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Understanding the first phases of star formation is highly dependent on understanding gravitational collapse, especially at small scales, where the protostar is forming. Thanks in particular to their spatial resolution, interferometers like the upcoming ALMA will undoubtedly bring much progress in this field. Although we are hindered by molecular depletion in order to study the densest regions of pre-stellar cores, we can fortunately rely on millimeter tracers which do not (or little) deplete, like
N2H+ and N2D+. These ions, which have high critical densities, are very promising for routine investigation of core and protostar infall motions with ALMA. We have started a program to study in details the kinematics of nearby pre-stellar cores, using several transitions of N2H+ and a radiative transfer model. Preliminary results show that one core of our sample, though not "chemically evolved" is undergoing fast infall: we find that the central parts of the core are collapsing at velocities slightly larger than sound speed. We discuss this result and its implications in the view of theoretical collapse models.
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