particles by the first-order Fermi mechanism; the particles then emit synchrotron emission in the post-shock gas. This process is of particular interest in the models used for the afterglow of gamma-ray bursts. In this paper we use recent results in the theory of particle acceleration at highly relativistic shocks to model the synchrotron emission in an evolving, inhomogeneous and highly relativistic flow. We have developed a numerical code that integrates the relativistic Euler equations for fluid dynamics with a general equation of state, together with a simple transport equation for the accelerated particles. We present tests of this code and, in addition, we use it to study the gamma-ray burst afterglow predicted by the fireball model, along with the hydrodynamics of a spherically-symmetric relativistic blast wave.
We find that, while broadly speaking the behaviour of the emission is similar to that already predicted with semi-analytic approaches, the detailed behaviour is somewhat different. The 'breaks' in the synchrotron spectrum behave differently with time, and the spectrum above the final break is harder than had previously been expected. These effects are due to the incorporation of the geometry of the (spherical) blast wave, along with relativistic beaming and adiabatic cooling of the energetic particles leading to a mix, in the observed spectrum, between recently injected 'uncooled' particles and the older 'cooled' population in different parts of the evolving, inhomogeneous flow.     

Critical Self-Organization of Astrophysical Shocks

There are two distinct regimes of the first-order Fermi acceleration of shocks. The first is a linear (test-particle) regime in which most of the shock energy goes into thermal and bulk motions of the plasma. The second is an efficient regime in which the shock energy goes into accelerated particles. Although the transition region between them is narrow, we identify the factors that drive the system toward a self-organized critical state between those two regimes. Using an analytic solution, we determine this critical state and calculate the spectra and maximum energy of accelerated particles.     

Gamma-Ray Emission from Pulsar Outer Magnetospheres

We investigate a stationary pair production cascade in the outer magnetosphere of an isolated, spinning neutron star. The charge depletion due to global flows of charged particles, causes a large electric field along the magnetic field lines. Migratory electrons and/or positrons are accelerated by this field to radiate gamma-rays via curvature and inverse-Compton processes. Some of such gamma-rays collide with the X-rays to materialize as pairs in the gap. The replenished charges partially screen the electric field, which is self-consistently solved together with the energy distribution of particles and gamma-rays at each point along the field lines. By solving the set of Maxwell and Boltzmann equations, we demonstrate that an external injection of charged particles at nearly Goldreich-Julian rate does not quench the gap but shifts its position and that the particle energy distribution cannot be described by a power-law. The injected particles are accelerated in the gap and escape from it with large Lorentz factors. We show that such escaping particles migrating outside of the gap contribute significantly to the gamma-ray luminosity for young pulsars and that the soft gamma-ray spectrum between 100 MeV and 3 GeV observed for the Vela pulsar can be explained by this component. We also discuss that the luminosity of the gamma-rays emitted by the escaping particles is naturally proportional to the square root of the spin-down luminosity.     

Cosmic ray acceleration

This review describes the basic theory of cosmic ray acceleration by shocks including the plasma instabilities confining cosmic rays near the shock, the effect of the magnetic field orientation, the maximum cosmic ray energy and the shape of the cosmic ray spectrum. Attention is directed mainly towards Galactic cosmic rays accelerated by supernova remnants.     

Non-linear diffusive shock acceleration with free-escape boundary

We present here a semi-analytical solution of the problem of particle acceleration at non-linear shock waves with a free-escape boundary at some location upstream. This solution, besides allowing us to determine the spectrum of particles accelerated at the shock front, including the shape of the cutoff at some maximum momentum, also allows us to determine the spectrum of particles escaping the system from upstream. This latter aspect of the problem is crucial for establishing a connection between the accelerated particles in astrophysical sources, such as supernova remnants, and the cosmic rays observed at the Earth. An excellent approximate solution, which leads to a computationally fast calculation of the structure of shocks with an arbitrary level of cosmic ray modification, is also obtained.     

Bidirectional distribution of energetic protons in the 18 February 1979 interplanetary shock event

Pitch angle distributions of 35–1000 keV protons in the 18 January 1979 interplanetary shock event are presented and discussed in this paper. The observed, apparently unusual distributions can be well explained by assuming a magnetic bottle configuration of the interplanetary magnetic field immediately prior to the shock passage. The implication of such a magnetic configuration on the acceleration of charged particles by shock waves is also discussed.     

On the acceleration of cosmic rays

The intensive acceleration of energetic charged particles in perpendicular shock waves which has been known to take place in the interplanetary medium has been utilized in this work in order to account for the energization of cosmic rays. It is proposed that cosmic rays can be accelerated up to 10¹⁴–10¹⁵ eV in successive perpendicular shock waves which appear inside supernova shells in our Galaxy.     

Fundamentals of collisionless shocks for astrophysical application, 2. Relativistic shocks

In this concise review of the recent developments in relativistic shock theory in the Universe we restrict ourselves to shocks that do not exhibit quantum effects. On the other hand, emphasis is given to the formation of shocks under both non-magnetised and magnetised conditions. We only briefly discuss particle acceleration in relativistic shocks where much of the results are still preliminary. Analytical theory is rather limited in predicting the real shock structure. Kinetic instability theory is briefed including its predictions and limitations. A recent self-similar relativistic shock theory is described which predicts the average long-term shock behaviour to be magnetised and to cause reasonable power-law distributions for energetic particles. The main focus in this review is on numerical experiments on highly relativistic shocks in (i) pair and (ii) electron-nucleon plasmas and their limitations. These simulations do not validate all predictions of analytic and self-similar theory and so far they do not solve the injection problem and the self-modification by self-generated cosmic rays. The main results of the numerical experiments discussed in this review are: (i) a confirmation of shock evolution in non-magnetised relativistic plasma in 3D due to either the lepton-Weibel instability (in pair plasmas) or to the ion-Weibel instability; (ii) the sensitive dependence of shock formation on upstream magnetisation which causes suppression of Weibel modes for large upstream magnetisation ratios σ>10⁻³; (iii) the sensitive dependence of particle dynamics on the upstream magnetic inclination angle θ _Bn , where particles of θ _Bn >34° cannot escape upstream, leading to the distinction between ‘subluminal’ and ‘superluminal’ shocks; (iv) particles in ultra-relativistic shocks can hardly overturn the shock and escape to upstream; they may oscillate around the shock ramp for a long time, so to speak ‘surfing it’ and thereby becoming accelerated by a kind of SDA; (v) these particles form a power-law tail on the downstream distribution; their limitations are pointed out; (vi) recently developed methods permit the calculation of the radiation spectra emitted by the downstream high-energy particles; (vii) the Weibel-generated downstream magnetic fields form large-amplitude vortices which could be advected by the downstream flow to large distances from the shock and possibly contribute to an extended strong field region; (viii) if cosmic rays are included, Bell-like modes can generate upstream magnetic turbulence at short and, by diffusive re-coupling, also long wavelengths in nearly parallel magnetic field shocks; (ix) advection of such large-amplitude waves should cause periodic reformation of the quasi-parallel shock and eject large-amplitude magnetic field vortices downstream where they contribute to turbulence and to maintaining an extended region of large magnetic fields.