I discuss the shape of the high energy end of the spectrum of particles arising from diffusive shock acceleration in the presence of (i) additional diffusive escape from the accelerator, (ii) continuous energy losses, (iii) energy changes arising from interactions. The form of the spectrum near cut-off is sensitive to these processes as well as to the momentum-dependence of the diffusion coefficients and the compression ratio, and so the spectrum of any radiation emitted by the accelerated particles may reflect the physical conditions of the acceleration region. Results presented in this paper have applications in interpreting the spectral energy distributions of many types of astrophysical object including supernova remnants (SNR), active galactic nuclei (AGN) and acceleration sources of ultra high energy cosmic rays (UHE CR). Except for extremely nearby sources, spectral features imprinted on the spectrum of UHE CR during the acceleration process will be largely eroded during propagation, but the spectrum of UHE neutrinos produced in interactions of UHE CR with radiation, both during cosmic ray acceleration and subsequent propagation through the cosmic microwave background radiation, contains sufficient information to determine the cut-off momentum of the UHE CR just after acceleration for reasonable assumptions. Observation of these UHE neutrinos by the Pierre Auger Observatory may help in identifying the sources of the highest energy cosmic rays.
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