Non-stationarity parameters of open clusters

Estimates of a number of dynamical parameters of 103 open star clusters are presented: the density contrast in the cluster core, stellar velocity dispersion, taking into account the influence of the external field of the Galaxy and non-stationarity of the cluster, oscillation periods of the cluster and the cluster core, etc. Analytical solutions to the gross dynamical equations are obtained for simple cluster models. These solutions are used to estimate a number of quantities characterizing the degree of non-stationarity of the cluster, such as the amplitude of oscillations of the cluster virial coefficient and of the radius of the cluster core, etc. Astrophysical applications of these results are discussed.     

Dynamical models of stellar motions at the peripheries of open clusters

A model for a uniform, gravitating, ellipsoidal star cluster moving in a circular orbit around the Galactic center is considered. Three independent isolated integrals of stellar motion are written for this model. The characteristic features of the motion of a cluster star according to these three integrals are analyzed. Retrograde stellar motions dominate at the periphery of the model cluster, and the distribution of the stellar velocities is elongated along the direction of the cluster motion. A phase-space density function that depends on two of the integrals of motion is constructed. The distribution of the stellar velocities is constructed for the case of a three-integral phase-space density. Possible applications of the results are discussed.     

Peculiarities of the abundances of neutron-capture elements in Galactic open clusters

The properties of the relative abundances of rapid and slow neutron-capture elements are studied using a catalog containing spectroscopic abundance determinations for 14 elements produced in various nuclear-synthesis processes for 90 open clusters. The catalog also contains the positions, ages, velocities, and elements of the Galactic orbits of the clusters. The relative abundances of both r-elements (Eu) and s-elements (Y, Ba, La, and Ce) in clusters with high, elongated orbits and in field stars of the Galactic thin disk display different dependences on metallicity, age, Galactocentric distance, and the elements of the Galactic orbits, supporting the view that these objects have different natures. In young clusters, not only barium, but also the three other studied s-elements display significantly higher relative abundances than field stars of the same metallicity. The relative abundances of Eu are lower in highmetallicity clusters ([Fe/H] > -0.1) with high, elongated orbits than in field giants, on average, while the [Eu/Fe] ratios in lower-metallicity clusters are the same as those in field stars, on average, although with a large scatter. The metallicity dependence of the [O, Mg/Eu] ratios in clusters with high, elongated orbits and in field stars are substantially different. These and other described properties of the Eu abundances, together with the properties of the abundances of primary a-elements, can be understood in a natural way if clusters with high, elongated orbits with different metallicities formed as a result of interactions of two types of high-velocity clouds with the interstellar medium of the Galactic disk: low-metallicity highvelocity clouds that formed from “primordial” gas, and high-metallicity clouds with intermediate velocities that formed in “Galactic fountains.”     

Timescales for mechanisms for the dynamical evolution of open star clusters

We have obtained the stellar velocity dispersion in three mutually perpendicular directions in the halos and cores of clusters as a function of time for several non-stationary open-cluster models. During the dynamical evolution of the open-cluster models, the velocity dispersions undergo oscillations that do not decay during 5–10 violent-relaxation timescales, τ _vr . We estimated the time for synchronization of the rotation of the open-cluster models and their motion around the center of the Galaxy, t _s , which, depending on the model parameters, is t _s ? (5–27)τ _vr . Synchronization mechanisms for the models are discussed. The disruption of the systems in the force field of the Galaxy is strongly affected by tidal friction. We have also estimated the time for the formation of a spherical stellar-velocity distribution in the cluster models, t _σ ? (6 ? 25)τ _vr . The impact of instability in the stellar motions in a cluster on the formation of a spherical velocity distribution in the open-cluster models is discussed. We have noted a tendency for a weakening of the dependence of the coarse phase density of the cluster on small initial perturbations of the stellar phase coordinates in the model cluster cores for times about five times longer than the violent-relaxation time.     

