Effect of Coriolis force on the shapes of rotating stars and stars in binary systems

In this paper an attempt has been made to determine the effect of Coriolis force on the shapes of Roche equipotential surfaces of rotating stars and stars in binary systems. Equations of Roche equipotential surfaces have been obtained for rotating and binary stars which take into account the effects of Coriolis force besides the centrifugal and gravitational forces. Shapes of Roche equipotentials and values of Roche limits are obtained for different values of angular velocity of rotation for rotating stars and for different values of mass ratios for the binary stars. The obtained results have been compared with the corresponding results in which the effect of Coriolis force has not been considered.     

Effect of Coriolis force on the equilibrium structures of rotating stars and stars in binary systems

Kopal (Adv. Astron. Astrophys. 9:1–65, 1972) introduced the concept of Roche equipotentials to incorporate the effects of rotation and tidal distortions on the equilibrium structure and periods of small oscillations of rotating stars and stars in binary systems. However his expression for the Roche equipotential accounts for only the effects of centrifugal and gravitational forces and does not take into account the effect of Coriolis force. In this paper we have suitably modified Kopal’s expression for Roche equipotentials to incorporate into it the effect of Coriolis force as well. The modified expression for the Roche equipotential has then been used to compute the equilibrium structures and shapes of polytropic models of rotating stars and stars in binary systems.     

On the validity of series expansion being used for the position of a point on a Roche equipotential

The concept of Roche equipotentials has been frequently used in literature to study the problems of rotating stars and stars in binary systems. However in spite of using this simplifying concept, it is still not possible to express the position of a point in the potential field of such a system in a closed analytic form. In order to carry out further analytic studies, Kopal (Adv. Astron. Astrophys. 9:1–65, 1972), therefore, developed a series expansion for it. The series expansion of Kopal has often been used in the analysis of the problem of equilibrium structure and the periods of oscillations of rotating stars and stars in binary systems, but its validity and convergence has not been analytically established. It is important that this aspect of the problem is checked so that one is sure of the correctness of subsequent analysis and results based on this series expansion. In the present brief note, we have addressed ourselves to this problem and validated the correctness of the numerical results obtained through the use of this series expansion.     

On the Use of Roche Equipotentials in Analysing the Problems of Binary and Rotating Stars

Kopal (Adv. Astron. Astrophys., 9, 1, 1972) introduced the concept of Roche equipotentials to analyse the effects of rotational and tidal distortions in case of stars in binary systems. In this approach a mathematical expression for the potential of a star in a binary system is obtained by approximating its inner structure with Roche model. This expression for the potential has been used in subsequent analysis by various authors to analyse the problems of structures and oscillations of synchronous and nonsynchronus binary stars as well as single rotating stars. Occasionally, doubts have been expressed regarding the validity of the use of this approach for analysing nonsynchronous binaries and rotationally and tidally distorted single stars. In this paper we have tried to clarify these doubts.     

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.     

Eigenfrequencies of rotationally and tidally distorted white dwarf models of stars

In the present paper we have studied the eigenfrequencies of small adiabatic barotropic pseudo-radial and nonradial modes of oscillations of the white dwarf models of rotating stars in binary systems. In this work the methodology of Mohan and Saxena (in Astrophys. Space Sci. 113:155, 1985) has been used that utilizes the averaging technique of Kippenhahn and Thomas (in Proc. IAU Colloq., vol. 4, p. 20, 1970) and certain results on Roche equipotential as that given by Kopal (in Advances in Astronomy and Astrophysics, Academic Press, 1972). The objective of this study is to investigate the effects of rotation and/or tidal distortion on the periods of oscillations of rotationally and/or tidally distorted white dwarf models of stars assuming it to be the primary component of the binary system and rotating uniformly. The results of present study show that the eigenfrequencies (both radial and nonradial modes) of the rotationally distorted and rotationally and tidally distorted white dwarf model of stars in binary systems tend to decrease under the influence of rotational distortions and rotational and tidal distortions, respectively. However, results are contrary for tidally distorted white dwarf model of stars.     

