Accretion Tori around Black Holes

The work presented in this Thesis has been developed along two complementary lines: the construction of numerical relativistic codes to follow the non-linear dynamics of compact objects and the study of the oscillation properties of relativistic tori using linear perturbative analysis. In what follows, a brief overview on how this work has been carried out will be presented. In particular, Chapter 1 serves as an introduction to the oscillation properties of accretion discs and the runaway instability of tori around black holes since these issues are the main motivation behind the work presented in this Thesis. In Chapter 2 of this Thesis we investigate the oscillation properties of nonselfgravitating, relativistic tori orbiting around black holes. We extend the work done in a Schwarzschild background and consider the axisymmetric oscillations of vertically integrated tori in a Kerr spacetime. The tori are modelecl with a number of different non-Keplerian distributions of specific angular momentum and we discuss how the oscillation properties depend on these and on the rotation of the central black hole. We first consider a local analysis to highlight the relations between acoustic and epicyclic oscillations in a Kerr spacetime and subsequently perform a global eigenmocle analysis to compute the axisymmetric p-mocles. In analogy with what was found in a Schwarzschilcl background, these modes behave as sound waves and are globally trapped in the torus. For constant specific angular momentum distributions, the eigenfrequencies appear in a sequence 2:3:4:... which is essentially independent of the size of the disc and of the black hole rotation. For non-constant angular momentum distributions, on the other hand, the sequence depends on the properties of the disc and on the spin of the black hole, becoming harmonic for sufficiently large tori. We also compare the linear perturbative approach with non-linear hydrodynamic simulations of geometrically thick discs performed with a 2D general relativistic hydrodynamic code. Next, we present estimates of the gravitational wave emission clue to the oscillations of high density tori. We also comment on how p-modes in low-density tori could explain the high frequency quasi-periodic oscillations observed in low-mass X-ray binaries with a black hole candidate and the properties of an equivalent model in Newtonian physics. An introduction to numerical relativistic hydrodynamics is presented in Chapter 3. The first part of this Chapter is devoted to the 3+ 1 decomposition of the Einstein equations and a conformal traceless reformulation of this system of equations. Next, we concentrate on the high resolution shock capturing schemes which are the most advanced methods for solving the hydrodynamic equations. In the final part of this Chapter, we concentrate on three dimensional general relativistic hydrodynamic simulations and the development of a new 3D, parallel and general relativistic hydrodynamic code, the Whisky code. This code implements high resolution shock capturing methods and exploits the spacetime evolution provided by the Cactus code. We describe our contributions to the Whisky code, several numerical tests we have performed and simulations of relativistic tori orbiting around Schwarzschild black holes we have carried out. We also note that this code has recently been used to investigate the dynamics of the gravitational collapse of rotating neutron stars to fonn Kerr black holes. The following two chapters are devoted to the study of relativistic compact objects in axisymmetry by means of numerical simulations and to the development of the computational tools needed to solve the combined system of Einstein equations and the general relativistic hydrodynamic equations. With the aim of investigating the dynamics of geometrically thick self-gravitating tori orbiting around black holes and assessing whether these systems may be subject to the runaway instability, we have developed a new two dimensional, fully relativistic and non-vacuum code, the Nada code for the study of axisymmetric systems. The Nada code implements high resolution shock capturing methods and the Einstein equations are cast into a system of constraint and evolution equations in the 3+ 1 decomposition of this system of equations. In particular, Chapter 4 focusses on the description of the Nada code. We first present the method we have implemented to impose the axisymmetry condition while using Cartesian coordinates. Next, we describe the boundary conditions and gauge conditions implemented in our code and we conclude the Chapter with an outline of the formulation of the relativistic hydrodynamic equations used and the schemes to integrate in time the coupled system of the Einstein equations and the relativistic hydrodynamic equations. In the following Chapter, Chapter 5, we present several tests performed to assess the accuracy of the code and results from simulations of spherical relativistic stars.