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Injection of $κ$-like Suprathermal Particles into Diffusive Shock Acceleration

We consider a phenomenological model for the thermal leakage injection in the diffusive shock acceleration (DSA) process, in which suprathermal protons and electrons near the shock transition zone are assumed to have the so-called $κ$-distributions produced by interactions of background thermal particles with pre-existing and/or self-excited plasma/MHD waves or turbulence. The $κ$-distribution has a power-law tail, instead of an exponential cutoff, well above the thermal peak momentum. So there are a larger number of potential seed particles with momentum, above that required for participation in the DSA process. As a result, the injection fraction for the $κ$-distribution depends on the shock Mach number much less severely compared to that for the Maxwellian distribution. Thus, the existence of $κ$-like suprathermal tails at shocks would ease the problem of extremely low injection fractions, especially for electrons and especially at weak shocks such as those found in the intracluster medium. We suggest that the injection fraction for protons ranges $10^{-4}-10^{-3}$ for a $κ$-distribution with $10 < κ_p < 30$ at quasi-parallel shocks, while the injection fraction for electrons becomes $10^{-6}-10^{-5}$ for a $κ$-distribution with $κ_e < 2$ at quasi-perpendicular shocks. For such $κ$ values the ratio of cosmic ray electrons to protons naturally becomes $K_{e/p}\sim 10^{-3}-10^{-2}$, which is required to explain the observed ratio for Galactic cosmic rays.