Properties of quasi-relaxed stellar systems in an external tidal field

In a previous paper, we have constructed a family of self-consistent triaxial models of quasi-relaxed stellar systems, shaped by the tidal field of the hosting galaxy, as an extension of the well-known spherical King models. For a given tidal field, the models are characterized by two physical scales (such as total mass and central velocity dispersion) and two dimensionless parameters (the concentration parameter and the tidal strength). The most significant departure from spherical symmetry occurs when the truncation radius of the corresponding spherical King model is of the order of the tidal radius, which, for a given tidal strength, is set by the maximum concentration value admitted. For such maximally extended (or "critical") models the outer boundary has a generally triaxial shape, given by the zero-velocity surface of the relevant Jacobi integral, which is basically independent of the concentration parameter. In turn, the external tidal field can give rise to significant global departures from spherical symmetry (as measured, for example, by the quadrupole of the mass distribution of the stellar system) only for low-concentration models, for which the allowed maximal value of the tidal strength can be relatively high. In this paper we describe in detail the intrinsic and the projected structure and kinematics of the models, covering the entire parameter space, from the case of sub-critical (characterized by "underfilling" of the relevant Roche volume) to that of critical models. The intrinsic properties can be a useful starting point for numerical simulations and other investigations that require initialization of a stellar system in dynamical equilibrium. The projected properties are a key step in the direction of a comparison with observed globular clusters and other candidate stellar systems.Comment: 13 pages, 11 figures, uses emulateapj.cls with apjfonts.sty. Accepted for publication in The Astrophysical Journa

A dynamical study of Galactic globular clusters under different relaxation conditions

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^(\nu) 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. 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. 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^(\nu) 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.Comment: 18 pages, 7 figures. Accepted for publication in Astronomy & Astrophysic

Rotating Globular Clusters

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 omega 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 omega 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.Comment: 19 pages, 14 figures, accepted for publication in the Astrophysical Journa

Biases in the determination of dynamical parameters of star clusters: today and in the Gaia era

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/over-estimated up to 50% 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 center 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.Comment: 14 pages, 8 figures, accepted for publication by MNRA

On the density profile of the globular cluster M92

We present new number density and surface brightness profiles for the globular cluster M92 (NGC 6341). These profiles are calculated from optical images collected with the CCD mosaic camera MegaCam at the Canada-France-Hawaii-Telescope and with the Advanced Camera for Surveys on the Hubble Space Telescope. The ground-based data were supplemented with the Sloan Digital Sky Survey photometric catalog. Special care was taken to discriminate candidate cluster stars from field stars and to subtract the background contamination from both profiles. By examining the contour levels of the number density, we found that the stellar distribution becomes clumpy at radial distances larger than about 13 arcminutes, and there is no preferred orientation of contours in space. We performed detailed fits of King and Wilson models to the observed profiles. The best-fit models underestimate the number density inside the core radius. Wilson models better represent the observations, in particular in the outermost cluster regions: the good global agreement of these models with the observations suggests that there is no need to introduce an extra-tidal halo to explain the radial distribution of stars at large radial distances. The best-fit models for the number density and the surface brightness profiles are different, even though they are based on the same observations. Additional tests support the evidence that this fact reflects the difference in the radial distribution of the stellar tracers that determine the observed profiles (main sequence stars for the number density, bright evolved stars for the surface brightness).Comment: 18 pages, 10 figures, Accepted by A

Ongoing hierarchical massive cluster assembly: the LISCA II structure in the Perseus complex

We report on the identification of a massive ($\sim10^5$ M$_\odot$) sub-structured stellar system in the Galactic Perseus complex likely undergoing hierarchical cluster assembly. Such a system comprises nine star clusters (including the well-known clusters NGC 654 and NGC 663) and an extended and low-density stellar halo. Gaia-DR3 and available spectroscopic data show that all its components are physically consistent in the 6D phase-space (position, parallax, and 3D motion), homogeneous in age (14 $-$ 44 Myr), and chemical content (half-solar metallicity). In addition, the system's global stellar density distribution is that of typical star clusters and shows clear evidence of mass segregation. We find that the hierarchical structure is mostly contracting towards the center with a speed of up to $\simeq4-5$ km s$^{-1}$, while the innermost regions expand at a lower rate (about $\simeq1$ km s$^{-1}$) and are dominated by random motions. Interestingly, this pattern is dominated by the kinematics of massive stars, while low-mass stars ($M<2$ M$_\odot$) are characterized by contraction across the entire cluster. Finally, the nine star clusters in the system are all characterized by a relatively flat velocity dispersion profile possibly resulting from ongoing interactions and tidal heating. We show that the observational results are generally consistent with those found in $N$-body simulations following the cluster violent relaxation phase strongly suggesting that the system is a massive cluster in the early assembly stages. This is the second structure with these properties identified in our Galaxy and, following the nomenclature of our previous work, we named it LISCA II.Comment: 21 pages, 24 figures, 1 table; accepted for publication in A&

