On the stability of massive stars

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On the basis of the equations for stellar structure together with an equation of state, stellar models may be constructed. Using these stellar models, the effects of different parameters on the evolution and internal structure of stars can be studied. The stability of stellar models is investigated by applying perturbations to the set of dependent variables. For infinitesimally small perturbations higher order terms can be neglected and a linear stability problem is obtained (linear approximation). The mathematical problem then poses a boundary eigenvalue problem with the complex eigenfrequencies of the stellar models considered as eigenvalues. The real parts correspond to the inverse of the pulsation periods whereas the imaginary parts indicate the growth rate of an unstable mode or damping of a damped mode respectively. Thus this approach provides an estimate of possible pulsation periods which may be compared with observed periods of a star with parameters close to that of the model considered. A linear approach can never predict the amplitude of an oscillation. Therefore nonlinear simulations are required to determine, e.g., the final velocity amplitude or the final variation of the brightness of the object. Moreover, the final period in the nonlinear regime might be different from the linear period due to nonlinear effects. In many models studied here it has been found that the linearly determined periods substantially differ from the periods obtained by nonlinear simulations. Therefore nonlinear simulations are inevitable if theoretically determined periods are to be compared with observed periods in stars. In this thesis, linear stability analyses together with nonlinear simulations have been performed for a variety of models for massive stars. The linear pulsation equations are solved using the Riccati method which has the advantage that the frequencies and eigenfunctions of even high order modes can be calculated with prescribed high accuracy. In order to determine the final fate of unstable models, nonlinear simulations are performed. If these simulations are to be meaningful, they have to satisfy an extremely high accuracy, since the energies of interest (e.g., the kinetic energy) are by several orders of magnitude smaller than the dominant energies (gravitational potential and internal thermal energy). The requirements are met by the fully conservative numerical scheme adopted. Full conservativity is achieved by implicit time integration. For the stability analysis envelope models for zero age main sequence stars with solar chemical composition in the mass range between 50 M$_{\odot}$ and 150 M$_{\odot}$ have been constructed. The linear stability analysis of these models reveals instabilities above 58 M$_{\odot}$. The pulsation periods of unstable modes lie in the range between 3 hours and 1 day. Nonlinear simulations of unstable models indicate that their final state is associated with pulsationally driven mass loss and mass loss rates of the order of 10$^{-7}$ M$_{\odot}$/yr. Recent observations of the B-type supergiant 55 Cygni, reveal that this star pulsates with periods in the range between 2.7 hours and 22.5 days. The authors identify the pulsations with pressure, gravity and strange modes. Motivated by the observations we have performed a linear stability analysis of corresponding stellar models together with nonlinear simulations of unstable models. As a result we find that the mass of 55 Cygni lies below 28 M$_{\odot}$. The pulsation periods derived from nonlinear simulations lie well within the range of observed periods and the mass loss estimated from the simulations is consistent with observed mass loss. In the linear stability analysis of zero age main sequence models a set of non oscillatory (monotonically) unstable modes has been identified. Such modes are present both for radial and nonradial perturbations. Their growth rates vary with the harmonic degree and their kinetic energies show a secondary maximum very close to the surface of the models which may indicate the possibility of an observational identification. In the thesis, we present an attempt to understand the behaviour and origin of these modes.