Cosmic-ray spectra at spherical termination shocks

We discuss the nature of the steady-state spectra of particles accelerated at stationary spherical shocks, such as the solar wind termination shock. In addition to the two well-known spectral regions characterized by a power-law momentum dependence and a high-energy cutoff, a new region can be identified. This consists of an enhancement of the cosmic-ray intensity (or a "bump") just below the cutoff. Similar features have been seen previously in multidimensional models and cosmic-ray modified shocks, where they were explained by acceleration and drift in the latitudinal direction along the shock face and decreasing effective shock compression ratio, respectively. We show that a similar bump may be obtained in a purely spherically symmetric geometry with no drifts, and that this effect may also have contributed to the previous results. We attribute this effect to increased shock acceleration efficiency at certain energies. We also demonstrate that a one-dimensional planar shock with a reflecting wall upstream can give a similar effect. We conclude that care is necessary in interpreting observed bumps in any given situation.