Oscillations of vertically integrated relativistic tori - II. Axisymmetric modes in a Kerr space-time

This is the second in a series of papers investigating the oscillation properties of relativistic, nonself-gravitating tori orbiting around black holes. Extending the work done in a Schwarzschild background, here we consider the axisymmetric oscillations of vertically integrated tori in a Kerr space-time. The tori are modelled 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 space-time, and subsequently perform a global eigenmode analysis to compute the axisymmetric p modes. In analogy with what has been found in a Schwarzschild background, these modes behave as sound waves that are modified by rotation and are globally trapped in the torus. For constant distributions of specific angular momentum, 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 distributions of angular momentum, 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 comment on how p modes 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.