mass accretion rate will naturally drop to negligible values after some time. In this case, the final stellar mass may be essentially determined at the pre-collapse stage. Direct observational constraints on the radial density structure at the onset of protostellar collapse have been obtained through (sub)millimeter dust continuum mapping of a large number of starless cores and Class 0/Class I protostellar envelopes with the JCMT and IRAM 30 m telescopes (e.g. Ward-Thompson et al. 1994; André, Ward-Thompson, & Motte 1996 – hereafter AWM96; Ward-Thompson, Motte, & André 1999 – hereafter WMA99). General trends have been found: while protostellar envelopes are always highly centrally condensed, with radial density profiles consistent with $\rho \propto r^{-1.5}$ or $\rho \propto r^{-2}$ (e.g. Ladd et al. 1991, Motte & André 1999), starless cores have radial profiles that flatten out near their centers and become much flatter than $\rho \propto r^{-2}$ at radii less than a few thousand AUs (see André, Ward-Thompson, & Barsony 2000 for a review). It is however necessary to bear in mind the limitations of (sub)millimeter continuum emission maps: since the dust [*emission*]{} from the outer parts of cores is intrinsically very weak, (sub)millimeter mapping is mostly sensitive to the inner density 