Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

Exact MHD solutions for rotating,magnetic stars

Simple exact solutions of the magnetohydrodynamic equations are found for rotating, magnetic stars. The velocity and magnetic field are axisymmetric and purely toroidal, and the magnetic energy density equals the kinetic energy density. For constant mass density, the solution reduces to that of Chandrasekhar (1956), which is stable even against non-axisymmetric perturbations. For an ideal gas equation of state, the condition for radiative thermal equilibrium is solved to lowest order in the non-spherical perturbation. The velocity, magnetic field and non-spherical pressure and temperature perturbations all vanish within cones centered around the rotation axis, |cos |>x _i a zero of a Legendre polynomial. Low-order, long-period stellar oscillations may be excited by MHD instabilities near the equatorial region which become damped near the axis.     

2-D hydrodynamical simulations of wind compressed disks

We present results of 2-D hydrodynamical simulations of a radiatively driven stellar wind from a rapidly rotating Be-star. These generally confirm predictions of the semi-analytic Wind-Compressed-Disk model recently proposed by Bjorkman and Cassinelli to explain the circumstellar disks inferred observationally to exist around such rapidly rotating stars. However, our numerical simulations are able to incorporate several important effects not accounted for in the simple model, including a dynamical treatment of the outward radiative driving and gas pressure, as well as a rotationally distorted, oblate stellar surface. This enables us to model quantitatively the compressed wind and shock that forms the equatorial disk. The simulation results thus do differ in several important details from the simple model, showing, for example, an inner diskinflow not possible in the heuristic approach of assuming a fixed outward velocity law. There is also no evidence for the predicted detachment of the disk that arises in the fixed outflow picture. The peak equatorward velocity in the dynamical models is furthermore about a factor of two smaller than the analytically predicted value of 50% the stellar equatorial rotation speed. As a result, the dynamical disks are somewhat weaker than predicted, with a wider opening angle, lower disk/pole density ratio, and smaller shock velocity jump (each by roughly the same factor of two).     

Non-radial oscillations and energy transport in rotating solar (stellar) wind

The propagation of non-radial, small amplitude perturbations superposed on a zero-order, stationary, non-magnetic, polytropic, rotating stellar wind is studied in the limit of the local theory, i.e. for k r 1, k being the module of the wave vector and r the characteristic scale of the zero-order flow. The resulting dispersion equation is of the 3rd order in (complex) frequency and the possible modes correspond to two acoustic type waves, and to a gravity-shear wave with strongly anisotropic propagation properties, due to coupling between the internal gravity waves and shear motion. The gravity-shear mode allows velocity differences in the medium to exist with no corresponding density fluctuations and hence with no shock wave formation. It is suggested that this mode corresponds to some of the fast-slow velocity streams observed in the interplanetary medium and may provide means for wave energy being transported outwards with the zero-order flow, with little dissipation in the inner region of the solar wind.     

High resolution fabry-perot and multi-pupil spectral observations of the genuine blue compact dwarf galaxy IZW18

We present the observations of the BCDG IZW18 performed with a Fabry-Perot interferometer at the CFH 3.6 m telescope and with a multi-pupil spectrograph at the SAO (Russia) 6 m telescope. The morphological structure of the galaxy in emission lines and in continuum, the velocity field of the ionized gas, and [OIII]/H ratio distribution along the NW component have been investigated. Besides the NW and SE HII components, we find a population of small HII regions. Continuum maps show that the peaks of the stellar light distribution are displaced with respect to the emission lines maxima. the velocity field shows peculiar motions superposed on an approximately regular background implying solid body rotation. Emission line profiles exhibit an asymmetric structure, except for the NW compact component. The [OIII]/H ratio decreases from the center of the NW component to its edge with the gradient of 1.86 kpc^–1.Published in Astrofizika, Vol. 38, No. 4, pp. 602–615, October–December, 1995.     

