Efficiency of cosmic ray reflections from an ultrarelativistic shock wave

The process of cosmic ray acceleration up to energies in excess of 10²⁰ eV at relativistic shock waves with large Lorentz factors, Γ≫1, requires ∼Γ² particle energy gains at single reflections from the shock (cf. Gallant & Achterberg). In the present Letter , by applying numerical simulations we address an efficiency problem arising for such models. The actual efficiency of the acceleration process is expected to be substantially lower than the estimates of previous authors.     

Acceleration time-scale in an ultrarelativistic shock

The acceleration mechanism in ultrarelativistic shocks is investigated using Monte Carlo simulations. We apply a method of discrete small-amplitude particle momentum scattering to reproduce highly anisotropic conditions at the shock and describe the acceleration mechanism carefully. The obtained acceleration time equals 1.0 r _g c if the spectral index reaches the value of 2.2, independent of physical conditions in the shock. Some other parameters of the acceleration process are also provided.     

Shock acceleration of solar cosmic rays

We investigated the acceleration of solar cosmic rays (SCRs) by the shock waves produced by coronal mass ejections. We performed detailed numerical calculations of the SCR spectra produced during the shock propagation in the solar corona in terms of a model based on the diffusive transport equation using a realistic set of physical parameters for the corona. The resulting SCR energy spectrum N(ε) ∝ ε^?γ exp [? (ε/ε_max)^α] is shown to include a power-law portion with an index γ?2 that ends with an exponential tail with α ? 2.5 ? β, where β is the spectral index of the background Alfvén turbulence. The maximum SCR energy lies within the range ε_max = 1–300 MeV, depending on the shock velocity. Because of the steep spectrum of the SCRs, their backreaction on the shock structure is negligible. The decrease in the Alfvén Mach number of the shock due to the increase in the Alfvén velocity with heliocentric distance r causes the efficient SCR acceleration to terminate when the shock reaches a distance of r = 2–3R_⊙. Since the diffusive SCR propagation in this case is faster than the shock expansion, SCR particles intensively escape from the shock vicinity. A comparison of the calculated SCR fluxes expected near the Earth’s orbit with available experimental data indicates that the theory satisfactorily explains all of the main observed features.     

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.     

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.     

Small-scale anisotropy of cosmic rays with energies above 3×1018 eV based on data from the Yakutsk EAS facility

We analyze the directions of the arrival of cosmic rays with energies E ₀≥3×10¹⁸ eV and zenith angles θ≤45° recorded by the Yakutsk extensive air shower (EAS) facility during 1974–2000. They are shown to have a small-scale structure with scale sizes of 5°–10°. Enhanced particle fluxes compared to the expected levels for random distributions at (4–5)σ are observed from the Galactic and Supergalactic planes.     

The Influences of Shock Thickness and Alfven Waves on the Particle Acceleration by Diffusive Shocks

The influences of the shock thickness and Alfven waves on the particle acceleration by diffusive shock waves are numerically studied through solving one-dimensional diffusive equation including the second-order Fermi effect. It is shown that the spectral index of the energetic particles strongly depends on the shock thickness. For example, the spectral index increases from 2.1 to 3.7 in the low energy range of 3—10 MeV and from 2.5 to 5.0 in the high energy range of 20—60 MeV as the thickness increases. The spectral index decreases from 4.3 to 3.1 as the particle injection energy increases. The spectral index decreases from 4.0 to 1.8 at the quasi-steady stage with the enhancement of the compression ratio from 2 to 4. The results indicate that under the influence of Alfven waves, the energetic particle spectrum at lower energy becomes flat and the spectral index decreases from 2.5 to 0.6 in the low energy range of 3—10 MeV and from 11.6 to 5.0 in the high energy range of 20—60 MeV. At the same time, the turning point energy reaches 19.6 MeV. The spectral index decreases from 5.8 to 2.9 as the energy density of Alfven waves increases. All these results are basically consistent with the theoretical models, as well as the observations of typical energetic particle events.     

Some properties of synchrotron radio and inverse-Compton gamma-ray images of supernova remnants

The synchrotron radio maps of supernova remnants (SNRs) in a uniform interstellar medium and interstellar magnetic field (ISMF) are analysed, allowing for different 'sensitivity' of the injection efficiency to the obliquity of the shock. The very-high-energy γ-ray maps arising from inverse Compton processes are also synthesized. The properties of images in these different wavelength bands are compared, with particular emphasis on the location of the bright limbs in bilateral SNRs. Recent High-Energy Stereoscopic System (HESS) observations of SN 1006 show that the radio and inverse Compton γ-ray limbs coincide, and we found that this may happen if (i) injection is isotropic but the variation of the maximum energy of electrons is rather fast to compensate for differences in the magnetic field, or (ii) the obliquity dependence of injection (either quasi-parallel or quasi-perpendicular) and the electron maximum energy are strong enough to dominate the magnetic field variation. In the latter case, the obliquity dependences of the injection and the maximum energy should not be opposite. We argue that the position of the limbs alone, and even their coincidence in radio, X-rays and γ-rays, as discovered by HESS in SN 1006, cannot be conclusive as regards the dependence of the electron injection efficiency, the compression/amplification of the ISMF and the electron maximum energy on the obliquity angle.     