The structure of rotating stars from an extension of the Kippenhahn-Thomas method

A method is presented whereby the structure of rotating stars may be determined from an initial guess at the geometry of equipotential surfaces. The method may be considered an extension of the work of Kippenhahn and Thomas in that a uniformly continuous geometry is defined in terms of the appropriate spherical model with Roche characteristics at the surface of the configuration and sphericity at the centre. A simple Cowling model in uniform rotation is employed to illustrate the technique and for comparison purposes with previous work.     

On the roche constants for Main-Sequence binaries

The ratiosC ₁/C ₂ of the constants defining the equipotential surfaces which describe the external forms of the components of a close binary system have been calculated on the basis of evolutionary models. Theoretical systems have been considered allowing for a wide range of input parameters (masses and separation) and taking into account the evolutionary effects on the radii of the stars during their Main-Sequence lifetime. The systems have not undergone any transfer of matter and are representative of detached binaries with Main-Sequence components. The ratios of the constants are confined in limited intervals and, for the highest values of the mass-ratio, they are clustered around the unit.     

Analytic formulas for the mass-transfer rate and the evolution of a close binary system of neutron (degenerate) stars

We derive approximate analytic relations between the mass-transfer rate in a close binary system described in terms of the Roche potential and its basic parameters, such as the total mass of the binary, the radius of its circular orbit, the mass of the mass-losing component, and the degree of its Roche lobe overfilling. Using simplifying assumptions (conservative mass transfer, a short relaxation time of matter on the mass-gaining component compared to the mass-transfer time scale, adiabaticity and quasi-stationarity of the mass flow through the Lagrangian point L ₁) allows the evolution of a binary system of neutron (degenerate) stars to be described in terms of two ordinary differential equations. This makes it possible to qualitatively analyze the evolution process, which is useful in those cases where the evolution of a close binary system must be investigated in general terms, for example, in terms of the scenario for the transformation of the collapse of a rotating presupernova core into a supernova explosion proposed by Imshennik and Nadyozhin (1992) and Imshennik (1992).     

Tidal interactions in binary stellar systems

The tidal interactions in binary stellar systems are studied under the assumption that the orbital motion of the binary is negligible in comparison with the stellar motion. By integrating over time the tidal forces acting on the stars, the energy changes are derived. These are used to obtain simple analytical expressions for the rates of disruption and merging. This method gives appropriate value for the Roche density ρ_r and it is found that the disruption rate of a satellite of density ρ changes drastically at ρ ρ_R A comparison is made with earlier results obtained under the simplifying assumption that stellar motion is negligible in comparison with the orbital motion of the binary and its implications are discussed.     

Building the cosmic distance scale: from Hipparcos to Gaia

Hipparcos, the first ever experiment of global astrometry, was launched by ESA (European Space Agency) in 1989 and its results published in 1997 (Perryman et al. in Astron. Astrophys. 323:L49, 1997; Perryman & ESA (eds.) in The Hipparcos and Tycho catalogues, ESA SP-1200, 1997). A new reduction was later performed using an improved satellite attitude reconstruction leading to an improved accuracy for stars brighter than 9th magnitude (van Leeuwen & Fantino in Astron. Astrophys. 439:791, 2005; van Leeuwen in Astron. Astrophys. 474:653, 2007a). The Hipparcos Catalogue provided an extended dataset of very accurate astrometric data (positions, trigonometric parallaxes and proper motions), enlarging by two orders of magnitude the quantity and quality of distance determinations and luminosity calibrations. The availability of more than 20 000 stars (22 000 for the original catalogue, 30 000 for the re-reduction) with a trigonometric parallax known to better than 10% opened the way to a drastic revision of our 3-D knowledge of the solar neighbourhood and to a renewal of the calibration of many distance indicators and age estimations. The prospects opened by Gaia, the next ESA cornerstone, planned for launch in 2013 (Perryman et al., in Astron. Astrophys. 369:339, 2001), are still much more dramatic: a billion objects with systematic and quasi simultaneous astrometric, spectrophotometric and spectroscopic observations, about 150 million stars with expected distances to better than 10%, all over the Galaxy. All stellar distance indicators, in very large numbers, will be directly measured, providing a direct calibration of their luminosity and making possible detailed studies of the impacts of various effects linked to chemical element abundances, age or cluster membership. With the help of simulations of the data expected from Gaia, obtained from the mission simulator developed by DPAC (Gaia Data Processing and Analysis Consortium), we will illustrate what Gaia can provide with some selected examples.     