The Restricted Three-Body Problem: Comments on the Rotational Effect

The synchronization between the orbital motion and axial rotation of the two component stars of a binary system is reviewed. Some previous published papers are mentioned and the general conclusion is outlined: If we shall use a rotating coordinate system synchronous with one of the two stellar axial rotations, it is not possible to obtain a Jacobi integral and the Roche geometry cannot be further analyzed. In addition, a theoretical approach is summarized in order to use the axial rotations of the two component stars, even if the constants of the stellar structure (k₂)₁, (k₂)₂must be taken into consideration. So it is found that if the stellar angular velocities are higher than the corresponding Keplerian angular velocity (ω_i≫ ω_k, i= ), the problem of the rotational effect could be of practical consideration. Finally, a theoretical relationship between the two constants (k₂)₁and (k₂)₂of the stellar structure is established.     

Slowly rotating cosmological viscous fluid universe in Brans-Dicke theory

The Brans-Dicke field equations for a viscous distribution representing slowly rotating fluid spheres are investigated. Exact solutions are obtained for differential rotation by imposing physical restrictions on the matter rotation (r,t). The physical properties are discussed fork=±1.     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

Slowly rotating cosmological perfect fluids in Brans-Dicke theory

The Brans-Dicke field equations for a perfect fluid distribution representing slowly rotating fluid spheres are investigated. Exact solutions are obtained for differential rotation and perfect dragging by imposing some physical restrictions on the matter rotation (r,t). The physical properties are discussed fork=±1.     

Mass loss from coronae and its effect upon stellar rotation

The acoustic energy-generation rate from the convective zone was calculated for various models. Results show that chromosphere and corona can be expected around stars with temperature lower than 8000K at the main sequence, and lower than 6500K at logg=2.When a star is rotating rapidly, mass loss from its corona is large, and can be an effective mechanism of braking the stellar rotation. If this mechanism is effective, we can explain the slow rotation of stars later than F2 to be the result of the loss of the angular momentum through a stellar wind that is effective in their main sequence phase. Stars with massM>1.5M lose mass through a stellar wind during their contraction phase. The mass-loss rate is larger than the solar value because of the larger energy input into the chromosphere-corona system and because of the smaller gravitational potential at the surface. T Tauri stars may be the observational counterparts for such stars. As the duration of contraction phase is very short (less than 10⁷ years), the braking mechanism works only in the presence of a strong magnetic field (Ap) or in the presence of a companion (Am).Presented at the Trieste Colloquium on Mass Loss from Stars, September 12–16, 1968.     

Equilibrium Structures of Differentially Rotating Primary Components of Binary Stars

In this paper a method is proposed for computing the equilibrium structures and various other observable physical parameters of the primary components of stars in binary systems assuming that the primary is more massive than the secondary and is rotating differentially about its axis. Kippenhahn and Thomas averaging approach (1970) is used in a manner earlier used by Mohan, Saxena and Agarwal (1990) to incorporate the rotational and tidal effects in the equations of stellar structure. Explicit expressions for the distortional terms appearing in the stellar structure equations have been obtained by assuming a general law of differential rotation of the typeω² = b ₀+b ₁ s ²+b ₂ s ⁴, where ω is the angular velocity of rotation of a fluid element in the star at a distance s from the axis of rotation, and b ₀, b ₁, b ₂ are suitably chosen numerical constants. The expressions incorporate the effects of differential rotation and tidal distortions up to second order terms. The use of the proposed method has been illustrated by applying it to obtain the structures and observable parameters of certain differentially rotating primary components of the binary stars assuming the primary components to have polytropic structures. This revised version was published online in July 2006 with corrections to the Cover Date.     

The effects of viscous friction on the precession and nutation of celestial bodies

