Kinematic analysis of stellar radial velocities by the spherical harmonics

A method for a kinematic analysis of stellar radial velocities using spherical harmonics is proposed. This approach does not depend on the specific kinematic model and allows both low-frequency and high-frequency kinematic radial velocity components to be analyzed. The possible systematic variations of distances with coordinates on the celestial sphere that, in turn, are modeled by a linear combination of spherical harmonics are taken into account. Theoretical relations showing how the coefficients of the decomposition of distances affect the coefficients of the decomposition of the radial velocities themselves have been derived. It is shown that the larger the mean distance to the sample of stars being analyzed, the greater the shift in the solar apex coordinates, while the shifts in the Oort parameter A are determined mainly by the ratio of the second zonal harmonic coefficient to the mean distance to the stars, i.e., by the degree of flattening of the spatial distribution of stars toward the Galactic plane. The distances to the stars for which radial velocity estimates are available in the CRVAD-2 catalog have been decomposed into spherical harmonics, and the existing variations of distances with coordinates are shown to exert no noticeable influence on both the solar motion components and the estimates of the Oort parameter A, because the stars from this catalog are comparatively close to the Sun (no farther than 500 pc). In addition, a kinematic component that has no explanation in terms of the three-dimensional Ogorodnikov-Milne model is shown to be detected in the stellar radial velocities, as in the case of stellar proper motions.     

Application of vector spherical harmonics for kinematic analysis of stars from zonal catalogues

We solve the problem on a kinematic analysis of the three-dimensional velocity field of stars from zonal catalogues, i.e., catalogues in which the stars are presented at all right ascensions in some declination zones. We have constructed a system of vector spherical harmonics with the properties of completeness and orthogonality for a chosen declination zone. We suggest a method that allows the Ogorodnikov-Milne model parameters in the Galactic coordinate system to be estimated by analyzing the proper motions and radial velocities of stars in the equatorial coordinate system. The vector spherical harmonics are shown to have the following advantages over the standard approach based on a direct leastsquares estimation of the parameters for a specific model. First, in contrast to the standard approach, the new method can reveal all systematic components of the velocity field irrespective of a particular model. Second, it allows one to get rid of the correlation between the sought-for parameters, which presents a serious problem for the conventional method in the case of zonal catalogues. Third, the method of vector spherical harmonics allows the kinematic parameters to be estimated at least by two techniques. Comparison of these two solutions makes it possible to test the standard kinematic model for compatibility with the observational data. The developed method has been tested on the basis of numerical experiments and applied for a kinematic analysis of the proper motions of Tycho-2 stars in the southern hemisphere for which the parallaxes can be estimated using data from the Tycho-2 Spectral Type Catalogue.     

Retrograde precession of stellar orbits and gravitational loss-cone instability in galactic centers

We consider disk and spherical subsystems of stars with nearly radial orbits under conditions when the well-known radial orbit instability is not possible. This requires that the precession of stellar orbits be retrograde, i.e., in the direction opposite to the orbital rotation of stars. We show that an instability that is an analogue of the loss-cone instability known in plasma physics can then develop in the presence of a “loss cone” in the angular momentum distribution of stars, which ensures a deficit or even absence of stars with low angular momenta. Examples of systems with a loss cone are the centers of galaxies or star clusters with massive black holes. The instability can produce a flux of stars onto the galactic center, i.e., it can serve as a mechanism of fueling the nuclear activity of galaxies. Mathematically, the problem is reduced to analyzing simple characteristic equations that describe small perturbations in a disk and a sphere of radially highly elongated stellar orbits. In turn, these characteristics equations are derived through a number of successive simplifications of the general linearized Vlasov equations (i.e., the system that includes the collisionless Boltzmann kinetic equation and the Poisson equation) in action—angle variables.     

