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 ~ { } { 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 .