#### 3.4.2. Regular Epicyclic Motion and Explanation of HF QPOs in Microquasars by Magnetically Modified Geodesic Models

Although the motion of charged particles around magnetized black holes is generally chaotic, according to the Kolmogorov–Moser–Arnold theorem in the vicinity of the minima of the effective potential, i.e., for energies sufficiently close to the energy of stable circular orbit with given axial angular momentum, the charged particle can follow an epicyclic motion in radial and vertical directions, governed by the equations of fully regular linear harmonic motion [

32]. The frequencies of the harmonic oscillatory motion around a stable circular orbits are governed by the second derivatives of the effective potential at the position of the static circular orbit, as shown in [

31,

32].

6 Here, we demonstrate an alternative approach to determine the epicyclic frequencies, based on direct application of the equations of motion and their perturbations [

22].

The four-velocity of the circular orbits has two non-zero components,

${u}^{\mu}=({u}^{t},0,0,{u}^{\varphi})$. For the maximally charged magnetized Kerr black holes (with

${Q}_{W}$), the Lorentz equation implies the radial component of the equatorial motion in the form

while the normalization condition for the four-velocity gives

Determination of a circular orbit at a given radius

r by the four-velocity components

${u}^{t}$ and

${u}^{\varphi}$ is an alternative to determination by the motion constants

$\mathcal{L}$ and

$\mathcal{E}$. The orbital frequency at the circular orbit

${\mathsf{\Omega}}_{\varphi}$, related to distant observers and called also Keplerian frequency,

${\mathsf{\Omega}}_{K}$, and the Larmor frequency, related purely to the external magnetic field,

${\mathsf{\Omega}}_{L}$, are determined by the relations:

The epicyclic motion of a charged particle, given by the world-line

${x}^{\mu}\left(\tau \right)$ and concentrated around the “equilibrium” stable circular orbit with

${x}_{0}^{\mu}\left(\tau \right)$ governed by the canonical axial angular momentum of the charged particle, is determined by deviation vector

${\xi}^{\mu}\left(\tau \right)={x}^{\mu}\left(\tau \right)-{x}_{0}^{\mu}\left(\tau \right)$. Introducing the deviation vector into the Lorentz equation, we arrive in the first order expansion at small deviations to the equations of linear harmonic oscillations in the vertical and radial directions [

25]

where

${\mathsf{\Omega}}_{\theta}$ (

${\mathsf{\Omega}}_{r}$) is the angular frequency of the particle vertical (latitudinal) or radial epicyclic motion related to the coordinate time

t, i.e., measured by distant static observers. For the maximally charged magnetized Kerr black holes with the maximal Wald charge

${Q}_{W}$, we arrive to the relations [

22]

and

where

The frequencies determining the orbital and epicyclic motion of charged particles represent generalization of the frequencies related to the epicyclic geodesic motion. As the frequencies are given in dimensionless form, we have to use the formula

where

$i=(\varphi ,r,\theta )$, if we are interested in directly observable frequencies given in physical units.

The radial profiles of the orbital and epicyclic frequencies of the charged particle motion in the field of magnetized black holes have several properties fundamentally different in comparison with the radial profiles of the frequencies characterizing the geodesic epicyclic motion—however, the crucial effect of vanishing of the radial epicyclic frequency

${\nu}_{r}$ at the marginally stable circular orbit holds in both cases [

22]. The “magnetically induced” modifications of the radial profiles of the orbital and epicyclic frequencies of the charged particle motion enable more efficient application of the magnetically modified geodesic models of the the twin HF QPOs observed in some microquasars [

102] or active galactic nuclei [

103].

7 We thus present a short overview of the special properties of the radial profiles of the “magnetized” orbital and epicyclic frequencies. The properties are related to the four families of the circular orbits of charged particles orbiting a magnetized Kerr black hole as introduced in [

32].

