Weak nonlinear coupling between epicyclic modes in slender tori

TL;DR

This paper investigates the weak nonlinear coupling of epicyclic modes in slender tori near compact objects, identifying a dominant 3:2 resonance and conditions for parametric excitation of oscillations.

Contribution

It demonstrates the weak coupling of epicyclic modes in slender tori and characterizes the resonances, especially the 3:2 resonance, including symmetry constraints and excitation conditions.

Findings

Strongest resonance is 3:2 between axisymmetric modes.

Resonances between axisymmetric and non-axisymmetric modes are excluded.

Parametric excitation occurs only at high radial oscillation amplitudes.

Abstract

We examine nonlinear oscillations of slender tori in the vicinity of black holes and compact stars. These tori represent useful probes of the complicated, nonlinear dynamics of real accretion disks and provide at least qualitative understanding of their oscillations. We demonstrate that epicyclic modes of such tori are weakly coupled due to the pressure and gravitational forces. We explore all possible resonances between two epicyclic modes up to the fourth order. We show that the strongest resonance between axisymmetric modes is 3:2. In addition, any resonance between an axisymmetric and a non-axisymmetric mode is excluded due to axial and equatorial-plane symmetries of the equilibrium torus. We examine a parametric excitation of vertical axisymmetric oscillations by radial oscillations in the 3:2 resonance. We show that the resonance may be significant only for high-amplitude radial oscillations.