Free motion around black holes with discs or rings: Between integrability and chaos - I

Abstract

Geodesic dynamics is regular in the fields of isolated stationary black holes. However, due to the presence of unstable periodic orbits, it easily becomes chaotic under various perturbations. Here, we examine what amount of stochasticity is induced in Schwarzschild space-time by the presence of an additional source. Following astrophysical motivation, we specifically consider thin rings or discs lying symmetrically around the hole, and describe the total field in terms of exact static and axially symmetric solutions of Einstein's equations. The growth of chaos in time-like geodesic motion is illustrated on Poincaré sections, on time series of position or velocity and their Fourier spectra, and on time evolution of the orbital 'latitudinal action'. The results are discussed in terms of dependence on the mass and position of the ring/disc and on geodesic parameters (energy and angular momentum). In the Introduction, we also add an overview of the literature. © 2010 The Authors. Journal compilation © 2010 RAS.