Distortion of the stellar velocity field parameters due to systematic variations of parallaxes over the celestial sphere

We study the effect of systematic variations in stellar parallaxes over the celestial sphere on the results of a kinematic analysis of stellar proper motions. Our approach is based on the representation of stellar parallaxes by scalar spherical harmonics and on the decomposition of stellar proper motions into a system of vector spherical harmonics. We derive theoretical relations that relate the coefficients of the decomposition of stellar proper motions into toroidal and spheroidal harmonics to the coefficients of the decomposition of stellar parallaxes into scalar spherical harmonics. We have established that the systematic variations of parallaxes over the celestial sphere distort all parameters of the linear Ogorodnikov-Milne model and can be responsible for the appearance of beyond-the-model harmonics. We have performed a kinematic analysis of the proper motions of blue-white and red giants based on Hipparcos data. The parallaxes of blue-white giants show a strong dependence on Galactic latitude (with predominant contraction along the Galactic equator). In contrast, the deviations of the parallaxes from the mean for red giants are localized only in two regions of the celestial sphere. For these samples, the effect of parallax variations over the celestial sphere on kinematic parameters has turned out to be comparable to their rms errors. The global solutions performed using both samples have revealed strong beyond-the-model kinematic effects described by second-order toroidal harmonics and third-order spheroidal harmonics. Using the solutions performed separately in the northern and southern Galactic hemispheres, we have established that not the systematic variations of parallaxes over the celestial sphere but the retardation of Galactic rotation with increasing distance of stars from the principal Galactic plane is mainly responsible for the appearance of these harmonics. Based on these samples of stars, we have estimated the magnitude of the vertical Galactic rotation velocity gradient to be 18.0±2.9 and 22.7±2.2 km s^?1 kpc^?1, respectively.     

Two-integral distribution functions for axisymmetric systems

Some formulae are presented for finding two-integral distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. They come from a combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multiplied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.     

Method of ignoring the systematic variations of stellar parallaxes over the celestial sphere in a kinematic analysis of stellar proper motions

A method for determining the velocity field parameters free from the distortions due to the systematic variations of stellar parallaxes over the celestial sphere is proposed. The method is based on the approximation of parallaxes as a function of coordinates on the sphere using spherical harmonics and can be applied in those cases where the solar motion cannot be eliminated from the stellar proper motions. Numerical experiments have shown that our method is able to obtain accurate coordinates of the solar apex and to calculate the kinematic parameters of the Ogorodnikov-Milne model to within three coefficients of the decomposition of parallaxes into first-order spherical harmonics. Examples of applying the method to the stellar proper motions of the Hipparcos catalogue, which admits checking the results using trigonometric parallaxes, are provided. Such a check has been found to yield a positive result only for nearby stars at heliocentric distances that do not exceed 400 pc and for which the parallaxes were determined with a relative error of at least 30%. An interesting feature of this method is the possibility to construct the shape of the figure which is formed by the deviations of the parallaxes from the sphere corresponding to the average parallaxes of the stars under consideration. It should be specially emphasized that all of this is done in the complete absence of information about the stellar parallaxes. The “solar terms” of the stellar proper motions that are formed by the products of the parallaxes by the solar motion components relative to the centroid of stars are the main source of information about the parallaxes here.     

A two-component model of a spherical stellar system

As a generalization of the King's widely-used model for a spherical stellar system containing stars of identical masses, a two-component model of an equilibrium stellar system consisting of stars of two different masses is built proceeding from a proper generalization of the distribution function. In such a system, the principle of thermal equilibrium, generally speaking, does not hold, i.e. equipartition of mean kinetic energy between light and heavy stars is absent.The radial distribution of partial density and velocity dispersion for each of the components and of the total density and velocity dispersion is obtained. The profiles of both the total density and velocity dispersion differ considerably from those given by the King's one-component model, especially in the centre of the system. In particular, the velocity dispersion of stars may there have a non-monotonous behaviour. The possibility of applying the many-component model for the explanation of the recently discovered luminosity cusps in the central regions of some elliptical galaxies is briefly discussed.     

Effects of external tidal field on the evolution of the outer regions of multi-mass star clusters

