Measurement of the limb darkening coefficient of stars

This paper introduces a new method of describing the limb darkening phenomenon of stellar atmospheres in the 1st and 2nd order approximations. The limb darkening coefficients are given by direct measurements of relevant physical quantities or measurements of flux and a supplementary quantity, namely, the star's surface temperature. The comparison either with solar observations or with the method of numerical simulation of the atmosphere shows that the new method can very accurately describe the star's limb darkening and determine its relevant coefficients. The accuracy of the new direct method is assessed using 176 observational data points of the Sun, and the mean fractional differences is found to be 0.38% for the 1st-, and 0.26% for the 2nd-order approximations. The mean differences of the indirect method are close to, and slightly larger than these values. When compared with numerical models of the atmosphere, the differences are in the range from under 1% to under 3%.     

Orbital structure of self-consistent triaxial stellar systems

We used a multipolar code to create, through the dissipationless collapses of systems of 1,000,000 particles, three self-consistent triaxial stellar systems with axial ratios corresponding to those of E4, E5 and E6 galaxies. The E5 and E6 models have small, but significant, rotational velocities although their total angular momenta are zero, that is, they exhibit figure rotation; the rotational velocity decreases with decreasing flattening of the models and for the E4 model it is essentially zero. Except for minor changes, probably caused by unavoidable relaxation effects, the systems are highly stable. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary for the non-rotating model and stationary in the rotating frame for the rotating ones. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the different systems, allowing the recognition of regular and partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. As it could be expected, the percentages of chaotic orbits increase with the flattening of the system. As one goes from E6 through E4, the fraction of partially chaotic orbits relative to that of fully chaotic ones increases, with the former surpassing the latter in model E4; the likely cause of this behavior is that triaxiality diminishes from E6 through E4, the latter system being almost axially symmetric. We especulate that some of the partially chaotic orbits may obey a global integral akin to the long axis component of angular momentum. Our results show that is perfectly possible to have highly stable triaxial models with large fractions of chaotic orbits, but such systems cannot have constant axial ratios from center to border: a slightly flattened reservoir of highly chaotic orbits seems to be mandatory for those systems.     

Regular and chaotic orbits in a self-consistent triaxial stellar system with slow figure rotation

We created a self-consistent triaxial stellar system through the cold disipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. The resulting system rotates slowly even though its total angular momentum is zero, i.e., it offers an example of figure rotation. The potential of the system was subsequently approximated with interpolating formulae yielding a smooth potential stationary in the rotating frame. The Lyapunov exponents could then be computed for a randomly selected sample of 3,472 of the bodies that make up the system, allowing the recognition of regular and partially and fully chaotic orbits. The regular orbits were Fourier analyzed and classified using their locations on the frequency map. A comparison with a similar non-rotating model showed that the fraction of chaotic orbits is slightly but significantly enhanced in the rotating model; alternatively, there are no significant differences between the corresponding fractions neither of partially and fully chaotic orbits nor of long axis tubes, short axis tubes, boxes and boxlets among the regular orbits. This is a reasonable result because the rotation causes a breaking of the symmetry that may increase chaotic effects, but the rotation velocity is probably too small to produce any other significant differences. The increase in the fraction of chaotic orbits in the rotating system seems to be due mainly to the effect of the Coriolis force, rather than the centrifugal force, in good agreement with the results of other investigations.     

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.     

N-body simulations of tidal encounters between stellar systems

N-Body simulations have been performed to study the tidal effects of a primary stellar system on a secondary stellar system of density close to the Roche density. Two hyperbolic, one parabolic and one elliptic encounters have been simulated. The changes in energy, angular momentum, mass distribution, and shape of the secondary system have been determined in each case. The inner region containing about 40 per cent of the mass was found to be practically unchanged and the mass exterior to the tidal radius was found to escape. The intermediate region showed tidal distension. The thickness of this region decreased as we went from hyperbolic encounters to the elliptic encounter keeping the distance of closest approach constant. The numerical results for the fractional change in energy have been compared with the predictions of the available analytic formulae and the usefulness and limitations of the formulae have been discussed.     

Color functions of stellar systems

Model calculations of the photometric evolution of rather dense stellar systems, such as globular clusters, are presented. On “luminosity-effective temperature” diagrams of these systems, low-mass stars are concentrated near the minimum and maximum temperatures for a given luminosity and are deficient in the intermediate region. This sort of double-peaked distribution of the stars can be avoided in open models with ejection of excess metals into the surrounding medium. The distributions of the stars with respect to effective temperature on a “ luminosity-effective temperature” diagram are sensitive to the history of star formation in the system and to possible time variations in the initial mass function. In open systems with a single-peak distribution function, the asymmetry in the distribution varies over wide limits with the lower bound for the initial mass function and this can be used to establish whether the first generations of stars might have been more massive than in the present epoch. __________ Translated from Astrofizika, Vol. 49, No. 1, pp. 139–150 (February 2006).     

