DYNAMICS OF GLOBULAR CLUSTERS

Context and motivation: Globular star clusters have long been considered the ideal astrophysical systems for the study of stellar dynamics. For such stellar systems, the relevant two-body relaxation times are typically shorter than their age, so that it can be argued that they are close to a thermodynamically relaxed state. Indeed, as a zeroth-order dynamical description, the class of models defined as a truncated Maxwellian distribution function (King models), supplemented by the assumption of spherical symmetry, have had remarkable success in the application to observed globular clusters. In fact, the great progress recently made in the acquisition of detailed photometric and kinematic information on the structure of globular clusters as well as the improvements in computational speed of the codes for performing N-body simulations and the availability of accelerator hardware call for a renewed effort in theoretical modeling. Main results: Driven by these motivations, the present Thesis is devoted to the study of such quasi-relaxed stellar systems, with the aim of providing a more realistic dynamical paradigm in which fundamental physical ingredients such as the external tidal field, internal rotation, and weak anisotropy in the velocity space are properly taken into account. The main results can be summarized as follows: (i) Self-consistent triaxial tidal models: As a generalization of the above mentioned spherical King models, we constructed a family of triaxial models in which the deviations from sphericity are induced by the presence of an external tidal field, taken into account self-consistently. By considering the simple case of a cluster in circular orbit within a host galaxy, the equilibrium distribution function is obtained from the one describing the spherical models by replacing the energy integral with the relevant Jacobi integral in the presence of the stationary tidal field. The construction of the models requires the solution of a singular perturbation problem for the relevant Poisson equation. A full characterization of the resulting configurations in terms of the relevant intrinsic and projected properties has been given and the range of the predicted flattening is consistent with that observed in most Galactic globular clusters. (ii) Self-consistent axisymmetric rotating models: By following general statistical mechanics considerations, we constructed a family of rigidly rotating models defined as an extension of the King models to the case of axisymmetric equilibria, flattened by solid-body rotation. The relevant distribution function depends only on the Jacobi integral associated to the internal rotation; the structure of the models is determined by solving the relevant Poisson equation with the same perturbation method discussed for the tidal models, since the corresponding singular perturbation problem is formally equivalent. In addition, we also considered a second family of models characterized by differential rotation, designed to be rigid in the central regions and to vanish in the outer parts. In this case the relevant Poisson equation is solved by a spectral iteration method, based on the Legendre expansion of the density and the potential. A full description of the photometric and kinematic observables has been provided and the models in the moderate rotation regime seem particularly suited to the description of the observed rotating star clusters. For general interest in stellar dynamics, we also studied the models in the strong rotation regime, which tend to show a central toroidal structure. (iii) Dynamical stability of rotating stellar systems: By means of specifically designed N-body simulations with a direct numerical code (Starlab), a full stability analysis of the family of differentially rotating models has been performed. Configurations in the rigid and moderate differential rotation regime are found to be dynamically stable; curiously, there also exists an intermediate rotation regime in which the systems exhibit a central toroidal structure and are dynamically stable. In turn, a new dynamical instability, characterized by a variety of unstable Fourier modes of the density distribution, is observed in models with strong rotation and high degree of shear, in striking analogy with recent stability analyses of differentially rotating fluids with polytropic equations of state. The excitation of an unstable mode seems to be triggered by the presence of the relevant corotation point inside the rotating configuration. This result may help to clarify the physical motivation of the ``empirical'' Ostriker & Peebles stability criterion for rotating stellar systems. (iv) Long-term evolution of rotating stellar systems: The long-term dynamical evolution of the differentially rotating models, studied as isolated systems, has been investigated by means of a comprehensive survey of N-body simulations. This study clarifies how the presence of global angular momentum affects the evolution of stellar systems with respect to the traditional paradigm for the dynamical evolution of nonrotating models and enriches the results obtained in the context of Fokker-Planck evolutionary models with rotation. In particular, by comparing the evolution of several rotating models with selected nonrotating models, characterized by the same initial structural properties, we found that rotating configurations reach core collapse more rapidly. Following early investigations, we also interpreted the evolution of a rotating system by distinguishing between a (short) initial phase, in which the gravo-gyro instability takes place and subsequently levels off, and a second phase in which the residual rotation no longer affects the dynamical evolution of the system, which experiences the gravothermal catastrophe and reaches core collapse, as it happens for nonrotating configurations. (v) Observational signatures of internal rotation in Galactic globular clusters: We successfully applied the family of differentially rotating models to the interpretation of the structure and kinematics of three Galactic globular clusters, characterized by the presence of internal rotation, namely omega Cen, 47 Tuc, and M15. The selection of the relevant models has been performed by a method which combines a number of physically-based kinematic criteria with a statistically rigorous best-fit procedure for the determination of the relevant dimensionless parameters and physical scales of the configuration, respectively. (vi) Pressure anisotropy as signature of partial relaxation in Galactic globular clusters: We have carried out a photometric and kinematic study of sample of Galactic globular clusters in different relaxation conditions, by means of King and f_nu models. The latter is a family of radially-biased spherical models, explicitly constructed for violently relaxed elliptical galaxies. The study suggests that less relaxed clusters tend to conform to the picture of formation via incomplete ``violent relaxation'', that is, the process associated to the rapid fluctuations of the gravitational potential during the early collapse phase of a self-gravitating system.