STRUCTURE OSCILLATIONS AND STABILITY OF DIFFERENTIALLY ROTATING AND TIDALLY DISTORTED STELLAR MODELS

Abstract

The present thesis deals with the problems of determining the equilibrium structure, oscillations and stability of differentially rotating and tidally distorted theoretical models of the stars. Such a study has practical relevance in astrophysics where it is expected to help in understanding the problems of stellar stability and stellar variability of differentially rotating stars, as well as of differentially rotating stars in binary and multiple systems. Theoretically a star can be considered as a gaseous sphere both in hydrostatic and thermal equilibrium. Observations show that some of the stars are rotating about their axes. This rotation can be a solid body rotation as well as a differential rotation in which different parts of the star are rotating with different angular velocities (for instance "Further Evidence for Differential Rotation in V1057 Cygni," Allan D. Welty et al. Astrophys J. Vol.349, pp. 328-334, 1990). Equilibrium structures of rotating stars get distorted by rotational forces. Some of the stars are also observed as members of binary and multiple systems. Equilibrium models of such stars get distorted by the tidal effects alone if the star is not rotating and by the combined tidal and rotational effects if the star in the binary, or multiple systems, is rotating as well. Some of rotating stars and stars in binary systems are also known to be variable stars (for instance 53 Per, 1Mon, 12Lac, 16Lac etc.). The equilibrium structures and eigenfrequencies of oscillations of such stars get effected by the rotational forces in the case of stars which are only rotating, and by the combined effects of rotational and tidal forces in the case of the rotating stars in binary ii and multiple systems. Analytic study of the problem of determining the equilibrium structures, oscillations and stability, of rotationally and tidally distorted stellar models, is quite complex. The problem becomes still more complex if the rotation is differential. Attempts have been made in literature to investigate these problems in some approximate ways. In one such attempts Mohan, Saxena and Agarwal ('Equilibrium Structures of Rotationally and Tidally Distorted Stellar Models', Astrophys. Space Sci., Vol.163, pp. 23-39, 1990) used Kippenhahn and Thomas averaging technique, in conjunction with Kopal's results on Roche equipotentials, to determine the combined effects of rotation and tidal distortions, on the equilibrium structure and eigenfrequencies of small adiabatic barotropic modes of oscillations of the theoretical models of stars. They also demonstrated the use of this approach in case of certain main-sequence stars. This approach has also been earlier used by Mohan and Saxena (Astrophys. and Space Sci., Vol.389, p.95, 1983) and Mohan and Agarwal (Astrophys and Space Sci., Vol.129, pp.73-74, 1987) to determine the equilibrium structures and periods of oscillations of the theoretical models of other types of rotationally and tidally distorted of stars. However, mostijhis work pertains to stars having solid body rotation. In the present thesis we have considered the possibility of using Mohan, Saxena and Agarwal approach to determine the effects of differential rotation on the equilibrium structures, stability and the eigenfrequencies of the radial and nonradial modes of oscillations of theoretical models of stars. Mathematical models of differentially iii rotating stars have been developed in general and used to compute the equilibrium structures and eigenfrequencies of small adiabatic modes of oscillations of certain types of differentially rotating polytropic and white dwarf models. Effects of tidal distortions, caused by the presence of a companion star in a binary system, on the equilibrium structure and periods of oscillations of differentially rotating stars have also been considered. The study is expected to give some insight into the effects of rotation and tidal distortions on the equilibrium structure and periods of oscillations of stars. It is also expected to help in better understanding the limitations and scope of Mohan, Saxena and Agarwal approach of determining the effects of rotation and tidal distortions on the equilibrium structures and periods of small adiabatic modes of oscillations of the theoretical models of the stars. The thesis consists of nine chapters. Chapter one is introductory in nature. In this chapter we first briefly discuss the astrophysical significance of the problem of determining the equilibrium structures, oscillations and stability of differentially rotating as well as differentially rotating and tidally distorted stellar models of the stars. Mohan, Saxena and Agarwal 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 star is also discussed in this chapter. A ' brief survey of the literature available on the subject, and summary of the work presented in the succeeding chapters of the thesis, also appears in this chapter. v In chapter II we first discuss some of the laws of differential rotation which have been commonly used by various :authors to depict angular velocity S2 as a function of distance s measured from the axis of rotation (0 = D(s)). Keeping these laws in view we have assumed a law of differential rotation of the type w2 = bo+bis2+b2s4, where w is nondimensional measure of the angular velocity of rotation of a fluid element of the star distances (in nondimensional form from the axis of rotation) and bo'bb2 are suitable numerical constants. This law may be regarded as Taylor's series expansion of w2 = f(s2)) in which terms upto second order of smallness in s2 are retained. The law is thus applicable to differentially rotating stars in which angular velocity of rotation is not too large. We have preferred this law of differential rotation in our present study since by giving suitable values to bo' b1 and b2 it can generate a variety of differential rotations expected in the case of stars. The law is also in a form in which it can be conveniently subjected to mathematical operations which have to performed in the analytic studies of differentially rotating stars. Important types of differential rotations which can be generated in stars using this law of differential rotation are also discussed in this chapter. The stability of various types of differential rotations generated by this law are checked using a stability criterion commonly used for this purpose. The system of differential equations governing the equilibrium structures of differentially rotating stellar models, obeying this law of differential rotation, is next obtained in this chapter using Mohan, Saxena and Agarwal approach. The analysis includes the effects of rotational terms in the equations of stellar structure upto second order of smallness in the distortional parameters bo, bl and b2. The possibility of using this approach to numerically compute the equilibrium structures of various types of differentially rotating models of the stars is also discussed.