Mass Function Of Star Clusters Research Articles

The Present-day Mass Function of Star Clusters in the Solar Neighborhood

This work analyzes the present-day mass function (PDMF) of 93 star clusters utilizing Gaia Data Release 3 data, with membership determined by the StarGo machine-learning algorithm. The impact of unresolved binary systems on mass estimation is rigorously assessed, adopting three mass ratio profiles for correction. The PDMF is characterized by the power-law index, α, derived through a robust maximum likelihood method that avoids biases associated with data binning. The value of α for stars between the completeness limited mass of Gaia (with a mean 0.3 M ⊙ for our cluster samples) and 2 M ⊙ exhibits stability for clusters younger than 200 Myr, decreasing for older clusters, particularly when considering stars within the half-mass radius. The PDMF of these star clusters is consistent with a dynamically evolved Kroupa initial mass function via the loss of low-mass stars. Cluster morphology shows a correlation with α, as α values exhibit a decreasing trend from filamentary to tidal-tail clusters, mirroring the sequence of increasing cluster age. The dependence of α on the total cluster mass is weak, with a subtle increase for higher-mass clusters, especially outside the half-mass radius. We do not observe a correlation between α and the mean metallicity of the clusters. Younger clusters have lower metallicity compared to their older counterparts, which indicates that the older clusters might have migrated to the solar neighborhood from the inner disk. A comparison with numerical models incorporating a black hole population suggests the need for observations of distant, older, massive open clusters to determine whether or not they contain black holes.

Pre-supernova feedback sets the star cluster mass function to a power law and reduces the cluster formation efficiency

Context. The star cluster initial mass function is observed to have an inverse power law exponent around 2, yet there is no consensus on what determines this distribution, and why some variation is observed in different galaxies. Furthermore, the cluster formation efficiency (CFE) covers a range of values, particularly when considering different environments. These clusters are often used to empirically constrain star formation and as fundamental units for stellar feedback models. Detailed galaxy models must therefore accurately capture the basic properties of observed clusters to be considered predictive. Aims. We study how feedback mechanisms acting on different timescales and with different energy budgets affect the star cluster mass function and CFE. Methods. We use hydrodynamical simulations of a dwarf galaxy as a laboratory to study star cluster formation. We test different combinations of stellar feedback mechanisms, including stellar winds, ionizing radiation, and supernovae (SNe). Results. Each feedback mechanism affects the CFE and cluster mass function. Increasing the feedback budget by combining the different types of feedback decreases the CFE by reducing the number of massive clusters. Ionizing radiation is found to be especially influential. This effect depends on the timing of feedback initiation, as shown by comparing early and late feedback. Early feedback occurs from ionizing radiation and stellar winds with onset immediately after a massive star is formed. Late feedback occurs when energy injection only starts after the main-sequence lifetime of the most massive SN progenitor, a timing that is further influenced by the choice of the most massive SN progenitor. Late feedback alone results in a broad, flat mass function, approaching a log-normal shape in the complete absence of feedback. Early feedback, on the other hand, produces a power-law cluster mass function with lower CFE, albeit with a steeper slope than that usually observed.