The Algol-type eclipsing binaries RW CrB and VZ Leo: new RI photometric study and search for pulsations

In this study we present the new R and I light curve solutions for the eclipsing binary systems RW CrB and VZ Leo, which for VZ Leo is the first one in the literature. Our new observations have been analyzed using the Wilson-Devinney code from which new geometric and photometric elements are derived. The geometry of both stars is that of a semi-detached binary system where the secondary component fills its Roche lobe while the primary component is well inside. In the case of RW CrB, asymmetry in the light curve was explained by a spot model. The orbital period changes of both systems were studied and the results indicated a period decrease which can be explained by angular momentum loss. We also investigated the possibility of pulsations of the primary components since these systems are mentioned as candidates of δ Sct type pulsation. However, a time-series analysis of the residual curves in the filter I does not indicate any evidence of periodic light variation for both systems. Finally, we compared the results obtained for both binary stars to those of similar systems.     

Magellanic Cloud WC/WO Wolf–Rayet stars – I. Binary frequency and Roche lobe overflow formation

A nearly complete sample of 24 Magellanic Cloud WC/WO subclass Wolf–Rayet stars is studied spectroscopically and photometrically to determine its binary frequency. Theory predicts the Roche lobe overflow produced Wolf–Rayet binary frequency to be 52±14 per cent in the Large Magellanic Cloud and 100 per cent in the Small Magellanic Cloud, not counting non-Roche lobe overflow Wolf–Rayet binaries. Lower ambient metallicity ( Z ) leads to lower opacity, preventing all but the most massive (hence luminous) single stars from reaching the Wolf–Rayet stage. However, theory predicts that Roche lobe overflow even in binaries of modest mass will lead to Wolf–Rayet stars in binaries with periods below approximately 200 d, for initial periods below approximately 1000 d, independent of Z . By examining their absolute continuum magnitudes, radial velocity variations, emission-line equivalent widths and full widths at half-maximum, a WC/WO binary frequency of only 13 per cent, significantly lower than the prediction, is found in the Large Magellanic Cloud. In the unlikely event that all of the cases with a less certain binary status actually turn out to be binary, current theory and observation would agree. (The Small Magellanic Cloud contains only one WC/WO star, which happens to be a binary.) The three WC+O binaries in the Large Magellanic Cloud all have periods well below 1000 d. The large majority of WC/WO stars in such environments apparently can form without the aid of a binary companion. Current evolutionary scenarios appear to have difficulty explaining either the relatively large number of Wolf–Rayet stars in the Magellanic Clouds, or the formation of Wolf–Rayet stars in general.     

The Roche coordinates for the rotational problem

Curvilinear coordinates in three dimensions associated with the Roche model distorted by centrifugal force alone constitute a Lamé family, of which one (-) coordinate can be defined by equipotential surfaces which are known in closed algebraic form; the other () becomes identical with the meridional planes of the rotationally distorted Roche model; while the third () then follows from the requirements of orthogonality to the others. The explicit form of such coordinates in terms of the polar or cartesian systems has already been established by the author (Kopal, 1970) correctly to quantities of the first order in superficial distortion of the respective Roche model. In the present paper this latter restriction on accuracy will be removed, and expressions constructed for the -coordinate in the form of infinite series which are exact and converge rapidly for any distortion below that which entails equatorial break-up.     

