The evolution of the binary population in globular clusters: a full analytical computation

I present a simplified analytical model that simulates the evolution of the binary population in a dynamically evolving globular cluster. A number of simulations have been run spanning a wide range in initial cluster and environmental conditions by taking into account the main mechanisms of formation and destruction of binary systems. Following this approach, I investigate the evolution of the fraction, the radial distribution, the distribution of mass ratios and periods of the binary population. According to these simulations, the fraction of surviving binaries appears to be dominated by the processes of binary ionization and evaporation. In particular, the frequency of binary systems changes by a factor of 1–5 depending on the initial conditions and on the assumed initial distribution of periods. The comparison with the existing estimates of binary fractions in Galactic globular clusters suggests that significant variations in the initial binary content could exist among the analysed globular cluster. This model has been also used to explain the observed discrepancy found between the most recent N -body and Monte Carlo simulations in the literature.     

The binary star population of the young cluster NGC 1818 in the Large Magellanic Cloud

We determine the binary star fraction as a function of radius in NGC 1818, a young rich cluster in the Large Magellanic Cloud, using Hubble Space Telescope images in bands F336W (∼ U ) and F555W (∼ V ). Our sample includes binaries with M _primary ∼ 2–5.5 M_⊙ and M _secondary ≳ 0.7 M_primary. The binary fraction increases towards the cluster centre, from ∼ 20 ± 5 per cent in the outer parts, to ∼ 35 ± 5 per cent inside the core. This increase is consistent with dynamical mass segregation and need not be primordial. We compare our results with expectations from N -body models, and discuss the implications for the formation and early evolution of such clusters.     

On the onset of runaway stellar collisions in dense star clusters – I. Dynamics of the first collision

We study the circumstances under which first collisions occur in young and dense star clusters. The initial conditions for our direct N -body simulations are chosen such that the clusters experience core collapse within a few million years, before the most massive stars have left the main sequence. It turns out that the first collision is typically driven by the most massive stars in the cluster. Upon arrival in the cluster core, by dynamical friction, massive stars tend to form binaries. The enhanced cross-section of the binary compared to a single star causes other stars to engage the binary. A collision between one of the binary components and the incoming third star is then mediated by the encounters between the binary and other cluster members. Due to the geometry of the binary–single star engagement the relative velocity at the moment of impact is substantially different than in a two-body encounter. This may have profound consequences for the further evolution of the collision product.     

Star clusters with primordial binaries – I. Dynamical evolution of isolated models

In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behaviour of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N -body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0–100 per cent, with values of N ranging from 256 to 16 384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters.
In this range, we find that the steady-state 'deuterium main sequence' is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff, although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10 per cent, the binary heating in the core during the post-collapse phase almost saturates (becoming nearly independent of f ), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker–Planck simulations by Gao et al., this is of little dynamical consequence.
At variance with the study of Gao et al., we see no sign of gravothermal oscillations before 150 half-mass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations.     

A stochastic Monte Carlo approach to model real star cluster evolution – II. Self-consistent models and primordial binaries

The new approach outlined in Paper I to follow the individual formation and evolution of binaries in an evolving, equal point-mass star cluster is extended for the self-consistent treatment of relaxation and close three- and four-body encounters for many binaries (typically a few per cent of the initial number of stars in the cluster mass). The distribution of single stars is treated as a conducting gas sphere with a standard anisotropic gaseous model. A Monte Carlo technique is used to model the motion of binaries, their formation and subsequent hardening by close encounters, and their relaxation (dynamical friction) with single stars and other binaries. The results are a further approach towards a realistic model of globular clusters with primordial binaries without using special hardware. We present, as our main result, the self-consistent evolution of a cluster consisting of 300 000 equal point-mass stars, plus 30 000 equal-mass binaries over several hundred half-mass relaxation times, well into the phase where most of the binaries have been dissolved and evacuated from the core. The cluster evolution is about three times slower than found by Gao et al. Other features are rather comparable. At every moment we are able to show the individual distribution of binaries in the cluster.     