Star cluster formation and survival in the first galaxies

ABSTRACT Using radiation-hydrodynamic cosmological simulations, we present a detailed (0.1 pc resolution), physically motivated portrait of a typical-mass dwarf galaxy before the epoch of reionization, resolving the formation, and evolution of star clusters into individual 10 M⊙ star particles. In the rest-frame ultraviolet, the galaxy has an irregular morphology with no bulge or disc, dominated by light emitted from numerous, compact, and gravitationally-bound star clusters. This is especially interesting in light of recent James Webb Space Telescope observations that − aided by the magnifying power of gravitational lenses – have imaged, at parsec-scale resolution, individual young star clusters forming in similar galaxies at z> 6. Because of their low metallicities and high temperatures, star-forming gas clouds in this galaxy have densities ∼100 times higher than typical giant molecular clouds; hence, their expected star formation efficiencies (SFEs) are high enough (around 10 − 70 per cent) to produce a sizeable population of potential globular cluster progenitors, but typically smaller (a few 100 − 2 ×104M⊙, half-mass radii of up to 3 pc) and of lower metallicities (10−3.5– 10−2.5 Z⊙). The initial mass function of the star-forming clouds is log-normal, whereas the bound star cluster mass function is a power-law with a slope that depends mainly on SFE but also on the temporal proximity to a major starburst. We find slopes between −0.5 and −2.5 depending on the assumed sub-grid SFE. Star formation is self-regulated on galactic scales; however, the multimodal metallicity distribution of the star clusters and the fraction of stars locked into surviving bound star clusters depends on SFE.

The Panchromatic Hubble Andromeda Treasury: Triangulum Extended Region (PHATTER). III. The Mass Function of Young Stellar Clusters in M33

Abstract We measure the star cluster mass function (CMF) for the Local Group galaxy M33. We use the catalog of stellar clusters selected from the Panchromatic Hubble Andromeda Treasury: Triangulum Extended Region survey. We analyze 711 clusters in M33 with 7.0 < log ( Age / yr ) < 8.5 , and log(M/M ⊙) > 3.0 as determined from color–magnitude diagram fits to individual stars. The M33 CMF is best described by a Schechter function with power-law slope α = − 2.06 − 0.13 + 0.14 , and truncation mass log(M c /M ⊙) = 4.24 − 0.13 + 0.16 . The data show strong evidence for a high-mass truncation, thus strongly favoring a Schechter function fit over a pure power law. M33's truncation mass is consistent with the previously identified linear trend between M c , and star formation rate surface density, ΣSFR. We also explore the effect that individual cluster mass uncertainties have on derived mass function parameters, and find evidence to suggest that large cluster mass uncertainties have the potential to bias the truncation mass of fitted mass functions at the 1σ level.

Realistic modelling of wind and supernovae shocks in star clusters: addressing 22Ne/20Ne and other problems in Galactic cosmic rays

ABSTRACT Cosmic ray (CR) sources leave signatures in the isotopic abundances of CRs. Current models of Galactic CRs that consider supernovae (SNe) shocks as the main sites of particle acceleration cannot satisfactorily explain the higher 22Ne/20Ne ratio in CRs compared to the interstellar medium. Although stellar winds from massive stars have been invoked, their contribution relative to SNe ejecta has been taken as a free parameter. Here, we present a theoretical calculation of the relative contributions of wind termination shocks (WTSs) and SNe shocks in superbubbles, based on the hydrodynamics of winds in clusters, the standard stellar mass function, and stellar evolution theory. We find that the contribution of WTSs towards the total CR production is at least $25{{\ \rm per\ cent}}$, which rises to $\gtrsim 50{{\ \rm per\ cent}}$ for young (≲10 Myr) clusters, and explains the observed 22Ne/20Ne ratio. We argue that since the progenitors of apparently isolated supernovae remnants (SNRs) are born in massive star clusters, both WTS and SNe shocks can be integrated into a combined scenario of CRs being accelerated in massive clusters. This scenario is consistent with the observed ratio of SNRs to γ-ray bright (Lγ ≳ 1035 erg s−1) star clusters, as predicted by star cluster mass function. Moreover, WTSs can accelerate CRs to PeV energies, and solve other long-standing problems of the standard SN paradigm of CR acceleration.

Evolution of the Black Hole Mass Function in Star Clusters from Multiple Mergers

Abstract We investigate the effects of black hole (BH) mergers in star clusters on the black hole mass function (BHMF). As BHs are not produced in pair-instability supernovae, it is suggested that there is a dearth of high-mass stellar BHs. This dearth generates a gap in the upper end of the BHMF. Meanwhile, parameter fitting of X-ray binaries suggests the existence of a gap in the mass function under 5 solar masses. We show, through evolving a coagulation equation, that BH mergers can appreciably fill the upper mass gap, and that the lower mass gap generates potentially observable features at larger mass scales. We also explore the importance of ejections in such systems and whether dynamical clusters can be formation sites of intermediate-mass BH seeds.

