Aims . We test the use of three common molecular ions, ortho-H 2 D + (oH 2 D + ), N 2 H + , and DCO + , as probes of the internal structure and kinematics of a dense, starless molecular cloud core. Methods . The pre-stellar core H-MM1 in Ophiuchus was mapped in the oH 2 D + (1 10 − N 2 H + (4 − 3), and DCO + (5 − 4) lines with the Large APEX sub-Millimeter Array (LAsMA) multi-beam receiver of the Atacama Pathfinder EXperiment (APEX) telescope. We also ran a series of chemistry models to predict the abundance distributions of the observed molecules, and to estimate the effect of the cosmic-ray ionisation rate on their abundances. Results . The three line maps show different distributions. The oH 2 D + map is extended and outlines the general structure of the core, N 2 H + mainly shows the density maxima, and the DCO + emission peaks are shifted towards one edge of the core where a region of enhanced desorption had previously been found. According to the chemical simulation, the fractional oH 2 D + abundance remains relatively high in the centre of the core, and its column density correlates strongly with the cosmic-ray ionisation rate, ζ H 2 . Simulated line maps constrain the cosmic-ray ionisation rate to be low, between 5 × 10 −18 s −1 and 1 × 10 −17 s −1 in the H-MM1 core. This estimate agrees with the gas temperature measured in the core. Conclusions . The present observations show that very dense, cold gas in molecular clouds can be traced by mapping the ground-state line of oH 2 D + and high- J transitions of DCO + and N 2 H + , despite the severe depletion of the latter two molecules. Modelling line emission of oH 2 D + provides a straightforward method of determining the cosmic-ray ionisation rate in dense clouds, where the primary ion, H 3 + , is not observable.
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