Models Of Open Clusters Research Articles

Open-cluster density profiles derived using a kernel estimator

Surface and spatial radial density profiles in open clusters are derived using a kernel estimator method. Formulae are obtained for the contribution of every star into the spatial density profile. The evaluation of spatial density profiles is tested against open-cluster models from N-body experiments with N = 500. Surface density profiles are derived for seven open clusters (NGC 1502, 1960, 2287, 2516, 2682, 6819 and 6939) using Two-Micron All-Sky Survey data and for different limiting magnitudes. The selection of an optimal kernel half-width is discussed. It is shown that open-cluster radius estimates hardly depend on the kernel half-width. Hints of stellar mass segregation and structural features indicating cluster non-stationarity in the regular force field are found. A comparison with other investigations shows that the data on open-cluster sizes are often underestimated. The existence of an extended corona around the open cluster NGC 6939 was confirmed. A combined function composed of the King density profile for the cluster core and the uniform sphere for the cluster corona is shown to be a better approximation of the surface radial density profile.The King function alone does not reproduce surface density profiles of sample clusters properly. The number of stars, the cluster masses and the tidal radii in the Galactic gravitational field for the sample clusters are estimated. It is shown that NGC 6819 and 6939 are extended beyond their tidal surfaces.

Wavenumber spectra of oscillations of open star cluster models

A correlation and spectral analysis of the phase-space density oscillations and oscillations of absolute values of stellar velocities is performed for six open cluster models (for cluster model 1 different smoothing parameters e for force functions in equations of stellar motions is considered). The mutual correlation functions are computed for the fluctuations of phase-space density f and absolute values of stellar velocities v. The spatial Fourier transform of the correlation functions with zero time shift is used to compute the wavenumber spectra of the oscillations of f and v. The wavenumber spectrum of phase-space density oscillations has a more simple structure than the wavenumber spectrum of oscillations of absolute velocities v. The most powerful oscillations of f and v are located in the domain of small wavenumbers k (and large wavelengths λ > 0.91–1.25 pc) with homologous cluster oscillations contributing significantly to the mean power of oscillations of f and v. Recurrent oscillations of v at equally spaced k are found in five cluster models. The wavenumber spectra of oscillations of f and v for cluster model 1 are found to depend on e, and the corresponding dependences are analyzed. These wavenumber spectra are found to be appreciably less dependent on e than the frequency spectra of oscillations of f. An increase of e has a weaker effect on the wavenumber spectra of oscillations of f than on those of the oscillations of absolute velocity v. The slope of the wavenumber spectrum increases, on the average, with increasing e because of the change of the radial structure of the cluster model. Astrophysical applications of the results (the dissimilarity of our spectra from the Kolmogorov spectrum, discrete structure of the wavelength spectrum of oscillations of v in five cluster models, widths of the spectra of the most powerful oscillations of cluster models, estimates of cluster tidal radii, etc.) are discussed.

Correlations, spectra, and instability of phase-space density fluctuations in open-cluster models

The dynamical evolution of six open star cluster models is analyzed using the correlation and spectral analysis of phase-space density fluctuations. The two-time and mutual correlation functions are computed for the fluctuations of the phase-space density of cluster models. The data for two-time and two-particle correlations are used to determine the correlation time for phase-space density fluctuations ((0.1–1) τ v.r., where τ v.r. is the violent relaxation time of the model) and the average phase velocities of the propagation of such fluctuations in cluster models. These velocities are 2–20 times smaller than the root mean square velocities of the stars in the cluster core. The power spectra and dispersion curves of phase-space density fluctuations are computed using the Fourier transform of mutual correlation functions. The results confirm the presence of known unstable phase-space density fluctuations due to homologous fluctuations of the cluster cores. The models are found to exhibit a number of new unstable phase-space density fluctuations (up to 32–41 pairs of fluctuations with different complex conjugate frequencies in each model; the e-folding time of the amplitude growth of such fluctuations is (0.4–10) τ v.r. and their phases are distributed rather uniformly). Astrophysical applications of the obtained results (irregular structure of open star clusters, formation and decay of quasi-stationary states in such clusters) 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.

Phase-Density Fluctuations at the Centers of Six Open Clusters

Instability of proper fluctuations of the phase density at the centers of six open clusters is demonstrated. The boundaries of the region of instability in terms of the mean mass densities of the cluster cores are determined. Estimates for a number of dynamical parameters (the potential, velocity dispersion of the stellar motions, mass density, phase-density fluctuation period, time for the development of instability of the fluctuations, etc.) are obtained for the central parts of these clusters, as well as for numerical dynamical models for open clusters. Equation that can be used to estimate the dynamical mass of a cluster taking into account the influence of the external field of the Galaxy and the non-stationarity of the cluster is derived. Allowance for these effects makes it possible to decrease the dynamical masses of clusters compared to the virial masses assuming isolated systems. Astrophysical applications of the results 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 γ1, corresponding to the growth increment γ2 for this instability. Estimates of γ1 and γ2 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, τvr. We estimated the time for synchronization of the rotation of the open-cluster models and their motion around the center of the Galaxy, ts, which, depending on the model parameters, is ts ≃ (5–27)τvr. 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, tσ ≃ (6 − 25)τvr. 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.

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.

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.

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.

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 τvr. The stars form an equilibrium distribution in (ɛ, ɛζ, l) space, where ɛ and l are the energy and angular momentum per unit stellar mass in the combined field of the Galaxy and cluster and ɛζ 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>τvr. 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.