Carsten Weidner

Galactic-Field Initial Mass Functions of Massive Stars

Over the past years observations of young and populous star clusters have shown that the stellar IMF appears to be an invariant featureless Salpeter power-law with an exponent alpha=2.35 for stars more massive than a few Msun. A consensus has also emerged that most, if not all, stars form in stellar groups and star clusters, and that the mass function of young star clusters in the solar-neighborhood and in interacting galaxies can be described, over the mass range of a few 10 Msun to 10^7 Msun, as a power-law with an exponent near beta=2. These two results imply that galactic-field IMFs for early-type stars cannot, under any circumstances, be a Salpeter power-law, but that they must have a steeper exponent alpha_field>2.8. This has important consequences for the distribution of stellar remnants and for the chemo-dynamical and photometric evolution of galaxies.Over the past years observations of young and populous star clusters have shown that the stellar initial mass function (IMF) appears to be an invariant featureless Salpeter power law with an exponent ? = 2.35 for stars more massive than a few M?. A consensus has also emerged that most, if not all, stars form in stellar groups and star clusters and that the mass function of young star clusters in the solar neighborhood and in interacting galaxies can be described, over the mass range of a few 10 to 107 M?, as a power law with an exponent ? ? 2. These two results imply that galactic-field IMFs for early-type stars cannot, under any circumstances, be a Salpeter power law, but that they must have a steeper exponent, ?field 2.8. This has important consequences for the distribution of stellar remnants and for the chemodynamical and photometric evolution of galaxies.

The maximum stellar mass, star-cluster formation and composite stellar populations

We demonstrate that the mass of the most massive star in a cluster correlates non-trivially with the cluster mass. A simple algorithm, according to which a cluster is filled up with stars that are chosen randomly from the standard initial mass function (IMF) but sorted with increasing mass, yields an excellent description of the observational data. Algorithms based on random sampling from the IMF without sorted adding are ruled out with a confidence larger than 0.9999. A physical explanation of this would be that a cluster forms by more-massive stars being consecutively added until the resulting feedback energy suffices to revert cloud contraction and stops further star formation. This has important implications for composite populations. For example, 10 4 clusters of mass 10 2 M ⊙ will not produce the same IMF as one cluster with a mass of 10 6 M ⊙ . It also supports the notion that the integrated galaxial stellar IMF (IGIMF) should be steeper than the stellar IMF and that it should vary with the star formation rate of a galaxy.

Evidence for a fundamental stellar upper mass limit from clustered star formation

The observed masses of the most massive stars do not surpass about 150 M ○. . This may either be a fundamental upper mass limit which is defined by the physics of massive stars and/or their formation, or it may simply reflect the increasing sparsity of such very massive stars, so that the observation of even higher mass stars becomes unlikely in the Galaxy and the Magellanic Clouds. It is shown here that if the stellar initial mass function (IMF) is a power law with a Salpeter exponent (a = 2. 35) for massive stars then the richest very young cluster R136 seen in the Large Magellanic Cloud (LMC) should contain stars with masses larger than 750 M ○. . If, however, the IMF is formulated by consistently incorporating a fundamental upper mass limit then the observed upper mass limit is arrived at readily even if the IMF is invariant. An explicit down-turn or cut-off of the IMF near 150 M ○. is not required: our formulation of the problem contains this implicitly. We are therefore led to conclude that a fundamental maximum stellar mass near 150 M ○. exists, unless the true IMF has a > 2.8.

The variation of integrated star initial mass functions among galaxies

The integrated galaxial initial mass function (IGIMF) is the relevant distribution function containing the information on the distribution of stellar remnants, the number of supernovae, and the chemical enrichment history of a galaxy. Since most stars form in embedded star clusters with different masses, the IGIMF becomes an integral of the assumed (universal or invariant) stellar IMF over the embedded star cluster mass function (ECMF). For a range of reasonable assumptions about the IMF and the ECMF we find the IGIMF to be steeper (containing fewer massive stars per star) than the stellar IMF, but below a few solar masses it is invariant and identical to the stellar IMF for all galaxies. However, the steepening sensitively depends on the form of the ECMF in the low-mass regime. Furthermore, observations indicate a relation between the star formation rate of a galaxy and the most massive young stellar cluster in it. The assumption that this cluster mass marks the upper end of a young-cluster mass function leads to a connection of the star formation rate and the slope of the IGIMF above a few solar masses. The IGIMF varies with the star formation history of a galaxy. Notably, large variations of the IGIMF are evident for dE, dIrr, and LSB galaxies with a small to modest stellar mass. We find that for any galaxy the number of supernovae per star (NSNS) is suppressed relative to that expected for a Salpeter IMF. Dwarf galaxies have a smaller NSNS than massive galaxies. For dwarf galaxies the NSNS varies substantially depending on the galaxy assembly history and the assumptions made about the low-mass end of the ECMF. The findings presented here may be of some consequence for the cosmological evolution of the number of supernovae per low-mass star and the chemical enrichment of galaxies of different mass.

