On the escape of particles from cosmic ray modified shocks

Stationary solutions to the problem of particle acceleration at shock waves in the non-linear regime, when the dynamical reaction of the accelerated particles on the shock cannot be neglected, are known to show a prominent energy flux escaping from the shock towards upstream infinity. On physical grounds, the escape of particles from the upstream region of a shock has to be expected in all those situations in which the maximum momentum of accelerated particles,   p _max  , decreases with time, as is the case for the Sedov–Taylor phase of expansion of a shell supernova remnant, when both the shock velocity and the cosmic ray induced magnetization decrease. In this situation, at each time t , particles with momenta larger than   p _max( t )  leave the system from upstream, carrying away a large fraction of the energy if the shock is strongly modified by the presence of cosmic rays. This phenomenon is of crucial importance for explaining the cosmic ray spectrum detected at the Earth. In this paper, we discuss how this escape flux appears in the different approaches to non-linear diffusive shock acceleration, and especially in the quasi-stationary semi-analytical kinetic ones. We apply our calculations to the Sedov–Taylor phase of a typical supernova remnant, including in a self-consistent way particle acceleration, magnetic field amplification and the dynamical reaction on the shock structure of both particles and fields. Within this framework, we calculate the temporal evolution of the maximum energy reached by the accelerated particles and of the escape flux towards upstream infinity. The latter quantity is directly related to the cosmic ray spectrum detected at the Earth.     

Acceleration of Pick-up Ions at the Solar Wind Termination Shock

It is generally accepted that pick-up ions act as a seed population for anomalous cosmic rays originating at the solar wind termination shock. We believe that the ion pre-acceleration process operating in the heliosphere up to the termination shock can be very important to inject the ions into the shock acceleration process. The pick-up ions pre-accelerated by solar wind turbulences have already a pronounced high energy tail when they reach the shock. Some fraction of these ions can experience further acceleration up to energies of anomalous cosmic rays by means of shock drift and diffusive acceleration. In the present paper the shock drift acceleration of pick-up ions suffering multiple reflection due to abrupt changes in both the strength and direction of the magnetic field through the shock is considered. The reflection process operates for high velocity particles different from the reflection by the electric cross-shock potential. During the first reflection the mean kinetic energy of pick-up ions increases by approximately a factor of 10. Reflected particles have highly anisotropic velocity distribution. Subsequent excursion of the particles in the turbulent upstream flow leads to diffusion in pitch-angle space and, as a result, the particles can return to the shock again suffering, thus, multiple encounters. In order to describe the motion of particles in the upstream and down streamparts of the flow we solve the Fokker-Plank transport equation for anisotropic velocity distribution function. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Cosmic-ray diffusive acceleration at shock waves with finite upstream and downstream escape boundaries

In the present paper we discuss the modifications introduced into the first-order Fermi shock acceleration process due to a finite extent of diffusive regions near the shock or due to boundary conditions leading to an increased particle escape upstream and/or downstream of the shock. In the simple example of the planar shock wave considered we idealize the escape phenomenon by imposing a particle escape boundary at some distance from the shock. The presence of such a boundary (or boundaries) leads to coupled steepening of the accelerated particle spectrum and decreasing of the acceleration time scale. It allows for a semi-quantitative evaluation and, in some specific cases, also for modelling of the observed steep particle spectra as a result of the first-order Fermi shock acceleration. We also note that the particles close to the upper energy cut-off are younger than the estimate based on the respective acceleration time scale. In Appendix A we present a new time-dependent solution for infinite diffusive regions near the shock allowing for different constant diffusion coefficients upstream and downstream of the shock.     

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.     

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.     

