Holger Baumgardt

Dynamical evolution of star clusters in tidal fields

We report the results of a large set of N-body calculations aimed at studying the evolution of multimass star clusters in external tidal fields. Our clusters start with the same initial mass functions, but varying particle numbers, orbital types and density profiles. Our main focus is to study how the stellar mass function and other cluster parameters change under the combined influence of stellar evolution, two-body relaxation and the external tidal field. We find that the lifetimes of star clusters moving on similar orbits scale as T ∼ T x , where T rh is the relaxation time, and the exponent x depends on the initial concentration of the cluster and is around x ≈ 0.75. The scaling law does not change significantly if one goes from circular orbits to eccentric ones. From the results for the lifetimes, we predict that between 53 and 67 per cent of all Galactic globular clusters will be destroyed within the next Hubble time. Low-mass stars are preferentially lost and the depletion is strong enough to turn initially increasing mass functions into mass functions that decrease towards the low-mass end. The details of this depletion are insensitive to the starting condition of the cluster and can be characterized as a function of a single variable, such as, for example, the fraction of time spent until total cluster dissolution. The preferential depletion of low-mass stars from star clusters leads to a decrease of their mass-to-light ratios except for a short period close to final dissolution, when the mass fraction in the form of compact remnants starts to dominate. The fraction of compact remnants increases throughout the evolution. They are more strongly concentrated towards the cluster cores than main-sequence stars and their mass fraction in the centre can reach 95 per cent or more around and after core collapse. For a sample of Galactic globular clusters with well-observed parameters, we find a correlation between the observed slope of the mass function and the lifetimes predicted by us. It seems possible that Galactic globular clusters started with a mass function similar to what one observes for the average mass function of the Galactic disc and bulge.

Formation of massive black holes through runaway collisions in dense young star clusters

A luminous X-ray source is associated with MGG 11—a cluster of young stars ∼200 pc from the centre of the starburst galaxy M 82 (refs 1, 2). The properties of this source are best explained by invoking a black hole with a mass of at least 350 solar masses (350 M[circdot]), which is intermediate between stellar-mass and supermassive black holes. A nearby but somewhat more massive cluster (MGG 9) shows no evidence of such an intermediate-mass black hole, raising the issue of just what physical characteristics of the clusters can account for this difference. Here we report numerical simulations of the evolution and motion of stars within the clusters, where stars are allowed to merge with each other. We find that for MGG 11 dynamical friction leads to the massive stars sinking rapidly to the centre of the cluster, where they participate in a runaway collision. This produces a star of 800–3,000 M[circdot], which ultimately collapses to a black hole of intermediate mass. No such runaway occurs in the cluster MGG 9, because the larger cluster radius leads to a mass segregation timescale a factor of five longer than for MGG 11.

A comprehensive set of simulations studying the influence of gas expulsion on star cluster evolution

We have carried out a large set of N-body simulations studying the effect of residual-gas expulsion on the survival rate, and final properties of star clusters. We have varied the star formation efficiency (SFE), gas expulsion time-scale and strength of the external tidal field, obtaining a three-dimensional grid of models which can be used to predict the evolution of individual star clusters or whole star cluster systems by interpolating between our runs. The complete data of these simulations are made available on the internet.

An analytical description of the disruption of star clusters in tidal fields with an application to Galactic open clusters

