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.     

Single star scattering on an open cluster

We examine the scattering of single stars from an open star cluster. The probability of the capture of a star by a star cluster is dependent on the velocity and mass of the star, and the stars that are not captured experience a velocity change. For low-velocity stars there is an exponential decrease of the capture probability with the initial velocity, and the velocity change decreases almost linearly. For high-velocity stars there is a v ⁻⁶ dependence for the capture probability, and a v ⁻¹ dependence for the velocity change. Analytical estimations, Monte Carlo and full N -body simulations are all in good agreement.     

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.     

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.     

The role of cluster evolution in disrupting planetary systems and discs: the Kozai mechanism

We examine the effects of dynamical evolution in clusters on planetary systems or protoplanetary discs orbiting the components of binary stars. In particular, we look for evidence that the companions of host stars of planetary systems or discs could have their inclination angles raised from zero to between the threshold angles (39.23° and 140.77°) that can induce the Kozai mechanism. We find that up to 20 per cent of binary systems have their inclination angles increased to within the threshold range. Given that half of all extrasolar planets could be in binary systems, we suggest that up to 10 per cent of extrasolar planets could be affected by this mechanism.     

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.     

金属丰度对球状星团动力学演化的影响

动力学过程和恒星演化及二者的互相影响都会对球状星团的演化产生重要影响.由于金属丰度会影响恒星的演化轨迹,与之相伴随的恒星质量损失率的变化也会对球状星团的动力学过程造成影响.通过一系列N体模拟研究金属丰度对球状星团的质量损失率、半径等的影响,并分析其原因,同时研究了大质量恒星以及星团初始数密度分布的影响.模拟中采用的球状星团模型初始成员星数目N=50000,运行于类银河系的引力势中并考虑成员星的演化.结果显示,由于低金属丰度恒星拥有较快的演化时标,所以贫金属球状星团在早期会拥有较高的质量损失,但与此同时它们的核塌缩时间会比后者显著推迟,因此在核塌缩之后其质量损失会被富金属星团反超.另外由于大质量恒星演化导致的质量损失较大,所以大质量星的存在会使金属丰度更加显著地影响球状星团早期的扩张以及随后的核塌缩过程,同时星团的初始数密度分布也对该效应有着不可忽视的影响.     

The formation of a bound star cluster: from the Orion nebula cluster to the Pleiades

Direct N -body calculations are presented of the formation of Galactic clusters using GasEx , which is a variant of the code Nbody6 . The calculations focus on the possible evolution of the Orion nebula cluster (ONC) by assuming that the embedded OB stars explosively drove out 2/3 of its mass in the form of gas about 0.4 Myr ago. A bound cluster forms readily and survives for 150 Myr despite additional mass loss from the large number of massive stars, and the Galactic tidal field. This is the very first time that cluster formation is obtained under such realistic conditions. The cluster contains about 1/3 of the initial 10⁴ stars, and resembles the Pleiades cluster to a remarkable degree, implying that an ONC-like cluster may have been a precursor of the Pleiades. This scenario predicts the present expansion velocity of the ONC, which will be measurable by upcoming astrometric space missions. These missions should also detect the original Pleiades members as an associated expanding young Galactic-field subpopulation. The results arrived at here suggest that Galactic clusters form as the nuclei of expanding OB associations.
The results have wide implications, also for the formation of globular clusters and the Galactic-field and halo stellar populations. In view of this, the distribution of binary orbital periods and the mass function within and outside the model ONC and Pleiades is quantified, finding consistency with observational constraints. Advanced mass segregation is evident in one of the ONC models. The calculations show that the primordial binary population of both clusters could have been much the same as is observed in the Taurus–Auriga star-forming region. The computations also demonstrate that the binary proportion of brown dwarfs is depleted significantly for all periods, whereas massive stars attain a high binary fraction.     

Monte Carlo simulations of star clusters – IV. Calibration of the Monte Carlo code and comparison with observations for the open cluster M67

We outline the steps needed in order to incorporate the evolution of single and binary stars into a particular Monte Carlo code for the dynamical evolution of a star cluster. We calibrate the results against N -body simulations, and present models for the evolution of the old open cluster M67 (which has been studied thoroughly in the literature with N -body techniques). The calibration is done by choosing appropriate free code parameters. We describe, in particular, the evolution of the binary, white dwarf and blue straggler populations, though not all channels for blue straggler formation are represented yet in our simulations. Calibrated Monte Carlo runs show good agreement with results of N -body simulations not only for global cluster parameters, but also for, for example, binary fraction, luminosity function and surface brightness. Comparison of Monte Carlo simulations with observational data for M67 shows that it is possible to get reasonably good agreement between them. Unfortunately, because of the large statistical fluctuations of the numerical data and uncertainties in the observational data the inferred conclusions about the cluster initial conditions are not firm.     

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.     

Roche volume filling and the dissolution of open star clusters

From direct N‐body simulations we find that the dynamical evolution of star clusters is strongly influenced by the Roche volume filling factor. We present a parameter study of the dissolution of open star clusters with different Roche volume filling factors and different particle numbers. We study both Roche volume underfilling and overfilling models and compare with the Roche volume filling case. We find that in the Roche volume overfilling limit of our simulations two‐body relaxation is no longer the dominant dissolution mechanism but the changing cluster potential. We call this mechanism “mass‐loss driven dissolution” in contrast to “two‐body relaxation driven dissolution” which occurs in the Roche volume underfilling regime. We have measured scaling exponents of the dissolution time with the two‐body relaxation time. In this experimental study we find a decreasing scaling exponent with increasing Roche volume filling factor. The evolution of the escaper number in the Roche volume overfilling limit can be described by a log‐logistic differential equation. We report the finding of a resonance condition which may play a role for the evolution of star clusters and may be calibrated by the main periodic orbit in the large island of retrograde quasiperiodic orbits in the Poincaré surfaces of section. We also report on the existence of a stability curve which may be of relevance with respect to the structure of star clusters. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Do binaries in clusters form in the same way as in the field?

We examine the dynamical destruction of binary systems in star clusters of different densities. We find that at high densities  (10⁴– 10⁵ M_⊙ pc⁻³)  almost all binaries with separations  >10³  au are destroyed after a few crossing times. At low densities [     ], many binaries with separations  >10³  au are destroyed, and no binaries with separations  >10⁴  au survive after a few crossing times. Therefore, the binary separations in clusters can be used as a tracer of the dynamical age and past density of a cluster.
We argue that the central region of the Orion nebula cluster was ∼100 times denser in the past with a half-mass radius of only 0.1–0.2 pc as (i) it is expanding, (ii) it has very few binaries with separations  >10³  au and (iii) it is well mixed and therefore dynamically old.
We also examine the origin of the field binary population. Binaries with separations  <10²  au are not significantly modified in any cluster, therefore at these separations the field reflects the sum of all star formation. Binaries with separations in the range  10²– 10⁴  au are progressively more and more heavily affected by dynamical disruption in increasingly dense clusters. If most star formation is clustered, these binaries must be overproduced relative to the field. Finally, no binary with a separation  >10⁴  au can survive in any cluster and so must be produced by isolated star formation, but only if all isolated star formation produces extremely wide binaries.