ALBERT

Description: The structural and dynamical properties of star clusters are generally derived by means of the comparison between steady-state analytic models and the available observables. With the aim of studying the biases of this approach, we fitted different analytic models to simulated observations obtained from a suite of direct N -body simulations of star clusters in different stages of their evolution and under different levels of tidal stress to derive mass, mass function and degree of anisotropy. We find that masses can be under/overestimated up to 50 per cent depending on the degree of relaxation reached by the cluster, the available range of observed masses and distances of radial velocity measures from the cluster centre and the strength of the tidal field. The mass function slope appears to be better constrainable and less sensitive to model inadequacies unless strongly dynamically evolved clusters and a non-optimal location of the measured luminosity function are considered. The degree and the characteristics of the anisotropy developed in the N -body simulations are not adequately reproduced by popular analytic models and can be detected only if accurate proper motions are available. We show how to reduce the uncertainties in the mass, mass function and anisotropy estimation and provide predictions for the improvements expected when Gaia proper motions will be available in the near future.

Description: Two-body interactions play a major role in shaping the structural and dynamical properties of globular clusters (GCs) over their long-term evolution. In particular, GCs evolve towards a state of partial energy equipartition that induces a mass dependence in their kinematics. By using a set of Monte Carlo cluster simulations evolved in quasi-isolation, we show that the stellar mass dependence of the velocity dispersion ( m ) can be described by an exponential function 2 exp (– m / m eq ), with the parameter m eq quantifying the degree of partial energy equipartition of the systems. This simple parametrization successfully captures the behaviour of the velocity dispersion at lower as well as higher stellar masses, that is, the regime where the system is expected to approach full equipartition. We find a tight correlation between the degree of equipartition reached by a GC and its dynamical state, indicating that clusters that are more than about 20 core relaxation times old, have reached a maximum degree of equipartition. This equipartition–dynamical state relation can be used as a tool to characterize the relaxation condition of a cluster with a kinematic measure of the m eq parameter. Vice versa, the mass dependence of the kinematics can be predicted knowing the relaxation time solely on the basis of photometric measurements. Moreover, any deviations from this tight relation could be used as a probe of a peculiar dynamical history of a cluster. Finally, our novel approach is important for the interpretation of state-of-the-art Hubble Space Telescope proper motion data, for which the mass dependence of kinematics can now be measured, and for the application of modelling techniques which take into consideration multimass components and mass segregation.

Description: Several self-consistent models have been proposed, aiming at describing the phase-space distribution of stars in globular clusters. This study explores the ability of the recently proposed limepy models to reproduce the dynamical properties of direct N -body models of a cluster in a tidal field, during its entire evolution. These dynamical models include prescriptions for the truncation and the degree of radially biased anisotropy contained in the system, allowing us to explore the interplay between the role of anisotropy and tides in various stages of the life of star clusters. We show that the amount of anisotropy in an initially tidally underfilling cluster increases in the pre-collapse phase, and then decreases with time, due to the effect of the external tidal field on its spatial truncation. This is reflected in the correspondent model parameters, and the best-fitting models reproduce the main properties of the cluster at all stages of its evolution, except for the phases immediately preceding and following core collapse. We also notice that the best-fitting limepy models are significantly different from isotropic King models, especially in the first part of the evolution of the cluster. Our results put limits on the amount of radial anisotropy that can be expected for clusters evolving in a tidal field, which is important to understand other factors that could give rise to similar observational signatures, such as the presence of an intermediate-mass black hole.

Description: The presence of an external tidal field often induces significant dynamical evolutionary effects on the internal kinematics of star clusters. Previous studies investigating the restricted three-body problem with applications to star cluster dynamics have shown that unbound stars on retrograde orbits (with respect to the direction of the cluster's orbit) are more stable against escape than prograde orbits, and predicted that a star cluster might acquire retrograde rotation through preferential escape of stars on prograde orbits. In this study, we present evidence of this prediction, but we also illustrate that there are additional effects that cannot be accounted for by the preferential escape of prograde orbits alone. Specifically, in the early evolution, initially underfilling models increase their fraction of retrograde stars without losing significant mass, and acquire a retrograde angular velocity. We attribute this effect to the development of preferentially eccentric/radial orbits in the outer regions of star clusters as they are expanding into their tidal limitation. We explore the implications of the evolution of the fraction of prograde and retrograde stars for the evolution of the cluster internal rotation, and its dependence on the initial structural properties. Although all the systems studied here evolve towards an approximately solid-body internal rotation with angular velocity equal to about half of the angular velocity of the cluster orbital motion around the host galaxy, the evolutionary history of the radial profile of the cluster internal angular velocity depends on the cluster initial structure.

Description: We explore the long-term evolution of the anisotropy in the velocity space of star clusters starting with different structural and kinematical properties. We show that the evolution of the radial anisotropy strength and its radial variation within a cluster contain distinct imprints of the cluster initial structural properties, dynamical history, and of the external tidal field of its host galaxy. Initially isotropic and compact clusters with small initial values of the ratio of the half-mass to Jacobi radius, r h / r J , develop a strong radial anisotropy during their long-term dynamical evolution. Many clusters, if formed with small values of r h / r J , should now be characterized by a significant radial anisotropy increasing with the distance from the cluster centre, reaching its maximum at a distance between 0.2 r J and 0.4 r J , and then becoming more isotropic or mildly tangentially anisotropic in the outermost regions. A similar radial variation of the anisotropy can also result from an early violent relaxation phase. In both cases, as a cluster continues its evolution and loses mass, the anisotropy eventually starts to decrease and the system evolves towards an isotropic velocity distribution. However, in order to completely erase the strong anisotropy developed by these compact systems during their evolution, they must be in the advanced stages of their evolution and lose a large fraction of their initial mass. Clusters that are initially isotropic and characterized by larger initial values of r h / r J , on the other hand, never develop a significant radial anisotropy.