The most important differences between the radial profiles of the orbital and epicyclic frequencies in the magnetized and purely geodesic cases occur for the frequencies

${\nu}_{r}$ and

${\nu}_{\varphi}={\nu}_{K}$—due to the strong effect of the Lorentz force in the radial direction. The Lorentz force increases the radial epicyclic frequency

${\nu}_{r}$, namely for the PALO and RALO orbits, but the axial (Keplerian) frequency

${\nu}_{K}$ decreases for such orbits. Inversely, the radial profile of the latitudinal (vertical) epicyclic frequency

${\nu}_{\theta}$ cannot be significantly influenced by the magnetic field because of the direction of the Lorentz force. The other crucial difference corresponds to the relation of the magnitudes of the three considered frequencies—for the geodesic motion the relation

${\nu}_{r}<{\nu}_{\theta}<{\nu}_{\varphi}$ holds at all radii of the radial profiles around non-extreme black holes

8, for the magnetized orbits, the interrelations between the three frequency radial profiles strongly depend on the type of the orbit. Nevertheless, the difference most relevant from the point of view of astrophysics occurs for all four types of the circular orbits—the point of intersection of the

${\nu}_{r}$ and the

${\nu}_{\theta}$ profiles occurs above the ISCO radius. For the PALO and RALO orbits, there is intersection of the

${\nu}_{\varphi}$ and the

${\nu}_{r}$ radial profiles, and for the PALO orbits even the crossing of the

${\nu}_{\theta}$ and the

${\nu}_{\varphi}$ radial profiles can occur.

The crossings of the orbital and epicyclic frequency radial profiles have important astrophysical consequences that could lead to significant observable signatures as vanishing of the nodal shift of orbiting matter. In the case of the RALO orbits, we observe possibility that at some radii the orbital frequency is much smaller than both the epicyclic frequencies,

${\nu}_{\varphi}<<{\nu}_{r}\sim {\nu}_{\theta}$ enabling the toroidal type of the charged particle motion required in the kinetic model [

106]. In

Figure 10, we present typical types of the radial profiles of the “magnetized” frequencies—notice that the magnetized frequency radial profiles are similar to radial profiles of the frequencies of the string loop oscillations [

100,

101].

The so-called geodesic models proposed to explain the twin HF QPOs observed in microquasars, i.e., binary systems composed of a black hole and a companion star, use the frequencies of the orbital and epicyclic geodesic motion—-for detailed overview, see [

104]. The twin HF QPOs are two peaks of the X-ray power density observed in the case of microquasars in the frequency range

$\nu \sim 100$–400 Hz; we can thus assume their creation in the innermost parts of the Keplerian accretion disks as the observed frequencies are close to the frequency of the orbital Keplerian motion near the ISCO of the stellar mass black holes with mass in the range

$M\sim 5$–

$15\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}$ [

107]. In the most frequently studied microquasars, the HF QPOs are usually detected as twin with the frequency ratio close to 3:2 invoking relevance of the resonance phenomena [

108,

109].

9Twin HF QPOs with the frequency ratio 3:2 indicating the strongest effect of the so-called parametric resonance [

109] are extensively studied in the case of three microquasars related to the sources GRS 1915+105, XTE 1550-564 and GRO 1655-40 [

111,

112]. The mass of the black holes in the microquasars are estimated by optical measurements based on the weak field gravity effects, being independent of the strong gravity effects that are governing the twin HF QPOs and profiled spectral lines or spectral continuum relevant in estimates of the spin of the black holes in microquasars [

113]. Both the spectral methods are independent of the HF QPO measurements, so the predictions of the spin by the spectral methods have to be confronted with the predictions of the spin by the various variants of the geodesic model of the twin HF QPOs, thus making separation of the acceptable variants giving predictions in accordance with the spectral measurements.

