Cosmic ray protons in the energy range $10^{16}-10^{18.5}$ eV: stochastic gyroresonant acceleration in hypernova shocks?

The hypernovae (HNe) associated with Gamma-ray Bursts (GRBs) may have a fairly steep energy-velocity distribution, i.e., $E(\geq \beta)\propto \beta^{-q}$ for $q<2$ and $\beta\geq \beta_o$, where $\beta$ is the velocity of the material and $\beta_o \sim 0.1$ is the velocity of the slowest ejecta of the HN explosion, both in units of the speed of light $(c)$. The cosmic ray protons above the second knee but below the ankle may be accelerated by the HN shocks in the velocity range of $\beta \sim \beta_o - 4\beta_o$. When $\beta \leq 4\beta_o$, the radius of the shock front to the central engine is very large and the medium decelerating the HN outflow is very likely to be homogeneous. With this argument, we show that for $q\sim 1.7$, as inferred from the optical modelling of SN 2003lw, the stochastic gyroresonant acceleration model can account for the spectrum change of high energy protons around the second knee. The self-magnetized shock acceleration model, however, yields a too much steep spectrum that is inconsistent with the observation unless, the medium surrounding the HN is a free wind holding up to a (unrealistic large) radius $\sim 1-10 {\rm kpc}$ or alternatively the particle acceleration mainly occurs in a narrow "dense" shell that terminates the free wind at a radius $\sim 10^{19}$ cm.