On the general properties of the secondaries of cataclysmic variable stars

The mass-radius, mass-orbital period, spectral type-orbital period and absolute magnitude-orbital period relationships for the secondary star components of cataclysmic variable (CV) binary stars are discussed. By comparing these with those expected for systems containing Roche lobe-filling main sequence stars, using empirical data for low-mass main sequence stars, it is shown thatas a group the CV secondaries do not differ significantly from main sequence stars.     

Revised photometric elements of the eclipsing binary AI Cru

Using Wood's (1972) model we have re-analyzed the photoelectric light curve of the eclipsing binary AI Cru, published and unsatisfactorily discussed by Ollongren (1956), in order to obtain reliable photometric elements. Our photometric solution differs considerably from Ollongren's (1956). For our photometric mass ratio,q=0.8, both components, though detached from their Roche lobes, are fairly close to a contact configuration.AI Cru, composed of a brighter and larger B5 primary and a fainter and smaller (around B9.5) secondary, is probably not a normal Main Sequence system, since the ratios of the secondary's mass and radius over the primary's turn out to be larger than expected for main sequence stars of the same spectral types.     

Effect of rotational and tidal distortions on the periods of small adiabatic oscillations of stellar models

Kopal's method of representing the inner structure of a rotationally and tidally-distorted star by a rotationally-and tidally-distorted Roche model has been used in conjunction with an averaging concept introduced by Kippenhahn and Thomas of representing the equipotential surfaces of a rotationally and tidally distorted stellar model by the equipotential surfaces of an equivalent spherical model to determine the combined effects of rotation and tidal distortion on the periods of small adiabatic oscillations of a stellar model.     

Thermal imbalance effects on pulsational stability in stellar models

The treatment followed by Coxet al. (1973) for the estimation of the effects of thermal imbalance on the pulsational stability of stars is applied to Roche and point source (Chandrika Prashad) model. The calculations suggest that the eigenvalues obtained for the above model are correct to 0(t _ff/t_s). The expressions for stability coefficients and corresponding eigenfunctions for the stars in thermal imbalance for the above models have also been derived.     

The gravity-darkening of highly distorted stars in close binary systems

From a quantitative analysis of the photometric ellipticity effect, values of the exponent of gravity-darkening have been empirically deduced for the components of sixteen W UMa systems with accurate cross-correlation spectroscopic mass ratios. In the analysis, the exponents for both components of a W UMa system are assumed to be the same. In view of the generally unstable nature of their light curves and the consequent difficulty to find an unique set of the fractional radii of both components, -values have been derived for various equipotential surfaces with boundary cases set at the inner and outer Roche lobes.The results indicate that if the fill-out factors are less than 0.5 (as is so for most of the actual cases), the deduced -values are all greater than Lucy's conventional value of 0.32. The empirical -values as a whole are found to increase with the mean effective temperature of the components and, furthermore, they show a tendency to decrease with increasing mass ratio, particularly for A-type systems. Thus, a W UMa system consisting of two components of quite different masses has a large empirical -value. These observational evidences indicate that the structure of the photosphere of contact components of W UMa systems is significantly different from that of normal Main-Sequence stars of similar spectral type and may provide a clue to solve puzzles of W UMa stars.     

Stellar surface imaging: mapping brightness and magnetic fields

Rapidly rotating late‐type stars typically display signs of magnetic activity that exceed those seen on the Sun by over two orders of magnitude. The techniques of Doppler imaging and Zeeman Doppler imaging have been instrumental in unveiling magnetic activity patterns at the photospheres of these active stars. Essentially, these techniques work by inverting time‐series of high resolution spectra to produce temperature, brightness and/or magnetic field maps at the surfaces of stars. I will describe how these techniques work and review what they have taught us about the nature of magnetic activity in rapid rotators over the last 20 years. Finally, I will conclude by outlining the capabilities of these techniques in light of new instrumentation that is now becoming available. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)