The Origin of Cosmic Rays from Supernova Remnants

The origin of cosmic rays is one of the key questions in high-energy astrophysics. Supernovae have been always considered as the dominant sources of cosmic rays below the energy spectrum knee. Multi-wavelength observations indeed show that supernova remnants are capable for accelerating particles into sub-PeV (10${}^{15}$ eV) energies. Diffusive shock acceleration is considered as one of the most efficient acceleration mechanisms of astrophysical high-energy particles, which may just operate effectively in the large-scale shocks of supernova remnants. Recently, a series of high-precision ground and space experiments have greatly promoted the study of cosmic rays and supernova remnants. New observational features challenge the classical acceleration model by diffusive shock and the application to the scenario of supernova remnants for the origin of Galactic cosmic rays, and have deepened our understanding to the cosmic high-energy phenomena. In combination with the time evolution of radiation energy spectrum of supernova remnants, a time-dependent particle acceleration model is established, which can not only explain the anomalies in cosmic-ray distributions around 200 GV, but also naturally form the cosmic-ray spectrum knee, even extend the contribution of supernova particle acceleration to cosmic ray flux up to the spectrum ankle. This model predicts that the high-energy particle transport behavior is dominated by the turbulent convection, which needs to be verified by future observations and plasma numerical simulations relevant to the particle transport.     

Piston shock and Forbush effect

A model of a piston shock produced by a sharp jump in the velocity of the solar wind with a helical magnetic field is suggested to explain the origin of the Forbush decrease of cosmic rays with a hard energy spectrum at solar activity minimum and, in some cases, at solar activity maximum. The contact velocity, the compression ratio, the contact position, the ratio of the magnetic field near the contact to the seed field, the maximum momentum of the particles subjected to modulation, and the modulation depth for the neutron and muon cosmic ray components have been calculated. The theoretical calculations are compared with experimental data.     

Charged particle acceleration in the front of the shock wave bounding supersonic solar wind

The process of cosmic ray acceleration in the front of the spherical shock wave bounding the supersonic solar wind is studied. On the basis of our analytical solution of the transport equation, the energy and spatial distributions of cosmic ray intensity and anisotropy are investigated. It is shown that the shape of accelerated particle spectrum is determined by the medium compressibility at the shock front and by cosmic ray modulation parameters.     

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.     

The physics of particle acceleration at the heliospheric termination shock

Cosmic rays are ubiquitous in space, and the essential similarity of their energy spectra in many different regions places significant general constraints on the mechanisms for their acceleration and confinement. Diffusive shock acceleration is at present the most successful acceleration mechanism proposed, and, together with transport in Kolmogorov turbulence, can account for the universal specta. A unique laboratory for studying the acceleration and transport of charged particles is the outer heliosphere, including the solar wind termination shock and heliosheath.

A widely accepted paradigm for the transport and acceleration of energetic particles in the heliosphere has evolved over the last few decades. This picture has successfully explained many features of the modulation of galactic cosmic rays and the transport and acceleration of anomalous cosmic rays at the solar-wind termination shock. Recent Voyager observations near and beyond the termination shock have revealed new, and in some cases, unexpected phenomena which have led to questions concerning the established paradigm. The physical interpretation of the observations requires a blunt termination shock, rapid inward motion of the shock and temporal variations over time scales ranging from hours to 22 years. Incorporation of these into the physics has promise of explaining most, if not, all of the observed phenomena while retaining the advantages of the termination shock paradigm for both galactic and anomalous cosmic rays.     

Reflection of pre-accelerated pick-up ions at the solar wind termination shock: The seed for anomalous cosmic rays