Internal stellar rotation and orbital period modulation in close binary systems

The orbital period modulation, observed in close binary systems with late-type secondary stars, is considered in the framework of a general model that allows us to test the hypothesis proposed by Applegate. It relates the orbital period variation to the modulation of the gravitational quadrupole moment of their magnetically active secondary stars produced by angular momentum exchanges within their convective envelopes. By considering the case of RS CVn binary systems, it is found that the surface angular velocity variation of the secondary component required by Applegate's hypothesis is between 4 and 12 per cent, i.e. too large to be compatible with the observations and that the kinetic energy dissipated in its convection zone ranges from 4 to 43 times that supplied by the stellar luminosity along one cycle of the orbital period modulation. Similar results are obtained for other classes of close binary systems by applying a scaling relationship based on a simplified internal structure model. The effect of rapid rotation is briefly discussed finding that it is unlikely that the rotational quenching of the turbulent viscosity may solve the discrepancy. Therefore, the hypothesis proposed by Applegate is not adequate to explain the orbital period modulation of close binary systems with a late-type secondary.     

Degenerate equilibrium states of collisionless stellar systems

We study spherically symmetrical equilibrium states of collisionless stellar systems confined to a spherical box. These equilibrium states correspond to the statistics introduced by Lynden-Bell in his theory of 'violent relaxation', and are described by a Fermi–Dirac distribution function. We compute the corresponding equilibrium diagram and show that a global entropy maximum exists for any accessible control parameter. This equilibrium state shows a pronounced separation between a degenerate core and a halo. We therefore check that degeneracy is able to stop the gravitational collapse (of a collisionless system), and we propose a simple model for the 'core–halo' structure. We also discuss the relevance of our study for real galaxies or other astrophysical systems such as massive neutrinos.     

Dynamical evolution of rotating stellar systems – II. Post-collapse, equal-mass system

We present the first post-core-collapse models of initially rotating star clusters, using the numerical solution of an orbit-averaged 2D Fokker–Planck equation. Based on the code developed by Einsel & Spurzem, we have improved the speed and the stability and included the steady three-body binary heating source. We have confirmed that rotating clusters, whether they are in a tidal field or not, evolve significantly faster than non-rotating ones. Consequences for the observed shapes, density distribution and kinematic properties of young and old star clusters are discussed. The results are compared with gaseous and 1D Fokker–Planck models in the non-rotating case.     

Undamped oscillations of collisionless stellar systems: Spheres,spheroids and discs

The collisionless Boltzmann equation governing self-gravitating systems such as galaxies has recently been shown to admit exact oscillating solutions with planar and spherical symmetry. The relation of the spherically symmetric solutions to the Virial theorem, as well as generalizations to non-uniform spheres, uniform spheroids and discs form the subject of this paper. These models generalize known families of static solutions. The case of the spheroid is worked out in some detail. Quasiperiodic as well as chaotic time variation of the two axes is demonstrated by studying the surface of section for the associated Hamiltonian system with two degrees of freedom. The relation to earlier work and possible implications for the general problem of collisionless relaxation in self gravitating systems are also discussed     

The magnetohydrodynamics of stellar structures

The equation governing the equilibrium of a centrally-symmetric, self-gravitating distribution of matter is obtained by assuming that matter is described by magnetofluid and it is shown that the equation of stellar structure described by an ideal fluid is recovered when magnetic fieldh is vanishing     

恒星自转的引力昏暗效应对谱线的影响

对于快速自转的恒星,谱线的等值宽度（ＥＷ）受引力昏暗效应的影响,即使自转速度达到临界也是很小的;而谱线半宽（ＦＷＨＭ）受引力昏暗效应影响相对较大,在（ω≈０．９９）时,相对变化达到８％．对于光谱型在Ｂ０～Ｂ９范围内的主序星,谱线半宽的相对变化在自转速度较小时（ω＜０．８）与光谱型无关;在自转速度较大时（ω≈０．９９）,谱线半宽的相对变化随光谱次型的增加而下降．     

Measurement and analysis of rotation in close binaries. I. Observations and results

A set of close binaries have been observed using the same instrumental configuration at Kitt Peak National Observatory, USA. The rotation velocities for 99 components in 75 binaries, derived by two methods, are given here, including 55 components for the first time. These high-accuracy velocities provide a reliable observational basis for investigating synchronism in binaries and the various synchronizing mechanisms.     

Nonlinear simulations of magnetic instabilities in stellar radiation zones: The role of rotation and shear

Using the 3-dimensional ASH code, we have studied numerically the instabilities that occur in stellar radiation zones in presence of large-scale magnetic fields, rotation and large-scale shear. We confirm that some configurations are linearly unstable, as predicted by Tayler and collaborators, and we determine the saturation level of the instability. We find that rotation modifies the peak of the most unstable wave number of the poloidal instability but not its growth rate as much as in the case of the m = 1 toroidal instability for which it is changed to σ = /Ω. Further in the case with rotation and shear, we found no sign of the dynamo mechanism suggested recently by Spruit even though we possess the essential ingredients (Tayler's m = 1 instability and a large scale shear) supposedly at work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two dimensional spectral classification of stars in the neighborhood of the stellar cluster NGC 2175

Spectral classifications of 1224 O-M stars in different luminosity classes in the neighborhood of NGC 2175 are presented. The classifications are based on low dispersion objective prism spectra. B-V and U-B data for 106 of the 1224 stars are used to obtain MK classifications using the photometric Q method. __________ Translated from Astrofizika, Vol. 51, No. 2, pp. 339–341 (May 2008).