Analysis of the three-dimensional stellar velocity field using vector spherical functions

We apply vector spherical functions to problems of stellar kinematics. Using these functions allows all of the systematic components in the stellar velocity field to be revealed without being attached to a specific physical model. Comparison of the theoretical decomposition coefficients of the equations for a particular kinematical model with observational data can provide precise information about whether the model is compatible with the observations and can reveal systematic components that are not described by this model. The formalism of vector spherical functions is particularly well suited for analyzing the present and future (e.g., GAIA) catalogs containing all three velocity vector components: the propermotions in both coordinates and the radial velocity. We show that there are systematic components in the proper motions of Hipparcos stars that cannot be interpreted in terms of the linear Ogorodnikov-Milne model. The same result is also confirmed by an analysis of the radial velocities for these stars.     

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.     

Rigorous computation of proper motions and their effects on star positions

New rigorous formulas are given for the computation of the effects of proper motions and radial velocities on star positions, and for the transformation of proper motion components and radial velocities from one epoch to another. These expressions depend explicitly only on the values of the star's coordinates and distance, proper motion components and radial velocity, at the initial epoch.     

Photographic Observations of the Minor Planet (1) Ceres in Tashkent in 1951–1994

A procedure is described for remeasuring photographic plates with images of the minor planet (1) Ceres obtained in 1951–1994 with the Tashkent normal astrograph. To determine the observed spherical coordinates, the PPM catalog, based on the FK5 system, was used as a reference catalog. The calculation of the orbit of the minor planet (1) Ceres and the determination of the PPM zero-points and periodic errors on the basis of these observations yielded a unit weight error of 0.270. The derived coordinates and velocity components of (1) Ceres for J.D. = 2441600.5 are presented together with PPM zero-point and periodic-error-coefficient corrections. One of the results of the work is the high precision of the remeasured images, which makes it possible to use these observations, along with no less accurate observations of other selected minor planets, for determining corrections to the stellar coordinate system.     

The roche coordinates and their use in hydrodynamics or celestial mechanics

The aim of the present paper will be to introduce a new system of curvilinear coordinateshereafter referred to as Roche coordinates-in which spheres of constant radius are replaced by equipotential surfaces of a rotating gravitational dipole (which consists of two discrete points of finite mass, revolving around their common center of gravity); while the remaining coordinates are orthogonal to the equipotentials. It will be shown that the use of such coordinates offers a new method of approach to the solution of certain problems of particle dynamics (such as, for instance, the construction of certain types of trajectories in the restricted problem of three bodies); as well as of the hydrodynamics of gas streams in close binary systems, in which the equipotential surfaces of their components distorted by axial rotation and mutual tidal interaction constitute essential boundary conditions.Following a general outline of the problem in Section 1, the Roche coordinates associated with the equipotentials of a rotating gravitational dipole will be constructed in the plane case (Section 2), and their geometrical properties discussed. In Section 3, we shall transform the fundamental equations of hydrodynamics to their forms appropriate in the curvilinear Roche coordinates. The metric coefficients of this transformation will be formulated in a closed form in Section 4 in terms of the respective partial derivatives of the potential; while in Section 5 analytic expressions for the Roche coordinates will be given in the orbital plane of the dipole, which are exact as far as the distortion of the equipotential curves from circular form can be described by the second, third and, fourth harmonics.The concluding Section 6 will be devoted to a formulation of the equations of a mass-point in the restricted problem of three bodies in the Roche coordinates. Three special cases will be considered: (a) motion in the neighborhood of the equipotential curves; (b) motion in the direction normal to such curves; and (c) motion in the neighbourhood of the Lagrangian points. It will be shown that motion in one coordinate is possible only in limiting cases which will be enumerated; but twodimensional motions in which one velocity component is very much smaller than the other invite further study.A generalization of the plane Roche coordinates to three dimensions, with application to additional classes of problems, is being postponed for a subsequent paper.     

