Analytic and Computational Studies of Structures and Oscillations of Rotating Stars and Stars in Binary Systems

Abstract

Abstract

In the present thesis an attempt has been made to extend the work of Mohan et al (108, 109) and Lal (70). This work relates to the analytic studies of the problems of computing the equilibrium structure and the periods of small adiabatic oscillations of rotating stars and stars in binary systems. While computing the effects of rotational and tidal distortions on the equilibrium structures and the eigenfrequencies of radial and nonradial modes of oscillations of rotating stars and stars in binary systems, Mohan and Saxena (99, 100) and Mohan et al (108, 109, 104, 107) did not explicitly account for the effect of Coriolis force and they analyze these problems in inertial frame of reference only. In the present study we have tried to investigate the effects of inclusion of Coriolis force on the equilibrium structures as well as periods of small adiabatic barotropic modes of oscillations of rotating stars and stars in binary systems. In the case of some of the observed rotating stars and stars in binary systems, rotation is not only expected to be uniform but also differential in which angular velocity of rotation varies from point to point. In this thesis we have also investigated the effect of Coriolis force on the equilibrium structures and periods of small adiabatic barotropic oscillations of differentially rotating stars. Thesis consists of seven chapters. Chapter I is introductory in nature. In this chapter we briefly discuss the astrophysical significance of the problem of determining the equilibrium structures and periods of small adiabatic oscillations of rotationally and tidally distorted stellar models of the stars. Mohan, Saxena and Agarwal (108, 109) approach of determining the effects of rotation and/or tidal distortions on the equilibrium structure and the eigenfrequencies of radial and nonradial modes of oscillations of the theoretical models of the stars is also discussed. A brief survey of the literature available on the subject and summary of the work presented in the succeeding chapters of the thesis are also given in this chapter. In chapter II we first develop an expression for the Roche equipotential of a rotating star in a binary system in a rotating frame of reference to explicitly include the effect of Coriolis force besides the centrifugal and gravitational forces. Next, we use it to determine the effects of Coriolis force on the shapes of Roche equipotential surfaces and position of Roche limit for different types of binary stars. Numerical computations have been performed to determine the shapes of Roche equipotentials and the values of Roche limits for different values of mass ratio of the companion stars and the values of angular velocity of rotation of the primary star. The results thus obtained have been compared with the earlier corresponding results in which the effect of Coriolis force had not been considered. For comparison the shapes of various Roche equipotential surfaces of rotating stars and stars in binary systems have also been drawn in certain cases both in the presence as well as absence of Coriolis force. Analytic expression for Roche equipotentials developed in chapter II which incorporates the effects of Coriolis force also, has been used in chapter III in conjunction with Mohan, Saxena and Aggarwal (108) approach to obtain the system of differential equations governing the equilibrium structures of rotationally and tidally distorted stellar models. The method has been applied to obtain the equilibrium structures and certain other observable parameters of rotationally and tidally distorted polytropic models of indices 1.5 and 3.0 for different choices of the values of rotational and tidal distortion parameters. The results thus obtained are compared with the results earlier computed by Mohan and Saxena (99) in which the effect of Coriolis force had not been considered. Shapes of the outer most surfaces of certain polytropic models of stars have also been drawn to highlight the effect of Coriolis force on the shapes of stars. In chapter IV we develop eigenvalued boundary value problem which determines the periods of small adiabatic barotropic modes of oscillations of rotating stars and stars in binary systems incorporating the effects of Coriolis force besides the gravitational and centrifugal forces. The method has been then used to determine the effects of Coriolis force on the eigenfrequencies of various pseudo – radial and nonradial modes of oscillations of polytropic models of stars. The eigenvalued boundary value problem, determining the radial and nonradial modes of oscillations have then been solved numerically for various polytropic models of indices 1.5 and 3.0. The results thus obtained are compared with the results earlier obtained by Mohan and Saxena (100) and certain other authors. In chapter V we have used the methodology developed in chapters II and chapter III to determine the effect of Coriolis force on the equilibrium structures of differentially rotating stars in binary systems using the law of differential rotation earlier used by Clement (15). Expressions for the volumes, surface areas and other physical parameters of polytropic models, rotating differentially according to the above law have also been obtained. Numerical computations have also been performed to compute the equilibrium structures of certain differentially rotating polytropic models of stars of indices 1.5 and 3.0. The numerical results thus obtained are compared with the results earlier obtained by Mohan et al (104) in which the effects of Coriolis force were not considered. In chapter VI we study the effect of Coriolis force on the eigenfrequencies of small adiabatic barotropic modes of oscillations of differentially rotating stars in binary systems assuming a law of differential rotation used in chapter V. The eigenvalued boundary value problems, determining the eigenfrequencies of small adiabatic pseudo radial and nonradial modes of oscillations of differentially rotating stellar models have been formulated taking into account the effects of Coriolis force. The eigenvalued problems have then been solved numerically to compute the eigenfrequencies of pseudo radial and nonradial modes of oscillations of certain differentially rotating polytropic models of the stars whose equilibrium structures were earlier obtained in chapter V. The numerical results so obtained have been compared with the results earlier reported in Mohan et al (107) which do not explicitly account for the effects of Coriolis force. Conclusions based on the present study are finally drawn in the concluding chapter VII. The limitations and scope for future work are also discussed in this chapter. A research paper entitled “Effect of Coriolis force on the shapes of rotating stars and stars in binary systems” based on the contents of the work presented in chapter II has been published in the International Journal of Astrophysics and Space Science, Springer, 319, 45 – 53, 2009. A research paper entitled “Effect of Coriolis force on the equilibrium structures of rotating stars and stars in binary systems” based on the contents of the work presented in chapter III has been published in the International Journal of Astrophysics and Space Science, Springer, 315, 157 – 165, 2008.