Basic Tools for Studies on the Formation and Disruption of Star Clusters: the Luminosity Function

The luminosity function (LF) of young star clusters is often approximated by a power law function. For clusters in a wide range of galactic environments this has resulted in fit indices near -2, but on average slightly steeper. A fundamental property of the -2 power law function is that the luminosity of the brightest object (L_max) scales linearly with the total number of clusters, which is close to what is observed. This suggests that the formation of Young Massive Clusters (YMCs) is a result of the size of the sample, i.e. when the SFR is high it is statistically more likely to form YMCs, but no particular physical conditions are required. In this contribution we provide evidence that the LF of young clusters is not a -2 power law, but instead is curved, showing a systematic decrease of the (logarithmic) slope from roughly -1.8 at low luminosities to roughly -2.8 at high luminosities. The empirical LFs can be reproduced by model LFs using an underlying cluster IMF with a Schechter type truncation around M*=2x10^5 M_sun. This value of M* can not be universal since YMCs well in excess of this M* are known in merging galaxies and merger remnants. Therefore, forming super massive clusters (>10^6 M_sun) probably requires conditions different from those in (quiescent) spiral galaxies and hence is not only the result of a size-of-sample effect. From the vertical offset a cluster formation efficiency of ~10% is derived. We find indications for this efficiency to be higher when the SFR is higher.Comment: 6 pages, 4 figures. To appear in the proceedings of "Galaxy Wars: Stellar Populations and Star Formation in Interacting Galaxies" (Tennessee July 2009

Conference summary: Mass loss from stellar clusters

This conference dealt with the mass loss from stars and from stellar clusters. In this summary of the cluster section of the conference, I highlight some of the results on the formation and the fundamental properties of star clusters (Sect. 2), the early stages of their evolution (Sect. 3) and go into more detail on the subsequent mass evolution of clusters (Sect. 4). A discussion on how this may, or may not, depend on mass is given in Sect. 5. Obviously, there will be a bias towards the topics where Henny Lamers has contributed. Some of the contributions to these proceedings have already reviewed extensively the topics of clusters mass loss and disruption, so I will try to fit these in a general framework as much as possible.Comment: 6 pages, To appear in "Mass loss from stars and the evolution of stellar clusters". Proc. of a workshop held in honour of H.J.G.L.M. Lamers, Lunteren, The Netherlands. Eds. A. de Koter, L. Smith and R. Waters (San Francisco: ASP

The effect of giant molecular clouds on star clusters

We study the encounters between stars clusters and giant molecular clouds (GMCs). The effect of these encounters has previously been studied analytically for two cases: 1) head-on encounters, for which the cluster moves through the centre of the GMC and 2) distant encounters, where the encounter distance p > 3*R_n, with p the encounter parameter and R_n the radius of the GMC. We introduce an expression for the energy gain of the cluster due to GMC encounters valid for all values of p and R_n. This analytical result is confronted with results from N-body simulations and excellent agreement is found. From the simulations we find that the fractional mass loss is only 25% of the fractional energy gain. This is because stars escape with velocities much higher than the escape velocity. Based on the mass loss, we derive a disruption time for star clusters due to encounters with GMCs of the form t_dis [Gyr] = 2.0*S*(M_c/10^4 M_sun)^gamma, with S=1 for the solar neighbourhood and inversely proportional with the global GMC density and gamma=1-3lambda, with lambda the index that relates the cluster half-mass radius to the cluster mass (r_h ~ M_c^lambda). The observed shallow relation between cluster radius and mass (e.g. lambda=0.1), makes the index (gamma=0.7) similar to the index found both from observations and from simulations of clusters dissolving in tidal fields (gamma=0.62). The constant of 2.0 Gyr, which is the disruption time of a 10^4 M_sun cluster in the solar neighbourhood, is close to the value of 1.3 Gyr which was empirically determined from the age distribution of open clusters. This suggests that the combined effect of GMC encounters, stellar evolution and galactic tidal field can explain the lack of old open clusters in the solar neighbourhood.Comment: 2 pages, 2 figures, contribution to "Globular Clusters: Guides to Galaxies", March 6th-10th, 200

Integrated properties of mass segregated star clusters

In this contribution we study integrated properties of dynamically segregated star clusters. The observed core radii of segregated clusters can be 50% smaller than the ``true'' core radius. In addition, the measured radius in the red filters is smaller than those measured in blue filters. However, these difference are small ($\lesssim10$), making it observationally challenging to detect mass segregation in extra-galactic clusters based on such a comparison. Our results follow naturally from the fact that in nearly all filters most of the light comes from the most massive stars. Therefore, the observed surface brightness profile is dominated by stars of similar mass, which are centrally concentrated and have a similar spatial distribution.Comment: 2 pages, 2 figures. To appear in proceedings of the 246th IAU symposium on "Dynamical evolution of dense stellar systems"; acknowledgements include

The role of tidal forces in star cluster disruption

Star clusters are subject to density irregularities in their host galaxy, such as giant molecular clouds (GMCs), the galactic disc and spiral arms, which are largely ignored in present day (N-body) simulations of cluster evolution. Time dependent external potentials give rise to tidal forces that accelerate stars leading to an expansion and more rapid dissolution of the cluster. I explain the basic principles of this tidal heating in the impulse approximation and show how related disruption time-scales depend on properties of the cluster.Comment: 2 pages, To appear in "Mass loss from stars and the evolution of stellar clusters". Proc. of a workshop held in honour of H.J.G.L.M. Lamers, Lunteren, The Netherlands. Eds. A. de Koter, L. Smith and R. Waters (San Francisco: ASP

What determines the mass of the most massive star cluster in a galaxy: statistics, physics or disruption?

