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Observational studies are showing that the galaxy-wide stellar initial mass function are top-heavy in galaxies with high star-formation rates (SFRs). Calculating the integrated galactic stellar initial mass function (IGIMF) as a function of the SFR of a galaxy, it follows that galaxies which have or which formed with SFRs > 10 Msol yr^-1 would have a top-heavy IGIMF in excellent consistency with the observations. Consequently and in agreement with observations, elliptical galaxies would have higher M/L ratios as a result of the overabundance of stellar remnants compared to a stellar population that formed with an invariant canonical stellar initial mass function (IMF). For the Milky Way, the IGIMF yields very good agreement with the disk- and the bulge-IMF determinations. Our conclusions are that purely stochastic descriptions of star formation on the scales of a pc and above are falsified. Instead, star formation follows the laws, stated here as axioms, which define the IGIMF theory. We also find evidence that the power-law index beta of the embedded cluster mass function decreases with increasing SFR. We propose further tests of the IGIMF theory through counting massive stars in dwarf galaxies., 13 pages, 9 figures, 3 tables; accepted for publication by MNRAS; references updated; typo corrected

The current knowledge on the stellar IMF is documented. It appears to become top-heavy when the star-formation rate density surpasses about 0.1Msun/(yr pc^3) on a pc scale and it may become increasingly bottom-heavy with increasing metallicity and in increasingly massive early-type galaxies. It declines quite steeply below about 0.07Msun with brown dwarfs (BDs) and very low mass stars having their own IMF. The most massive star of mass mmax formed in an embedded cluster with stellar mass Mecl correlates strongly with Mecl being a result of gravitation-driven but resource-limited growth and fragmentation induced starvation. There is no convincing evidence whatsoever that massive stars do form in isolation. Various methods of discretising a stellar population are introduced: optimal sampling leads to a mass distribution that perfectly represents the exact form of the desired IMF and the mmax-to-Mecl relation, while random sampling results in statistical variations of the shape of the IMF. The observed mmax-to-Mecl correlation and the small spread of IMF power-law indices together suggest that optimally sampling the IMF may be the more realistic description of star formation than random sampling from a universal IMF with a constant upper mass limit. Composite populations on galaxy scales, which are formed from many pc scale star formation events, need to be described by the integrated galactic IMF. This IGIMF varies systematically from top-light to top-heavy in dependence of galaxy type and star formation rate, with dramatic implications for theories of galaxy formation and evolution., 167 pages, 37 figures, 3 tables, published in Stellar Systems and Galactic Structure, Vol.5, Springer. This revised version is consistent with the published version and includes additional references and minor additions to the text as well as a recomputed Table 1. ISBN 978-90-481-8817-8

The question whether or not massive stars can form in isolation or only in star clusters is of great importance for the theory of (massive) star-formation as well as for the stellar initial mass function of whole galaxies (IGIMF-theory). While a seemingly easy question it is rather difficult to answer. Several physical processes (e.g. star-loss due to stellar dynamics or gas expulsion) and observational limitations (e.g. dust obscuration of young clusters, resolution) pose severe challenges to answer this question. In this contribution we will present the current arguments in favour and against the idea that all O stars form in clusters., Comment: 6 pages, 3 Figures, to appear in the proceedings of "Stellar Clusters and Associations - A RIA workshop on GAIA", 23-27 May 2011, Granada, Spain

We develop a four-phase galaxy evolution model in order to study the effect of accretion of extra-galactic gas on the star formation rate (SFR) of a galaxy. Pure self-regulated star formation of isolated galaxies is replaced by an accretion-regulated star formation mode. The SFR settles into an equlibrium determined entirely by the gas accretion rate on a Gyr time scale., To appear in "UP: Have Observations Revealed a Variable Upper End of the Initial Mass Function?", Astronomical Society of the Pacific Conference Series

The functional form of the galaxy-wide stellar initial mass function is of fundamental importance for understanding galaxies. So far this stellar initial mass function has been assumed to be identical to the IMF observed directly in star clusters. But because stars form predominantly in embedded groups rather than uniformly distributed over the whole galaxy, the galaxy-wide IMF needs to be calculated by adding all IMFs of all embedded groups. This integrated galactic stellar initial mass function (IGIMF) is steeper than the canonical IMF and steepens with decreasing SFR, leading to fundamental new insights and understanding of star forming properties of galaxies. This contribution reviews the existing applications of the IGIMF theory to galactic astrophysics, while the parallel contribution by Weidner, Pflamm-Altenburg & Kroupa (this volume) introduces the IGIMF theory., To appear in "UP: Have Observations Revealed a Variable Upper End of the Initial Mass Function?", Astronomical Society of the Pacific Conference Series

