Though pick-up ions (PUIs) are a well known phenomenon in the inner heliosphere their phase-space distribution nevertheless is a theoretically unsettled problem. Especially the question of how pick-up ions form their suprathermal tails, extending to
far above their injection energies, still now is unsatistactorily answered. Though Fermi-2 velocity diffusion theories have revealed that such tails are populated, they nevertheless show that resulting population densities are much less than seen in observations showing power-laws with a velocity index of -5. We first investigate here, whether or not observationally suggested power-laws can be the result of a quasi-equilibrium state between suprathermal ions and magnetohydrodynamic turbulences in energy exchange with eachother. We demonstrate that such an equilibrium cannot be established, since it would require too high pick-up ion pressures enforcing a shock-free deceleration of the solar wind. We furthermore show that Fermi-2 type energy diffusion in the outer heliosphere is too inefficient to determine the shape of the distribution function there. As we can show, however, power-laws beyond the injection threshold can be established, if the injection takes place at higher energies of the order of 100 keV. As we demonstrate here, such an injection is connected with modulated anomalous cosmic ray (ACR) particles at the lower end of their spectrum when they again start being convected outwards with the solar wind. Therefore, we refer to these particles as ACR-PUIs. In our quantitative calculation of the pick-up ion spectrum resulting under such conditions we in fact find again power-laws, however with a velocity power index of -4 and fairly distance-independent spectral intensities.