Optimization of lunar occultations for determining the positions of point-like X-ray sources

In view of the scheduled satellite mission EXOSAT (European X-Ray Observatory Satellite) of ESA (European Space Agency) the lunar occultation technique to determine the position of point-like X-ray sources is investigated. An error analysis for the source coordinates resulting from this technique is presented and an occultation strategy is proposed to achieve optimum lunar occultations. The analysis takes into account the errors of the space coordinates of the satellite and the Moon, the unevenness of the lunar surface, the intensities of source and background, the apparent angular velocity of the Moon as seen from the satellite, the finite sizes of the preoccultation position error boxes of the X-ray sources and the inaccuracies in the satellite orbit correction manoeuvres necessary to achieve the occultations.     

Maunder convection mode on the Sun and long solar activity minima

A model of velocity field oscillations in the solar convective zone is suggested. The system of convective equations is investigated for a thin rotating spherical envelope when the rotation velocity is depended on the coordinates. It is shown that two different structures of convective cells (longitudinal, or latitudinal) can exist in the envelope depending on gradients values of the rotation velocity and Prandtl number. It is supposed that two different regimes of convection (stationary and autofluctuating) are possible in the envelope when the angular velocity gradients are determined by the convection itself. In the case of autofluctuating regime the alternation of longitudinal and latitudinal structure of convection is realized. If one assumes that on the Sun there exists an autooscillating convection regime, then the periods of the existence of latitudinal convection structure may be associated with long periods of activity minima since according to Cowling's theorem, the action of the axisymmetric magnetic field generation mechanism is impossible under conditions of axisymmetric velocity structures.     

Effects of rotation on internal structure of the stars

The aim of the present paper will be to establish the explicit form of the equations which govern the internal structure of stars rotating with constant angular velocity formulated in terms of Clairaut coordinates (cf. Kopal, 1980) in which the radial coordinate is replaced by the total potential, which for equilibrium configurations remains constant over distorted level surfaces. The introductory Section 1 contains an account of previous work on rotating stars, commencing with Milne (1923), von Zeipel (1924) and Chandrasekhar (1933), who all employed orthogonal coordinates for their analysis. In Section 2 we shall apply to this end the curvilinear Clairaut coordinates introduced already in our previous work (cf. Kopal, 1980, 1981); and although these are not orthogonal, this disadvantage is more than offset by the fact that, in their terms, the fundamental equation of our problem will assume the form of ordinary differential equations, subject to very simple boundary conditions. The explicit form of these equations — exact to terms of fourth order in surficial distortion caused by centrifugal force—will be obtained in Section 3; while in the concluding Section 4 these will be particularized (for the sake of comparison with work of previous investigators) to stars of initially polytropic structure. These will prove to be much simpler in Clairaut coordinates than they were in any previously used frame of reference. Lastly, in Appendix A we shall present the explicit forms, in Clairaut coordinates, of the differential operators which were needed to establish the results given in Sections 3–4; while Appendix B will summarize other auxiliary algebraic relations of which use was made to formulate our fourth-order theory developed in Section 3.     

Effects of variations of parallel angular velocity and vorticity on the oscillations of compressible homogeneous rotating ellipsoids

Earlier work on the oscillations of an ellipsoid is extended to investigate the behaviour of a nonequilibrium compressible homogeneous rotating gaseous ellipsoid, with the components of the velocity field as linear functions of the coordinates, and with parallel angular velocity and uniform vorticity. The dynamical behaviour of the ellipsoid is obtained by numerically integrating the relevant differential equations for different values of the initial angular velocity and vorticity. This behaviour is displayed by the (a ₁,a ₂) and (a ₁,a ₃) phase plots, where thea _i's (i = 1, 2, 3) are the semi-diameters, and by the graphs ofa ₁,a ₂,a ₃, the volume, and the angular velocity as functions of time.The dynamical behaviour of the nonequilibrium ellipsoid depends on the deviation of the angular momentum from its equilibrium value; for larger deviations, the oscillations are more nonperiodic with larger amplitudes.An initially ellipsoidal configuration always remains ellipsoidal, but it cannot become spheroidal about its rotation axis, though it may become spheroidal instantaneously about either one of the other two principal axes.For an ellipsoid approaching axisymmetry about its axis of rotation, the angular velocity can suddenly increase by a large amount. Thus if an astrophysical object can be modelled by a nonequilibrium ellipsoid, it may occasionally undergo sudden large increases of angular velocity.     

