Oscillation modes of relativistic slender tori

Abstract

Accretion flows with pressure gradients permit the existence of standing waves which may be responsible for observed quasi-periodic oscillations (QPO's) in X-ray binaries. We present a comprehensive treatment of the linear modes of a hydrodynamic, non-self-gravitating, polytropic slender torus, with arbitrary specific angular momentum distribution, orbiting in an arbitrary axisymmetric spacetime with reflection symmetry. We discuss the physical nature of the modes, present general analytic expressions and illustrations for those which are low order, and show that they can be excited in numerical simulations of relativistic tori. The mode oscillation spectrum simplifies dramatically for near Keplerian angular momentum distributions, which appear to be generic in global simulations of the magnetorotational instability. We discuss our results in light of observations of high frequency QPO's, and point out the existence of a new pair of modes which can be in an approximate 3:2 ratio for arbitrary black hole spins and angular momentum distributions, provided the torus is radiation pressure dominated. This mode pair consists of the axisymmetric vertical epicyclic mode and the lowest order axisymmetric breathing mode.Comment: submitted to MNRA

Oscillations of tori in the pseudo-Newtonian potential

“…Zanotti et al 2005;and Blaes et al 2006a), and (following Lee et al 2004) the radial and vertical positions of the centre of mass of the torus, as well as snapshots of the density, internal energy and velocity field inside the torus.…”

“…The residual ρu resembles that of the breathing mode (see Fig. 1 in Blaes et al 2006a). 3:2 parametric resonance may only happen in a limited range of the parameter space, the Arnold tongue (see e.g. Arnold 1978Arnold , 1983).…”

“…The strong peak at 1.5 Ω K is present in the spectrum at this frequency regardless of the location of the torus, while the less prominent peak at approximately 0.86 Ω K slightly changes its position depending on the location of the torus. Blaes et al (2006a) have calculated the lowest-order slender torus modes. Two of these modes, the incompressible "plusmode" and the acoustic "breathing" mode have frequencies of 0.87 Ω K and 1.54 Ω K for the torus of model A2 (see their Table 2 for explicit expressions).…”

“…It is of interest since its frequency forms a ratio of 3:2 with the Keplerian frequency Ω K , which is also the vertical epicyclic frequency. It can perhaps be identified as the lowest-order acoustic wave mode (see Blaes et al 2006a), which has a frequency close to 1.5 Ω K , but with some dependence on the radial location of the torus. It was suggested by Blaes et al (2006a) that this mode and the vertical epicyclic mode might be responsible for the 3:2 commensurability observed in the HF QPOs.…”

The Polish doughnuts revisited

“…Sikora 1971;Madau 1988;Szuszkiewicz et al 1996) and in analytic calculations of small-amplitude oscillations of accretion structures in connection with QPOs (see e.g. Blaes et al 2006). In the same context, numerical studies of their oscillation properties for different angular momentum distributions were first carried out by Rezzolla et al (2003a,b).…”