Determination of the orbital parameters of binary pulsars

We present a simple, novel method for determining the orbital parameters of binary pulsars. This method works with any sort of orbital sampling, no matter how sparse, provided that information on the period derivatives is available with each measurement of the rotational period of the pulsar, and it is applicable to binary systems with nearly circular orbits. We use the technique to estimate precisely the hitherto unknown orbital parameters of two binary millisecond pulsars in the globular cluster 47 Tucanae, 47 Tuc S and T. The method can also be used more generally to make first-order estimates of the orbital parameters of binary systems using a minimal number of data.     

Evolution of globular cluster systems in three galaxies of the Fornax cluster

We studied and compared the radial profiles of globular clusters and of the stellar bulge component in three galaxies of the Fornax cluster observed with the WFPC2 of the Hubble Space Telescope ( HST ). The stars are more concentrated toward the galactic centres than globular clusters, in agreement with what has already been observed in many other galaxies: if the observed difference is the result of evolution of the globular cluster systems starting from initial profiles similar to those of the halo–bulge stellar components, a relevant fraction of their initial mass (74, 47 and 52 per cent for NGC 1379, 1399 and 1404, respectively) should have disappeared in the inner regions. This mass has probably contributed to the nuclear field population, local dynamics and high-energy phenomena in the primeval life of the galaxy. An indication in favour of the evolutionary interpretation of the difference between the globular cluster system and stellar bulge radial profiles is given by the positive correlation we found between the value of the mass lost from the globular cluster system and the central galactic black hole mass in the set of seven galaxies for which these data are available.     

Predictions for triple stars with and without a pulsar in star clusters

Though about 80 pulsar binaries have been detected in globular clusters so far, no pulsar has been found in a triple system in which all three objects are of comparable mass. Here, we present predictions for the abundance of such triple systems, and for the most likely characteristics of these systems. Our predictions are based on an extensive set of more than 500 direct simulations of star clusters with primordial binaries, and a number of additional runs containing primordial triples. Our simulations employ a number N _tot of equal-mass stars from   N _tot= 512  to  19 661  and a primordial binary fraction from 0 to 50 per cent. In addition, we validate our results against simulations with   N = 19 661  that include a mass spectrum with a turn-off mass at  0.8 M_⊙  , appropriate to describe the old stellar populations of Galactic globular clusters. Based on our simulations, we expect that typical triple abundances in the core of a dense cluster are two orders of magnitude lower than the binary abundances, which in itself already suggests that we do not have to wait too long for the first comparable-mass triple with a pulsar to be detected.     

Evolution of the mass function of the Galactic globular cluster system

In this work we investigate the evolution of the mass function of the Galactic globular cluster system (GCMF) taking into account the effects of stellar evolution, two-body relaxation, disc shocking and dynamical friction on the evolution of individual globular clusters. We have adopted a lognormal initial GCMF and considered a wide range of initial values for the dispersion, σ, and the mean value, 〈log  M 〉. We have studied in detail the dependence on the initial conditions of the final values of σ, 〈log  M 〉, the fraction of the initial number of clusters surviving after one Hubble time and the difference between the properties of the GCMF of clusters closer to the Galactic Centre and those of clusters located in the outer regions of the Galaxy. In most of the cases considered, evolutionary processes alter significantly the initial population of globular clusters and the disruption of a significant number of globular clusters leads to a flattening in the spatial distribution of clusters in the central regions of the Galaxy. The initial lognormal shape of the GCMF is preserved in most cases and if a power-law in M is adopted for the initial GCMF, evolutionary processes tend to modify it into a lognormal GCMF. The difference between initial and final values of σ and 〈log  M 〉 as well as the difference between the final values of these parameters for inner and outer clusters can be positive or negative depending on initial conditions. A significant effect of evolutionary processes does not necessarily give rise to a strong trend of 〈log  M 〉 with the galactocentric distance. The existence of a particular initial GCMF able to keep its initial shape and parameters unaltered during the entire evolution through a subtle balance between disruption of clusters and evolution of the masses of those which survive, suggested by Vesperini, is confirmed.     

