The effect of a finite mass reservoir on the collapse of spherical isothermal clouds and the evolution of protostellar accretion

Motivated by recent observations that detect an outer boundary for starless cores, and evidence for time-dependent mass accretion in the Class 0 and Class I protostellar phases, we re-examine the case of spherical isothermal collapse in the case of a finite mass reservoir. The presence of a core boundary, implemented through a constant-volume approximation in our simulation, results in the generation of an inward-propagating rarefaction wave. This steepens the gas density profile from r(-2) (self-similar value) to r(-3) or steeper. After a protostar forms, the mass accretion rate M evolves through three distinct phases: (1) an early phase of decline in M, which is a non-self-similar effect due to rapid and spatially non-uniform infall in the pre-stellar phase; (2) for large cores, an intermediate phase of near-constant. M from the infall of the outer part of the self-similar density profile, which has low (subsonic) infall speed in the pre-stellar phase; and (3) a late phase of rapid decline in. M when accretion occurs from the region affected by the inward-propagating rarefaction wave. Our model clouds of small to intermediate size make a direct transition from phase (1) to phase (3) above. Both the first and second phase (if the latter is indeed present) are characterized by a temporally increasing bolometric luminosity L-bol, while L-bol is decreasing in the third (final) phase. We identify the period of temporally increasing L-bol with the Class 0 phase, and the later period of terminal accretion and decreasing L-bol with the Class I phase. The peak in L-bol corresponds to the evolutionary time when 50 +/- 15 per cent of the cloud mass has been accreted by the protostar. This is in agreement with the classification scheme proposed in the early 1990s by Andre et al.; our model adds a physical context to their interpretation. We show how our results can be used to explain tracks of envelope mass M-env versus L-bol for protostars in Taurus and Ophiuchus. We also develop an analytic formalism that successfully reproduces the protostellar accretion rate from profiles of density and infall speed in the pre-stellar phase. It shows that the spatial gradient of infall speed that develops in the pre-stellar phase is a primary cause of the temporal decline in M during the early phase of protostellar accretion.