Gravitational Focusing and the Star Cluster Initial Mass Function

Abstract We discuss the possibility that gravitational focusing is responsible for the power-law mass function of star clusters . This power law can be produced asymptotically when the mass accretion rate of an object depends upon the mass of the accreting body, as . Although Bondi–Hoyle–Lyttleton accretion formally produces this dependence on mass in a uniform medium, realistic environments are much more complicated. However, numerical simulations in SPH that allow for sink formation yield such an asymptotic power-law mass function. We perform pure N-body simulations to isolate the effects of gravity from those of gas physics and to show that clusters naturally result with the power-law mass distribution. We also consider the physical conditions necessary to produce clusters on appropriate timescales. Our results help support the idea that gravitationally dominated accretion is the most likely mechanism for producing the cluster mass function.

On the link between energy equipartition and radial variation in the stellar mass function of star clusters

We make use of $N$-body simulations to determine the relationship between two observable parameters that are used to quantify mass segregation and energy equipartition in star clusters. Mass segregation can be quantified by measuring how the slope of a cluster's stellar mass function $\alpha$ changes with clustercentric distance r, and then calculating $\delta_\alpha = \frac{d \alpha(r)}{d ln(r/r_m)}$ where $r_m$ is the cluster's half-mass radius. The degree of energy equipartition in a cluster is quantified by $\eta$, which is a measure of how stellar velocity dispersion $\sigma$ depends on stellar mass m via $\sigma(m) \propto m^{-\eta}$. Through a suite of $N$-body star cluster simulations with a range of initial sizes, binary fractions, orbits, black hole retention fractions, and initial mass functions, we present the co-evolution of $\delta_\alpha$ and $\eta$. We find that measurements of the global $\eta$ are strongly affected by the radial dependence of $\sigma$ and mean stellar mass and the relationship between $\eta$ and $\delta_\alpha$ depends mainly on the cluster's initial conditions and the tidal field. Within $r_m$, where these effects are minimized, we find that $\eta$ and $\delta_\alpha$ initially share a linear relationship. However, once the degree of mass segregation increases such that the radial dependence of $\sigma$ and mean stellar mass become a factor within $r_m$, or the cluster undergoes core collapse, the relationship breaks down. We propose a method for determining $\eta$ within $r_m$ from an observational measurement of $\delta_\alpha$. In cases where $\eta$ and $\delta_\alpha$ can be measured independently, this new method offers a way of measuring the cluster's dynamical state.

Radial variation in the stellar mass functions of star clusters

A number of recent observational studies of Galactic globular clusters have measured the variation in the slope of a cluster's stellar mass function $\alpha$ with clustercentric distance $r$. In order to gather a deeper understanding of the information contained in such observations, we have explored the evolution of $\alpha(r)$ for star clusters with a variety of initial conditions using a large suite of $N$-body simulations. We have specifically studied how the time evolution of $\alpha(r)$ is affected by initial size, mass, binary fraction, primordial mass segregation, black hole retention, an external tidal field, and the initial mass function itself. Previous studies have shown that the evolution of $\alpha_G$ is closely related to the amount of mass loss suffered by a cluster. Hence for each simulation we have also followed the evolution of the slope of the cluster's global stellar mass function, $\alpha_G$, and have shown that clusters follow a well-defined track in the $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane. The location of a cluster on the $\alpha_G-d\alpha(r)/d(ln(r/r_m))$ plane can therefore constrain its dynamical history and, in particular, constrain possible variations in the stellar initial mass function. The $\alpha_G$-$d\alpha(r)/d(ln(r/r_m))$ plane thus serves as a key tool for fully exploiting the information contained in wide field studies of cluster stellar mass functions.

