The distribution of stellar masses that form together, the initial mass function (IMF), is one of the most important astrophysical distribution functions. The determination of the IMF is a very difficult problem because stellar masses cannot be measured directly and because observations usually cannot assess all stars in a population requiring elaborate bias corrections. Nevertheless, impressive advances have been achieved during the last decade, such that the shape of the IMF is reasonably well understood from low-mass brown dwarfs (BDs) to very massive stars. The case can be made for a rather universal form that can be well approximated by a two-part power-law function in the stellar regime. However, there exists a possible hint for a systematic variation with metallicity. From very elaborate observational surveys a picture is emerging according to which the binary properties of very-low-mass stars (VLMSs) and BDs may be fundamentally different from those of late-type stars implying the probable existence of a discontinuity in the IMF, but the surveys also appear to suggest the number of BDs per star to be independent of the physical conditions of current Galactic star formation. Star-burst clusters and thus globular cluster may, however, have a much larger abundance of BDs. Very recent advances have allowed the measurement of the physical upper stellar mass limit, which also appears to be disconcertingly robust to variations in metallicity. Furthermore, it now appears that star clusters are formed in a rather organised fashion from lowto high stellar masses, such that the most-massive stars just forming terminate further star-formation within the particular cluster. Populations formed from many star clusters, composite populations, would then have steeper IMFs (fewer massive stars per low-mass star) than the simple populations in the constituent clusters. A near invariant star-cluster mass function implies the maximal cluster mass to correlate with the galaxy-wide star-formation rate. This then leads to the result that the composite-stellar IMFs vary in dependence of galaxy type, with potentially dramatic implications for theories of galaxy formation and evolution.The simple and composite IMF 5 30 Dor cluster (R136) in the LMC, NGC 3603 in the MW, and the Arches cluster near the Galactic centre. The 30 Dor star-burst cluster (

Binary populations in young star clusters show multiplicity fractions both lower and up to twice as high as those observed in the Galactic field. We follow the evolution of a population of binary stars in dense and loose star clusters starting with an invariant initial binary population and a formal multiplicity fraction of unity, and demonstrate that these models can explain the observed binary properties in Taurus, ρ Ophiuchus, Chamaeleon, Orion, IC 348, Upper Scorpius A, Praesepe, and the Pleiades. The model needs to consider solely different birth densities for these regions. The evolved theoretical orbital-parameter distributions are highly probable parent distributions for the observed ones. We constrain the birth conditions (stellar mass, M ecl , and half-mass radius, r h ) for the derived progenitors of the star clusters and the overall present-day binary fractions allowed by the present model. The results compare very well with properties of molecular cloud clumps on the verge of star formation. Combining these with previously and independently obtained constraints on the birth densities of globular clusters, we identify a weak stellar mass -half-mass radius correlation for cluster-forming cloud clumps, r h /pc ∝ (M ecl /M ) 0.13 ± 0.04 . The ability of the model to reproduce the binary properties in all the investigated young objects, covering present-day densities from 1−10 stars pc −3 (Taurus) to 2 × 10 4 stars pc −3 (Orion), suggests that environment-dependent dynamical evolution plays an important role in shaping the present-day properties of binary populations in star clusters, and that the initial binary properties may not vary dramatically between different environments.

Residual‐gas expulsion after cluster formation has recently been shown to leave an imprint in the low‐mass present‐day stellar mass function (PDMF) which allowed the estimation of birth conditions of some Galactic globular clusters (GCs) such as mass, radius and star formation efficiency. We show that in order to explain their characteristics (masses, radii, metallicity and PDMF) their stellar initial mass function (IMF) must have been top heavy. It is found that the IMF is required to become more top heavy the lower the cluster metallicity and the larger the pre‐GC cloud‐core density are. The deduced trends are in qualitative agreement with theoretical expectation. The results are consistent with estimates of the shape of the high‐mass end of the IMF in the Arches cluster, Westerlund 1, R136 and NGC 3603, as well as with the IMF independently constrained for ultra‐compact dwarf galaxies (UCDs). The latter suggests that GCs and UCDs might have formed along the same channel or that UCDs formed via mergers of GCs. A Fundamental Plane is found which describes the variation of the IMF with density and metallicity of the pre‐GC cloud cores. The implications for the evolution of galaxies and chemical enrichment over cosmological times are expected to be major.

In this paper, we constrain the properties of primordial binary populations in Galactic globular clusters. Using the MOCCA Monte Carlo code for cluster evolution, our simulations cover three decades in present-day total cluster mass. Our results are compared to the observations of Milone et al. (2012) using the photometric binary populations as proxies for the true underlying distributions, in order to test the hypothesis that the data are consistent with an universal initial binary fraction near unity and the binary orbital parameter distributions of . With the exception of a few possible outliers, we find that the data are to first-order consistent with the universality hypothesis. Specifically, the present-day binary fractions inside the half-mass radius can be reproduced assuming either high initial binary fractions near unity with a dominant soft binary component as in the Kroupa distribution combined with high initial densities (10 4 -10 6 M ⊙ pc −3 ), or low initial binary fractions (∼ 5-10%) with a dominant hard binary component combined with moderate initial densities near their present-day values (10 2 -10 3 M ⊙ pc −3 ). This apparent degeneracy can potentially be broken using the binary fractions outside the half-mass radiusonly high initial binary fractions with a significant soft component combined with high initial densities can contribute to reproducing the observed anti-correlation between the binary fractions outside the half-mass radius and the total cluster mass. We further illustrate using the simulated present-day binary orbital parameter distributions and the technique first introduced in Leigh et al. (2012) that the relative fractions of hard and soft binaries can be used to further constrain both the initial cluster density and the initial mass-density relation. Our results favour an initial mass-density relation of the form r h ∝ M α clus with α < 1/3, corresponding to an initial correlation between cluster mass and density.