### 8. CLUSTER MASS FUNCTIONS

The cluster mass function is a power law and it is natural to look for explanations of this that are related to the other power laws in star formation, including the hierarchical structure. If we imagine a cloud divided hierarchically into clumps and sub-clumps, then the mass distribution function of the nodes in this hierarchy is dN / d log M ~ 1 / M, or dn / dM ~ 1 / M2, because there is an equal total mass in all levels (MN[M]d logM = constant). This is the same as the mass function for star clusters. Also in such a hierarchy, the probability that a mass between M and 2M is selected is proportional to 1 / M, as given by the number of levels and clouds at those levels having a mass in that range.

Cluster mass functions typically are a power law with a slope equal to this value, dn / dM ~ M- for ~ 2. This slope was found by Battinelli et al. (1994) for the solar neighborhood and Elmegreen & Efremov (1997) for the LMC, where the clusters were subdivided according to age. A second study of LMC clusters (Hunter et al. 2003) found the same slope for discrete cluster age intervals. It is important to consider clusters within a narrow age interval because older clusters are dimmer and the selection effects for clusters depend on their age. Zhang & Fall (1999) found = 1.95 ± 0.03 for young clusters in the Antenna galaxy, and = 2.00 ± 0.08 for old clusters. de Grijs & Anders (2006) looked at the LMC again and found = 1.85 ± 0.05 for various age intervals. de Grijs et al. (2003) found similar results in two other galaxies: = 2.04 ± 0.23 for NGC 3310 and = 1.96 ± 0.15 for NGC 6745.

The HII region luminosity function is about the same as the cluster mass function, having a slope of around -2 for linear intervals of luminosity. The first large study was by Kennicutt et al. (1989). Many other surveys have obtained about the same result (e.g., Banfi et al. 1993). Bradley et al. (2006) included 53 spiral galaxies and got a steeper slope at log L > 38.6 (for L in erg s-1), suggesting that larger HII regions were density bounded, and they also got a steep fall-off at log L > 40, suggesting an upper limit for cluster mass. The same general power law for HII regions has been obtained in detailed studies of individual galaxies (e.g., NGC 3389: Abdel-Hamid et al. (2003); M81: Lin et al. 2003; NGC 1569: Buckalew & Kobulnicky 2006; NGC 6384: Hakobyan et al. 2007; the Milky Way: Paladini et al. 2009).

There is growing evidence for an upper mass cutoff in the cluster mass function. In Gieles et al. (2006a, 2006b), mass functions in M51 were fit to a double power law, i.e., with an increased slope at higher mass, or to a power law with = 2 throughout and an upper mass cutoff of around 105 M. A power law with an exponential cutoff is a Schechter function, dN / dM = M-exp(-M / Mc) for cutoff mass Mc.

For several local galaxies, Larsen (2009) fit the brightest cluster and the 5th brightest cluster with a mass function having a cutoff. Larsen found that rich and poor spirals have about the same cluster mass functions, both with a cutoff, and that the cutoffs are independent of position in a galaxy. The origin of an upper mass limit for clustering is not known.