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.     

Phenomenological explanation for the shape of the spectrum of cosmic rays with energies <Emphasis Type="Italic">E</Emphasis> > 10 GeV

Assuming that the energy gain by cosmic-ray (CR) particles is a stochastic process with stationary increments, we derive expressions for the shape of their energy spectrum up to energies E ～ 10¹⁸ eV. In the ultrarelativistic case under study, the energy is proportional to the momentum, whose time derivative is the force. According to the Fermi mechanism, a particle accelerates when it passes through a system of shock waves produced by supernova explosions. Since these random forces act on time scales much shorter than the particle lifetime, we assume them to be delta-correlated in time. In this case, due to the linear energy-momentum relationship, the mean square of the energy (increments) is proportional to the differential scale τ(E) ～ Eτ(≥E), where τ (≥E) is the cumulative time it takes for a particle to gain an energy ≥E. The probability of finding a particle with energy ≥E somewhere in the system is inversely proportional to the time it takes to gain the energy E. To estimate an upper limit for the space number density of CR particles, we use estimates of the CR volume energy density, a quantity known for our Galaxy. It is taken to be constant in the range 10 GeV ≤ E ≤ 3 × 10⁶ GeV, where the index of the energy spectrum was found to be ?8/3 ≈ ?2.67 against its empirical value of ?2.7. In the range 3 × 10⁶ GeV ≤ E < 10⁹ GeV, the upper limit for the volume energy density is estimated by using the results from the previous range to be ?28/9 ≈ ?3.11 against its empirical value of ?3.1. The numerical coefficients in the suggested shapes of the spectrum can be determined by comparison with observational data. Thus, the CR energy spectrumis the result of the random walks of ultrarelativistic particles in energy/momentum space caused by the Fermi mechanism.     

Possible origin of clusters in ultra-high-energy cosmic rays

We estimate the detection rate of ultra-high-energy cosmic rays on ground-based arrays by assuming that the cosmic-ray sources are active galactic nuclei. We analyze the cases of detection of clusters, several particles that arrived, within the error limits, from the same area of the sky. The adopted model is shown to explain the detection rate of clusters on the AGASA array.     

宇宙线的超新星遗迹起源

宇宙线的起源是高能天体物理的核心问题之一.一直以来,超新星爆发被认为是能谱膝区以下宇宙线的主要来源.多波段观测表明,超新星遗迹有能力加速带电粒子至亚PeV (10~(15)eV)能量.扩散激波加速被认为是最有效的天体高能粒子加速机制之一,而超新星遗迹的大尺度激波正好为这一机制提供平台.近年来,一系列较高精度的地面和空间实验极大地推动了对宇宙线以及超新星遗迹的研究.新的观测事实挑战着传统的扩散激波加速模型以及其在银河系宇宙线超新星遗迹起源学说上的应用,深化了人们对宇宙高能现象的认识.结合超新星遗迹辐射能谱的时间演化特性,构建的时间依赖的超新星遗迹粒子加速模型,不仅能够解释200 GV附近宇宙线的能谱反常,还自然地形成能谱膝区,甚至可以将超新星遗迹粒子加速对宇宙线能谱的贡献延伸至踝区.该模型预期超新星遗迹中粒子的输运行为表现为湍流扩散,这需要未来的观测以及与粒子输运相关的等离子体数值模拟工作来进一步验证.     

Non-linear amplification of a magnetic field driven by cosmic ray streaming

One-, two- and three-dimensional numerical results of the non-linear interaction between cosmic rays and a magnetic field are presented. These show that cosmic ray streaming drives large-amplitude Alfvénic waves. The cosmic ray streaming energy is very efficiently transferred to the perturbed magnetic field of the Alfvén waves, and the non-linear time-scale of the growth of the waves is found to be very rapid, of the order of the gyro-period of the cosmic ray. Thus, a magnetic field of interstellar values, assumed in models of supernova remnant blast wave acceleration, would not be appropriate in the region of the shock. The increased magnetic field reduces the cosmic ray acceleration time and so increases the maximum cosmic ray energy, which may provide a simple and elegant resolution to the highest energy Galactic cosmic ray problem, where the cosmic rays themselves provide the fields necessary for their acceleration.     

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.