Particle acceleration at Astrophysical shocks: a detailed study of the energy spectrum in test-particle relativistic theory and non-linear Newtonian theory

Diffusive Shock Acceleration (DSA), applied in different astrophysical environment, has provided by far the most popular model for the origin of Cosmic Rays (CRs), both for what concern the Galactic CRs (from GeV to PeV energies), through the application of DSA at the outer front of expanding supernova remnants, as well as for extragalactic CRs (beyond PeV energies), applying the DSA to extragalactic sources like Gamma Ray Bursts, Active Galactic Nuclei and Radio Galaxies. Beyond the considerable successes that this theory has achieved in the past years, lots of obscure points still remains to be enlightened. In particular the explanation of ultra high energy CRs seems to require the extension of DSA for shock at relativistic speed. Moreover recent studies of nonthermal emission at young SNRs shock has revealed the importance of developing a fully non liner theory capable to include the dynamical effects of accelerated particles, but a full understanding of all the phenomenology related to the nonlinearity is far to be reached. In this work we study several aspects relating to DSA both for newtonian and for relativistic shocks. A mathematical approach to investigate particle acceleration at shock waves moving at arbitrary speed in a medium with arbitrary scattering properties was first discussed in \cite{Vietri:03} and \cite{Blasi-Vietri:05}. In the fist part of this thesis we use this method and somewhat extend it in order to include the effect of a large scale magnetic field in the upstream plasma, with arbitrary orientation with respect to the direction of motion of the shock. We also use this approach to investigate the effects of anisotropic scattering on spectra and anisotropies of the distribution function of the accelerated particles. A furter step in the analysis of the DSA process is put forward introducing a general equation of state to describe the shocked downstream plasma. More specifically we consider the effect of energy exchange between the electron and proton thermal components downstream, and the effect of generation of a turbulent magnetic field in the downstream plasma. The slope of the spectrum turns out to be appreciably affected by all these phenomena, especially in the Newtonian and trans-relativistic regime, while in the ultra-relativistic limit the universal spectrum $s\approx 4.3$ seems to be a very solid prediction. In the second part of the thesis we present the general solution for the non linear (time independent) theory of particle accelerated at Newtonian shocks in the presence of a pre-existing non-thermal particle population and for arbitrary diffusion coefficient. Using this solution we show that, in general, the contribution of a pre-existing energetic particle's flux, like the galactic CRs, cannot be neglected in determine the shock dynamics. The first consequence of this statement is that shocks like SNRs' ones, that propagates into the Galactic environment can evolve in a nonlinear way even if the injection of fresh particles were an inefficient process.