The geodesic model of twin HF QPOs is represented by a set of variants that assume a crucial role of the orbital and epicyclic frequencies of the circular geodesic motion and various combinations of these frequencies [

104,

112,

114]. The frequency ratio 3:2 (or other rational ratios as 3:1 or 2:1) of the observed twin frequencies in microquasars indicate resonance phenomena at work [

104,

108]. The most widely discussed are the epicyclic resonance [

109] and the hot-spot relativistic precession [

115] variants of the geodesic model, having simple identification of the twin observed frequencies (lower

${f}_{\mathrm{L}}$, and upper

${f}_{\mathrm{U}}$) with the frequencies of the variants of the geodesic model of twin HF QPOs—for the epicyclic resonance (ER) variant, the identification of the frequencies reads

while, in the relativistic precession (RP) variant of the geodesic model, there is

Modifications of the basic ER and RP variants of the geodesic model are presented and discussed in [

104,

112].

The geodesic twin HF QPO model can explain the observed twin frequencies with ratio 3:2 in all the three microquasars, GRS 1915+105, XTE 1550-564 and GRO 1655-40, but not by a single variant [

111]. However, the magnetic modification of the geodesic QPO model improves this discrepancy as a consequence of the more complex behavior of the radial profiles of the orbital and epicyclic frequencies. In the fitting of the observational frequencies to those predicted by variants of the magnetically modified geodesic QPO model, the calculated model frequencies

${\nu}_{r}(r,M,a,\mathcal{B})$,

${\nu}_{\theta}(r,M,a,\mathcal{B})$,

${\nu}_{\varphi}(r,M,a,\mathcal{B})$, and their combinations corresponding to a specific variant of the QPO model, are related to the observed QPO frequencies,

${\nu}_{u}={f}_{\mathrm{U}}$,

${\nu}_{l}={f}_{\mathrm{L}}$, thus implying relations between the radius

${r}_{3:2}$ and the parameters

$a,M,\mathcal{B}$. The crucial ingredient of the magnetic modification of the geodesic model is the presence of the crossing point of the epicyclic radial and latitudinal (or orbital) frequencies, implying that we have to consider two possible “resonance” radii

${r}_{3:2}$ and

${r}_{2:3}$ due to the resonance conditions

As the resonance conditions in the magnetically modified model are independent of the black hole mass

M, their solution has no dependence on mass and the method developed in [

104] can be used. The fitting gives relevant restrictions on the magnetic parameter combining the magnetic field intensity with the specific charge of the orbiting matter; knowing the magnetic field intensity in the region of the epicyclic motion of radiating matter, we can obtain relevant restriction on the specific charge of the matter, or vice versa, assuming specific charge of the radiating matter, we obtain restrictions on the intensity of the magnetic field. Both ways then enable direct comparison with the data obtained due to observations.

We briefly demonstrate the results of the fitting procedure in the case of the ER variant applied to the three microquasars GRS 1915+105, XTE 1550-564 and GRO 1655-40, with mass and spin limited by methods independent of the measurements of HF QPOs (for discussion of these methods see e.g., [

112]); the limits on mass and spin are given in

Table 3.

Introducing the resonance radius

${r}_{3:2}$ (

${r}_{2:3}$) to the equations for the frequencies

${\nu}_{u}$ or

${\nu}_{l}$, we express them in terms of the black hole mass, spin, and the magnetic parameter that give so-called fitting lines. This enables comparison with limits on the black hole mass and spin obtained by other methods. We have demonstrated that the fitting could be done well both for the epicyclic and the relativistic resonance models—for details, see [

22]. Limits on the magnetic parameters implied by the HF QPOs fitting procedure for all types of the orbits and the ER variant of the magnetized geodesic model are presented in

Table 4 .

#### 3.4.3. Chaotic Scattering

For charged particles with canonical axial angular momentum allowing stable circular motion, their fate depends on their energy related to the effective potential. If the energy is sufficiently small, the regular harmonic oscillations in the radial and vertical directions can occur, as discussed in the previous subsection. With increasing energy, the particles can enter the region of the effective potential open to the black hole horizon, where the particle directly falls into the black hole. The other possibility is that the motion remains bounded, but undergoes transition to purely chaotic motion. The final possibility is represented with energies corresponding to the effective potential open to infinity in the vertical direction corresponding to direction of the magnetic field lines. All the possible cases are treated in the following subsection devoted to the study of the fate of the ionized disks in dependence on the magnitude of the magnetic interaction parameter. We give some comments to the last case, when transmutation from the regular circular motion to the regular motion along the magnetic field lines occurs through an intermediate period of the so-called chaotic scattering [

33], see

Figure 11.

