Star Clusters: From the Milky Way to the Early Universe Proceedings IAU Symposium No. 351, 2019 A. Bragaglia, M. Davies, A. Sills & E. Vesperini, eds. c  International Astronomical Union 2020 doi:10.1017/S1743921319007749 The systematically varying stellar IMF Pavel Kroupa Helmholtz-Institut f¨ ur Strahlen- und Kernphysik, Universit¨ at Bonn Nussallee 14-16, 53115 Bonn, Germany Charles University in Prague, Faculty of Mathematics and Physics, Astronomical Institute V Holeˇ soviˇ ck´ach 2, CZ-18000 Praha, Czech Republic emails: pkroupa@uni-bonn.de, kroupa@sirrah.troja.mff.cuni.cz Abstract. Some ultra-compact dwarf galaxies have large dynamical mass to light (M/L) ratios and also appear to contain an overabundance of LMXB sources, and some Milky Way globular clusters have a low concentration and appear to have a deficit of low-mass stars. These observations can be explained if the stellar IMF becomes increasingly top-heavy with decreasing metallicity and increasing gas density of the forming object. The thus constrained stellar IMF then accounts for the observed trend of metallicity and M/L ratio found amongst M31 globular star clusters. It also accounts for the overall shift of the observationally deduced galaxy-wide IMF from top-light to top-heavy with increasing star formation rate amongst galaxies. If the IMF varies similarly to deduced here, then extremely young very massive star-burst clusters observed at a high redshift would appear quasar-like (Jerabkova et al. 2017). Keywords. stars: luminosity function, mass function; galaxies: stellar content; galaxies: star clusters; galaxies: starburst; galaxies: high-redshift; quasars: general 1. Introduction The stellar IMF is valid for a simple stellar population as emerges, on a time-scale of about one Myr, from a molecular cloud core (i.e. on its dynamical collapse time scale) in which forms an embedded cluster on a scale of a pc or less containing dozens to many millions of stars. The galaxy-wide IMF, gwIMF, in contrast is the composite IMF resulting from the addition of the individual IMFs forming throughout the system. The above is summarised in this contribution, with the observationally constrained gwIMF being used as a boundary condition on the independently deduced variation of the IMF. Definitions: • The stellar IMF (IMF), ξ (m)= dN/dm, where dN is the infinitesimal number of stars born together (in one embedded cluster) with individual masses in the interval m to m + dm. The canonical IMF has the shape ξ (m) ∝ m -αi , with α 2 ≈ 2.3, 0.5 M ⊙ <m< m max being the Salpeter power-law index and α 1 ≈ 1.3,m< 0.5 M ⊙ . Here, m max is the mass of the most massive star forming in an embedded cluster with a total stellar mass of M ecl (Weidner et al. 2013). Since stars typically form as binary systems (Goodwin & Kroupa 2005), the molecular cloud filament density variations which fragment to stars (Andr´ e et al. 2010) are related to the initial system mass function (eq. 4-57 in Kroupa et al. 2013). • The galaxy-wide IMF (gwIMF) is the sum of all IMFs over a whole galaxy (Yan et al. 2017; Jerabkova et al. 2018; Hopkins 2018). The calculation of the gwIMF is performed, in the integrated galaxy IMF (IGIMF) theory, by integrating over all embedded clusters formed in the galaxy over a period of about 10 Myr. The 10 Myr time-scale (see Schulz et al. 2015; Yan et al. 2017; Jerabkova et al. 2018 for discussions) is given by the time-scale for local collapse of the interstellar medium (ISM), the thickness of the gas disk being 117 https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921319007749 Downloaded from https://www.cambridge.org/core. IP address: 3.238.51.201, on 15 Jan 2022 at 14:58:13, subject to the Cambridge Core terms of use, available at