Over the past years observations of young and populous star clusters have shown that the stellar initial mass function (IMF) can be conveniently described by a two-part power-law with an exponent alpha_2 = 2.3 for stars more massive than about 0.5 Msol and an exponent of alpha_1 = 1.3 for less massive stars. A consensus has also emerged that most, if not all, stars form in stellar groups and star clusters, and that the mass function of these can be described as a power-law (the embedded cluster mass function, ECMF) with an exponent beta ~2. These two results imply that the integrated galactic IMF (IGIMF) for early-type stars cannot be a Salpeter power-law, but that they must have a steeper exponent. An application to star-burst galaxies shows that the IGIMF can become top-heavy. This has important consequences for the distribution of stellar remnants and for the chemo-dynamical and photometric evolution of galaxies. In this contribution the IGIMF theory is described, and the accompanying contribution by Pflamm-Altenburg, Weidner & Kroupa (this volume) documents the applications of the IGIMF theory to galactic astrophysics., 8 pages, 3 figures, to appear in the proceedings of the conference "UP: Have Observations Revealed a Variable Upper End of the Initial Mass Function?", ASP Conference Series

Star formation rates (SFR) larger than 1000 Msun/ yr are observed in extreme star bursts. This leads to the formation of star clusters with masses > 10^6 Msun in which crowding of the pre-stellar cores may lead to a change of the stellar initial mass function (IMF). Indeed, the large mass-to-light ratios of ultra-compact dwarf galaxies and recent results on globular clusters suggest the IMF to become top-heavy with increasing star-forming density. We explore the implications of top-heavy IMFs in these very massive and compact systems for the integrated galactic initial mass function (IGIMF), which is the galaxy-wide IMF, in dependence of the star-formation rate of galaxies. The resulting IGIMFs can have slopes, alpha_3, for stars more massive than about 1 Msun between 1.5 and the Salpeter slope of 2.3 for an embedded cluster mass function (ECMF) slope (beta) of 2.0, but only if the ECMF has no low-mass clusters in galaxies with major starbursts. Alternatively, beta would have to decrease with increasing SFR >10 Msun/ yr such that galaxies with major starbursts have a top-heavy ECMF. The resulting IGIMFs are within the range of observationally deduced IMF variations with redshift., Accepted for publication in MNRAS, reference added

Star-formation rates (SFRs) of galaxies are commonly calculated by converting the measured Halpha luminosities (L_Halpha) into current SFRs. This conversion is based on a constant initial mass function (IMF) independent of the total SFR. As recently recognised the maximum stellar mass in a star cluster is limited by the embedded total cluster mass and, in addition, the maximum embedded star cluster mass is constrained by the current SFR. The combination of these two relations leads to an integrated galaxial initial stellar mass function (IGIMF, the IMF for the whole galaxy) which is steeper in the high mass regime than the constant canonical IMF, and is dependent on the SFR of the galaxy. Consequently, the L_Halpha-SFR relation becomes non-linear and flattens for low SFRs. Especially for dwarf galaxies the SFRs can be underestimated by up to three orders of magnitude. We revise the existing linear L_Halpha-SFR relations using our IGIMF notion. These are likely to lead to a revision of the cosmological star formation histories. We also demonstrate that in the case of the Sculptor dwarf irregular galaxies the IGIMF-formalism implies a linear dependence of the total SFR on the total galaxy gas mass. A constant gas depletion time scale of a few Gyrs results independently of the galaxy gas mass with a reduced scatter compared to the conventional results. Our findings are qualitatively independent of the explicit choice of the IGIMF details and challenges current star formation theory in dwarf galaxies., Accepted by ApJ

The recent finding that the IGIMF (integrated galaxial initial stellar mass function) composed of all newly formed stars in all young star clusters has, in dependence of the SFR, a steeper slope in the high mass regime than the underlying canonical IMF of each star cluster offers new insights into the galactic star formation process: The classical linear relation between the SFR and the produced H$��$ luminosity is broken and SFRs are always underestimated. Our new relation is likely to lead to a revision of the cosmological SFH., 2 pages, 4 figures, To be published in the proceedings of "Pathways Through an Eclectic Universe", ed. T. Mahoney, A. Vazdekis, and J. Knapen