Red giants in the vicinity of open clusters. Field stars

We present a comparative analysis of the atmospheric abundances of red giants in the vicinity of open clusters. The atmospheric parameters, atmospheric abundances, masses, ages, Galactic velocities, and elements of the Galactic orbits are derived for all the studied stars. We have discovered high metal abundances (close to 0.3dex) for five stars, which we classify as super-metal-rich stars. Several stars have lower [Na/Fe] than normal red giants with similar atmospheric parameters. The kinematic characteristics of these stars are somewhat different from those for objects in the Galactic thin disk. We suggest that the observed effect can be explained by inhomogeneity of the chemical composition of gas-dust clouds, which could be due to different rates of SNe II supernovae in different regions of the Galaxy.     

Analysis of density fluctuations in models of open clusters

The frequencies for homologous and non-homologous density flucutations in unisolated, ellipsoidal models of open clusters are determined. These fluctuations are stable. In the cases considered, the frequencies of non-homologous fluctuations are higher than those for homologous fluctuations of the ellipsoidal models. The instability of the proper fluctuations of the phase density at the center of a homogeneous ellipsoid whose central characteristics correspond to the core parameters of a dynamical, numericalmodel cluster is demonstrated. The development of instability of the phase-density flucutations at the center of such an ellipsoid leads to the domination of fluctuations with the frequency γ ₁, corresponding to the growth increment γ ₂ for this instability. Estimates of γ ₁ and γ ₂ agree with estimates obtained for the core of the corresponding dynamical, numerical model cluster. Astrophysical applications of the results are discussed.     

Stellar fluxes in numerical dynamical models of open clusters

Fluxes of stars in models of open clusters that are nonstationary in the regular field of the cluster are analyzed. An equilibrium distribution function describes the state of these models from the beginning of their dynamical evolution. Violent relaxation in the open-cluster models proceeds under equilibrium conditions in phase space and does not result in virial equilibrium. The cluster relaxation times in the one-dimensional spaces of a number of stellar-motion parameters are estimated. The stellar fluxes are anisotropic in some two-dimensional parameter spaces. Such open-cluster models exhibit a number of manifestations of self-organization (an energy flux toward the center of the cluster and a transfer of energy from large-scale to small-scale motions, periodic decreases in the entropies of the cluster models with a period equal to that of oscillations of the regular field of the cluster, etc.). It is concluded that violent relaxation represents one form of self-organization in such systems.     

Proper motions of open star clusters and the rotation rate of the galaxy

The mean proper motions of 167 Galactic open clusters with radial-velocity measurements are computed from the data of the Tycho-2 catalog using kinematic and photometric cluster membership criteria. The resulting catalog is compared to the results of other studies. The new proper motions are used to infer the Galactic rotation rate at the solar circle, which is found to be ω₀=+24.6±0.8 km s^?1 kpc^?1. Analysis of the dependence of the dispersion of ω₀ estimates on heliocentric velocity showed that even the proper motions of clusters with distances r>3 kpc contain enough useful information to be used in kinematic studies demonstrating that the determination of proper motions is quite justified even for very distant clusters.     

Comparison of the luminosity functions of open clusters based on USNO-A1 data

The luminosity and mass functions of a group of Galactic open clusters are constructed by applying a statistical method to photometric data from the USNO-A1 catalog. Despite some limitations, this catalog can be used for statistical analyses in Galactic astronomy. Pairwise comparisons of the derived cluster luminosity functions are performed for five age intervals. The differences between the luminosity functions of the open clusters are not statistically significant in most cases. It is concluded that the luminosity functions are approximately universal throughout a large volume in the solar neighborhood. Combined luminosity and mass functions are constructed for six age intervals. The slope of the mass spectrum may vary somewhat from cluster to cluster, and the mean slope may be somewhat higher than the Salpetervalue.     

Equilibrium after violent relaxation in numerical dynamical models of open clusters

Models of open clusters in a state of equilibrium in the space of the three parameters of the stellar motion and simultaneously far from virial equilibrium are analyzed. A formula for the phase-space stellar number density in such cluster models is derived, as well as formulas for the integrated and differential energy distributions of the phase-space volume occupied by cluster stars per unit stellar mass. These three quantities are computed for several times exceeding the time for violent relaxation of the cluster model. The phase-space density function obtained is used to compute the distribution of the absolute values of stellar velocities for the cluster model considered.     

