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Context. The phase space transport of pick-up ions (PUIs) in the heliosphere has been studied for the cases that these particles are experiencing a second-order Fermi process, i.e. velocity diffusion, a convection with the solar wind, and adiabatic or “magnetic” cooling, i.e. cooling connected with the conservation of the magnetic moment. Aims. The study aims at a quantification of the process of “magnetic cooling” that has recently been introduced as a modification of adiabatic cooling in the presence of frozen-in magnetic fields. Methods. The isotropic PUI velocity distributions are obtained as numerical solutions of a Fokker-Planck phase space transport equation. Results. It is demonstrated that this newly discussed process is, like adiabatic energy changes, not limited to cooling but can also, depending on the shape of the distribution function, result in heating. For pure cooling with negligible velocity diffusion a v −5 velocity power law is found for magnetic cooling, thus confirming earlier analytical results. For non-negligible second-order Fermi acceleration, the tails of the distribution functions exhibit different shapes, which in special cases are also close to the prominent v −5 behaviour, which is often found in observations. Conclusions. The existence of an exact v −5 power law of PUI distribution functions can be confirmed for insignificant velocity diffusion and its approximate validity for specific choices of the velocity dependence of the diffusion coefficient.

Pick-up ion transport under conservation of particle invariants: how important are velocity diffusion and cooling processes?