We present a simple analytical description of the disruption of star clusters in a tidal field. The cluster disruption time, defined as tdis = {dln M/dt} −1 , depends on the mass M of the cluster as tdis = t0(M/M� ) γ with γ = 0.62 for clusters in a tidal field, as shown by empirical studies of cluster samples in different galaxies and by N-body simulations. Using this simple description we derive an analytic expression for the way in which the mass of a cluster decreases with time due to stellar evolution and disruption. The result agrees very well with those of detailed N-body simulations for clusters in the tidal field of our galaxy. The analytic expression can be used to predict the mass and age histograms of surviving clusters for any cluster initial mass function and any cluster formation history. The method is applied to explain the age distribution of the open clusters in the solar neighbourhood within 600 pc, based on a new cluster sample that appears to be unbiased within a distance of about 1 kpc. From a comparison between the observed and predicted age distributions in the age range between 10 Myr to 3 Gyr we find the following results: (1) The disruption time of a 10 4 Mcluster in the solar neighbourhood is about 1.3 ± 0.5 Gyr. This is a factor of 5 shorter than that derived from N-body simulations of clusters in the tidal field of the galaxy. Possible reasons for this discrepancy are discussed. (2) The present star formation rate in bound clusters within 600 pc of the Sun is 5.9 ± 0.8 × 10 2 MMyr −1 , which corresponds to a surface star formation rate of bound clusters of 5.2 ± 0.7 × 10 −10 Myr −1 pc −2 . (3) The age distribution of open clusters shows a bump between 0.26 and 0.6 Gyr when the cluster formation rate was 2.5 times higher than before and after. (4) The present star formation rate in bound clusters is about half that derived from the study of embedded clusters. The difference suggests that about half of the clusters in the solar neighbourhood become unbound within about 10 Myr. (5) The most massive clusters within 600 pc had an initial mass of about 3 × 10 4 M� . This is in agreement with the statistically expected value based on a cluster initial mass function with a slope of −2, even if the physical upper mass limit for cluster formation is as high as 10 6 M� .

Dynamical Formation of Close Binary Systems in Globular Clusters

We know from observations that globular clusters are very efficient catalysts in forming unusual short-period binary systems or their offspring, such as low-mass X-ray binaries (LMXBs; neutron stars accreting matter from low-mass stellar companions), cataclysmic variables (white dwarfs accreting matter from stellar companions), and millisecond pulsars (rotating neutron stars with spin periods of a few milliseconds). Although there has been little direct evidence, the overabundance of these objects in globular clusters has been attributed by numerous authors to the high densities in the cores, which leads to an increase in the formation rate of exotic binary systems through close stellar encounters. Many such close binary systems emit X-radiation at low luminosities (LX 1034 ergs s-1) and are being found in large numbers through observations with the Chandra X-Ray Observatory. Here we present conclusive observational evidence of a link between the number of close binaries observed in X-rays in a globular cluster and the stellar encounter rate of the cluster. We also make an estimate of the total number of LMXBs in globular clusters in our Galaxy.

Star cluster disruption by giant molecular clouds

We investigate encounters between giant molecular clouds (GMCs) and star clusters. We propose a single expression for the energy gain of a cluster due to an encounter with a GMC, valid for all encounter distances and GMC properties. This relation is verified with N-body simulations of cluster-GMC encounters, where the GMC is represented by a moving analytical potential. Excellent agreement is found between the simulations and the analytical work for fractional energy gains of Delta E/vertical bar E-0 vertical bar < 10, where vertical bar E-0 vertical bar is the initial total cluster energy. The fractional mass loss from the cluster scales with the fractional energy gain as (Delta M/M-0) = f(Delta E/vertical bar E-0 vertical bar), where f similar or equal to 0.25. This is because a fraction 1 - f of the injected energy goes to the velocities of escaping stars, that are higher than the escape velocity. We therefore suggest that the disruption time of clusters, t(dis), is best defined as the time needed to bring the cluster mass to zero, instead of the time needed to inject the initial cluster energy. We derive an expression for t(dis) based on the mass loss from the simulations, taking into account the effect of gravitational focusing by the GMC. Assuming spatially homogeneous distributions of clusters and GMCs with a relative velocity dispersion of sigma(cn), we find that clusters lose most of their mass in relatively close encounters with high relative velocities (similar to 2 sigma(cn)). The disruption time depends on the cluster mass (M-c) and half-mass radius (r(h)) as t(dis) = 2.0 S(M-c/10(4) M-circle dot)(3.75 pc/r(h))(3) Gyr, with S equivalent to 1 for the solar neighbourhood and S scales with the surface density of individual GMCs (Sigma(n)) and the global GMC density (rho(n)) as S proportional to (Sigma(n)rho(n))(-1). Combined with the observed relation between r(h) and M-c, that is, r(h) proportional to M-c(lambda), t(dis) depends on M-c as t(dis)proportional to M-c(gamma). The index gamma is then defined as gamma= 1 - 3 lambda. The observed shallow relation between cluster radius and mass (e.g. lambda similar or equal to 0.1), makes the value of the index gamma = 0.7 similar to that found from observations and from simulations of clusters dissolving in tidal fields (gamma similar or equal to 0.62). The constant of 2.0 Gyr, which is the disruption time of a 10(4) M circle dot cluster in the solar neighbourhood, is about a factor of 3.5 shorter than that found from earlier simulations of clusters dissolving under the combined effect of Galactic tidal field and stellar evolution. It is somewhat higher than the observationally determined value of 1.3 Gyr. It suggests, however, that the combined effect of tidal field and encounters with GMCs can explain the lack of old open clusters in the solar neighbourhood. GMC encounters can also explain the (very) short disruption time that was observed for star clusters in the central region of M51, since there rho(n) is an order of magnitude higher than that in the solar neighbourhood.

