Nonlinear effects in Pulsations of Compact Stars and Gravitational Waves

Abstract

This thesis investigates in the time domain a particular class of second order perturbations of a perfect fluid non-rotating compact star: those arising from the coupling between first order radial and non-radial perturbations. Radial perturbations of a non-rotating star, by themselves not emitting gravitational waves, produce a peculiar gravitational signal at non-linear order through the coupling with the non-radial perturbations. The information contained in this gravitational signal may be relevant for the interpretation of the astrophysical systems, e.g. proto-neutron stars and accreting matter on neutron stars, where both radial and non-radial oscillations are excited. Expected non-linear effects in these systems are resonances, composition harmonics, energy transfers between various mode classes.The coupling problem has been treated by developing a gauge invariant formalism based on the 2-parameter perturbation theory (Sopuerta, Bruni and Gualtieri, 2004), where the radial and non-radial perturbations have been separately parameterized. Our approach is based on the gauge invariant formalism for non-radial perturbations on a time-dependent and spherically symmetric background introduced in and Gundlach & M. García (2000). It consists of further expanding the spherically symmetric and time-dependent spacetime in a static background and radial perturbations and working out the consequences of this expansion for the non-radial perturbations. As a result, the non-linear perturbations are described by quantities which are gauge invariant for second order gauge transformations where the radial gauge has been fixed. This method enables us to set up a boundary initial-value problem for studying the coupling between the radial pulsations and both the axial (Passamonti et al.,2006) and polar (Passamonti at el., 2004) non-radial oscillations. These non-linear perturbations obey inhomogeneous partial differential equations, where the structure of the differential operator is given by the previous perturbative orders and the source terms are quadratic in the first order perturbations. In the exterior spacetime the sources vanish, thus the gravitational wave properties are completely described by the second order Zerilli and Regge-Wheeler functions.The dynamical and spectral properties of the non-linear oscillations have been studied with a numerical code based on finite differencing methods and standard explicit numerical algorithms.The main initial configuration we have considered is that of a first order differentially rotating and radially pulsating star, where the initial profile of the stationary axial velocity has been derived by expanding in tensor harmonics the relativistic j-constant rotation law. For this case we have found a new interesting gravitational signal, whose wave forms show a periodic signal which is driven by the radial pulsations through the sources. The spectra confirm this picture by showing that the radial normal modes are precisely mirrored in the gravitational signal at non-linear ...

Generic behavior of nonlinear sound waves near the surface of a star: Smooth solutions

“…Finally, for smooth solutions, sufficiently close to the surface, the entropy gradient can be neglected compared to the density gradient in determining the pressure gradient. We can therefore approximate the ideal gas as isentropic, with equation of state (2). (This last approximation would not hold if a shock reached the surface.…”

“…Our main question is what kinematic boundary conditions can be used in a numerical simulation to represent the free boundary at the surface of the star. This has been addressed in general relativity by Sperhake [1] for nonlinear spherical perturbations, using Lagrangian coordinates, and by Passamonti [2] for linear non-spherical perturbations. Here we consider the nonlinear case in Eulerian coordinates.…”

Gravitational Radiation from Neutron Stars and Black Holes

“…We have evolved these perturbative equations for two types of initial configurations: (i) a differentially rotating and radially pulsating star, and (ii) the scattering of a gravitational wave by a radially pulsating star. We will focus on the former configuration, as the dynamics of coupling oscillations is more interesting (for a complete description see [20]). …”