The oxygen-bearing molecular ions OH+, H2O+, and H3O+ are key species that probe the ionization rate of (partially) molecular gas that is ionized by X-rays and cosmic rays permeating the interstellar medium. We report Herschel far-infrared and submillimeter spectroscopic observations of OH+ in Mrk 231, showing both ground-state P-Cygni profiles, and excited line profiles with blueshifted absorption wings extending up to ~1000 km s^{-1}. In addition, OH+ probes an excited component peaking at central velocities, likely arising from the torus probed by the OH centimeter-wave megamaser. Four lines of H2O+ are also detected at systemic velocities, but H3O+ is undetected. Based on our earlier OH studies, we estimate an abundance ratio of OH/OH+~5-10 for the outflowing components and ~20 for the torus, and an OH+ abundance relative to H nuclei of ~>10^{-7}. We also find high OH+/H2O+ and OH+/H3O+ ratios, both are ~>4 in the torus and ~>10-20 in the outflowing gas components. Chemical models indicate that these high OH+ abundances relative to OH, H2O+, and H3O+ are characteristic of gas with a high ionization rate per unit density, zeta/n_H~(1-5)x10^{-17} cm^3 s^{-1} and ~(1-2)x10^{-16} cm^3 s^{-1} for the above components, respectively, and an ionization rate of zeta~(0.5-2)x10^{-12} s^{-1}. X-rays appear to be unable to explain the inferred ionization rate, and thus we suggest that low-energy (10-400 MeV) cosmic-rays are primarily responsible for the ionization with dot{M}_{CR}~0.01 M_{sun} yr^{-1} and dot{E}_{CR}~10^{44} erg s^{-1}, the latter corresponding to 1% of the AGN luminosity and similar to the energetics of the molecular outflow. We suggest that cosmic-rays accelerated in the forward shock associated with the molecular outflow are responsible for the ionization, as they diffuse through the outflowing molecular phase downstream.
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