Equilibrium phase-space density distribution in numerical dynamical models of open clusters

In dynamical models for open clusters, virial equilibrium is not achieved over the violent relaxation time scale τ_vr. The stars form an equilibrium distribution in (?, ?_ζ, l) space, where ? and l are the energy and angular momentum per unit stellar mass in the combined field of the Galaxy and cluster and ?_ζ is the energy of motion perpendicular to the Galactic plane per unit mass of cluster stars in the gravitational field of the Galaxy. This distribution of stars changes little when t>τ_vr. The stellar phase-space distribution corresponding to this type of equilibrium and the regular cluster potential vary periodically (or quasi-periodically) with time. This phase-space equilibrium is probably possible due to an approximate balance in the stellar transitions between phase-space cells over times equal to the oscillation period for the regular cluster field.     

A study of red giants in the fields of open clusters. Cluster members

We present the results of a comparative analysis of the atmospheric chemical abundances of red giants in several open clusters: the Hyades, Collinder 350, NGC 6475, and Ruprecht 147. We determined the atmospheric parameters of all the stars and the elemental abundances in their atmospheres, as well as their masses, Galactic velocities, and the elements of their orbits in the Galaxy. The observed excess [Na/Fe] and [Eu/Fe] abundances in the atmospheres of Hyades giants suggests that matter later used for star formation had been enriched in the ejecta from type II supernovae.     

The motion of halo stars in dynamical numerical models of open clusters

Equations of motion containing a small parameter μ are derived for stars at the peripheries of open star clusters. The parameter μ is obtained for six numerical open-cluster models. The general analytical solution of these equations of motion for μ = 0 is found. An iterative method is used to derive the frequencies of the stellar motions for first-order expansions in μ of the solutions of the equations of motion for stars at the cluster periphery. Applications of the results are discussed.     

Estimates of relaxation times in numerical dynamical models of open star clusters

Relaxation times in the spaces of several stellar-motion parameters are obtained for a number of open-cluster models. The differences between the relaxation times in these spaces increase with the degree of nonstationarity of the cluster models. In the course of the cluster’s evolution, the relaxation times increase in all the spaces considered. During violent relaxation, the stars occupy all domains accessible to them, first in absolute velocity and then in clustercentric distance. The dependence of the coarse-grained phase-space density of the cluster on small initial perturbations of the phase-space coordinates of its constituent stars tends to decrease at times exceeding the time scales for violent and “collisional” relaxation.     

Properties of stellar trajectories in numerical dynamical models of open star clusters

Stellar trajectories in models of open star clusters that are nonstationary in the regular field of the cluster are analyzed. The maximum characteristic Lyapunov exponents λ of the trajectories of the stellar motions in the open cluster are estimated. The mean λ in the open-cluster models considered are \(\bar \lambda \simeq ({\rm M}yr)^{ - 1} \). Cluster cores and halos are regions of highly stochastic and more ordered stellar motions, respectively. The mean Lyapunov exponent, \(\bar \lambda \), increases with the cluster density, as does the size of the highly stochastic region in the cluster core. The stellar trajectories in phase space are “glued” to a domain with a given λ. A Fourier analysis of the stellar trajectories in the open-cluster models is performed. The distributions of the periods of the stellar trajectories with the highest power-spectrum levels are constructed. The distributions of the periods corresponding to the most significant oscillations of the stellar trajectories exhibit peaks with periods commensurable with (or close to) those of the most significant oscillations of the regular field of the system. Specific features of the distributions of the periods of the most significant oscillations of the stellar trajectories and the origins of the formation of these features in the open-cluster models are discussed.     

