## Injection models

In the semi-analytic model a free injection parameter, E,sa , determines the fraction of total particles injected into the acceleration mechanism and the injection momentum, pinj-. Specifically, pinj = %SApth , where pth =V2mpkTDS (4.24.1)

( Tds is the downstream temperature). The fraction, J]SA, of un-shocked particles crossing the shock which become super-thermal in the semi-analytic model is nsA = (V3n12 ( - exp(- ), (4.24.2)

where rsub is the sub-shock compression ratio. The fraction rjSA, which is approximately the number of particles in the Maxwellian defined by Tds with momentum p > pinj, is determined by requiring the continuity, at pinj, of the Maxwellian and the super-thermal distribution (Blasi et al., 2005). Since pth depends on the injected fraction, the solution must be obtained by iteration.

In the Monte Carlo model, the injection depends on the scattering assumptions. We assume that particles pitch-angle scatter elastically and isotropically in the local plasma frame and that the mean free path is proportional to the gyro-radius, i.e., A<x rg, where rg = pc/qB . With these assumptions, the injection is purely statistical with those 'thermal' particles which manage to diffuse back upstream gaining additional energy and becoming super-thermal. Note that in this scheme the viscous sub-shock is assumed to be transparent to all particles, even thermal ones, and that any downstream particle with v > U2 has a chance to be injected (here U2 is the downstream flow speed). For comparison with the semi-analytic model, we have included an additional parameter, v^^,,, to limit injection in the Monte Carlo simulation. Only downstream particles with v > vthh^es are injected, i.e., allowed to re-cross the shock into the upstream region and become super-thermal. In our previous Monte Carlo results, with the sole exception of paper Ellison (1985), we have taken v^^,*, = 0 .