Diffusive compression acceleration of energetic particles with an application to shock acceleration near injection energies

The behavior of energetic charged particles accelerated by a continuous plasma compression profile is explored in the framework of diffusion. At high enough energies, the accelerated particles have a power-law spectrum with a slope generally steeper than a shock spectrum with the same compression ratio. The spectral slope depends on the ratio of the diffusion coefficient to the product of the upstream plasma speed and the thickness of the compression region. In the limit of large diffusion, the spectrum becomes identical to that of diffusive shock acceleration, and when diffusion is small enough, it is consistent with adiabatic acceleration by a single passage through the compression region. For particles with a diffusion coefficient that increases with energy, the spectral shape of the accelerated particles changes from an adiabatic compressional acceleration spectrum at low energies to a shock spectrum at high energies. The flux level of the high energy asymptotic shock spectrum is generally lower than that from a calculation with diffusive shock acceleration. When this result is applied to particles accelerated to high energies by shocks, the particle injection energy and efficiency can be determined once the energy dependence of the diffusion coefficient and the spectrum of the source particles are known. A threshold criterion for particle injection to shock acceleration can be set from the injection efficiency calculation.