Kinematics of SDSS subdwarfs: structure and substructure of the Milky Way halo

We construct a new sample of ∼1700 solar neighbourhood halo subdwarfs from the Sloan Digital Sky Survey (SDSS), selected using a reduced proper-motion diagram. Radial velocities come from the SDSS spectra and proper motions from the light-motion curve catalogue of Bramich et al. Using a photometric parallax relation to estimate distances gives us the full phase-space coordinates. Typical velocity errors are in the range  30–50 km s⁻¹  . This halo sample is one of the largest constructed to date and the disc contamination is at a level of ≲1 per cent. This enables us to calculate the halo velocity dispersion to excellent accuracy. We find that the velocity dispersion tensor is aligned in spherical polar coordinates and that  (σ _r , σ_φ, σ_θ) = (143 ± 2, 82 ± 2, 77 ± 2) km s⁻¹  . The stellar halo exhibits no net rotation, although the distribution of   v _φ  shows tentative evidence for asymmetry. The kinematics are consistent with a mildly flattened stellar density falling with distance like   r ^−3.75  .
Using the full phase-space coordinates, we look for signs of kinematic substructure in the stellar halo. We find evidence for four discrete overdensities localized in angular momentum and suggest that they may be possible accretion remnants. The most prominent is the solar neighbourhood stream previously identified by Helmi et al., but the remaining three are new. One of these overdensities is potentially associated with a group of four globular clusters (NGC 5466, NGC 6934, M2 and M13) and raises the possibility that these could have been accreted as part of a much larger progenitor.     

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.The reduction of order is performed in these new coordinates by using the angular velocity vector or the Euler angles.The Eulerian case of collinear motion is revisited from our own perspective.The value of the auxiliar angle is computed for the Sun–Earth–Moon system.     

General solution for interplanetary magnetic field produced from stationary sources on the sun

It is well known that interplanetary space contains Parker's Archimedean spiral magnetic field, the components of which are respectively radial and longitudinal in the solar polar coordinates (r, θ, φ) and are intimately related to each other, depending on the solar wind velocity. In this paper, we present a general solution of the interplanetary magnetic field which is produced from time-independent sources fixed on the solar surface and contains the Parker field as a particular solution. The field is first classified broadly into two types called the φ-dependent (or nonzonal) and the φ-independent (or zonal) fields. The former field is further subdivided into two types, one is the so-called Parker type and the other is the vortex type which has no radial component. The resultant of these two fields exhibits the helical (or twisted) structure in space, tentatively introduced by Lee and Fisk. The zonal field is also subclassified into two; one is the radial-type zonal field and the other is the toroidal field. These two fields are mutually independent and therefore their resultant does not always coincide in direction with the Parker field.     

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 high-resolution radio study of neutral gas in the starburst galaxy NGC 520

We present subarcsec angular resolution observations of the neutral gas in the nearby starburst galaxy NGC 520. The central kpc region of NGC 520 contains an area of significantly enhanced star formation. The radio continuum structure of this region resolves into ∼10 continuum components. By comparing the flux densities of the brightest of these components at 1.4 GHz with published 15-GHz data we infer that these components detected at 1.4 and 1.6 GHz are related to the starburst and are most likely to be collections of several supernova remnants within the beam. None of these components is consistent with emission from an active galactic nuclei. Both neutral hydrogen (H  i ) and hydroxyl (OH) absorption lines are observed against the continuum emission, along with a weak OH maser feature probably related to the star formation activity in this galaxy. Strong H  i absorption  ( N _H∼ 10²² atoms cm⁻²)  traces a velocity gradient of 0.5 km s⁻¹ pc⁻¹ across the central kpc of NGC 520. The H  i absorption velocity structure is consistent with the velocity gradients observed in both the OH absorption and in CO emission observations. The neutral gas velocity structure observed within the central kpc of NGC 520 is attributed to a kpc-scale ring or disc. It is also noted that the velocity gradients observed for these neutral gas components appear to differ with the velocity gradients observed from optical ionized emission lines. This apparent disagreement is discussed and attributed to the extinction of the optical emission from the actual centre of this source hence implying that optical ionized emission lines are only detected from regions with significantly different radii to those sampled by the observations presented here.     

