  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 (Shu 1977), the
Turbulent Core model (McKee & Tan 2003), and an approximate representation of
the Competitive Accretion model (Bonnell et al. 1997, 2001a). 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 (Offner & McKee 2010).
