Oscillations of vertically integrated relativistic tori - I. Axisymmetric modes in a Schwarzschild space-time

This is the first of a series of papers investigating the oscillation properties of relativistic, non-self-gravitating tori orbiting around a black hole. In this initial paper we consider the axisymmetric oscillation modes of a torus constructed in a Schwarzschild space-time. To simplify the treatment and make it as analytical as possible, we build our tori with vertically integrated and vertically averaged quantities, thus transforming the eigenvalue problem into a set of coupled ordinary differential equations. The tori are also modelled with a number of different non-Keplerian distributions of specific angular momentum, and we discuss how the oscillation properties change when different distributions of angular momentum are considered. Our investigation progresses by steps. We first consider a local analysis in Newtonian gravity and determine the properties of acoustic wave propagation within these objects, as well as the relations between acoustic and epicyclic oscillations. Next, we extend the local analysis to a general relativistic framework. Finally, we perform a global analysis and determine both the eigenfunctions and the eigenfrequencies of the axisymmetric oscillations corresponding to the p modes of relativistic tori. These behave as sound waves globally trapped in the torus and possess eigenfrequencies appearing in the simple sequence 2:3:4:..., independently of the distribution of angular momentum considered. The properties of the modes investigated here are in good agreement with those observed in recent numerical simulations, and could have a number of different applications. In X-ray binary systems containing a black hole candidate, for instance, p-mode oscillations could be used to explain the harmonic relations in the high-frequency quasi-periodic oscillations observed. In systems comprising a massive torus orbiting a black hole, on the other hand, p-mode oscillations could be used to explain the development or the suppression of the runaway instability.