Description: The distinction between globular clusters and dwarf galaxies has been progressively blurred by the recent discoveries of several extended star clusters, with size (20–30 pc) and luminosity (–6 〈  M v  〈 –2) comparable to the one of the faint dwarf spheroidals. In order to explain their sparse structure, it has been suggested that they formed as star clusters in dwarf galaxy satellites that later accreted on to the Milky Way. If these clusters form in the centre of dwarf galaxies, they evolve in a tidally compressive environment where the contribution of the tides to the virial balance can become significant, and lead to a supervirial state and subsequent expansion of the cluster, once removed. Using N -body simulations, we show that a cluster formed in such an extreme environment undergoes a sizable expansion, during the drastic variation of the external tidal field due to the accretion process. However, we show that the expansion due to the removal of the compressive tides is not enough to explain the observed extended structure, since the stellar systems resulting from this process are always more compact than the corresponding clusters that expand in isolation due to two-body relaxation. We conclude that an accreted origin of extended globular clusters is unlikely to explain their large spatial extent, and rather favour the hypothesis that such clusters are already extended at the stage of their formation.

Description: We investigate the dynamical evolution of a star cluster in an external tidal field by using N -body simulations, with focus on the effects of the presence or absence of neutron star natal velocity kicks. We show that, even if neutron stars typically represent less than 2 per cent of the total bound mass of a star cluster, their primordial kinematic properties may affect the lifetime of the system by up to almost a factor of 4. We interpret this result in the light of two known modes of star cluster dissolution, dominated by either early stellar evolution mass-loss or two-body relaxation. The competition between these effects shapes the mass-loss profile of star clusters, which may either dissolve abruptly (‘jumping’), in the pre-core-collapse phase, or gradually (‘skiing’), after having reached core collapse.

Description: We study the effects of the external tidal field on the violent relaxation phase of star clusters dynamical evolution, with particular attention to the kinematical properties of the equilibrium configurations emerging at the end of this phase. We show that star clusters undergoing the process of violent relaxation in the tidal field of their host galaxy can acquire significant internal differential rotation and are characterized by a distinctive radial variation of the velocity anisotropy. These kinematical properties are the result of the symmetry breaking introduced by the external tidal field in the collapse phase and of the action of the Coriolis force on the orbit of the stars. The resulting equilibrium configurations are characterized by differential rotation, with a peak located between one and two half-mass radii. As for the anisotropy, similar to clusters evolving in isolation, the systems explored in this Letter are characterized by an inner isotropic core, followed by a region of increasing radial anisotropy. However, for systems evolving in an external tidal field, the degree of radial anisotropy reaches a maximum in the cluster intermediate regions and then progressively decreases, with the cluster outermost regions being characterized by isotropy or a mild tangential anisotropy. Young or old but less-relaxed dynamically young star clusters may keep memory of these kinematical fingerprints of their early dynamical evolution.

Description: We use radial velocities from spectra of giants obtained with the WIYN telescope, coupled with existing chemical abundance measurements of Na and O for the same stars, to probe the presence of kinematic differences among the multiple populations of the globular cluster (GC) M13. To characterize the kinematics of various chemical subsamples, we introduce a method using Bayesian inference along with a Markov chain Monte Carlo algorithm to fit a six-parameter kinematic model (including rotation) to these subsamples. We find that the so-called extreme population (Na-enhanced and extremely O-depleted) exhibits faster rotation around the centre of the cluster than the other cluster stars, in particular, when compared with the dominant ‘intermediate’ population (moderately Na-enhanced and O-depleted). The most likely difference between the rotational amplitude of this extreme population and that of the intermediate population is found to be ~4 km s –1 , with a 98.4 per cent probability that the rotational amplitude of the extreme population is larger than that of the intermediate population. We argue that the observed difference in rotational amplitudes, obtained when splitting subsamples according to their chemistry, is not a product of the long-term dynamical evolution of the cluster, but more likely a surviving feature imprinted early in the formation history of this GC and its multiple populations. We also find an agreement (within uncertainties) in the inferred position angle of the rotation axis of the different subpopulations considered. We discuss the constraints that these results may place on various formation scenarios.

Description: We study the evolution of star clusters on circular and eccentric orbits using direct N -body simulations. We model clusters with initially N = 8k and 16k single stars of the same mass, orbiting around a point-mass galaxy. For each orbital eccentricity that we consider, we find the apogalactic radius at which the cluster has the same lifetime as the cluster with the same N on a circular orbit. We show that then, the evolution of bound particle number and half-mass radius is approximately independent of eccentricity. Secondly, when we scale our results to orbits with the same semimajor axis, we find that the lifetimes are, to first order, independent of eccentricity. When the results of Baumgardt and Makino for a singular isothermal halo are scaled in the same way, the lifetime is again independent of eccentricity to first order, suggesting that this result is independent of the galactic mass profile. From both sets of simulations, we empirically derive the higher order dependence of the lifetime on eccentricity. Our results serve as benchmark for theoretical studies of the escape rate from clusters on eccentric orbits. Finally, our results can be useful for generative models for cold streams and cluster evolution models that are confined to spherical symmetry and/or time-independent tides, such as Fokker–Planck models, Monte Carlo models, and (fast) semi-analytic models.