Abstract
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 of the shock. In the simple example of the planar shock wave considered we idealize the escape phenomenon by imposing a particle escape boundary at some distance from the shock. The 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 of the shock.
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Ostrowski, M., Schlickeiser, R. Cosmic-ray diffusive acceleration at shock waves with finite upstream and downstream escape boundaries. Sol Phys 167, 381–394 (1996). https://doi.org/10.1007/BF00146347
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DOI: https://doi.org/10.1007/BF00146347