Determination of chaotic behaviour in time series generated by charged particle motion around magnetized Schwarzschild black holes

Abstract

We study behaviour of ionized region of a Keplerian disk orbiting a Schwarzschild black hole immersed in an asymptotically uniform magnetic field. In dependence on the magnetic parameter B, and inclination angle θ of the disk plane with respect to the magnetic field direction, the charged particles of the ionized disk can enter three regimes: a) regular oscillatory motion, b) destruction due to capture by the magnetized black hole, c) chaotic regime of the motion. In order to study transition between the regular and chaotic type of the charged particle motion, we generate time series of the solution of equations of motion under various conditions, and study them by non-linear (box counting, correlation dimension, Lyapunov exponent, recurrence analysis, machine learning) methods of chaos determination. We demonstrate that the machine learning method appears to be the most efficient in determining the chaotic region of the θ − r space. We show that the chaotic character of the ionized particle motion increases with the inclination angle. For the inclination angles θ ∼ 0 whole the ionized internal part of the Keplerian disk is captured by the black hole.

Influence of Cosmic Repulsion and Magnetic Fields on Accretion Disks Rotating around Kerr Black Holes

“…We study the influence of the dimensionless black hole spin a on the fate of the ionized Keplerian disks for given magnetic intensity parameter characterizing the ratio of the electromagnetic and gravitational forces acting on charged test particles. The special case of ionized Keplerian disks orbiting a Schwarzschild black hole (a = 0) was studied in detail in [35], back-reaction effect due to radiation of the charge particles on their motion was studied in [36,37].…”

Magnetized Black Holes: Ionized Keplerian Disks and Acceleration of Ultra-High Energy Particles

“…The dependence of the fate of the Keplerian disk on its inclination angle θ to the magnetic field lines of an uniform magnetic field is studied in detail in [24]-for the mediatelly strong magnetic field parameter |B| = 0.1 representing the reachest variety of possible behaviour we can summarize the following conclusions. For small inclination angles (0 < θ < 30 o ) there is a tendency to destroy the full inner region of the disk by its capture by the black hole, for mediate values of the inclination (30 o < θ < 75 o ) the chaotic regime fully enters the play creating thick accretion structures, while in the case of large inclination angles corresponding to disks nearly perpendicular to the magnetic field lines (75 o < θ < 90 o ), the chaotic regime allows for some exception of regularity corresponding to epicyclic motion.…”

“…In the second step of our discussion we consider a Keplerian disk around a Schwarzschild black hole immersed in one of the three considered types of the magnetic field, having an arbitrary inclination angle θ relative to the magnetic field lines that are perpendicular to the spacetime equatorial plane θ = π/2-see Figure 9. The case of Keplerian disks inclined to the external uniform magnetic field aligned with the z axis (vertical direction) [5] was considered in detail by [24]. The conditions at the moment of the ionization of the Keplerian disk are thus determined for the case of the arbitrary inclination of the disk in the uniform magnetic field or any other magnetic field, as they can be considered as fully local conditions that can be applied for the ionization event.…”

“…Here we discuss the fate of the ionized Keplerian disk around magnetized Schwarzschild black holes under general conditions, with special attention devoted to the dependence of the fate on the initial inclination of the disk to the magnetic field lines. Chaoticity of the motion of the ionized matter orbiting Schwarzschild black holes has been recently studied in detail in [24].…”

“…For small inclination angles (0 < θ < 30 o ) there is a tendency to destroy the full inner region of the disk by its capture by the black hole, for mediate values of the inclination (30 o < θ < 75 o ) the chaotic regime fully enters the play creating thick accretion structures, while in the case of large inclination angles corresponding to disks nearly perpendicular to the magnetic field lines (75 o < θ < 90 o ), the chaotic regime allows for some exception of regularity corresponding to epicyclic motion. Detailed study of the distribution of the chaoticity and regularity regions in dependence on the parameter B, the position of the original circular geodesic orbit, and its inclination, studied by a variety of non-linear methods of determination of chaotic motion, can be found in [24]. In Figure 9 we add an overview and comparison of the influence of the inclination angle in the case of the uniform, dipole and parabolic magnetic fields.…”