Assessing the accuracy of star formation rate measurements by direct star count in molecular clouds

Star formation estimates based on the counting of young stellar objects (YSOs) are commonly carried out for nearby star-forming regions in the Galaxy, and in principle could be extended to any star-forming region where direct star counts are possible. With this method, the SFRs are measured using the counts of YSOs in a particular class, a typical mass, and the lifetime associated with this class. Another variant of this method is to use the total number of YSOs found in a star-forming region along with a characteristic YSO timescale. However, the assumptions underlying the validity of this method, such as that of a constant star formation history (SFH), have never been fully tested, and it remains unclear as to whether or not the method is valid for all protostellar classes. In this work, we use Monte Carlo models to test the validity and robustness of the method. We build synthetic clusters in which stars form at times that are randomly drawn from a specified SFH distribution function. The latter is either constant or time dependent, with a burst like behavior. The masses of the YSOs are randomly drawn from a stellar initial mass function (IMF), which can be either similar to that of the Milky Way field or be variable within the limits of the variations observed among young stellar clusters in the Galaxy. For each star in every cluster, the lifetimes associated with the different protostellar classes are also randomly drawn from Gaussian distribution functions centered around their most likely value as suggested by the observations. We find that only the SFR derived using the Class 0 population can reproduce the true SFR at all epochs, and this is true irrespective of the shape of the SFH. For a constant SFH, the SFR derived using the more evolved populations of YSOs (Class I, Class F, Class II, and Class III) reproduce the real SFR only at later epochs, which correspond to epochs at which their numbers have reached a steady state. For a time-dependent burst-like SFH, all SFR estimates based on the number counts of the evolved populations fail to reproduce the true SFR. We show that these conclusions are independent of the IMF. We argue that the SFR based on the Class 0 alone can yield reliable estimates of the SFR. We also show how the offsets between Class I- and Class II-based SFRs and the true SFR plotted as a function of the number ratios of Class I and Class II versus Class III YSOs can be used in order to provide information on the SFH of observed molecular clouds.