High-energy γ-rays from globular clusters

It is expected that specific globular clusters (GCs) can contain up to a hundred of millisecond pulsars. These pulsars can accelerate leptons at the shock waves originated in collisions of the pulsar winds and/or inside the pulsar magnetospheres. Energetic leptons diffuse gradually through the GC Comptonizing stellar and microwave background radiation. We calculate the GeV–TeV γ-ray spectra for different models of injection of leptons and parameters of the GCs assuming reasonable, of the order of 1 per cent, efficiency of energy conversion from the pulsar winds into the relativistic leptons. It is concluded that leptons accelerated in the GC cores should produce well localized γ-ray sources which are concentric with these GCs. The results are shown for four specific GCs (47 Tuc, Ter 5, M13 and M15), in which significant population of millisecond pulsars have been already discovered. We argue that the best candidates, which might be potentially detected by the present Cherenkov telescopes and the planned satellite telescopes (AGILE, GLAST), are 47 Tuc on the Southern hemisphere, and M13 on the Northern hemisphere. We conclude that detection (or non-detection) of GeV–TeV γ-ray emission from GCs by these instruments put important constraints on the models of acceleration of leptons by millisecond pulsars.     

Effects of external tidal field on the evolution of the outer regions of multi-mass star clusters

We present N -body simulations (including an initial mass function) of globular clusters in the Galaxy in order to study effects of the tidal field systematically on the properties of the outer parts of globular clusters. Using nbody6 , which correctly takes into account the two-body relaxation, we investigate the development of tidal tails of globular clusters in the Galactic tidal field. For simplicity, we have employed only the spherical components (bulge and halo) of the Galaxy, and ignored the effects of stellar evolution which could have been important in the very early phase of the cluster evolution. The total number of stars in our simulations is about 20 000, which is much smaller than the realistic number of stars. All simulations had been done for several orbital periods in order to understand the development of the tidal tails. In our scaled-down models, the relaxation time is sufficiently short to show the mass segregation effect, but we did not go far enough to see the core collapse, and the fraction of stars lost from the cluster at the end of the simulations is only ∼10 per cent. The radial distribution of extra-tidal stars can be described by a power law with a slope around −3 in surface density. The directions of tidal tails are determined by the orbits and locations of the clusters. We find that the length of tidal tails increases towards the apogalacticon and decreases towards the perigalacticon. This is an anti-correlation with the strength of the tidal field, caused by the fact that the time-scale for the stars to respond to the potential is similar to the orbital time-scale of the cluster. The escape of stars in the tidal tails towards the pericentre could be another reason for the decrease of the length of tidal tails. We find that the rotational angular velocity of tidally induced clusters shows quite different behaviour from that of initially rotating clusters.     

Direct N-body modelling of stellar populations: blue stragglers in M67

We present a state-of-the-art N -body code which includes a detailed treatment of stellar and binary evolution as well as the cluster dynamics. This code is ideal for investigating all aspects relating to the evolution of star clusters and their stellar populations. It is applicable to open and globular clusters of any age. We use the N -body code to model the blue straggler population of the old open cluster M67. Preliminary calculations with our binary population synthesis code show that binary evolution alone cannot explain the observed numbers or properties of the blue stragglers. On the other hand, our N -body model of M67 generates the required number of blue stragglers and provides formation paths for all the various types found in M67. This demonstrates the effectiveness of the cluster environment in modifying the nature of the stars it contains, and highlights the importance of combining dynamics with stellar evolution. We also perform a series of N =10 000 simulations in order to quantify the rate of escape of stars from a cluster subject to the Galactic tidal field.     

Monte Carlo simulations of star clusters – VI. The globular cluster NGC 6397

We describe Monte Carlo models for the dynamical evolution of the nearby globular cluster NGC 6397. The code includes treatments of two-body relaxation, most kinds of three- and four-body interactions involving primordial binaries and those formed dynamically, the Galactic tide and the internal evolution of both single and binary stars. We arrive at a set of initial parameters for the cluster which, after 12 Gyr of evolution, gives a model with a fairly satisfactory match to the surface brightness profile, the velocity dispersion profile and the luminosity function in two fields. We describe in particular those aspects of the evolution which distinguish this cluster from M4, which has a roughly similar mass and Galactocentric distance, but a qualitatively different surface brightness profile. Within the limitations of our modelling, we conclude that the most plausible explanation for the difference is fluctuations: both clusters are post-collapse objects, but sometimes have resolvable cores and sometimes not.     

