The ionization fraction gradient across the Horsehead edge: an archetype for molecular clouds

Context. The ionization fraction (i. e., the electron abundance) plays a key role in the chemistry and dynamics of molecular clouds. Aims. We study the (HCO+)-C-13, DCO+ and HOC+ line emission towards the Horsehead, from the shielded core to the UV irradiated cloud edge, i. e., the photodissociation region (PDR), as a template to investigate the ionization fraction gradient in molecular clouds. Methods. We analyze an IRAM Plateau de Bure Interferometer map of the (HCO+)-C-13 J = 1-0 line at a 6.8 '' x 4.7 '' resolution, complemented with IRAM-30m (HCO+)-C-13 and DCO+ higher-J line maps and new HOC+ and CO+ observations. We compare self-consistently the observed spatial distribution and line intensities with detailed depth-dependent predictions of a PDR model coupled with a non-local radiative transfer calculation. The chemical network includes deuterated species, C-13 fractionation reactions and HCO+/ HOC+ isomerization reactions. The role of neutral and charged PAHs in the cloud chemistry and ionization balance is investigated. Results. The detection of the HOC+ reactive ion towards the Horsehead PDR proves the high ionization fraction of the outer UV irradiated regions, where we derive a low [HCO+]/[HOC+] similar or equal to 75-200 abundance ratio. In the absence of PAHs, we reproduce the observations with gas-phase metal abundances, [Fe+Mg+...], lower than 4 x 10(-9) (with respect to H), and a cosmic-ray ionization rate of zeta = (5 +/- 3) x 10(-17) s(-1). The inclusion of PAHs modifies the ionization fraction gradient and increases the required metal abundance. Conclusions. The ionization fraction in the Horsehead edge follows a steep gradient, with a scale length of similar to 0.05 pc (or similar to 25 ''), from [e(-)] similar or equal to 10(-4) (or n(e) similar to 1-5 cm(-3)) in the PDR to a few times similar to 10(-9) in the core. PAH(-) anions play a role in the charge balance of the cold and neutral gas if substantial amounts of free PAHs are present ([PAH] > 10(-8)).