On minimum spanning trees and the intergradation of clusters

Cluster analysis groups samples, but does not generally show gradations between clusters. To illustrate these relationships, principal-coordinate analysis and multidimensional scaling can be employed, but these methods may not be appropriate due to structural distortion. A minimum spanning tree (MST) computes a point-to-point path through the original matrix, and clusters are mapped on the MST diagram. The MST linkages, therefore, provide a natural continuum between the clusters, without distortion. Forty-six Silurian Tonoloway Limestone samples were analyzed for 16 sedimentary features indicative of subaerial exposure. Cluster analysis identified nine clusters which were reduced to five final, ordered groups by MST.     

On the dynamics of scree slopes

SummaryOn the Dynamics of Scree Slopes It is shown that the following dynamical phenomena occur on a scree slope: Collapse because of lack of stability, rapid slides of debris material, slow sliding of large blocks, jumping and rolling of large stones, and particle-separation due to a sieving effect. The mechanisms of these phenomena are investigated in detail; it is shown that the stability and the slow sliding of blocks can be explained by soil mechanics methods, the debris slides by the law of dry friction, the rolling of stones by the laws or rolling of spheres, and the sieving effect by a theory analogous to that encountered in the filtration of suspensions through porous media.

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ZusammenfassungZur Dynamik der Schutthalden Es wird gezeigt, daß folgende dynamische Vorgänge auf einer Schutthalde stattfinden: Zusammenbruch mangels Stabilität, schnelles Rutschen von Schuttmaterial, langsames Gleiten von großen Blöcken, Hüpfen und Rollen von großen Steinen und Teilchenseparation durch einen Siebeffekt. Die Mechanismen dieser Phänomene werden im Detail untersucht; es wird gezeigt, daß die Instabilität und das langsame Gleiten von großen Blöcken durch bodenmechanische Theorien erklärt werden können; das schnelle Rutschen von Schutt durch die Gesetze der trockenen Reibung; das Rollen von Steinen durch die Rollgesetze von Kugeln und der Siebeffekt durch eine Theorie, die derjenigen der Filtration von Suspensionen durch poröse Stoffe analog ist.

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RésuméPhenomènes dynamiques d'un talus d'éboulis Les phénomènes dynamiques d'un talus d'éboulis décrits ci-après sont des glissements rapides de masses tombées d'une paroi, des glissements lents de gros blocs, de blocs qui roulent ou sautent et de la séparation des particules par effet de tamisage. Il est montré que la stabilité et le glissement lent peuvent être expliqués par les méthodes de la mécanique des sols, le glissement d'éboulis par la loi du frottement sec, le roulement de blocs par les lois de sphères roulantes et l'effet de tamisage par une théorie analogue à celle qui explique la filtration de la matière en suspension par un milieu poreux.

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Report from 22nd Geomechanic-Colloquium, October 1973, Salzburg.

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With 12 Figures     

On the thermodynamics of open systems in geology

Semipermeable membranes are necessary conditions for the realisation of equilibrium open systems in the Earth's crust. Membraneless cases, as at the boundary Earth-Atmosphere, are rather exceptional. The application of the D. S. Korzhinskii's thermodynamics of open systems to mineral equilibria in rocks may be justified only if the perfect mobility of components is proved independently viz: by means other than the analysis of mineral paragenesis alone.     

The subsystem of open clusters in the post-hipparcos era: Cluster structural parameters and proper motions

The wide neighborhoods of 401 open clusters are analyzed using the modern, high-precision, homogeneous ASCC-2.5 all-sky catalog. More than 28000 possible cluster members (including about 12500 most probable members) are identified using kinematic and photometric criteria. Star counts with the ASCC-2.5 and USNO-A2.0 catalogs are used to determine the angular and linear radii of the cluster cores and coronas, which exceed the previously published values by factors of two to three. The segregation (differing central concentration) of member stars by magnitude is observed. The mean proper motions are determined directly in the Hipparcos system for 401 clusters, for 183 of them for the first time. The heliocentric distances of 118 clusters are determined for the first time based on color-magnitude diagrams for their identified members.     

On measuring time-averaged velocities in open channel flow

The problem of finding the necessary measuring time to determine the mean velocity in open channel flow is treated. The result shows an ‘on the safe side’ relation between measuring time and standard deviation of measured velocities.