Exploring the structure and dynamics of cold starless clouds is necessary to understand the different steps leading to the formation of protostars. Because clouds evolve slowly, many of them must be studied in detail to pick up different moments of a clouds lifetime. We study here L1506C in the Taurus region, a core with interesting dust properties which have been evidenced with the PRONAOS balloon-borne telescope. To trace the mass content of L1506C and its kinematics, we mapped the dust emission, and the line emission of two key species, C18O and N2H+ (plus 13CO and C17O). This cloud shows peculiar features: i) a large envelope traced solely by 13CO holding a much smaller core with a strong C18O depletion in its center despite a low maximum opacity (Av~20 mag), ii) extremely narrow C18O lines indicating a low, non-measurable turbulence, iii) contraction traced by C18O itself (plus rotation), iv) unexpectedly, the kinematical signature from the external envelope is opposite to the core one: the 13CO and C18O velocity gradients have opposite directions and opposite profiles (C18O blue peaked, 13CO red peaked). The core is large (r = 3E4 AU) and not very dense (n(H2) ? 5E4 cm-3 or less). This core is therefore not prestellar yet. All these facts suggest that the core is kinematically detached from its envelope and in the process of forming a prestellar core. This is the first time that the dynamical formation of a prestellar core is witnessed. The extremely low turbulence could be the reason for the strong depletion of this core despite its relatively low density and opacity in contrast with undepleted cores such as L1521E which shows a turbulence at least 4 times as high.
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