In order to understand the chaotic scattering phenomenon, it is useful to consider the asymptotic behavior of the effective potential of the charged particle motion in the field of a Kerr black hole immersed in a homogeneous magnetic field orthogonal to the equatorial plane of the geometry. In the

z-direction corresponding to the direction of the magnetic field lines, the effective potential takes the asymptotic form [

33]

Clearly, the behavior of the effective potential demonstrates binding of the particle in the

x-direction due to the term

$x\left(\mathcal{B}\right)$, while there is a minimum of the effective potential implying minimal energy of the particle that can reach infinity in the

z-direction given by the relations

A particle can reach infinity, if its energy

$\mathcal{E}\ge {\mathcal{E}}_{\mathrm{min}}$, however, the energy measured at infinity has to be modified by the “magnetic" factor to

${\mathcal{E}}_{\mathrm{infty}}=\mathcal{E}-2a\mathcal{B}$, as the effective potential for the motion in the homogeneous magnetic field in the Minkowski spacetime do not contain the term

$2a\mathcal{B}$. In the asymptotically flat region of the magnetized Kerr black holes, we can write the equations of motion of charged particle in the cylindrical coordinates

$t,\rho ,\theta ,\varphi $
where

$\tau $ denotes the proper time of the moving particle; we have introduced the separation to the energy of the translational (longitudal) motion

and the transverse (perpendicular) motion mixing the radial oscillatory motion with the orbital motion

In the Minkowski spacetime, all the energies

$\mathcal{E}$,

${\mathcal{E}}_{\infty}$,

${\mathcal{E}}_{z}$, and

${\mathcal{E}}_{0}$ are constants of the motion, however, in the strong gravity of the Kerr black hole, the motion constants are only the total energy parameters

$\mathcal{E}$ and

${\mathcal{E}}_{\infty}$, while there is possible transmutation of the energies

${\mathcal{E}}_{z}$, and

${\mathcal{E}}_{0}$, representing the translantional and the perpendicular motion due to the generally chaotic character of the particle motion—for details, see [

33].

The escaping particle can be characterized by the Lorentz factor

and its part corresponding to the translational motion. The maximum of the translational Lorentz factor has a different form in dependence on the sign of the magnetic parameter and corresponding velocity of the circular (azimuthal) motion [

33]

and

For the configurations with

$\mathcal{B}>0$, there is possible full transmutation of the transverse motion to the translational motion, and the particle can follow the magnetic field lines. On the other hand, the configurations with

$\mathcal{B}<0$ do not allow for the complete transmutation and minimal Larmor orbital motion must be completed to the translational motion, with minimum axial angular velocity given by

${u}^{\varphi}=2\mathcal{B}\mathcal{L}$. The Lorentz factors obtained in this way can be of order of 10 even for mediate values of the magnetic parameter magnitude,

$\mathcal{B}\sim \phantom{\rule{3.33333pt}{0ex}}1$, as demonstrated in [

33]. However, for elementary particles in the vicinity of realistic magnetized black holes, the magnetic parameter magnitude can be by many orders higher, implying much more efficient acceleration of elementary particles by the electromagnetic field, much larger energy of these particles and related Lorentz factor reaching many orders of 10 [

39].

The ionization of Keplerian disks by irradiation, or other ways as neutron decay on electrons and protons, can be treated as a magnetic Penrose process (MPP) that in combination with the chaotic scattering can be considered as a very simple model of creation of jets in the close vicinity of the horizon of magnetized black holes—in the MPP, the energy of the orbital motion ${\mathcal{E}}_{0}$ can be enormous due to the action of the electric part of the electromagnetic field governed by the potential component ${A}_{t}$, and the chaotic scattering near the black hole horizon can transform the whole energy of the ${\mathcal{E}}_{0}$ mode into the energy of the translational motion ${\mathcal{E}}_{z}$, giving escaping particles with Lorentz factor $\gamma >>1$.