The aim of the present study has been to set the system of differential equations which govern the precession and nutation of self-gravitating globes of compressible viscous fluid, due to the attraction exerted on the rotating configuration by its companion; and to construct their approximate solution which are correct to terms of the second order in small dependent variables of the problem. Section 2 contains an explicit formulation of the effects of viscosity arising in this connection, given exactly as far as the viscosity remains a function of radial distancer only; but irrespective of its magnitude. In Section 3 the equations of motion will be linearized for the case of near-circular orbits and small inclinations andi of the equator of the rotating configuration, and of its orbital plane, to the invariable plane of the system; while in Section 4 further simplifications will be introduced which are legitimate for studies of secular (or long-periodic) motions of the nodes and inclinations. The actual solutions of so simplified a system of equations are constructed in Section 5; and these represent a generalization of the results obtained in our previous investigation (Kopal, 1969) of the inviscid case.The physical significance of the new results will be discussed in the concluding Section 6. It is demonstrated that the axes of rotation of deformable components in close binary systems are initially inclined to the orbital plane, viscous dissipation produced by dynamical tides will tend secularly to rectify their positions until perpendicularity to the orbital plane has been established, and the equators as well as orbit made to coincide with the invariable plane of the system-in a similar manner as other effects of tidal friction are bound eventually to synchronize the velocity of axial rotation with that of orbital revolution in the course of time.An application of the results of the present study to the dynamics of the Earth-Moon system discloses that the observed inclination of 1°.5 of the lunar equator to the ecliptic cannot be regarded as being secularly constant, but representing the present deviations from perpendicularity of oscillatory motion of very long period.The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR-09-051-001 with the National Aeronautics and Space Administration. This paper constitutes the Lunar Science Institute Contribution No. 85.     

Effect of partial mixing of matter on the hydrodynamic angular momentum transport processes in massive main-sequence stars

We consider the evolution of a rotating star with a mass of 16M _⊙ and an angular momentum of 3.25 × 10⁵² g cm² s^?1, along with the hydrodynamic transport of angular momentum and chemical elements in its interiors. When the partial mixing of matter of the turbulent radiative envelope and the convective core is taken into account, the efficiency of the angular momentum transport by meridional circulation in the stellar interiors and the duration of the hydrogen burning phase increase. Depending on the Schmidt number in the turbulent radiative stellar envelope, the ratio of the equatorial rotational velocity to the circular one increases with time in the process of stellar evolution and can become typical of early-type Be stars during an additional evolution time of the star on the main sequence. Partial mixing of matter is a necessary condition under which the hydrodynamic transport processes can increase the angular momentum of the outer stellar layer to an extent that the equatorial rotational velocity begins to increase during the second half of the evolutionary phase of the star on the main sequence, as shown by observations of the brightest stars in open star clusters with ages of 10–25 Myr. When the turbulent Schmidt number is 0.4, the equatorial rotational velocity of the star increases during the second half of the hydrogen burning phase in the convective core from 330 to 450 km s^?1.     

Comparison between a radiative wind model for be stars and high angular resolution data from the GI2T interferometer

I present theoretical line profiles and intensity maps from an axi-symmetric radiative wind model from a rapidly rotating Be star. The introduction of a viscosity parameter in the latitude-dependent hydrodynamic code enables us to consider the effects of the viscous force in the azimuthal component of momentum equations (Araújo et al. 1994). Both velocity field and density law derived from the hydrodynamic equations have been used for solving the statistical equilibrium equations. By adopting the Sobolev approximation, we could easily obtain a good estimate of both electronic density and hydrogen level populations throughout the envelope. The numerical calculation was performed for parameters characterisic of the Be star Cassiopeiae.     

Stellar analogs of solar magnetic activity

The techniques and principal results of observational studies of stellar activity are summarized. Both chromospheric and coronal emission clearly track surface magnetic field properties, but it is not well known how the detailed relation between the emission and surface magnetic fields varies with spectral type. For lower Main-Sequence stars of the same spectral type, there is clear evidence of a close relationship between mean activity level and rotation period P _rot. There is also less definitive evidence for a similar dependence on convective overturn time _c , such that activity depends on the single parameter Ro = P _rot/ _c . For single stars, stellar rotation, and magnetic activity both decline smoothly with age. This implies a feedback between angular momentum loss rate and activity level. Temporal variations in mean stellar activity level mimic the solar cycle only for old stars like the Sun, being much more irregular for younger stars. The characteristic timescale of the variations (the cycle period) appears to depend on Ro for old stars, but shows no clear dependence on either rotation rate or spectral type for younger stars. Further data on mean activity and its variation for a large number of lower Main-Sequence stars should contribute significantly to our understanding of the causes of stellar magnetic activity.     