Dynamic status of the wide multiple system α Centauri + Proxima

We study the dynamics of a wide multiple system α Centauri + Proxima. The total energy of the system was estimated according to the available observational data on masses, coordinates, proper motions, and radial velocities of its components. To account for the effect of the observational data errors on the result, we have implemented the Monte Carlo method. From N = 10⁶ statistical tests we show that with the probability of about 90% the motion is hyperbolic, i.e., α Cen AB and Proxima will after a while diverge from each other by a considerable distance. We also perform numerical modeling of dynamic evolution of the wide pair α Cen AB + Proxima in the regular field of the Galaxy. The trajectory of relative motion is constructed. The components diverge from each other by a distance of 20 pc over the time scale of about 200 Myr. The critical parameter for determining the dynamic status of the system is the radial velocity of the C component (Proxima), known with an error of 200 ms^?1. For a reliable determination of the nature of motions in the system, we have to decrease the radial velocity error by at least an order of magnitude.     

Evolution of multimass globular clusters in the Galactic tidal field with the effects of velocity anisotropy

We study the evolution of globular clusters with mass spectra under the influence of the steady Galactic tidal field, including the effects of velocity anisotropy. Similarly to single-mass models, velocity anisotropy develops as the cluster evolves, but the degree of anisotropy is much smaller than in isolated clusters. Except for very early epochs of the cluster evolution, the velocity distributions of nearly all mass components become tangentially anisotropic at the outer parts. We examine how the mass function (MF) changes in time. Specifically, we find that the power-law index of the MF decreases monotonically with the total mass of the cluster, in agreement with previous findings based on isotropic models or N -body studies. This is also consistent with the behaviour of the observed slopes of MFs for a limited number of clusters. We attempt to compare our results with multimass King models, although it is almost impossible to fit the entire density profiles for all mass components. When the MF is fixed, the central densities of individual components show significant differences between Fokker–Planck and King models. We obtain 'best-fitting' multimass King models, for which the central density of individual components as well as the total density distribution agrees with the Fokker–Planck models by adjusting the MF. The MFs obtained in this way closely resemble the MF within the half-mass radius of the Fokker–Planck result. Also, we find that the local MFs predicted by Fokker–Planck calculations vary more rapidly with radius than best-fitting multimass King models. The projected velocity profiles for anisotropic models show significant flattening toward the tidal radius compared with the isotropic model. This is caused by the fact that the tangential velocity dispersion becomes dominant at the outer parts. Such a behaviour of velocity profile appears to be consistent with the observed profiles of the collapsed cluster M15.     

Statistical Characteristics of Meteoroid Parameters in the Earth’s Atmosphere

Statistical characteristics of meteoroids with kinetic energy from 0.1 to 440 kt TNT are estimated based on NASA satellite observations made in 1994–2016. The distributions of the number of falling meteoroids are constructed and analyzed based on the values of their initial kinetic energy, initial velocity, initial mass, altitude, geographic coordinates of the maximum total radiated energy region, and the year of the fall. Correlation dependences “mass–initial kinetic energy,” “maximum total radiated energy region altitude–initial kinetic energy,” and “maximum total radiated energy region altitude–initial velocity (the square of the initial velocity)” are constructed.     

A special case of the problem of two fixed centers

This paper deals with the asymptotic case of the problem of two fixed centers. The equations of motion are given in paraboloidal coordinates and then they are solved by the Hamilton-Jacobi method. The surfaces of zero velocity and the qualitative properties of possible motions are also discussed.     

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)