Extremely efficient acceleration of elementary particles is possible, leading to their ultra-high energy, if the magnetized black hole is rotating, generating the electric part of the electromagnetic potential; contrary to the case of the well known Blanford–Znajek process [

42], it is not necessary to have a near-extreme rotating black hole—any rotating black hole is sufficient.

10 No such electric part of the potential exists in the case of non-rotating magnetized Schwarzschild black holes and no acceleration is possible in this case, but the transmutation effects works as well as in the case of the Kerr black holes, implying possibility of creation of winds, flows of particles orthogonal to the plane of the disks orbiting Schwarzschild black holes, but unable to escape to infinity [

33].

Note that Lorentz factors

$\gamma \le \phantom{\rule{3.33333pt}{0ex}}1$, corresponding to ultra-high energy elementary particles, could be obtained due to the electromagnetic field of magnetars, i.e., rotating neutron stars having extremely strong magnetic fields that reach near their surface values of

$B\sim \phantom{\rule{3.33333pt}{0ex}}{10}^{15}$ G, overcoming by seven orders the typical magnetic field

$B\sim \phantom{\rule{3.33333pt}{0ex}}{10}^{8}$ G of standard neutron stars observed in the atoll sources. However, these highly accelerated elementary particles cannot reach the distant observers due to the enormous friction caused by the radiation reaction related to their synchrotron radiation in such extremely strong magnetic fields that is by many orders more efficient in comparison with the radiation reaction in the magnetic fields around black holes as we demonstrate below [

36].

#### 3.4.4. Modeling of Ionized Keplerian Disks around Magnetized Kerr Black Holes

For simplicity, we assume that the magnetic field lines are asymptotically parallel to the rotation axis of the Kerr black hole spacetime (i.e., to the vertical direction given by the

z-axis) [

23]. For the corotating Keplerian disk in the equatorial plane, a neutral test particle follows a circular geodesic corotating orbit with the covariant specific energy

$\mathcal{E}$ and the specific axial angular momentum

$\mathcal{L}$ given as [

46]

However, the purely equatorial circular geodesic motion remains equatorial after ionization and is not entering the chaotic scattering regime, if there is no perturbation influence in the latitudinal (vertical) direction.

11 In order to enter the chaotic scattering regime in the case of precisely equatorial disk, we have to assume its non-zero thickness allowing for an infinitesimal shift of the ionized particle in the perpendicular direction, or some infinitesimal impulsion in the perpendicular direction to the charged particles created by the ionization process. In fact, such a very small influence in the perpendicular direction can be caused by the irradiating photons.

On the other hand, it is natural to consider a Keplerian accretion disk having initially a very small but nonzero inclination to the equatorial (

x-

y) plane of the spacetime [

33]—the slightly inclined disk represents another natural possibility to guarantee the entering to the chaotic scattering regime. The external uniform magnetic field remains aligned with the

z-axis (vertical direction), being perpendicular to the equatorial plane of the spacetime.

Ionization of a neutral particle can be realized by irradiation of an atom by an incident photon resulting in an ion and electron, or we can consider a free neutron decay resulting in a proton and electron (plus an antineutrino, not considered here)—such a model corresponds naturally to the MPP [

38,

116] with the original 1st neutral particle being split into two charged particles—2nd and 3rd. Conservation of charge and canonical momentum of the particles entering the process reads

Of course, at the moment of ionization, the conservation of the kinetic momentum also holds

as the electromagnetic contributions cancel each other out [

10,

33]. In the realistic scenarios considered here, i.e., the neutral atom ionization by irradiation or neutron

$\beta $ decay, one of the created charged particles is much more massive than the other one, say

${m}_{2}/{m}_{3}\gg \phantom{\rule{3.33333pt}{0ex}}1$, due to the mass ratio of proton to electron or ion to electron. The more massive charged particle clearly takes almost all the momentum of the original neutral particle and the dynamical influence of the lighter charged particle can be neglected

Another realistic scenario is based on the idea of the accretion disk created by plasma considered as a quasi-neutral soup of charged particles, electrons and ions (protons), orbiting the black hole on circular orbits. For the matter of the disk dense enough, the main free path of the charged particles is very short in comparison with the length of the circular orbits, and the charged particles move collectively as a neutral body. Near the inner edge of the disk, the plasma density decreases substantially and the charged particles are not longer suppressed by their neighbours and start to move freely, being fully influenced by the electromagnetic field.

