The scale-free character of the cluster mass function and the universality of the stellar IMF

Abstract

Our recent determination of a Salpeter slope for the IMF in the field of 30 Doradus (Selman and Melnick 2005) appears to be in conflict with simple probabilistic counting arguments advanced in the past to support observational claims of a steeper IMF in the LMC field. In this paper we re-examine these arguments and show by explicit construction that, contrary to these claims, the field IMF is expected to be exactly the same as the stellar IMF of the clusters out of which the field was presumably formed. We show that the current data on the mass distribution of clusters themselves is in excellent agreement with our model, and is consistent with a single spectrum {\it by number of stars} of the type $n^\beta$ with beta between -1.8 and -2.2 down to the smallest clusters without any preferred mass scale for cluster formation. We also use the random sampling model to estimate the statistics of the maximal mass star in clusters, and confirm the discrepancy with observations found by Weidner and Kroupa (2006). We argue that rather than signaling the violation of the random sampling model these observations reflect the gravitationally unstable nature of systems with one very large mass star. We stress the importance of the random sampling model as a \emph{null hypothesis} whose violation would signal the presence of interesting physics.