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The Protostellar Luminosity Function

The protostellar luminosity function (PLF) is the present-day luminosity function of the protostars in a region of star formation. It is determined using the protostellar mass function (PMF) in combination with a stellar evolutionary model that provides the luminosity as a function of instantaneous and final stellar mass. As in McKee & Offner (2010), we consider three main accretion models: the Isothermal Sphere model, the Turbulent Core model, and an approximation of the Competitive Accretion model. We also consider the effect of an accretion rate that tapers off linearly in time and an accelerating star formation rate. For each model, we characterize the luminosity distribution using the mean, median, maximum, ratio of the median to the mean, standard deviation of the logarithm of the luminosity, and the fraction of very low luminosity objects. We compare the models with bolometric luminosities observed in local star forming regions and find that models with an approximately constant accretion time, such as the Turbulent Core and Competitive Accretion models, appear to agree better with observation than those with a constant accretion rate, such as the Isothermal Sphere model. We show that observations of the mean protostellar luminosity in these nearby regions of low-mass star formation suggest a mean star formation time of 0.3$\pm$0.1 Myr. Such a timescale, together with some accretion that occurs non-radiatively and some that occurs in high-accretion, episodic bursts, resolves the classical "luminosity problem" in low-mass star formation, in which observed protostellar luminosities are significantly less than predicted. An accelerating star formation rate is one possible way of reconciling the observed star formation time and mean luminosity.