We present a simple model of nonlinear di †usive shock acceleration (also called Ðrst-order Fermi shock acceleration) that determines the shock modiÐcation, spectrum, and efficiency of the process in the plane-wave, steady state approximation as a function of an arbitrary injection parameter, g. The model, which uses a threeÈpower-law form for the accelerated particle spectrum and contains only simple algebraic equations, includes the essential elements of the full nonlinear model and has been tested against Monte Carlo and numerical kinetic shock models. We include both adiabatic and wave heating Alfve n of the upstream precursor. The simplicity and ease of calculation make this model useful for studying the basic properties of nonlinear shock acceleration, as well as providing results accurate enough for many astrophysical applications. It is shown that the shock properties depend upon the shock speed with u 0 respect to a critical value which is a function of the injection rate g and maximum acceleru 0 * P gp max 1@4

Abstract. It is shown that amplification of the magnetic field in supernova remnants (SNRs) occurs in all six objects where morphological measurements are presently available in the hard X-ray continuum at several keV. For the three archetypical objects (SN 1006, Cas A and Tycho's SNR) to which nonlinear time-dependent acceleration theory has been successfully applied up to now, the global theoretical and the local observational field strengths agree very well, suggesting in addition that all young SNRs exhibit the amplification effect as a result of very efficient acceleration of nuclear cosmic rays (CRs) at the outer shock. Since this appears to be empirically the case, we may reverse the argument and consider field amplification as a measure of nuclear CR acceleration and it has indeed been argued that acceleration in the amplified fields allows the CR spectrum from SNRs to reach the knee in the spectrum or, in special objects, even beyond. The above results are furthermore used to investigate the time evolution of field amplification in young SNRs. Although the uncertainties in the data do not allow precise conclusions regarding this point, they rather clearly show that the ratio of the magnetic field energy density and the kinetic energy density of gas flow into the shock is of the order of a few percent if the shock speed is high enough V s > 10 3 km s −1 , and this ratio remains nearly constant during the SNR evolution. The escape of the highest energy nuclear particles from their sources becomes progressively important with age, reducing also the cutoff in the π 0 -decay gamma-ray emission spectrum with time after the end of the sweep-up phase. Simultaneously the leptonic gamma-ray channels will gain in relative importance with increasing age of the sources.

Abstract. The time-dependent nonlinear kinetic theory for cosmic ray (CR) acceleration in supernova remnants (SNRs) is applied studying the properties of the synchrotron emission from SNRs, in particular, the surface brightness-diameter (Σ − D) relation. Detailed numerical calculations are performed for the expected range of the relevant physical parameters, namely the ambient density and the supernova explosion energy. The magnetic field in SNRs is assumed to be significantly amplified by the efficiently accelerating nuclear CR component. Due to the growing number of accelerated CRs the expected SNR luminosity increases during the free expansion phase, reaches a peak value at the beginning of the Sedov phase and then decreases again, since in this stage the overall CR number remains nearly constant, whereas the effective magnetic field diminishes with time. The theoretically predicted brightness-diameter relation in the radio range in the Sedov phase is close to Σ R ∝ D −17/4 . It fits the observational data in a very satisfactory way. The observed spread of Σ R at a given SNR size D is the result of the spread of supernova explosion energies and interstellar medium densities.

Abstract. The nonlinear kinetic model of cosmic ray (CR) acceleration in supernova remnants (SNRs) is used to describe the properties of the remnant of SN 1006. It is shown, that the theory fits the existing data in a satisfactory way within a set of parameters which is consistent with the idea that SN 1006 is a typical source of Galactic CR nucleons, although not necessarily of CR electrons. The adjusted parameters are those that are not very well determined by present theory or not directly amenable to astronomical observations. The calculated expansion law and the radio-, X-ray and γ-ray emissions produced by the accelerated CRs in SN 1006 agree quite well with the observations. A rather large interior magnetic field B d ≈ 100 µG is required to give a good fit for the radio and X-ray synchrotron emission. In the predicted TeV γ-ray flux from SN 1006, the π 0 -decay γ-rays, generated by the nuclear CR component, dominate over the inverse Compton (IC) γ-rays, generated by the CR electrons in the cosmic microwave background. The predicted source morphology in high energy γ-rays roughly corresponds to that of the synchrotron emission. The predicted integral γ-ray flux F γ ∝ −1 γ extends up to energies ∼100 TeV if CR diffusion is as strong as the Bohm limit. Only if the interior magnetic field is much lower in the SNR, B d ≈ 10 µG, then the observed γ-ray emission can be due to the accelerated electron component alone. In this case, not plausible physically in our view, the lowest permissible value of the electron to proton ratio is rather high, and the maximum individual energy and total energy content of accelerated nucleons so small, that SN 1006 can not be considered as a typical source of the nuclear Galactic CRs.

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