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The theory of non-linear diffusive shock acceleration (NLDSA) predicts the formation of a precursor upstream of the shock, where accelerated particles diffuse and induce magnetic field amplification (MFA) through streaming instability. The non-detection of this precursor in X-rays in Chandra observations of the north-eastern region of SN 1006 (G329.6+14.6) led the authors of some previous work to impose an upper limit to the X-ray emission generated by accelerated electrons diffusing in this precursor, at an emissivity level of <1.5 per cent of the emission from the downstream region. This has been used as an argument against Fermi acceleration at this shock. Here, we calculate the spectrum and spatial distribution of accelerated particles in SN 1006 and show that Chandra results (including more recent data) are in perfect agreement with the predictions of NLDSA suggesting efficient particle acceleration and MFA upstream of the shock by a factor of ∼10. [ABSTRACT FROM AUTHOR]

Diffusive acceleration at collisionless shock waves remains one of the most promising acceleration mechanisms for the description of the origin of cosmic rays at all energies. A crucial ingredient to be taken into account is the reaction of accelerated particles on the shock, which in turn determines the efficiency of the process. Here we propose a semi-analytical kinetic method that allows us to calculate the shock modification induced by accelerated particles together with the efficiency for particle acceleration and the spectra of accelerated particles. The shock modification is calculated for arbitrary environment parameters (Mach number, maximum momentum, density) and for arbitrary diffusion properties of the medium. Several dependences of the diffusion coefficient on particle momentum and location are considered to assess the accuracy of the method. [ABSTRACT FROM AUTHOR]