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Pre-stellar cores are the first steps in the process of star and planet formation. However, the dynamical and chemical evolution of pre-stellar cores is still not well understood. We aim at estimating the central density of the pre-stellar core IRAS16293E and at carrying out an inventory of molecular species towards the density peak of the core. We observed high-\(J\) rotational transitions of N\(_2\)H\(^+\) and N\(_2\)D\(^+\), and several other molecular lines towards the dust emission peak using the Atacama Pathfinder EXperiment (APEX) telescope, and derived the density and temperature profiles of the core using far-infrared surface brightness maps from \(Herschel\). The N\(_2\)H\(^+\) and N\(_2\)D\(^+\) lines were analysed by non-LTE radiative transfer modelling. Our best-fit core model consists in a static inner region, embedded in an infalling envelope with an inner radius of approximately 3000 au (21" at 141 pc). The observed high-J lines of N\(_2\)H\(^+\) and N\(_2\)D\(^+\) (with critical densities greater than 10\(^6\) cm\(^{-3}\)) turn out to be very sensitive to depletion; the present single-dish observations are best explained with no depletion of N\(_2\)H\(^+\) and N\(_2\)D\(^+\) in the inner core. The N\(_2\)D\(^+\)/N\(_2\)H\(^+\) ratio that best reproduces our observations is 0.44, one of the largest observed to date in pre-stellar cores. Additionally, half of the molecules that we observed are deuterated isotopologues, confirming the high-level of deuteration towards this source. Non-LTE radiative transfer modelling of N\(_2\)H\(^+\) and N\(_2\)D\(^+\) lines proved to be an excellent diagnostic of the chemical structure and dynamics of a pre-stellar core. Probing the physical conditions immediately before the protostellar collapse is a necessary reference for theoretical studies and simulations with the aim of understanding the earliest stages of star and planet formation and the time scale of this process.