THE STAR CLUSTER MASS–GALACTOCENTRIC RADIUS RELATION: IMPLICATIONS FOR CLUSTER FORMATION

ABSTRACT Whether or not the initial star cluster mass function is established through a universal, galactocentric-distance-independent stochastic process, on the scales of individual galaxies, remains an unsolved problem. This debate has recently gained new impetus through the publication of a study that concluded that the maximum cluster mass in a given population is not solely determined by size-of-sample effects. Here, we revisit the evidence in favor and against stochastic cluster formation by examining the young ( ≲ ?> a few × 10 8 ?> year old) star cluster mass–galactocentric radius relation in M33, M51, M83, and the Large Magellanic Cloud. To eliminate size-of-sample effects, we first adopt radial bin sizes containing constant numbers of clusters, which we use to quantify the radial distribution of the first- to fifth-ranked most massive clusters using ordinary least-squares fitting. We supplement this analysis with an application of quantile regression, a binless approach to rank-based regression taking an absolute-value-distance penalty. Both methods yield, within the 1 &sgr; ?> to 3 &sgr; ?> uncertainties, near-zero slopes in the diagnostic plane, largely irrespective of the maximum age or minimum mass imposed on our sample selection, or of the radial bin size adopted. We conclude that, at least in our four well-studied sample galaxies, star cluster formation does not necessarily require an environment-dependent cluster formation scenario, which thus supports the notion of stochastic star cluster formation as the dominant star cluster-formation process within a given galaxy.

26Al kinematics: superbubbles following the spiral arms?

High energy resolution spectroscopy of the 1.8 MeV radioactive decay line of 26Al with the SPI instrument on board the INTEGRAL satellite has recently revealed that diffuse 26Al has large velocities in comparison to other components of the interstellar medium in the Milky Way. 26Al shows Galactic rotation in the same sense as the stars and other gas tracers, but reaches excess velocities up to 300 km/s. We investigate if this result can be understood in the context of superbubbles, taking into account the statistics of young star clusters and H I supershells, as well as the association of young star clusters with spiral arms. We derive energy output and 26Al mass of star clusters as a function of the cluster mass via population synthesis from stellar evolutionary tracks of massive stars. [...] We link this to the size distribution of HI supershells and assess the properties of likely 26Al-carrying superbubbles. 26Al is produced by star clusters of all masses above about 200 solar masses, roughly equally contributed over a logarithmic star cluster mass scale, and strongly linked to the injection of feedback energy. The observed superbubble size distribution cannot be related to the star cluster mass function in a straight forward manner. In order to avoid that the added volume of all superbubbles exceeds the volume of the Milky Way, individual superbubbles have to merge frequently. If any two superbubbles merge, or if 26Al is injected off-centre in a bigger HI supershell we expect the hot 26Al-carrying gas to obtain velocities of the order of the typical sound speed in superbubbles, about 300 km/s before decay. [...]

DEPENDENCY OF DYNAMICAL EJECTIONS OF O STARS ON THE MASSES OF VERY YOUNG STAR CLUSTERS

Massive stars can be efficiently ejected from their birth star clusters through encounters with other massive stars. We study how the dynamical ejection fraction of O star systems varies with the masses of very young star clusters, , by means of direct N-body calculations. We include diverse initial conditions by varying the half-mass radius, initial mass segregation, initial binary fraction, and orbital parameters of the massive binaries. The results show robustly that the ejection fraction of O star systems exhibits a maximum at a cluster mass of for all models, even though the number of ejected systems increases with cluster mass. We show that lower mass clusters () are the dominant sources for populating the Galactic field with O stars by dynamical ejections, considering the mass function of embedded clusters. About 15% (up to ≈38%, depending on the cluster models) of O stars of which a significant fraction are binaries, and which would have formed in a ≈10 Myr epoch of star formation in a distribution of embedded clusters, will be dynamically ejected to the field. Individual clusters may eject 100% of their original O star content. A large fraction of such O stars have velocities up to only 10 km s−1. Synthesising a young star cluster mass function, it follows, given the stellar-dynamical results presented here, that the observed fractions of field and runaway O stars, and the binary fractions among them, can be well understood theoretically if all O stars form in embedded clusters.

