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. Our partial knowledge of the chemical and physical structure of pre-stellar cores, as well as how they are fed and influenced by the surrounding environment, limits the level of knowledge that we can achieve at later stages in the star and planet formation process, from protostellar cores to exoplanets. Our aims are to estimate the central density of the pre-stellar core IRAS16293E and to carry out an inventory of molecular species towards the density peak of the core. We observed high-J rotational transitions of N_2H^+ and N_2D^+, 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_2H^+ and N_2D^+ lines were analysed by non-local thermodynamic equilibrium (non-LTE) radiative transfer modelling. Our best-fit core model consists of 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_2H^+ and N_2D^+ (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_2H^+ and N_2D^+ in the inner core. The N_2D^+/N_2H^+ ratio that best reproduces our observations is 0.44, one of the highest 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_2H^+ and N_2D^+ lines proved to be an excellent diagnostic of the chemical structure (i.e. molecular freeze-out) and dynamics (infall velocity profile) 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 timescale of this process.
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