Implications for the formation of star clusters from extragalactic star formation rates

Observations indicate that young massive star clusters in spiral and dwarf galaxies follow a relation between luminosity of the brightest young cluster and the star formation rate (SFR) of the host galaxy, in the sense that higher SFRs lead to the formation of brighter clusters. Assuming that the empirical relation between maximum cluster luminosity and SFR reflects an underlying similar relation between maximum cluster mass (M ecl,max ) and SFR, we compare the resulting SFR(M ecl,max ) relation with different theoretical models. The empirical correlation is found to suggest that individual star clusters form on a free-fall time-scale with their pre-cluster molecular-cloud-core radii typically being a few parsecs independent of mass. The cloud cores contract by factors of 5-10 while building up the embedded cluster. A theoretical SFR(M ecl,max ) relation in very good agreement with the empirical correlation is obtained if the CMF of a young population has a Salpeter exponent of β 2.35 and if this cluster population forms within a characteristic time-scale of a 1-10 Myr. This short time-scale can be understood if the interstellar medium is pressurized, thus precipitating rapid local fragmentation and collapse on a galactic scale. Such triggered star formation on a galactic scale is observed to occur in interacting galaxies. With a global SFR of 3-5 M ○. yr -1 , the Milky Way appears to lie on the empirical SFR(M ecl,max ) relation, given the recent detections of very young clusters with masses near 10 5 M ○. in the Galactic disc. The observed properties of the stellar population of very massive young clusters suggests that there may exist a fundamental maximum cluster mass, 10 6 < M ecl,max */M ○. < 10 7 .

A possible origin of the mass–metallicity relation of galaxies

Observations show that galaxies follow a mass-metallicity relation over a wide range of masses. One currently favoured explanation is that less massive galaxies are less able to retain the gas and stellar ejecta and thus may lose the freshly produced metals in the form of galactic outflows. Galaxies with a low current star formation rate have been found to contain star clusters up to a lower mass limit. Since stars are predominately born in clusters, and less massive clusters have been found to be less likely to contain very massive stars, this implies that in environments or at times of low star formation, the stellar initial mass function does not extend to as high masses as during high star formation epochs. It is found that the oxygen yield is reduced by a factor of 30 when the star formation rate is decreased by 3 to 4 orders of magnitude. With this concept, chemical evolution models for galaxies of a range of masses are computed and shown to provide an excellent fit to the mass-metallicity relation derived recently by Tremonti et al. Furthermore, the models match the relation between galaxy mass and effective yield. Thus, the scenario of a variable integrated stellar initial mass function, which is based on the concept of formation of stars in clusters, may offer an attractive alternative or partial explanation of the mass-metallicity relation in galaxies.

The stellar and sub-stellar IMF of simple and composite populations

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.

Variations of the IMF

(abridged) The {stellar IMF} has been found to be essentially invariant. While some apparent differences are seen, the uncertainties inherent to this game do not allow a firm conclusion to be made that the IMF varies systematically with conditions. The IMF integrated over entire galaxies, however, is another matter. Chemical and photometric properties of various galaxies do hint at {galaxial IMFs} being steeper than the stellar IMF, as is also deduced from direct star-count analysis in the MW. These results are sensitive to the modelling of stellar populations and to corrections for stellar evolution, and are thus also uncertain. However, by realising that galaxies are made from dissolving star clusters, star clusters being viewed as {the fundamental building blocks of galaxies}, the result is found that galaxial IMFs must be significantly steeper than the stellar IMF, because the former results from a folding of the latter with the star-cluster mass function. Furthermore, this notion leads to the important insight that galaxial IMFs must vary with galaxy mass, and that the galaxial IMF is a strongly varying function of the star-formation history for galaxies that have assembled only a small mass in stars. Cosmological implications of this are discussed.

Evidence for a fundamental stellar upper mass limit from clustered star formation and some implications thereof

Theoretical considerations lead to the expectation that stars should not have masses larger than about m_{max*}=60-120Msun, while the observational evidence has been ambiguous. Only very recently has a physical stellar mass limit near 150Msun emerged thanks to modern high-resolution observations of local star-burst clusters. But this limit does not appear to depend on metallicity, in contradiction to theory. Important uncertainties remain though. It is now also emerging that star-clusters limit the masses of their constituent stars, such that a well-defined relation between the mass of the most massive star in a cluster and the cluster mass, m_{max}=F(M_ecl) \le m_{max*}\approx 150Msun, exists. One rather startling finding is that the observational data strongly favour clusters being built-up by consecutively forming more-massive stars until the most massive stars terminate further star-formation. The relation also implies that composite populations, which consist of many star clusters, most of which may be dissolved, must have steeper composite IMFs than simple stellar populations such as are found in individual clusters. Thus, for example, 10^5 Taurus--Auriga star-forming groups, each with 20 stars, will ever only sample the IMF below about 1Msun. This IMF will therefore not be identical to the IMF of one cluster with 2 times 10^6 stars. The implication is that the star-formation history of a galaxy critically determines its integrated galaxial IMF and thus the total number of supernovae per star and its chemical enrichment history. Galaxy formation and evolution models that rely on an invariant IMF would be wrong.

Monte-Carlo Experiments on Star Cluster Induced Integrated-GALaxy IMF Variations

As most if not all stars are born in stellar clusters the shape of the mass function of the field stars is not only determined by the initial mass function of stars (IMF) but also by the cluster mass function (CMF). In order to quantify this Monte-Carlo simulations were carried out by taking cluster masses randomly from a CMF and then populating these clusters with stars randomly taken from an IMF. Two cases were studied. Firstly the star masses were added randomly until the cluster mass was reached. Secondly a number of stars, given by the cluster mass divided by an estimate of the mean stellar mass and sorted by mass, were added until the desired cluster mass was reached. Both experiments verified the analytical results of Kroupa & Weidner (2003) that the resulting integrated stellar initial mass function is a folding of the IMF with the CMF and therefore steeper than the input IMF above 1 Msol.