We present N -body simulations (including an initial mass function) of globular clusters in the Galaxy in order to study effects of the tidal field systematically on the properties of the outer parts of globular clusters. Using nbody6 , which correctly takes into account the two-body relaxation, we investigate the development of tidal tails of globular clusters in the Galactic tidal field. For simplicity, we have employed only the spherical components (bulge and halo) of the Galaxy, and ignored the effects of stellar evolution which could have been important in the very early phase of the cluster evolution. The total number of stars in our simulations is about 20 000, which is much smaller than the realistic number of stars. All simulations had been done for several orbital periods in order to understand the development of the tidal tails. In our scaled-down models, the relaxation time is sufficiently short to show the mass segregation effect, but we did not go far enough to see the core collapse, and the fraction of stars lost from the cluster at the end of the simulations is only ∼10 per cent. The radial distribution of extra-tidal stars can be described by a power law with a slope around −3 in surface density. The directions of tidal tails are determined by the orbits and locations of the clusters. We find that the length of tidal tails increases towards the apogalacticon and decreases towards the perigalacticon. This is an anti-correlation with the strength of the tidal field, caused by the fact that the time-scale for the stars to respond to the potential is similar to the orbital time-scale of the cluster. The escape of stars in the tidal tails towards the pericentre could be another reason for the decrease of the length of tidal tails. We find that the rotational angular velocity of tidally induced clusters shows quite different behaviour from that of initially rotating clusters.     

Kinematics of hypervelocity stars in the triaxial halo of the Milky Way

Hypervelocity stars (HVSs) ejected by the massive black hole at the Galactic Centre have unique kinematic properties compared to other halo stars. Their trajectories will deviate from being exactly radial because of the asymmetry of the Milky Way potential produced by the flattened disc and the triaxial dark matter halo, causing a change of angular momentum that can be much larger than the initial small value at injection. We study the kinematics of HVSs and propose an estimator of dark halo triaxiality that is determined only by instantaneous position and velocity vectors of HVSs at large Galactocentric distances ( r ≳ 50 kpc). We show that, in the case of a substantially triaxial halo, the distribution of deflection angles (the angle between the stellar position and velocity vector) for HVSs on bound orbits is spread uniformly over the range 10°–180°. Future astrometric and deep wide-field surveys should measure the positions and velocities of a significant number of HVSs, and provide useful constraints on the shape of the Galactic dark matter halo.     

Radial heating of a galactic disc by multiple spiral density waves

We consider a differentially rotating, 2D stellar disc perturbed by two steady-state spiral density waves moving at different pattern speeds. Our investigation is based on direct numerical integration of initially circular test-particle orbits. We examine a range of spiral strengths and spiral speeds and show that stars in this time-dependent gravitational field can be heated (their random motions increased). This is particularly noticeable in the simultaneous propagation of a two-armed spiral density wave near the corotation resonance (CR), and a weak four-armed one near the inner and outer 4:1 Lindblad resonances. In simulations with two spiral waves moving at different pattern speeds, we find: (i) the variance of the radial velocity,  σ² _R   , exceeds the sum of the variances measured from simulations with each individual pattern; (ii)  σ² _R   can grow with time throughout the entire simulation; (iii)  σ² _R   is increased over a wider range of radii compared to that seen with one spiral pattern; and (iv) particles diffuse radially in real space, whereas they do not when only one spiral density wave is present. Near the CR with the stronger, two-armed pattern, test-particles are observed to migrate radially. These effects take place at or near resonances of both spirals, so we interpret them as the result of stochastic motions. This provides a possible new mechanism for increasing the stellar velocity dispersion in galactic discs. If multiple spiral patterns are present in the Galaxy, we predict that there should be large variations in the stellar velocity dispersion as a function of radius.     

Winds from clusters with non-uniform stellar distributions

We present the analytic and numerical models of the 'cluster wind' resulting from the multiple interactions of the winds ejected by the stars of a dense cluster of massive stars. We consider the case in which the distribution of stars (i.e. the number of stars per unit volume) within the cluster is spherically symmetric, has a power-law radial dependence, and drops discontinuously to zero at the outer radius of the cluster. We carry out comparisons between an analytic model (in which the stars are considered in terms of a spatially continuous injection of mass and energy) and 3D gasdynamic simulations (in which we include 100 stars with identical winds, located in 3D space by statistically sampling the stellar distribution function). From the analytic model, we find that for stellar distributions with steep enough radial dependencies, the cluster wind flow develops a very high central density and a non-zero central velocity, and for steeper dependencies, it becomes fully supersonic throughout the volume of the cluster (these properties are partially reproduced by the 3D numerical simulations). Therefore, the wind solutions obtained for stratified clusters can differ dramatically from the case of a homogeneous stellar distribution (which produces a cluster wind with zero central velocity, and a fully subsonic flow within the cluster radius). Finally, from our numerical simulations, we compute predictions of X-ray emission maps and luminosities, which can be directly compared with observations of cluster wind flows.     