Monte Carlo simulations of star clusters – III. A million-body star cluster

A revision of Stodółkiewicz's Monte Carlo code is used to simulate the evolution of million-body star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The evolution of N -body systems influenced by the tidal field of a parent galaxy and by stellar evolution is presented. All models consist of 1 000 000 stars. The process of energy generation is realized by means of appropriately modified versions of Spitzer's and Mikkola's formulae for the interaction cross-section between binaries and field stars and binaries themselves. The results presented are in good agreement with theoretical expectations and the results of other methods. During the evolution, the initial mass function (IMF) changes significantly. The local mass function around the half-mass radius closely resembles the actual global mass function. At the late stages of evolution, the mass of the evolved stars inside the core can be as high as 97 per cent of the total mass in this region. For the whole system, the evolved stars can compose up to 75 per cent of the total mass. The evolution of cluster anisotropy strongly depends on initial cluster concentration, IMF and the strength of the tidal field. The results presented are the first step in the direction of simulating the evolution of real globular clusters by means of the Monte Carlo method.     

Black holes and core expansion in massive star clusters

In this study we present the results from realistic N -body modelling of massive star clusters in the Magellanic Clouds. We have computed eight simulations with   N ∼ 10⁵  particles; six of these were evolved for at least a Hubble time. The aim of this modelling is to examine in detail the possibility of large-scale core expansion in massive star clusters, and search for a viable dynamical origin for the radius–age trend observed for such objects in the Magellanic Clouds. We identify two physical processes which can lead to significant and prolonged cluster core expansion – mass-loss due to rapid stellar evolution in a primordially mass-segregated cluster, and heating due to a retained population of stellar mass black holes, formed in the supernova explosions of the most massive cluster stars. These two processes operate over different time-scales and during different periods of a cluster's life. The former occurs only at early times and cannot drive core expansion for longer than a few hundred Myr, while the latter typically does not begin until several hundred Myr have passed, but can result in core expansion lasting for many Gyr. We investigate the behaviour of each of these expansion mechanisms under different circumstances – in clusters with varying degrees of primordial mass segregation, and in clusters with varying black hole retention fractions. In combination, the two processes can lead to a wide variety of evolutionary paths on the radius–age plane, which fully cover the observed cluster distribution and hence define a dynamical origin for the radius–age trend in the Magellanic Clouds. We discuss in some detail the implications of core expansion for various aspects of globular cluster research, as well as the possibility of observationally inferring the presence of a significant population of stellar mass black holes in a cluster.     

Constraints on the initial conditions of globular clusters

N -body simulations are made with a variety of initial conditions, in particular clumpy and flattened distributions, to attempt to constrain the possible initial conditions of globular clusters, using the observations that young LMC globular clusters appear relaxed after only 20 to 40 Myr. It is found that violent relaxation is able to erase most of the initial substructure in only ≈ 6 crossing times. However, initially very clumpy distributions (≲ 100 clumps) form clusters that are too concentrated to resemble real globular clusters. Such clusters also often have large clumps in long-lasting (≳ 30 crossing times) orbits which do not appear in observed cluster profiles. It is also found that even modest amounts of initial flattening produce clusters that are too elliptical to resemble real globular clusters. In such a scenario, cloud–cloud collisions and similar energetic processes would be unlikely to produce sufficiently spherical globular clusters. It is suggested that globular clusters form from roughly spherical initial conditions with star formation occurring either smoothly or in many small clumps.     

