Reflection of pre-accelerated pick-up ions at the solar wind termination shock: The seed for anomalous cosmic rays

It has been hypothesized for quite some time that interplanetary pick-up ions due to energization taking place in the region close to the solar wind termination shock, at some fraction and as an outcome of a complicated chain of processes, eventually are converted into species of the anomalous cosmic-ray particles. For the actual conversion efficiency it is of great importance to know the energy distribution of these pick-up ions upon their arrival at the shock. It turns out that pre-acceleration of these ions during their passage through the heliosphere shall substantially increase their chances to become reflected at the shock into the upstream direction which is a prerequisite for a further climb-up in energy by virtue of Fermi-l acceleration processes. In this paper we start out from stochastically pre-accelerated pick-up ions and investigate their behaviour at the shock. With the use of adiabatic approaches in the de Hoffman-Teller frame of the shock, we calculate the energy distribution function of the reflected part of pick-up ions. From the calculated distribution functions it turns out that the reflected ions in the average suffer an energy increase by about a factor of 10, still not enough to let them move off the shock by spatial diffusion in the upstream direction. Thus, since convected back into the shock, they can undergo repeated reflections and energy gains till the diffusion-convection limit is reached. As we show in addition, the reflection probability for pick-up ions is about a factor of 10 higher than expected from the present literature and strongly varies with the off-axis angle, pointing to the fact that the termination shock represents a surface with a three-dimensionally varying source strength for the production of anomalous cosmic rays. The ACR source pattern is also expected to vary during the solar cycle and the relevant injection energies are expected to be larger by factors of 10 to 100 than the canonically adopted 1 keV nucl(-1).