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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

On the dynamics of open clusters

The kinematics of stars in open-cluster models are investigated. The central regions of these model clusters are fairly “cool, ” leading to gravitational instability in the cluster cores. A temporary virilization is observed in the model clusters, during which an appreciable fraction of the fluctuation energy of the cluster is temporarily transformed into the kinetic energy of the peculiar motions of stars. The duration of this stage can reach ∼10⁸ yr. The origins of this temporary virialization of the model clusters are discussed. The instability of the intrinsic fluctuations of the phase density in the centers of six clusters and model clusters is investigated. Several new regions of unstable phase-density fluctuations are found, in addition to those already known. The observed and model clusters can move toward a stable equilibrium state when the density of the cluster core both decreases and increases. The structure of regions of instability of the phase-density fluctuations in the centers of the six clusters is studied. Resonance curves for the amplitudes of steady-state phase-density fluctuations in the center of NGC 6705 are constructed. Analysis of the structure of the regions of instability indicates an appreciable rate of fluctuation-energy loss due to relaxation effects. The instability increments and widths of regions of instability fall off with increasing distance of the cluster from the Galactic center. An important role in the formation of regions of instability may be played by resonances between the frequency of the orbital motion of the cluster in the Galaxy and the frequencies of intrinsic phase-density fluctuations in the centers of the clusters considered.     

数值模型在地下水管理中的应用

为进一步推动模拟工作，并使模拟尽可能标准化和规范化，介绍国际上认可的建立模型的步骤、方法以及常用的模拟软件。引用作者在国外所做的两个实例，对建立模型的步骤、方法以及模型标定方法和标定结果作了具体的介绍。国内应鼓励使用已有的国际标准化模拟软件，进一步提高建模的专门知识和技巧。数值模型是真实世界的一个数学表达，对模型的产出应有一个现实的看法和专业的理解。模型建立前，应对现场数据作详细的专业分析，以确保概念模型的正确。在模型的建设中，应把注意力放在如何使模型在水文地质和物理过程方面去逼近真实世界。模型中采用的计算条件或方法都应有水文地质依据作支持，而不能仅仅为了逼近实测数据而随意地加入无依据的计算条件，这样的模型是不可靠的，其结果将是无实际意义的或是误导的。     

Evolution of dark-matter halos in numerical models

The properties of gravitationally bound clouds (halos) of dark matter derived via numerical simulations of the distribution of dark matter in the Universe are investigated. The analysis makes use of a catalog of halos obtained in the European “MareNostrum Universe” project, which has achieved a better balance between resolution and representativeness than catalogs used earlier for similar studies. This has made it possible to refine the main tendencies displayed by the evolution of the halo masses and the angular velocities and density profiles of the halos. The results are compared with the newest available observational data and with known results obtained earlier in numerical simulations with lower resolution and using smaller samples of halos, making it possible to trace the influence of these factors on the results obtained. Disagreements between observations and numerical models obtained in earlier studies are confirmed, and possible ways to explain them discussed.     

强夯动力性能的显式非线性数值分析

充分考虑碰撞中出现的材料非线性、几何非线性、接触非线性、运动非线性以及它们之间相互耦合的特性,利用显式瞬态非线性有限元分析技术分析了强夯对地基土的冲击碰撞过程,得到了夯锤与土体之间的撞击力时间历程、土体位移、应力的时间历程,并得到了落距、夯锤重量、夯锤底面积与强夯加固深度之间的相互关系,分析结果表明：土体位移滞后于土体应力,再现了夯锤与土体碰撞的整个过程,反映了强夯过程中的一般现象和基本规律。     

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.”     

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.     

河津凝灰岩广义应力松弛特性及数值模拟研究

在很多地下工程实践中,岩体既不是纯蠕变,也不是纯应力松弛,而是随着时间的增加,应力和应变同时发生变化,表现出时间依存性,导致岩体最终破坏,这种现象用一般的蠕变和应力松弛很难解释清楚。全面系统地阐述了广义应力松弛特性及其工程物理意义,借用非线性Maxwell模型,求解了广义应力松弛条件下可变模量本构方程解析解,特别提出了通过交替荷载速率试验和加、卸载试验,精确求解峰前区域荷载速率依从性系数n_1的新方法。进行了70%和90%应力水平下广义应力松弛试验,并讨论和分析了不同应力水平、不同方向系数下广义应力松弛试验的时间依存性,最后用基于非线性Maxwell模型的可变模量本构方程对70%和90%应力水平下的广义应力松弛试验进行了数值模拟,结果表明,计算和试验结果一致性较好,很好地解释了河津凝灰岩广义应力松弛特性。     

Benchmarking numerical models of brittle thrust wedges

We report quantitative results from three brittle thrust wedge experiments, comparing numerical results directly with each other and with corresponding analogue results. We first test whether the participating codes reproduce predictions from analytical critical taper theory. Eleven codes pass the stable wedge test, showing negligible internal deformation and maintaining the initial surface slope upon horizontal translation over a frictional interface. Eight codes participated in the unstable wedge test that examines the evolution of a wedge by thrust formation from a subcritical state to the critical taper geometry. The critical taper is recovered, but the models show two deformation modes characterised by either mainly forward dipping thrusts or a series of thrust pop-ups. We speculate that the two modes are caused by differences in effective basal boundary friction related to different algorithms for modelling boundary friction. The third experiment examines stacking of forward thrusts that are translated upward along a backward thrust. The results of the seven codes that run this experiment show variability in deformation style, number of thrusts, thrust dip angles and surface slope. Overall, our experiments show that numerical models run with different numerical techniques can successfully simulate laboratory brittle thrust wedge models at the cm-scale. In more detail, however, we find that it is challenging to reproduce sandbox-type setups numerically, because of frictional boundary conditions and velocity discontinuities. We recommend that future numerical-analogue comparisons use simple boundary conditions and that the numerical Earth Science community defines a plasticity test to resolve the variability in model shear zones.     

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.