The protostellar mass function (PMF) is the present-day mass function of the protostars in a region of star formation. It is determined by the initial mass function weighted by the accretion time. The PMF thus depends on the accretion history of protostars and in principle provides a powerful tool for observationally distinguishing different protostellar accretion models. We consider three basic models here: the isothermal sphere model, the turbulent core model, and an approximate representation of the competitive accretion model. We also consider modified versions of these accretion models, in which the accretion rate tapers off linearly in time. Finally, we allow for an overall acceleration in the rate of star formation. At present, it is not possible to directly determine the PMF since protostellar masses are not currently measurable. We carry out an approximate comparison of predicted PMFs with observation by using the theory to infer the conditions in the ambient medium in several star-forming regions. Tapered and accelerating models generally agree better with observed star formation times than models without tapering or acceleration, but uncertainties in the accretion models and in the observations do not allow one to rule out any of the proposed models at present. The PMF is essential for the calculation of the protostellar luminosity function, however, and this enables stronger conclusions to be drawn.
stars: formation - stars: luminosity function, mass function