The initial mass function of star clusters that form in turbulent molecular clouds

We simulate the formation and evolution of young star clusters using the combination of SPH simulations and direct N-body simulations. We start by performing SPH simulations of the giant molecular cloud with a turbulent velocity field, a mass of $4\times10^4$ to $5\times10^6M_{\odot}$, and a density between $1.7\times10^3$ and $170cm^{-3}$. We continue the SPH simulations for a free-fall time scale, and analyze the resulting structure of the collapsed cloud. We subsequently replace a density-selected subset of SPH particles with stars by adopting a local star-formation efficiency proportional to $\rho^{1/2}$. As a consequence, the local star formation efficiency exceeds 30 %, whereas globally only a few % of the gas is converted to stars. The stellar distribution by the time gas is converted to stars is very clumpy, with typically a dozen bound conglomerates that consist of 100 to $10^4$ stars. We continue to evolve the stars dynamically using the collisional N-body method, which accurately treats all pairwise interactions, stellar collisions and stellar evolution. We analyze the results of the N-body simulations when the stars have an age of 2 Myr and 10 Myr. During the dynamical simulations, massive clusters grow via hierarchical merging of smaller clusters. The shape of the cluster mass function that originates from an individual molecular cloud is consistent with a Schechter function with a power-law slope of -1.73 at 2 Myr and -1.67 at 10 Myr, which fits to observed cluster mass function of the Carina region. The superposition of mass functions have a power-law slope of < -2, which fits the observed mass function of star clusters in the Milky Way, M31 and M83. We further find that the mass of the most massive cluster formed in a single molecular cloud with a mass of $M_g$ scales with $6.1M_g^{0.51}$ which also agrees with recent observation of the GMC and young clusters in M51.

Dynamical modeling of the Arches cluster using Fokker-Planck calculations

AbstractThe Arches cluster is a young, compact, and massive star cluster located in ~30pc away from the Galactic Center in projection. The cluster is located in the extreme environment of the Galactic Center, making it an excellent target for understanding the effects of star-forming environment on the mass function of star clusters. In this study, we estimate the initial condition (mass, concentration parameter, and galactocentric radius) of the Arches cluster by comparing Fokker-Planck calculations with observed velocity dispersion, surface density and mass function data.

The effect of orbital eccentricity on the dynamical evolution of star clusters

We use N-body simulations to explore the influence of orbital eccentricity on the dynamical evolution of star clusters. Specifically we compare the mass loss rate, velocity dispersion, relaxation time, and the mass function of star clusters on circular and eccentric orbits. For a given perigalactic distance, increasing orbital eccentricity slows the dynamical evolution of a cluster due to a weaker mean tidal field. However, we find that perigalactic passes and tidal heating due to an eccentric orbit can partially compensate for the decreased mean tidal field by energizing stars to higher velocities and stripping additional stars from the cluster, accelerating the relaxation process. We find that the corresponding circular orbit which best describes the evolution of a cluster on an eccentric orbit is much less than its semi-major axis or time averaged galactocentric distance. Since clusters spend the majority of their lifetimes near apogalacticon, the properties of clusters which appear very dynamically evolved for a given galactocentric distance can be explained by an eccentric orbit. Additionally we find that the evolution of the slope of the mass function within the core radius is roughly orbit-independent, so it could place additional constraints on the initial mass and initial size of globular clusters with solved orbits. We use our results to demonstrate how the orbit of Milky Way globular clusters can be constrained given standard observable parameters like galactocentric distance and the slope of the mass function. We then place constraints on the unsolved orbits of NGC 1261,NGC 6352, NGC 6496, and NGC 6304 based on their positions and mass functions.