The effects of the tidal force and rotation on the velocity of stellar wind

The effects of tidal force and rotation on the velocity of stellar wind have been investigated. We have obtained the correct formula for the expansion of stellar wind. It is well known that the binary stars of Be type variable amount to more than 60% of their respective population, and their rotational velocity can be greater than 300 km s^–1. In such cases, we must take account of these effects.     

Lines formed in rotating and expanding atmospheres

Spectral lines formed in a rotating and expanding atmosphere have been computed in the frame of the observer at infinity. Two kinds of velocity laws are employed: (i) a uniform radial velocity of the gas and (ii) velocity increasing with radius (i.e. velocity gradients). The atmosphere has been assumed to be rotating with constant velocity. We have considered maximum radial and rotational velocities to be 10 and 3 mean thermal units respectively in an atmosphere whose geometrical thickness is 10 times the stellar radius. The total radial optical depth at line centre is taken to be about 100. In all cases, Doppler profile and a source function which is varying as 1/r ² have been used. Generally, the lines are broadened when rotation is introduced. However, when radial motion is also present, broadening becomes asymmetric and the red emission and blue absorption are enhanced.     

Libration effects for retrograde satellites in the restricted three-body problem

Numerical explorations of the restricted problem have shown that for stable large nonperiodic retrograde satellite orbits, the motion can be decomposed into a fast reference motion and a slow libration aroundB ₂ We study here this libration in the circular plane Hill's case, for which the reference motion is elliptic. We establish the equations of motion for the coordinates of the centre of this ellipse. We find two integrals of motion: the first is the semi-major axis of the ellipse; the second is essentially Jacobi's integral, translated into the new coordinates. We give a formula for the period of the libration and we find its limiting value for small libration amplitudes. A numerical verification gives very good agreement for all these results.     

Kinematics of Gas and Stars Along the Hubble Sequence

We present a comparison between the ionized gas and stellar kinematics for a sample of five early-to-intermediate disc galaxies. We measured the major axis V and σ radial profiles for both gas and stars, and the h ₃ and h ₄ radial profiles of the stars. We also derived from the R-band surface photometry of each galaxy the light contribution of their bulges and discs. In order to investigate the differences between the velocity fields of the sample galaxies we adopted the self-consistent dynamical model by Pignatelli and Galletta (1999), which takes into account the asymmetric drift effects, the projection effects along the line of sight and the non-Gaussian shape of the line profiles due to the presence of different components with distinct dynamical behaviour. We find for the stellar component a sizeable asymmetric drift effect in the inner regions of all the sample galaxies, as results from comparing their stellar rotation curves with the circular velocity predicted by the models. The galaxy sample is not wide enough to draw general conclusions. However, we have found a possible correlation between the presence of slowly rising gas rotation curves and the ratio of the bulge/disc half-luminosity radii, while there is no obvious correlation with the key parameter represented by the morphological classification, namely the bulge/disc luminosity ratio. Systems with a diffuse, dynamically hot component (bulge or lens) with a scale length comparable to that of the disc are characterized by slowly rising gas rotation curves. On the other hand, in systems with a small bulge the gas follows almost circular motions, regardless of the luminosity of the bulge itself. We noticed a similar behaviour also in the gas and stellar kinematics of the two early-type spiral galaxies modelled by Corsini et al.(1998). This revised version was published online in July 2006 with corrections to the Cover Date.     

The appearance of polarization in radiation from hot stars due to the faraday rotation effect as a possible method of determining stellar magnetic fields

The effect of Faraday rotation is shown to lead to the appearance of linear polarization of stellar radiation scattered in an optically-thin circumstellar electron-magnetized shell, even in the case when the shell is spherical. The spectral dependence of the polarization degree is evaluated for scattering in (i) a spherically-symmetric magnetized shell with a power-law radial dependence of the electron density, and (ii) a non-spherical ellipsoidal uniform envelope. The position of maximum in the polarization spectrum permits us to determine the magnetic field magnitude on a star surface. If the rotational and magnetic axes do not coincide, the periodic variability of the polarization will be observed with the period of stellar rotation. Some Be-stars, such as Cas, 48 Lib, EW Lac, Aqr, HD 45677, X Per, are proposed as candidates to be investigated for magnetic fields, as well as some stars of the T Tau-type. This method may be also applied to supernovae shells.