Colour and radial velocity variations in pulsating subluminous B stars

The utility of pulsations for the investigation of the structure and evolution of subdwarf B stars is considerable. However, the small number of detected modes generally limits the potential for a traditional seismological analysis such as that carried out for the Sun. Therefore, it is crucial to acquire additional primary data to characterise the stellar oscillations more completely. We review recent studies of radial velocity amplitudes and introduce new multi-colour photometry of small-amplitude sdBV stars. We also discuss a set of models for radial velocity and colour variations and demonstrate how these may be used to infer the spherical and azimuthal degrees of observed pulsations.     

Local galactic evolution

The evolution of the Solar neighbourhood is followed using a unique, magnitude complete and kinematically unbiased sample of 14,000 F, G, and K dwarfs. Metallicity, age, space motion and galactic orbits have been determined for all stars. The result is a detailed view of the complex evolution of the local Milky Way, which must be matched by any model for the chemical and dynamical evolution of the Galactic disk. E.g., such models must explain the shape and large scatter in the age-metallity relation as well as the overall metallicity distribution; the evolution of stellar kinematics with age; the distribution of stars in velocity space; and the contributions from the thick and thin disks to all these relations. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the problem of secular variability in the stellar initial mass function

The hypothesis of secular variations in the stellar initial mass function (IMF) is studied. It is found that neither the present-day mass function of the nearby main sequence field stars nor the velocity distribution of these stars are contradictory with this hypothesis. The luminosity functions of unbiased kinematically defined age-groups of the nearby stars also provide no strong constraints. Simple evolution models with time-dependent IMF and star formation rate enable to fit the data of blue irregular galaxies. Some problems with an universal IMF are pointed out.     

On various approaches to investigating the instability of stellar systems with highly elongated stellar orbits

We study the various approximations used to investigate the eigenmode spectrum for systems with highly elongated stellar orbits. The approximation in which the elongated orbits are represented by thin rotating spokes, with the rotation imitating the precession of real orbits, is the simplest and most natural one. However, we show that using this pictorial approximation does not allow the picture of stability to be properly presented. We show that for stellar systems with a plane disk geometry, this approach does not allow unstable spectral modes to be obtained even in the leading order in small parameter, which characterizes the spread of nearly radial orbits in angular momentum. For spherical systems, where the situation is more favorable, the spectrum can be determined but only in the leading order in this parameter. A rigorous approach based on the solution of more complex integral equations given here should be used to properly investigate the stability of stellar systems.     

Dark matter in dwarf spheroidals – II. Observations and modelling of Draco

We present stellar radial velocity data for the Draco dwarf spheroidal (dSph) galaxy obtained using the AF2/WYFFOS instrument combination on the William Herschel Telescope. Our data set consists of 186 member stars, 159 of which have good quality velocities, extending to a magnitude   V ≈19.5  with a mean velocity precision of ≈2 km s⁻¹. As this survey is based on a high-precision photometric target list, it contains many more Draco members at large radii. For the first time, this allows a robust determination of the radial behaviour of the velocity dispersion in a dSph.
We find statistically strong evidence of a rising velocity dispersion consistent with a dark matter halo that has a gently rising rotation curve. There is a <2 σ signature of rotation about the long axis, inconsistent with tidal disruption as the source of the rising dispersion. By comparing our data set with earlier velocities, we find that Draco probably has a binary distribution and fraction comparable to those in the solar neighbourhood.
We apply a novel maximum likelihood algorithm and fit the velocity data to a two parameter spherical model with an adjustable dark matter content and velocity anisotropy. Draco is best fit by a weakly tangentially anisotropic distribution of stellar orbits in a dark matter halo with a very slowly rising rotation law  ( v _circ∝ r ^0.17)  . We are able to rule out both a mass-follows-light distribution and an extended halo with a harmonic core at the 2.5 to 3 σ significance level, depending on the details of our assumptions about Draco's stellar binary population. Our modelling lends support to the idea that the dark matter in dwarf spheroidals is distributed in the form of massive, nearly isothermal haloes.     

