Cosmic ray acceleration in the presence of losses

Energy losses of cosmic rays (CR) due to e.g. adiabatic cooling, ionising, Coulomb-, or nuclear collisions, bremsstrahlung, synchrotron radiation, should play a role in limiting the acceleration, since most shocks are coupled with a loss region (HII regions, SNR's, galactic density wave, stellar wind shocks). In addition, for this process to be a local one in the galaxy, the magnetic irregularities must either be excited by the accelerated CR's or produced by a downstream source. This implies a finite wave build-up time during the shock life time. Nevertheless, in the loss free case, the time-asymptotic amplification is independent of the mean free path lambda (or the diffusion coefficient kappa) which only appears in the spatial scale for the CR intensity. To investigate the effects of losses, the CR diffusion equation is amended by a simple loss term f/tau, with an (energy dependent) loss time tau, f being the CR momentum distribution. (Auth.)