First- and Second-order Fermi Acceleration at Parallel Shocks

We report on a new Monte Carlo method for simulating diffusive shock acceleration (DSA) of solar energetic particles at upstream and downstream regions of quasi-parallel collisionless shock waves under the influence of self-generated turbulence. By way of example, we apply the model to a fast 1500 km \mathrm{s}^{-1} coronal mass ejection at ten solar radii. Results indicate that the maximum energies at outer corona are likely to be limited to few MeV, due to lack of suprathermal protons for appreciable wave growth, and insufficient time required acceleration. We find that the second-order Fermi acceleration, although being a too slow process to have a notable effect at the highest energies, significantly flattens energy spectra at low energy end. Simulations indicate that protons continue to damp waves efficiently several solar radii from the shock in the downstream region, which may be an important mechanism for heating suprathermals. Our simulations also suggest that models assuming a simple isotropic scattering are likely to predict too efficient acceleration.