  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.
