Interplanetary and Interacting Protons Accelerated in a Parallel Shock Wave

Abstract

We present a test-particle model of diffusive shock acceleration on open coronal field lines based on one-dimensional diffusion-convection equation with finite upstream and downstream diffusion regions. We calculate the energy spectrum of protons escaping into the interplanetary space and that of protons interacting with the subcoronal material producing observable secondary emissions. Our model can account for the observed power-law and broken power-law energy spectra as well as the values of the order of unity for the ratio of the interplanetary to interacting protons. We compare our model to Monte Carlo simulations of parallel shock acceleration including the effects of the diverging magnetic field. A good agreement between the models is found if (i) the upstream diffusion length is much smaller than the scale length LB of the large-scale magnetic field, κ1/U1 LB, where U1 is the upstream scattering center speed and κ1(p) is the momentum dependent upstream diffusion coefficent; (ii) the downstream diffusion length is much smaller than the length of the downstream diffusive region L2, for which L2 LB has to be satisfied; and (iii) most of the particles are injected to the acceleration process within a couple of LB's above the solar surface. We emphasize that concurrently produced interplanetary and interacting protons can be used as probes of turbulence in the vicinity of the shock; our model has two turbulence parameters, the scattering-center compression ratio at the shock and the number of diffusion lengths in the upstream region, that may be experimentally determined if the interplanetary and interacting proton spectra are measured.