It has been hypothesized for quite some time that interplanetary pick-up ions due to energization taking place in the region close to the solar wind termination shock, at some fraction and as an outcome of a complicated chain of processes, eventually are converted into species of the anomalous cosmic-ray particles. For the actual conversion efficiency it is of great importance to know the energy distribution of these pick-up ions upon their arrival at the shock. It turns out that pre-acceleration of these ions during their passage through the heliosphere shall substantially increase their chances to become reflected at the shock into the upstream direction which is a prerequisite for a further climb-up in energy by virtue of Fermi-1 acceleration processes. In this paper we start out from stochastically pre-accelerated pick-up ions and investigate their behaviour at the shock. With the use of adiabatic approaches in the de Hoffman-Teller frame of the shock, we calculate the energy distribution function of the reflected part of pick-up ions. From the calculated distribution functions it turns out that the reflected ions in the average suffer an energy increase by about a factor of 10, still not enough to let them move off the shock by spatial diffusion in the upstream direction. Thus, since converted back into the shock, they can undergo repeated reflections and energy gains till the diffusion-convection limit is reached. As we show in addition, the reflection probability for pick-up ions is about a factor of 10 higher than expected from the present literature and strongly varies with the off-axis angle, pointing to the fact that the termination shock represents a surface with a three-dimensionally varying source strength for the production of anomalous cosmic rays. The ACR source pattern is also expected to vary during the solar cycle and the relevant injection energies are expected to be larger by factors of 10 to 100 than the canonically adopted 1 keV nucl^–1.Institute for Problems of Mechanics of the Russian Academy of Sciences, Prospect Vernadskogo 101, 117526, Moscow, Russia.     

Cosmic ray propagation and the star formation history of NGC 1961

We present new radio continuum data at four frequencies for the supermassive, peculiar galaxy NGC 1961. These observations allow us to separate the thermal and non-thermal radio emission and to determine the non-thermal spectral index distribution. This spectral index distribution in the galactic disc is unusual: at the maxima of the radio emission the synchrotron spectrum is very steep, indicating aged cosmic ray electrons. Away from the maxima the spectrum is much flatter. The steep spectrum of the synchrotron emission at the maxima indicates that a strong decline of the star formation rate has taken place at these sites. The extended radio emission is a sign of recent cosmic ray acceleration, probably by recent star formation. We suggest that a violent event in the past, most likely a merger or a collision with an intergalactic gas cloud, has caused the various unusual features of the galaxy.     

Broad-band non-thermal emission from molecular clouds illuminated by cosmic rays from nearby supernova remnants

Molecular clouds are expected to emit non-thermal radiation due to cosmic ray interactions in the dense magnetized gas. Such emission is amplified if a cloud is located close to an accelerator of cosmic rays and if energetic particles can leave the accelerator site and diffusively reach the cloud. We consider here a situation in which a molecular cloud is located in the proximity of a supernova remnant which is efficiently accelerating cosmic rays and gradually releasing them in the interstellar medium. We calculate the multiwavelength spectrum from radio to gamma rays which is emerging from the cloud as the result of cosmic ray interactions. The total energy output is dominated by the gamma-ray emission, which can exceed the emission in other bands by an order of magnitude or more. This suggests that some of the unidentified TeV sources detected so far, with no obvious or very weak counterparts in other wavelengths, might be in fact associated with clouds illuminated by cosmic rays coming from a nearby source. Moreover, under certain conditions, the gamma-ray spectrum exhibits a concave shape, being steep at low energies and hard at high energies. This fact might have important implications for the studies of the spectral compatibility of GeV and TeV gamma-ray sources.     

Diffusive propagation of fast particles in the presence of a moving shock wave

Based on an analytical model, we determined the temporal dynamics of the spectral shape and spatial distribution of the particles that were impulsively (in time) injected with a specified spectrum in the vicinity of a moving plane shock front. We obtained a condition to determine the influence of the shock front on the particle propagation, where the spatial diffusion coefficient of the particles plays a major role. Diffusive shock acceleration is shown to strongly affect low-energy particles (the intensity maximum coincides spatially with the shock front; hard and soft spectral regions are formed in the spectrum) and weakly affect high-energy particles (the time at which the intensity reaches its maximum is well ahead of the shock arrival time; the spectral shape does not change). In events accompanied by a significant increase in the turbulence level, the influence of the shock front on high-energy particles can change from weak to strong. This change shows up in the spatial distribution and spectral shape of the particles. The dynamics of the particle intensity, calculated with the diffusion coefficients that were determined in accordance with the quasi-linear theory for measured turbulence levels, qualitatively corresponds to the observed solar energetic-particle intensity.     

