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Tides, rotation, and anisotropy : new self-consistent nonspherical models for globular clusters
No full textSpherical models of quasi-relaxed stellar systems provide a successful zeroth-order description of globular clusters. Yet, the great progress made in recent years in the acquisition of detailed information of the
structure of these stellar systems calls for a renewed effort on the side of modeling. In particular, more realistic analytical models would allow to address a number of key issues on both the theoretical and
observational side, such as the effects induced by different tidal environments, the dynamical interplay between internal rotation and two-body relaxation processes, and the physical origin of the deviations from spherical symmetry.
External tides, internal rotation and anisotropy in the velocity space are therefore the physical ingredients that should be added to the traditional paradigm in order to achieve a proper understanding of the internal dynamics of globular clusters. In this contribution, I will describe a recently presented family of triaxial models that incorporate in a self-consistent way the tidal effects of the host galaxy. I will then introduce two new families of axisymmetric rotating models, studied in collaboration with G. Bertin. The first one is an extension of the well-known family of King models to the case of axisymmetric equilibria flattened by
solid-body rotation, while the second family is characterized by differential rotation, designed to be rigid in the center and to vanish in the outer parts, where the imposed truncation in phase space becomes effective.
Preliminary results of an extensive survey of N-body simulations carried out in collaboration with E.
Vesperini and S. McMillan to explore the dynamical stability and the long-term evolution of these models will also be presented along with the comparison with observational data for selected Galactic globular clusters
Tides, Rotation Or Anisotropy? Self-consistent Nonspherical Models For Globular Clusters
No full textSpherical models of quasi-relaxed stellar systems provide a successful zeroth-order description of globular clusters. Yet, the great progress made in recent years in the acquisition of detailed information of the structure of these stellar systems calls for a renewed effort on the side of modeling. In particular, more general analytical models would allow to address the long-standing issue of the physical origin of the deviations from spherical symmetry of the globular clusters, that now can be properly measured. In fact, it remains to be established which is the cause of the observed flattening, among external tides, internal rotation, and pressure anisotropy.
In this paper we focus on the first two physical ingredients. We start by briefly describing a recently studied family of triaxial models that incorporate in a self-consistent way the tidal effects of the host galaxy, as a collisionless analogue of the Roche problem (Varri & Bertin ApJ 2009). We then present two new families of axisymmetric models in which the deviations from spherical symmetry are induced by the presence of internal rotation. The first one is an extension of the well-known family of King models to the case of axisymmetric equilibria flattened by solid-body rotation. The second family is characterized by differential rotation, designed to be rigid in the center and to vanish in the outer parts, where the imposed truncation in phase space becomes effective.
For possible application to globular clusters, models of interest should be those, in both families, characterized by low values of the rotation strength parameter and quasi-spherical shape. For general interest in stellar dynamics, we show that, for high values of that parameter, the differentially rotating models may exhibit unexpected morphologies, even with a toroidal core
Intrinsic and projected properties of quasi-relaxed stellar systems
No full textWe consider the construction of self-consistent models of collisionless but quasi-relaxed stellar systems, including the ingredients that lead to departures from spherical symmetry. A two-parameter family of triaxial tidal that extend the spherical King models to the case in which an external tidal field is taken into account explicitly is presented. We illustrate several properties that characterize the intrinsic and projected structure of the models. The analysis of the relevant parameter space reveals the existence of two tidal regimes and of a critical condition in which the models are maximally extended
The Construction of Non-Spherical Models of Quasi-Relaxed Stellar Systems
No full textSpherical models of collisionless but quasi-relaxed stellar systems have long been studied as a natural framework for the description of globular clusters. Here we consider the construction of self-consistent models under the same physical conditions, but including explicitly the ingredients that lead to departures from spherical symmetry. We take a stellar system on a circular orbit inside a galaxy represented as a ``frozen" external tidal field. The equilibrium distribution function is obtained from the one describing the spherical case by replacing the energy integral with the relevant Jacobi integral. The construction of the model requires the investigation of a singular perturbation problem for an elliptic partial differential equation with a free boundary, for which we provide an explicit solution to two orders. We outline the relevant parameter space, thus opening the way to a systematic study of the properties of a two-parameter family of physically justified non spherical models of quasi-relaxed stellar systems. The general method developed here can also be used to construct models for which the non-spherical shape is due to internal rotation. Eventually, the models will be a useful tool to investigate whether the shapes of globular clusters are primarily determined by internal rotation, by external tides, or by pressure anisotropy
DYNAMICS OF GLOBULAR CLUSTERS
Full text linkContext 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
Self-consistent models of quasi-relaxed rotating stellar systems
No full textWe present two new families of global equilibrium solutions of the Vlasov-Poisson equations, intended to model collisionless but quasi-relaxed stellar systems, such as globular clusters, characterized by the presence of internal rotation. The first one is an extension to the case of axisymmetric equilibria flattened by solid-body internal rotation of the well-known family of King models, defined by a quasi-Maxwellian distribution function. In turn, the second family is characterized by differential rotation, designed to be rigid in the center and to vanish in the outer parts of the stellar system, where the energy truncation is effective. The physical scenario that inspired the definition of the families is discussed in the more general context of the dynamical evolution of quasi-relaxed stellar systems and a preliminary analysis of their intrinsic properties is provided
Dynamical stability and long-term evolution of rotating stellar systems
No full textWe present the first results of an extensive survey of N-body simulations designed to investigate the dynamical stability and the long-term evolution of two new families of self-consistent stellar dynamical models, characterized by the presence of internal rotation.
The first family extends the well-known King models to the case of axisymmetric systems flattened by solid-body rotation while the second family is characterized by differential rotation. The equilibrium configurations thus obtained can be described in terms of two dimensionless parameters, which measure the concentration and the amount of rotation, respectively.
Slowly rotating configurations are found to be dynamically stable and we followed their long-term evolution, in order to evaluate the interplay between collisional relaxation and angular momentum transport. We also studied the stability of rapidly rotating models, which are characterized by the presence of a toroidal core embedded in an otherwise quasi-spherical configuration. In both cases, a description in terms of the radial and global properties, such as the ratio between the ordered kinetic energy and the gravitational energy of the system, is provided.
Because the role of angular momentum in the process of cluster formation is only partly understood, we also undertook a preliminary investigation of the violent relaxation of simple systems initially characterized by approximate solid-body rotation. The properties of the final equilibrium configurations thus obtained are compared with those of the above-described family of differentially rotating models
A dynamical study of Galactic globular clusters under different relaxation conditions
Full text linkAims. 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 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 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
Self-consistent models of quasi-relaxed rotating stellar systems
Full text linkAims: 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
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