The “injection problem” for quasiparallel shocks

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For a particle to be accelerated diffusively at a shock by the first-order Fermi acceleration mechanism, the particle must be sufficiently energetic that it can scatter across all the micro- and macrostructure of the shock, experiencing compression between the converging upstream and downstream states. This is the well-known “injection problem.” Here the interaction of ions with the ramp of a quasiparallel shock is investigated. Some ions incident on the shock experience specular reflection, caused either by the cross-shock electrostatic potential or by mirroring as the magnetic field is bent and compressed through the ramp. Scattering of reflected ions by self-generated and pre-existing turbulence in the region upstream of the shock then acts to trap backstreaming ions and return them to the ramp, where some experience further reflections. Such repeated reflections and scattering energize a subpopulation of ions up to energies sufficiently large that they can be diffusively shock accelerated. Two ion distributions are considered: pickup ions which are assumed to be described by a shell distribution, are thermal solar wind ions which may be described by a kappa distribution. Injection efficiencies are found analytically to be very high for pickup ions and much lower for thermal solar wind ions, suggesting that this injection mechanism, stochastic reflected ion or SRI acceleration, is a natural precursor for the acceleration of the anomalous cosmic ray component at a quasiparallel shock. While significantly less efficient, SRI acceleration is also viable for thermal solar wind ions described by a kappa distribution.