Scaling of N-body calculations

We report results of collisional N-body simulations aimed at studying the N dependence of the dynamical evolution of star clusters. Our clusters consist of equal-mass stars and are in virial equilibrium. Clusters moving in external tidal fields and clusters limited by a cut-off radius are simulated. Our main focus is to study the dependence of the lifetimes of the clusters on the number of cluster stars and the chosen escape condition.

Massive black holes in star clusters: II. Realistic cluster models

We have followed the evolution of multimass star clusters containing massive central black holes through collisional N-body simulations done on GRAPE6. Each cluster is composed of between 16,384 and 131,072 stars together with a black hole with an initial mass of MBH = 1000 M☉. We follow the evolution of the clusters under the combined influence of two-body relaxation, stellar mass loss, and tidal disruption of stars by the massive central black hole. We find that the (three-dimensional) mass density profile follows a power-law distribution ρ ~ r-α with slope α = 1.55 inside the sphere of influence of the central black hole. This leads to a constant-density profile of bright stars in projection, which makes it highly unlikely that core-collapse clusters contain intermediate-mass black holes (IMBHs). Instead, globular clusters containing massive central black holes can be fitted with standard King profiles. Because of energy generation in the cusp, star clusters with IMBHs expand. The cluster expansion is so strong that clusters that start very concentrated can end up among the least dense clusters. The amount of mass segregation in the core is also smaller compared to postcollapse clusters without IMBHs. Most stellar mass black holes with masses MBH > 5 M☉ are lost from the clusters within a few gigayears through mutual encounters in the cusp around the IMBH. Black holes in star clusters disrupt mainly main-sequence stars and giants and no neutron stars. The disruption rates are too small to form an IMBH out of a MBH ≈ 50 M☉ progenitor black hole even if all material from disrupted stars is accreted onto the black hole, unless star clusters start with central densities significantly higher than what is seen in present-day globular clusters. We also discuss the possible detection mechanisms for IMBHs. Our simulations show that kinematical studies can reveal 1000 M☉ IMBHs in the closest clusters. IMBHs in globular clusters are weak X-ray sources, since the tidal disruption rate of stars is low and the star closest to the IMBH is normally another black hole, which prevents other stars from undergoing stable mass transfer. For globular clusters, dynamical evolution can push compact stars near the IMBH to distances small enough that they become detectable sources of gravitational radiation. If 10% of all globular clusters contain IMBHs, extragalactic globular clusters could be one of the major sources of gravitational wave events for LISA.

The Nature of UCDs: Internal Dynamics from an Expanded Sample and Homogeneous Database

Context. The internal dynamics of ultra-compact dwarf galaxies (UCDs) has attracted increasing attention, with most of the UCDs studied to date located in the Virgo cluster.

On the Central Structure of M15

We present a detailed comparison between the latest observational data on the kinematical structure of the core of M15, obtained with the Hubble Space Telescope Space Telescope Imaging Spectrograph and Wide Field Planetary Camera 2 instruments, and the results of dynamical simulations carried out using the special purpose GRAPE-6 computer. The observations imply the presence of a significant amount of dark matter in the cluster core. In our dynamical simulations, neutron stars and/or massive white dwarfs concentrate to the center through mass segregation, resulting in a sharp increase in toward the center. While consistent with the presence of M/L a central black hole, the Hubble Space Telescope data can also be explained by this central concentration of stellar mass compact objects. The latter interpretation is more conservative, since such remnants result naturally from stellar evolution, although runaway merging leading to the formation of a black hole may also occur for some range of initial conditions. We conclude that no central massive object is required to explain the observational data, although we cannot conclusively exclude such an object at the level of. Our findings are similar to500-1000 M-circle dot. Our findings are unchanged when we reduce the assumed neutron star retention fraction in our simulations from 100% to 0%.