Close encounters in young stellar clusters: implications for planetary systems in the solar neighbourhood

The stars that populate the solar neighbourhood were formed in stellar clusters. Through N -body simulations of these clusters, we measure the rate of close encounters between stars. By monitoring the interaction histories of each star, we investigate the singleton fraction in the solar neighbourhood. A singleton is a star which formed as a single star, has never experienced any close encounters with other stars or binaries, or undergone an exchange encounter with a binary. We find that, of the stars which formed as single stars, a significant fraction is not singletons once the clusters have dispersed. If some of these stars had planetary systems, with properties similar to those of the Solar System, the planets' orbits may have been perturbed by the effects of close encounters with other stars or the effects of a companion star within a binary. Such perturbations can lead to strong planet–planet interactions which eject several planets, leaving the remaining planets on eccentric orbits. Some of the single stars exchange into binaries. Most of these binaries are broken up via subsequent interactions within the cluster, but some remain intact beyond the lifetime of the cluster. The properties of these binaries are similar to those of the observed binary systems containing extrasolar planets. Thus, dynamical processes in young stellar clusters will alter significantly any population of Solar System-like planetary systems. In addition, beginning with a population of planetary systems exactly resembling the Solar System around single stars, dynamical encounters in young stellar clusters may produce at least some of the extrasolar planetary systems observed in the solar neighbourhood.     

Ratios of star cluster core and half-mass radii: a cautionary note on intermediate-mass black holes in star clusters

There is currently much interest in the possible presence of intermediate-mass black holes (IMBHs) in the cores of globular clusters (GCs). Based on theoretical arguments and simulation results it has previously been suggested that a large core radius – or particularly a large ratio of the core radius to half-mass radius – is a promising indicator for finding such a black hole (BH) in a star cluster. In this study N -body models of 100 000 stars with and without primordial binaries are used to investigate the long-term structural evolution of star clusters. Importantly, the simulation data are analysed using the same processes by which structural parameters are extracted from observed star clusters. This gives a ratio of the core and half-mass (or half-light) radii that are directly comparable to the Galactic GC sample. As a result, it is shown that the ratios observed for the bulk of this sample can be explained without the need for an IMBH. Furthermore, it is possible that clusters with large core to half-light radius ratios harbour a BH binary (comprising stellar mass BHs) rather than a single massive BH. This work does not rule out the existence of IMBHs in the cores of at least some star clusters.     

On the possibility of observing the Shapiro effect for pulsars in globular clusters

For pulsars in globular clusters, we suggest using observations of the relativistic time delay of their radiation in the gravitational field of a massive body (the Shapiro effect) located close to the line of sight to detect and identify invisible compact objects and to study the distribution of both visible and dark matter in globular clusters and different components of the Galaxy. We have derived the dependences of the event probability on the Galactic latitude and longitude of sources for two models of the mass distribution in the Galaxy: the “classical” Bahcall—Soneira model and the more recent Dehnen—Binney model. Using three globular clusters (M15, 47 Tuc, Terzan 5) as an example, we show that the ratios of the probability of the events due to the passages of massive Galactic objects close to the line of sight to the parameter f ² for pulsars in the globular clusters 47 Tuc and M 15 are comparable to those for close passages of massive objects in the clusters themselves and are considerably higher than those for the cluster Terzan 5. We have estimated the rates of such events. We have determined the number of objects near the line of sight toward the pulsar that can produce a modulation of its pulse arrival times characteristic of the effect under consideration; the population of brown dwarfs in the Galactic disk, whose concentration is comparable to that of the disk stars, has been taken into account for the first time.     

Radio observations of NGC 2808 and other globular clusters: constraints on intermediate-mass black holes

We present the results of a deep radio observation of the globular cluster NGC 2808. We show that there are no sources detected within the core of the cluster, placing constraints on both the pulsar population of the cluster and the mass of a possible intermediate-mass black hole in NGC 2808. We compare the results for this cluster with other constraints on intermediate-mass black holes derived from accretion measures. With the exception of G1 in M 31 which has previously shown radio emission, even with considerably more conservative assumptions, only the clusters with the poorest of observational constraints are consistent with falling on the   M _BH–σ  relation. This result is interpreted in terms of the fundamental differences between galaxies and globular clusters.