Two-integral distribution functions for axisymmetric stellar systems with separable densities

We show different expressions of distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. The density of the system is required to be a product of functions separable in the potential and the radial coordinates, where the functions of the radial coordinate are powers of a sum of a square of the radial coordinate and its unit scale. The even part of the two-integral DF corresponding to this type of density is in turn a sum or an infinite series of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. A similar expression of its odd part can be also obtained under the assumption of the rotation laws. It can be further shown that these expressions are in fact equivalent to those obtained by using Hunter & Qian's contour integral formulae for the system. This method is generally computationally preferable to the contour integral method. Two examples are given to obtain the even and odd parts of their two-integral DFs. One is for the prolate Jaffe model and the other for the prolate Plummer model.
It can be also found that the Hunter–Qian contour integral formulae of the two-integral even DF for axisymmetric systems can be recovered by use of the Laplace–Mellin integral transformation originally developed by Dejonghe.     

Blue straggler stars in dwarf spheroidal galaxies – II. Sculptor and Fornax

The existence of blue straggler stars (BSSs) in dwarf spheroidal galaxies (dSphs) is still an open question. In fact, many BSS candidates have been observed in the Local Group dSphs, but it is unclear whether they are real BSSs or young stars. Shedding light on the nature of these BSS candidates is crucial in order to understand the star formation history of dSphs. In this paper, we consider BSS candidates in Sculptor and Fornax. In Fornax, there are strong hints that the BSS population is contaminated by young stars, whereas in Sculptor there is no clear evidence of recent star formation. We derive the radial and luminosity distribution of BSS candidates from wide field imaging data extending beyond the nominal tidal radius of these galaxies. The observations are compared with the radial distribution of BSSs expected from dynamical simulations. In Sculptor, the radial distribution of BSS candidates is consistent with that of red horizontal branch (RHB) stars and is in agreement with theoretical expectations for BSSs generated via mass transfer in binaries. On the contrary, in Fornax, the radial distribution of BSS candidates is more concentrated than that of all the considered stellar populations. This result supports the hypothesis that most of BSS candidates in Fornax are young stars, and this is consistent with previous studies.     

The resolved red giant branches of E/S0 galaxies

Formation paradigms for massive galaxies have long centered around two antipodal hypotheses – the monolithic-collapse and the accretion/merger scenarios. Empirical data on the stellar contents of galaxy halos is crucial in order to develop galaxy formation and assembly scenarios which have their root in observations, rather than in numerical simulations. The Hubble Space Telescope (HST) has enabled us to study directly individual stars in the nearby E/S0 galaxies Cen A, NGC 3115, NGC 5102, and NGC 404. We here present and discuss HST single-star photometry in V and I bands. Using color-magnitude diagrams and stellar luminosity functions, we gauge the galaxies' stellar contents. This can be done at more than one position in the halo, but data with deeper limiting magnitudes are desired to quantify the variation of metallicity with galactocentric radius. We here compare the color distributions of red giant stars with stellar isochrones, and we intercompare the galaxies' halo populations, noting that their total absolute V magnitudes cover the range from about –21.5 to –17.5. In the future, we plan to model the stellar metallicity distributions with the aim to constrain chemical enrichment scenarios, a step towards unravelling the evolutionary history of elliptical and lenticular galaxies. This revised version was published online in August 2006 with corrections to the Cover Date.     

Astrophysical supplements to the ASCC‐2.5. I. Radial velocity data

We present a catalogue of radial velocities of Galactic stars with high precision astrometric data CRVAD which is the result of the cross‐identification of star lists from the General Catalog of Average Radial Velocities (GCRV) and from the homogeneous All‐sky Compiled Catalogue of 2.5Million Stars (ASCC‐2.5). The CRVAD includes accurate J2000 equatorial coordinates, proper motions and trigonometric parallaxes in the Hipparcos system, Johnson's BV photometric data, spectral types, multiplicity and variability flags from the ASCC‐2.5, and radial velocities, stellar magnitudes and spectral types from the GCRV for 34553 ASCC‐2.5 stars. The CRVAD was used for the construction of a sample of standard stars with accurate astrometric, photometric and radial velocity data for the RAVE project. A second application of the CRVAD , the radial velocity determination for 292 open clusters (including 97 with previously unknown radial velocities), using their newly defined members from proper motions and photometry in the ASCC‐2.5, is briefly described. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)