In many different galactic environments the cluster initial mass function (CIMF) is well described by a power-law with index -2. This implies a linear relation between the mass of the most massive cluster (M_max) and the number of clusters. Assuming a constant cluster formation rate and no disruption of the most massive clusters it also means that M_max increases linearly with age when determining M_max in logarithmic age bins. We observe this increase in five out of the seven galaxies in our sample, suggesting that M_max is determined by the size of the sample. It also means that massive clusters are very stable against disruption, in disagreement with the mass independent disruption (MID) model presented at this conference. For the clusters in M51 and the Antennae galaxies the size-of-sample prediction breaks down around 10^6 M_sun, suggesting that this is a physical upper limit to the masses of star clusters in these galaxies. In this method there is a degeneracy between MID and a CIMF truncation. We show how the cluster luminosity function can serve as a tool to distinguish between the two.Comment: 6 pages, 3 figures, to appear in ``Young Massive Star Clusters - Initial Conditions and Environments'', 2008, Astrophysics & Space Science, eds. E. Perez, R. de Grijs, R. M. Gonzalez Delgad

The Star Cluster Population of M51

We present the age and mass distribution of star clusters in M51. The structural parameters are found by fitting cluster evolution models to the spectral energy distribution consisting of 8 HST-WFPC2 pass bands. There is evidence for a burst of cluster formation at the moment of the second encounter with the companion NGC5195 (50-100 Myr ago) and a hint for an earlier burst (400-500 Myr ago). The cluster IMF has a power law slope of -2.1. The disruption time of clusters is extremely short (< 100 Myr for a 10^4 Msun cluster).Comment: 2 pages, to appear in "The Formation and Evolution of Massive Young Star Clusters", 17-21 November 2003, Cancun (Mexico

Theoretical and Observational Agreement on Mass Dependence of Cluster Life Times

Observations and N-body simulations both support a simple relation for the disruption time of a cluster as a function of its mass of the form: t_dis = t_4 * (M/10^4 Msun)^gamma. The scaling factor t_4 seems to depend strongly on the environment. Predictions and observations show that gamma ~ 0.64 +/- 0.06. Assuming that t_dis ~ M^0.64 is caused by evaporation and shocking implies a relation between the radius and the mass of a cluster of the form: r_h ~ M^0.07, which has been observed in a few galaxies. The suggested relation for the disruption time implies that the lower mass end of the cluster initial mass function will be disrupted faster than the higher mass end, which is needed to evolve a young power law shaped mass function into the log-normal mass function of old (globular) clusters.Comment: 2 pages, to appear in "The Formation and Evolution of Massive Young Star Clusters", 17-21 November 2003, Cancun (Mexico

The Maximum Mass of Star Clusters

When an universal untruncated star cluster initial mass function (CIMF) described by a power-law distribution is assumed, the mass of the most massive star cluster in a galaxy (M_max) is the result of the size-of-sample (SoS) effect. This implies a dependence of M_max on the total number of star clusters (N). The SoS effect also implies that M_max within a cluster population increases with equal logarithmic intervals of age. This is because the number of clusters formed in logarithmic age intervals increases (assuming a constant cluster formation rate). This effect has been observed in the SMC and LMC. Based on the maximum pressure (P_int) inside molecular clouds, it has been suggested that a physical maximum mass (M_max[phys]) should exist. The theory predicts that M_max[phys] should be observable, i.e. lower than M_max that follows from statistical arguments, in big galaxies with a high star formation rate. We compare the SoS relations in the SMC and LMC with the ones in M51 and model the integrated cluster luminosity function (CLF) for two cases: 1) M_max is determined by the SoS effect and 2) M_max=M_max[phys]=constant. The observed CLF of M51 and the comparison of the SoS relations with the SMC and LMC both suggest that there exists a M_max[phys] of 5*10^5 M_sun in M51. The CLF of M51 looks very similar to the one observed in the ``Antennae'' galaxies. A direct comparison with our model suggests that there M_max[phys]=2*10^6 M_sun.Comment: 4 pages, contribution to "Globular Clusters: Guides to Galaxies", March 6th-10th, 200

The Star Cluster Population of M51: II. Age distribution and relations among the derived parameters

We use archival Hubble Space Telescope observations of broad-band images from the ultraviolet (F255W-filter) through the near infrared (NICMOS F160W-filter) to study the star cluster population of the interacting spiral galaxy M 51. We obtain age, mass, extinction, and effective radius estimates for 1152 star clusters in a region of ~7.3 × 8.1 kpc centered on the nucleus and extending into the outer spiral arms. In this paper we present the data set and exploit it to determine the age distribution and relationships among the fundamental parameters (i.e. age, mass, effective radius). We show the critical dependence of the age distribution on the sample selection, and confirm that using a constant mass cut-off, above which the sample is complete for the entire age range of interest, is essential. In particular, in this sample we are complete only for masses above 5× 104~M? for the last 1 Gyr. Using this dataset we find: i) that the cluster formation rate seems to have had a large increase ~50-70 Myr ago, which is coincident with the suggested second passage of its companion, NGC 5195; ii) a large number of extremely young (<10 Myr) star clusters, which we interpret as a population of unbound clusters of which a large majority will disrupt within the next ~10 Myr; and iii) that the distribution of cluster sizes can be well approximated by a power-law with exponent, -? = -2.2 ± 0.2, which is very similar to that of Galactic globular clusters, indicating that cluster disruption is largely independent of cluster radius. In addition, we have used this dataset to search for correlations among the derived parameters. In particular, we do not find any strong trends between the age and mass, mass and effective radius, nor between the galactocentric distance and effective radius. There is, however, a strong correlation between the age of a cluster and its extinction, with younger clusters being more heavily reddened than older clusters