• particle acceleration
• nonlinear theory
• astrofisica
• AGN
• accelerazione di particelle
• shock
• SNR
• supernova remnant

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.