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Cosmic ray diffusive acceleration at shock waves with finite upstream and downstream escape boundaries
In the present paper we discuss the modifications introduced into the
first-order Fermi shock acceleration process due to a finite extent of
diffusive regions near the shock or due to boundary conditions leading to an
increased particle escape upstream and/or downstream the shock. In the
considered simple example of the planar shock wave we idealize the escape
phenomenon by imposing a particle escape boundary at some distance from the
shock. Presence of such a boundary (or boundaries) leads to coupled steepening
of the accelerated particle spectrum and decreasing of the acceleration time
scale. It allows for a semi-quantitative evaluation and, in some specific
cases, also for modelling of the observed steep particle spectra as a result of
the first-order Fermi shock acceleration. We also note that the particles close
to the upper energy cut-off are younger than the estimate based on the
respective acceleration time scale. In Appendix A we present a new
time-dependent solution for infinite diffusive regions near the shock allowing
for different constant diffusion coefficients upstream and downstream the
shock.Comment: LaTeX, 14 pages, 4 postscript figures; Solar Physics (accepted
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