First Phase Space Portrait of a Hierarchical Stellar Structure in the Milky Way

We present the first detailed observational picture of a possible ongoing massive cluster hierarchical assembly in the Galactic disk as revealed by the analysis of the stellar full phase-space (3D positions and kinematics and spectro-photometric properties) of an extended area ($6^{\circ}$ diameter) surrounding the well-known $\it h$ and $\chi$ Persei double stellar cluster in the Perseus Arm. Gaia-EDR3 shows that the area is populated by seven co-moving clusters, three of which were previously unknown, and by an extended and quite massive ($M\sim10^5 M_{\odot}$) halo. All stars and clusters define a complex structure with evidence of possible mutual interactions in the form of intra-cluster over-densities and/or bridges. They share the same chemical abundances (half-solar metallicity) and age ($t\sim20$ Myr) within a small confidence interval and the stellar density distribution of the surrounding diffuse stellar halo resembles that of a cluster-like stellar system. The combination of these evidences suggests that stars distributed within a few degrees from $\it h$ and $\chi$ Persei are part of a common, sub-structured stellar complex that we named LISCA I. Comparison with results obtained through direct $N$-body simulations suggest that LISCA I may be at an intermediate stage of an ongoing cluster assembly that can eventually evolve in a relatively massive (a few $10^5 M_{\odot}$) stellar system. We argue that such cluster formation mechanism may be quite efficient in the Milky Way and disk-like galaxies and, as a consequence, it has a relevant impact on our understanding of cluster formation efficiency as a function of the environment and redshift.Comment: 19 pages, 8 figures, 1 table; accepted for publication in Ap

Forward and Back: Kinematics of the Palomar 5 Tidal Tails

The tidal tails of Palomar 5 (Pal 5) have been the focus of many spectroscopic studies in an attempt to identify individual stars lying along the stream and characterise their kinematics. The well-studied trailing tail has been explored out to a distance of 15^\text{o} from the cluster centre, while less than four degrees have been examined along the leading tail. In this paper, we present results of a spectroscopic study of two fields along the leading tail that we have observed with the AAOmega spectrograph on the Anglo-Australian telescope. One of these fields lies roughly 7^\text{o} along the leading tail, beyond what has been previously been explored spectroscopically. Combining our measurements of kinematics and line strengths with Pan-STARRS1 photometric data and Gaia EDR3 astrometry, we adopt a probabilistic approach to identify 16 stars with high probability of belonging to the Pal 5 stream. Eight of these stars lie in the outermost field and their sky positions confirm the presence of ``fanning'' in the leading arm. We also revisit previously-published radial velocity studies and incorporate Gaia EDR3 astrometry to remove interloping field stars. With a final sample of 109 {\it bona fide} Pal 5 cluster and tidal stream stars, we characterise the 3D kinematics along the the full extent of the system. We provide this catalogue for future modeling work.Comment: 13 pages, 8 figures, accepted for publication in MNRAS. The extended version of Table 4 is available as an ancillery file, and will be supplementary material in MNRA

The construction of non-spherical models of quasi-relaxed stellar systems

Spherical 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. In particular, we focus on the effects of the tidal field associated with the hosting galaxy. We then take a stellar system on a circular orbit inside a galaxy represented as a "frozen" external field. The equilibrium distribution function is obtained from the one describing the spherical case by replacing the energy integral with the relevant Jacobi integral in the presence of the external tidal field. Then 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 a method of solution to any desired order, with explicit solutions 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.Comment: AASTeX v5.2, 37 pages with 2 figures, accepted for publication in The Astrophysical Journa