Mass distributions of star clusters for different star formation histories in a galaxy cluster environment

Clusters of galaxies usually contain rich populations of globular clusters (GCs). We investigate how different star formation histories (SFHs) shape the final mass distribution of star clusters. We assume that every star cluster population forms during a formation epoch of length dt at a constant star-formation rate (SFR). The mass distribution of such a population is described by the embedded cluster mass function (ECMF), which is a pure power law extending to an upper limit M_max. Since the SFR determines M_max, the ECMF implicitly depends on the SFR. Starting with different SFHs, each SFH is divided into formation epochs of length dt at different SFRs. The requested mass function arises from the superposition of the star clusters of all formation epochs. An improved optimal sampling technique is introduced that allows generating number and mass distributions, both of which accurately agree with the ECMF. Moreover, for each SFH the distribution function of all involved SFRs, F(SFR), is computed. For monotonically decreasing SFHs, F(SFR) always follows a power law. With F(SFR), we develope the theory of the integrated galactic embedded cluster mass function (IGECMF). It describes the distribution function of birth stellar masses of star clusters that accumulated over a formation episode much longer than dt. The IGECMF indeed reproduces the mass distribution of star clusters created according to the superposition principle. Interestingly, all considered SFHs lead to a turn-down with increasing star cluster mass in their respective IGECMFs. In the past, a turn-down at the high-mass end has been observed for GC systems in different galaxy clusters and in the cluster initial mass function. This behavior can be explained naturally if the observed star cluster ensembles are superpositions of several individual star cluster populations that formed at different times at different SFRs.