Cosmic ray acceleration to very high energy through the non-linear amplification by cosmic rays of the seed magnetic field

The maximum energy for cosmic ray acceleration at supernova shock fronts is usually thought to be limited to around 10(14)-10(15) eV by the size of the shock and the time for which it propagates at high velocity. We show that the magnetic field can be amplified non-linearly by the cosmic rays to many times the pre-shock value, thus increasing the acceleration rate and facilitating acceleration to energies well above 10(15) eV. A supernova remnant expanding into a uniform circumstellar medium may accelerate protons to 10(17) eV and heavy ions, with charge Ze, to Z x 10(17) eV. Expansion into a pre-existing stellar wind may increase the maximum cosmic ray energy by a further factor of 10.