THE Paα LUMINOSITY FUNCTION OF H II REGIONS IN NEARBY GALAXIES FROMHST/NICMOS

The HII region luminosity function (LF) is an important tool for deriving the birthrates and mass distribution of OB associations, and is an excellent tracer of the newly formed massive stars and associations. To date, extensive work (predominantly in H{\alpha}) has been done from the ground, which is hindered by dust extinction and the severe blending of adjacent (spatially or in projection) HII regions. Reliably measuring the properties of HII regions requires a linear resolution <40 pc, but analyses satisfying this requirement have been done only in a handful of galaxies, so far. As the first space-based work using a galaxy sample, we have selected 12 galaxies from our HST NICMOS Pa{\alpha} survey and studied the luminosity function and size distribution of HII regions both in individual galaxies and cumulatively, using a virtually extinction-free tracer of the ionizing photon rate. The high angular resolution and low sensitivity to diffuse emission of NICMOS also offer an advantage over ground-based imaging by enabling a higher degree of de-blending of the HII regions. We do not confirm the broken power-law LFs found in ground-based studies. Instead, we find that the LFs, both individual and co-added, follow a single power law dN(L)/dlnL ~ L^-1, consistent with the mass function of star clusters in nearby galaxies, and in agreement with the results of the existing analyses with HST data. The individual and co-added size distribution of HII regions are both roughly consistent with dN(D)/dlnD ~ D^-3, but the power-law scaling is probably contaminated by blended regions or complexes.

Dynamical population synthesis: constructing the stellar single and binary contents of galactic field populations

[abridged] The galactic field's late-type stellar single and binary population is calculated on the supposition that all stars form as binaries in embedded star clusters. A recently developed tool (Marks, Kroupa & Oh) is used to evolve the binary star distributions in star clusters for a few Myr so that a particular mixture of single and binary stars is achieved. On cluster dissolution the population enters the galactic field with these characteristics. The different contributions of single stars and binaries from individual star clusters which are selected from a power-law embedded star cluster mass function are then added up. This gives rise to integrated galactic field binary distribution functions (IGBDFs) resembling a galactic field's stellar content (Dynamical Population Synthesis). It is found that the binary proportion in the galactic field of a galaxy is larger the lower the minimum cluster mass, the lower the star formation rate, the steeper the embedded star cluster mass function and the larger the typical size of forming star clusters in the considered galaxy. In particular, period-, mass-ratio- and eccentricity IGBDFs for the Milky Way are modelled. The afore mentioned theoretical IGBDFs agree with independently observed distributions. Of all late-type binaries, 50% stem from M<300M_sun clusters, while 50% of all single stars were born in M>10^4M_sun clusters. Comparison of the G-dwarf and M-dwarf binary population indicates that the stars formed in mass-segregated clusters. In particular it is pointed out that although in the present model all M-dwarfs are born in binary systems, in the Milky Way's Galactic field the majority ends up being single stars. This work predicts that today's binary frequency in elliptical galaxies is lower than in spiral and in dwarf-galaxies. The period and mass-ratio distributions in these galaxies are explicitly predicted.

