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Using one-dimensional hybrid expanding box model we investigate properties ofthe solar wind in the outer heliosphere. We assume a proton-electron plasmawith a strictly transverse ambient magnetic field and, beside the expansion, wetake into account influence of a continuous injection of cold pick-up protonsthrough the charge-exchange process between the solar wind protons and hydrogenof interstellar origin. The injected cold pick-up protons form a ringdistribution function that rapidly becomes unstable and generate Alfv\'encyclotron waves. The Alfv\'en cyclotron waves scatter pick-up protons to aspherical shell distribution function that thickens over that time owing to theexpansion-driven cooling. The Alf\'ven cyclotron waves heat solar wind protonsin the perpendicular direction (with respect to the ambient magnetic field)through the cyclotron resonance. At later times, the Alfv\'en cyclotron wavesbecome parametrically unstable and the generated ion acoustic waves heatprotons in the parallel direction through the Landau resonance. The resultingheating of the solar wind protons is efficient on the expansion time scale.

Proton heating by pick-up ion driven cyclotron waves in the outer heliosphere: Hybrid expanding box simulations