NONRADIAL OSCILLATIONS O F GASEOUS SPHERES

Abstract

The present thesis deals with the problem of nonradial oscillations of gassous spheres* Keeping In view the fact that nonradial oscillations of gassous spheres may be helpful in our understanding of soas of the observed features of stare, an attempt has been made in the present thesis to investigate some aspeots of ths problsa of nonradial oscillations of gaseous spheres. Ths thesis comprises of seven ohapters* Chapter one is introductory In nature* In this chapter ws first discuss the aatrophysloal importance of the mathematical problem of nonradial oscillations of gassous spheres* Then after g vlng a brief survey of the literature available on the subject, ws havs pointed cut some of ths important aspeots of the problem of nonradial oscillations which are at rsssnt ths main t pios of investigation* la chapter second vs first preeent in brief the mathematical formulation of the problem of nonradial oscillations of a gaseous sphsrs* Then we disouss ths details of ths numerical method for solving the eigen value problem of nonradial oscillations which ws havs prefered to follow In the course of our pressnt lnvestigatlona* The method is based on the use of Chebyehev (ii) polynon ais sad Is a variation of the method used by Hurley, fiobarte and Wright (1966). We have also shown that the method of Chebyshev polynomials as developed by us can be used to investigate the problsr. of nonradial oscillations of composite nodela as well* He have further shown that the present method assumes a much simple font and the numerical computations are considerably reduced when it Is applied to composite models i the approxi mation when perturbations In gravitational potential are neglected. We are not aware of any earlier use of Chebyshev polynomial technique in ths investigations of nonradial oscillations of composite models* In order to illustrate the salient features of the method and also to denonatr- to its accuracy, we have discussed in detail the application of the method to tl» homogeneous model* Ws have shown that the results obtained by the present method compare very well tilth the earlier results obtained for this model by other methods* As an appsndix to this chapter we give some of the inportant properties of Chebyshev poly nomials v; ion have been exploited in the development of the numerical method discussed in the chapter* In Chapter III the technique of numerical solution developed in chapter two is applied to study ths nonradial oscillations or the Prasad model* In this model density P varies according to the law P » p. (l - ^/R2), here P„ (ill) stands for the value of P at the centre and *r* is distance of an element from the centre of gaseous sphere of radius R*This model falls under the category of less centrally condensed models and can approximately represent the density distribution of a star in early stages of its evolution. The study has been undertaken to investigate the nature of the spectrum of nonradial oscillations of this model and also to explore the possibility of resonance between radial and nonradial modes of oscillations* To study the effect of perturbations in gravitational poten tial on the nonradial modes of oscillations, we have Investigated the nonradial oscillations of this model first by retaining, in the equations of nonradial oscillations* ths terms due to perturbations in gravitational potential and then by omitting these terms* In chapter IV we have investigated the problsr of nonradial oscillations of a series of oompoolts models usually refsrrsd to as generalised Roche models, Ths models consist of homogeneous cores surroundsd by envelopes in which density varies inversely as ths square of distance from the centre of mass* By taking the interface between ths core en& ths envelope at different points we get an interesting series of models in wlich not only the central condensation 9^/ Pvaries considerably but the parameter A (Schwareschild's criterion of oonvective stability) which Is known to vitally affect the epeotrum of modes of Ut) nonradial oscillations also behaves quits differently* In particular, we have obtained numsrlcal solutions for the nonradial modes of oscillations of three nodele* These models have the interface between the core and ths envelope at a distance from the centre which is *8, ,5 and *2 times ths radius of the model* Ths computations have been performed for bote the oases when ths offset of perturbations in gravitational potential is included in the equations of nonradial oscillations and also when this is neglected. In order to stuSy the effect of (order of spherical harmonic) and y (the ratio of speci fic heats), computations have also been performed for Gm 1, lm 2 and v m 5/3 and 10/7. The series of models discussed in Chapter I? lis between homogoneoua model at one extreme and inverse square model at the other extreme. In ease of inverse square model the density is assumed to vary inversely as the square of ths distance from ths centre. It folio under the category of highly centrally cond need models* wnereas ths nonradial oscillations of the homogeneous model have/been studied in detail, we are not aware of any earlier attempt to study the problem of nonradial oscilla tions of the inverse square model. In chapter ? we have tried to discuss in brief the problem of nonradial oscilla tions of the Inverse square model* It is shown that lias (r) Roehs's model and polytrope of index 5 inverse square model also has, in general, an irregular singularity and* therefore, regular solutions of ths problem of nonradial oscillations of this model compatible with the boundary conditions are not possible* It is further shown that for one particular value of y which is 1, tie Irregular nature of the singularity is removed* and thus solutions compatible with the boundary conditions are possible* However, y « l corresponds to an isothermal equilibrium* Therefore, for this particular value of 7* the solutions of nonradial oscillations of ths Invsrse square model have only limited astrophysical significance* Numerical solut ions fox a few modes of nonradial osdilations of this model for y » 1 have also been obtained* In chapter VI we have considered the nonradial osci llations of certain composite polytxoplc models. In this context we have considered two series of models* In ths first series we have considered models with polytroplo Index n * 3 in the envelope and n • 1*5 in the core* These correspond to models with radiative envelopss and oonvective cores.By taking interface between the core and ths envelope at various points we have investigated the effect of the sloe of ths oonvective oore on the nonradial oscillations of a stellar model* Ws have, in particular, oonsidered three models in this series by taking the dis tance from the centre of ths interface between the core end On) the envelope as .6250, .6160 and .2094 times ths radius of the model* These Investigations also enable us to investigate ths offset of keeping A, the Sehwarasohild's criterion of conveotlve stability* equal to aero upto different extents in the core of a model* because for a polytrope of index 1*5* A is zero if y, the ratio of speci fic heats, is taken to be 5/3* In the second series of models ws have considered a set of three additional composite models having different polytropic structures but having the same central condensa tion ?e/ p" (Pc being the density at the centre and P* the average density) as that of the third composite model of the first series (with polytxoplc index n « 1*5 in ths oore and a • 3*0 in the envelope and having interface at • 2094)* The three composits models considered in thU series aret (1) A composite model with polytroplc index n as 1,5 in the core, 3*0 in the envelope, the distance from the centre of the interface between the oore and ths envelope being *349 times the radius of the model* (11) A composite model with polytroplc index n as 0,5 in the oore, 4*0 In the envelope and having Inter face between the oore and the envelope at a dis- • 246 times the radius of the model* (vii) (iH) A composite model with polytroplc index a « 0.5 in the oore, 3#0 |a ths envelops and the (Hstanoo ot the interfaoe fron the centre being #146 times the radius of the model* The present eeries of model enable us to investigate the sffeet of A visa vis central condensation on the modes of nonradial oscillations of gaseous spheres. In view of the fact that there is considerable saving in computational time If we solve the equations of nonradial oscillations of composite models in the approximation when variations in gravitational potential are neglected, a majority of computations in the ease of ths above two series of models have been performed in the approximation when perturbations in gravitational potential are neglected* However, to study the sffeet of neglecting ths perturbations in gravitational potential on the modes of nonradial oscillations, sons computations have also been performed on the basis that ths variations in gravitational potential ere not neglected* Keeping in view ths interdependent nature of the problems considered in the present thesis* formal discussion of the results obtained in different chapters has been postponed till ths end* In chapter VII ws rive an overall discussion of the results obtained in earlier chapters and try to draw certain conclusions.