The evolution of the stellar mass function in star clusters

<i>Context. <i/>The dynamical escape of stars from star clusters affects the shape of the stellar mass function (MF) in these clusters, because the escape probability of a star depends on its mass. This is found in <i>N<i/>-body simulations and has been approximated in analytical cluster models by fitting the evolution of the MF. Both approaches are naturally restricted to the set of boundary conditions for which the simulations were performed.<i>Aims. <i/>The objective of this paper is to provide and to apply a simple physical model for the evolution of the MF in star clusters for a large range of the parameter space. It should also offer a new perspective on the results from <i>N<i/>-body simulations.<i>Methods. <i/>A simple, physically self-contained model for the evolution of the stellar MF in star clusters is derived from the basic principles of two-body encounters and energy considerations. It is independent of the adopted mass loss rate or initial mass function (IMF), and contains stellar evolution, stellar remnant retention, dynamical dissolution in a tidal field, and mass segregation.<i>Results. <i/>The MF evolution in star clusters depends on the disruption time, remnant retention fraction, initial-final stellar mass relation, and IMF. Low-mass stars are preferentially ejected after <i>t<i/> ~ 400 Myr. Before that time, masses around 15–20% of the maximum stellar mass are lost due to their rapid two-body relaxation with the massive stars that still exist at young ages. The degree of low-mass star depletion grows for increasing disruption times, but can be quenched when the retained fraction of massive remnants is large. The highly depleted MFs of certain Galactic globular clusters are explained by the enhanced low-mass star depletion that occurs for low remnant retention fractions. Unless the retention fraction is exceptionally large, dynamical evolution always decreases the mass-to-light ratio. The retention of black holes reduces the fraction of the cluster mass in remnants because white dwarfs and neutron stars have masses that are efficiently ejected by black holes.<i>Conclusions. <i/>The modeled evolution of the MF is consistent with <i>N<i/>-body simulations when adopting identical boundary conditions. However, it is found that the results from <i>N<i/>-body simulations only hold for their specific boundary conditions and should not be generalised to all clusters. It is concluded that the model provides an efficient method to understand the evolution of the stellar MF in star clusters under widely varying conditions.

NEW TESTS FOR DISRUPTION MECHANISMS OF STAR CLUSTERS: METHODS AND APPLICATION TO THE ANTENNAE GALAXIES

We present new tests for disruption mechanisms of star clusters based on the bivariate mass–age distribution g(M, τ). In particular, we derive formulae for g(M, τ) for two idealized models in which the rate of disruption depends on the masses of the clusters and one in which it does not. We then compare these models with our Hubble Space Telescope observations of star clusters in the Antennae galaxies over the mass–age domain in which we can readily distinguish clusters from individual stars: τ ≲ 107(M/104 M☉)1.3 yr. We find that the models with mass-dependent disruption are poor fits to the data, even with complete freedom to adjust several parameters, while the model with mass-independent disruption is a good fit. The successful model has the simple form g(M, τ) ∝ M−2τ−1, with power-law mass and age distributions, dN/dM ∝ M−2 and dN/dτ ∝ τ−1. The predicted luminosity function is also a power law, dN/dL ∝ L−2, in good agreement with our observations of the Antennae clusters. The similarity of the mass functions of star clusters and molecular clouds indicates that the efficiency of star formation in the clouds is roughly independent of their masses. The age distribution of the massive young clusters is plausibly explained by the following combination of disruption mechanisms: (1) removal of interstellar material by stellar feedback, τ ≲ 107 yr; (2) continued stellar mass loss, 107 yr ≲ τ ≲ 108 yr; (3) tidal disturbances by passing molecular clouds, τ ≳ 108 yr. None of these processes is expected to have a strong dependence on mass, consistent with our observations of the Antennae clusters. We speculate that this simple picture also applies—at least approximately—to the clusters in many other galaxies.

Gas removal and the initial mass function of star clusters

We explore how the expulsion of gas from star-forming cores due to supernova explosions affects the shape of the initial cluster mass function, that is, the mass function of star clusters when cluster infant weight-loss triggered by gas expulsion is over. We demonstrate that if the radii of cluster-forming gas ‘cores’ are roughly constant over the core mass range, as supported by observations, then more massive cores undergo slower gas expulsion. Therefore, for a given star-formation efficiency, more massive cores retain a larger fraction of stars after gas expulsion. The initial cluster mass function may thus substantially differ from the core mass function, with the final shape depending on the star-formation efficiency. A mass-independent star-formation efficiency of about 20% turns a power-law core mass function into a bell-shaped initial cluster mass function, while mass-independent efficiencies of order 40% preserve the shape of the core mass function.