PARTICLE ACCELERATION AT QUASI-PARALLEL SHOCK WAVES: THEORY AND OBSERVATIONS AT 1 AU
In this paper, we describe a theoretical model for accelerating an arbitrary upstream particle distribution. Only those particles that exceed a prescribed injection energy, E inj, are accelerated via the diffusive shock acceleration (DSA) mechanism, also known as first-order Fermi acceleration. We identify a set of quasi-parallel shocks at 1 AU and use the observed solar wind particle distribution information to construct our upstream distribution, which is then accelerated diffusively at the shock, assuming the observed shock parameters. The injection energy for particles to be accelerated diffusively at a quasi-parallel shock is discussed theoretically. By using the observed upstream solar wind distribution function and the observed shock parameters, we can compute the injection energy that matches the observed downstream accelerated particle spectrum. Like the previous studies of van Nes et al., Lario et al., and Ho et al., this analysis focuses on the acceleration of protons only via the first-order Fermi acceleration mechanism. However, our primary focus is on quasi-parallel shocks and the injection mechanism in the context of DSA with a background thermal solar wind modeled as a Maxwellian or kappa distribution. Our approach allows for a direct test of injection at interplanetary shocks. It has been proposed that an additional seed population of energetic particles is needed to explain the accelerated particle distribution downstream of quasi-parallel shocks. This conclusion is based typically on studies that address the acceleration of heavy ions primarily and do not characterize the injection of protons alone using the DSA mechanism. Through comparisons of Maxwellian and kappa upstream distributions, we find that DSA with injection directly from a thermal Maxwellian distribution, or weak departures therefrom, for protons is responsible for energetic solar particle events associated with quasi-parallel shocks.