The fluctuation origin of cosmic radiation

We have investigated the origin of cosmic radiation in terms of a sudden injection of particles in time, momentum and space. The appropriate boundary conditions for the various regions through which the particles pass were used. With all of the acceleration within a turbulent region, we find that the observed spectrum is explained by a continuous deceleration in which statistical fluctuations dominate. This is contradictory to the usual assumptions in which fluctuations do not play an important part. We find that the time dependent solution has an exponent of the power law spectrum that varies weakly with the momentum and time. The steady state solution shows the usual constant exponent.This work was supported by the National Aeronautics and Space Administration under Grant NsG-695.     

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.     

The effect of heavy ions on the formation and structure of cometary bow shocks

We investigate the interaction of heavy cometary ions with the solar wind and the formation of a bow shock in front of a comet by means of a hybrid (particle ion, fluid electron) simulation code that solves self-consistently for the electromagnetic fields and the motion of the charged particles. This kinetic treatment of the solar wind protons and the heavy cometary ions allows us to examine two important issues. One is the effect of the velocity distribution function of the heavy ions on the shock formation and structure, and the other is the degree of coupling between the two ion species. The result of this study indicate that at high Mach numbers the shock structure is highly dependent upon the velocity distribution of the heavy ions. For example, when the newly created ions comprise a ring distribution in the solar wind frame, most of them turn around downstream of the shock surface and reenter the upstream region to form a large foot that extends about a heavy ions gyroradius upstream of the shock. On the other hand, heavy ions which have been picked up by the solar wind and possesses a Maxwellian distribution can mostly penetrate the shock without returning upstream and affecting the shock structure as much. In either case, however, at high Mach numbers the shock strength is the same. At low Mach numbers, where the shock is weak, the velocity distribution of heavy ions has a smaller effect on the formation of shock and its structure. In this regime, the degree of coupling between the cometary ions and the solar wind protons and the corresponding critical Mach number (at which a shock should begin to form) are determined from a set of Rankine-Hugoniot relations. The results of the simulations suggest that some coupling does occur (evidently, through the electromagnetic fields, since there are no particle collisions in the calculations), but less than that expected from magnetohydrodynamics. For low Mach numbers, it is also shown that shocks have a transitory nature, where they are continuously formed by the protons and subsequently destroyed by the heavy ions.     

The role of the energy spectrum anisotropy in the variations of the cosmic-ray fluctuations power-spectrum before interplanetary medium disturbances

The variations in the form of the cosmic-ray fluctuation power spectrum as an interplanetary shock wave approaches the Earth have been calculated for different values of cosmic ray anisotropy. The relevant experimental estimates of the power spectra are inferred from the data of cosmic ray detection with the ground-based neutron monitors at cosmic-ray stations. A comparison between the theoretical and experimental estimates has demonstrated an important role of the cosmic ray anisotropy spectrum in the generation of the power spectrum as the latter is rearranged before the interplanetary medium disturbances.     

Pulsars and the origin of cosmic rays

The capabilities and limitations of pulsars as sources of cosmic rays are reviewed in the light of experimental observations. Pulsars can supply the cosmic ray power if they have rotational velocities in excess of 700 rad s^?1 at birth. Though this is theoretically possible, there is no experimental proof for the same. Pulsars can accelerate particles to the highest energies of 10²⁰ eV, but in general, the spectra on simple considerations, turn out to be flatter than the observed cosmic ray spectrum. At the highest energies, absorption processes due to fragmentation and photodisintegration dominate for heavy nuclei. The existence of a steady flux of cosmic rays of energy greater than 10¹⁷ eV demands acceleration of particles to last over fifty years, the time interval between supernovae outbursts, whereas the expected period of activity is less than a few years. Finally, the problem of anisotropy with relevance to pulsars as sources and the possibility of observing pulsar accelerated particles from galactic clusters is considered.