Anna Lisa Varri

ROTATING GLOBULAR CLUSTERS

Internal rotation is thought to play a major role in the dynamics of some globular clusters. However, in only a few cases has internal rotation been studied by the 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 interpreted very well 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.

A dynamical study of Galactic globular clusters under different relaxation conditions

Aims. We perform a systematic combined photometric and kinematic analysis of a sample of globular clusters under different relaxation conditions, based on their core relaxation time (as listed in available catalogs), by means of two well-known families of spherical stellar dynamical models. Systems characterized by shorter relaxation time scales are expected to be better described by isotropic King models, while less relaxed systems might be interpreted by means of non-truncated, radially-biased anisotropic f (ν) models, originally designed to represent stellar systems produced by a violent relaxation formation process and applied here for the first time to the study of globular clusters. Methods. The comparison between dynamical models and observations is performed by fitting simultaneously surface brightness and velocity dispersion profiles. For each globular cluster, the best-fit model in each family is identified, along with a full error analysis on the relevant parameters. Detailed structural properties and mass-to-light ratios are also explicitly derived. Results. We find that King models usually offer a good representation of the observed photometric profiles, but often lead to less satisfactory fits to the kinematic profiles, independently of the relaxation condition of the systems. For some less relaxed clusters, f (ν) models provide a good description of both observed profiles. Some derived structural characteristics, such as the total mass or the half-mass radius, turn out to be significantly model-dependent. The analysis confirms that, to answer some important dynamical questions that bear on the formation and evolution of globular clusters, it would be highly desirable to acquire larger numbers of accurate kinematic data-points, well distributed over the cluster field.

A novel look at energy equipartition in globular clusters

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 quasi isolation, we show that the stellar mass dependence of the velocity dispersion σ(m) can be described by an exponential function σ 2 ∝ exp(−m/meq), with the parameter meq 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 if 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 meq 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.

Self-consistent models of quasi-relaxed rotating stellar systems

Aims. Two new families of self-consistent axisymmetric truncated equilibrium models for the description of quasi-relaxed rotating stellar systems are presented. The first extends the well-known spherical King models to the case of solid-body rotation. The second is characterized by differential rotation, designed to be rigid in the central regions and to vanish in the outer parts, where the imposed energy truncation becomes effective. Methods. The models are constructed by solving the relevant nonlinear Poisson equation for the self-consistent mean-field potential. For rigidly rotating configurations, the solutions are obtained by an asymptotic expansion based on the rotation strength parameter, following a procedure developed earlier by us for the case of tidally generated triaxial models. The differentially rotating models are constructed by means of a spectral iterative approach, with a numerical scheme based on a Legendre series expansion of the density and the potential. Results. The two classes of models exhibit complementary properties. The rigidly rotating configurations are flattened toward the equatorial plane, with deviations from spherical symmetry that increase with the distance from the center. For models of the second family, the deviations from spherical symmetry are strongest in the central region, whereas the outer parts tend to be quasi-spherical. The relevant parameter spaces are thoroughly explored and the corresponding intrinsic and projected structural properties are described. Special attention is given to the effect of different options for the truncation of the distribution function in phase space. Conclusions. Models in the moderate rotation regime are best suited to applications to globular clusters. For general interest in stellar dynamics, at high values of the rotation strength the differentially rotating models tend to exhibit a toroidal core embedded in an otherwise quasi-spherical configuration. Physically simple analytical models of the kind presented here provide insights into dynamical mechanisms and may be a useful basis for more realistic investigations carried out with the help of N-body simulations.

Kinematical fingerprints of star cluster early dynamical evolution

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.

The inefficiency of satellite accretion in forming extended star clusters

ABSTRACT The distinction between globular clusters and dwarf galaxies has been progressivelyblurred by the recent discoveries of several extended star clusters, with size (20− 30pc) and luminosity (−6<M v <−2) comparable to the one of faint dwarf spheroidals.In order to explain their sparse structure, it has been suggested that they formed asstar clusters in dwarf galaxy satellites that later accreted onto the Milky Way. If theseclusters form in the centre of dwarf galaxies, they evolve in a tidally-compressiveenvironment where the contribution of the tides to the virial balance can becomesignificant, and lead to a super-virial state and subsequent expansion of the cluster,once removed. Using N-body simulations, we show that a cluster formed in such anextreme environment undergoes a sizable expansion, during the drastic variation of theexternal tidal field due to the accretion process. However, we show that the expansiondue to the removal of the compressive tides is not enough to explain the observedextended structure, since the stellar systems resulting from this process are alwaysmore compact than the corresponding clusters that expand in isolation due to two-body relaxation. We conclude that an accreted origin of extended globular clustersis unlikely to explain their large spatial extent, and rather favor the hypothesis thatsuch clusters are already extended at the stage of their formation.Key words: method: numerical, galaxy: globular clusters: general

Testing lowered isothermal models with direct N-body simulations of globular clusters

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.

Velocity anisotropy in tidally limited star 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,

Evolution of star clusters on eccentric orbits

r_h/r_J

Kinematical evolution of tidally limited star clusters: the role of retrograde stellar orbits

, develop a strong radial anisotropy during their long-term dynamical evolution. Many clusters, if formed with small values of