Internal rotation is considered to play a major role in the dynamics of some globular clusters. However, in only few cases it has been studied by quantitative application of realistic and physically justified global models. Here we present a dynamical analysis of the photometry and three-dimensional kinematics of ω Cen, 47 Tuc, and M15, by means of a recently introduced family of self-consistent axisymmetric rotating models. The three clusters, characterized by different relaxation conditions, show evidence of differential rotation and deviations from sphericity. The combination of line-of-sight velocities and proper motions allows us to determine their internal dynamics, predict their morphology, and estimate their dynamical distance. The well-relaxed cluster 47 Tuc is very well interpreted by our model; internal rotation is found to explain the observed morphology. For M15, we provide a global model in good agreement with the data, including the central behavior of the rotation profile and the shape of the ellipticity profile. For the partially relaxed cluster ω Cen, the selected model reproduces the complex three-dimensional kinematics; in particular the observed anisotropy profile, characterized by a transition from isotropy, to weakly-radial anisotropy, and then to tangential anisotropy in the outer parts. The discrepancy found for the steep central gradient in the observed line-of-sight velocity dispersion profile and for the ellipticity profile is ascribed to the condition of only partial relaxation of this cluster and the interplay between rotation and radial anisotropy.

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 toward 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 quasiisolation, we show that the stellar mass dependence of the velocity dispersion σ(m) can be described by an exponential function σ 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 modeling techniques which take into consideration multi-mass components and mass segregation.

High-precision proper motions of the globular cluster 47 Tuc have allowed us to measure for the first time the cluster rotation in the plane of the sky and the velocity anisotropy profile from the cluster core out to about 13 ′ . These profiles are coupled with prior measurements along the line of sight and the surface-brightness profile, and fit all together with self-consistent models specifically constructed to describe quasi-relaxed stellar systems with realistic differential rotation, axisymmetry and pressure anisotropy. The best-fit model provides an inclination angle i between the rotation axis and the line-of-sight direction of 30 • , and is able to simultaneously reproduce the full three-dimensional kinematics and structure of the cluster, while preserving a good agreement with the projected morphology. Literature models based solely on line-of-sight measurements imply a significantly different inclination angle (i = 45 • ), demonstrating that proper motions play a key role in constraining the intrinsic structure of 47 Tuc. Our best-fit global dynamical model implies an internal rotation higher than previous studies have shown, and suggests a peak of the intrinsic V /σ ratio of ∼0.9 at around two half-light radii, with a non-monotonic intrinsic ellipticity profile reaching values up to 0.45. Our study unveils a new degree of dynamical complexity in 47 Tuc, which may be leveraged to provide new insights into the formation and evolution of globular clusters.

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 toward 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.

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 characterise the kinematics of various chemical subsamples, we introduce a method using Bayesian inference along with an MCMC 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 to 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% 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.

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.

We present the first results of the Multi-Instrument Kinematic Survey of Galactic Globular Clusters, a project aimed at exploring the internal kinematics of a representative sample of Galactic globular clusters from the radial velocity of individual stars, covering the entire radial extension of each system. This is achieved by exploiting the formidable combination of multiobject and integral field unit spectroscopic facilities of the ESO Very Large Telescope. As a first step, here we discuss the results obtained for 11 clusters from high and medium resolution spectra acquired through a combination of FLAMES and KMOS observations. We provide the first kinematical characterization of NGC 1261 and NGC 6496. In all the surveyed systems, the velocity dispersion profile declines at increasing radii, in agreement with the expectation from the King model that best fits the density/luminosity profile. In the majority of the surveyed systems we find evidence of rotation within a few half-mass radii from the center. These results are in general overall agreement with the predictions of recent theoretical studies, suggesting that the detected signals could be the relic of significant internal rotation set at the epoch of the cluster's formation.

No abstract