Soft Cosmic Ray Spectra from Supernova Remnants*

It is becoming increasingly evident that the spectra of cosmic rays accelerated in supernova remnants (SNRs) are noticeably softer than the commonly used diffusive shock acceleration (DSA) theory predicts. We propose a simple physical explanation for this discrepancy by considering a spherical shock expanding into an interstellar medium with a quasi-homogeneous magnetic field. Particle injection into the DSA is efficient only on two opposite sides of a spherical shock where its normal does not strongly deviate from either parallel or antiparallel field direction. These regions (cusps) grow in time, as the shock expands, thus continually adding particles with a shorter acceleration history and, therefore, lower energies. A proper integration over their acceleration history, indeed, gives considerably softer spectra than what the standard DSA predicts for strong shocks. We also argue that similar spectral softening mechanism is pertinent to shocks and magnetic fields with more complicated geometry where the field-shock normal angle, \vartheta_{nB}, varies on the shock surface.