All the possible realistic ionization scenarios imply a simple model of ionized Keplerian disks based on the MPP process: a neutral particle becomes charged while its mechanical momentum remains conserved and starts to feel the influence of the electromagnetic field. This model was introduced in [

33] where particle escape velocities and structure of escape zones were explored that were later studied also in [

10,

34,

39]. This ionization model was also used to study the regular regime of the epicyclic oscillations [

32] and its relation to the HF QPOs [

22], or the fate of the Keplerian disks in dependence on the inclination angle in the case of non-rotating black holes [

10,

35].

The fate of the ionized Keplerian disks orbiting a magnetized Kerr black hole can be summarized describing the influence of the two basic parameters, the magnetic parameter

$\left(\mathcal{B}\right)$ and the black hole spin

a, on the fate of the ionized Keplerian disks, i.e., of the regular regime represented by the epicyclic motion, the regime of chaotic motion transforming the Keplerian disk into a toroidal accretion structure, destruction regime represented by direct infall of ionized matter into the black hole, and destruction jet regime represented by the escape of charged particles along the magnetic field lines to infinity. In the fourth case, the ultra-high energy particles (protons and ions) can be generated [

39].

For the inclined Keplerian disk, we assume a special off-equatorial motion realized along the spherical trajectories with the orbit radius

${r}_{0}$ remaining constant, and with varying latitude

$\theta $. The ionization event can be thus assumed at some

${\theta}_{0}\ne \pi /2$ enabling the entering of the chaotic regime of the ionized particle motion—the charged test particle will be initially, at the ionization event, located on the spherical orbit with initial position

${x}^{\alpha}$ and four-velocity

${u}_{\alpha}$
the motion constants of the particle are governed by the specific energy

$\mathcal{E}$ and specific angular momentum

$\mathcal{L}$ of the electrically neutral particles following spherical orbits [

117]

where the

${R}^{2}={r}^{2}+{a}^{2}{cos}^{2}\theta $ and the functions

$Q,S$ are given by the relations

Equations (

134) and () reduce to the expressions for the specific energy and specific angular momentum of the equatorial geodesic circular (Keplerian) orbits (

128) when

$\theta =\pi /2$. The inner edge of the spherically modified Keplerian disks is located at the innermost stable spherical orbit (ISSO) with radius implicitly given by

This relation reduces for the motion confined in the equatorial plane to those governing the ISCO.

In order to consider situations relevant from the point of view of astrophysics, we assume the ionized part of the corotating Keplerian disks to be located near its inner edge, and we follow the fate of the ionized part by integrating motion of the charged particles in the background characterized by the magnetic parameter

$\mathcal{B}$ and the black hole spin

a. Typical examples of trajectories of the charged particles created by the ionization, initially orbiting on spherical orbits with radius

${r}_{0}>{r}_{\mathrm{ISSO}}$, can be found in

Figure 12,

Figure 13,

Figure 14 and

Figure 15 constructed for characteristic values of the black hole spin

$a=\mathrm{0.0.3.0.7.0.998}$ and for typical values of the magnetic parameter

$\mathcal{B}=\pm 0.001,\pm 0.01,\pm 0.1,\pm 1$, assuming the inclination of the Keplerian disk to be

${\theta}_{0}=1.5$. In the case of the magnetized Schwarzschild black holes and arbitrarily inclined Keplerian disks detailed results were presented in [

10,

35]. Here, we present the main results and differences between the situation around non-rotating and rotating black holes. The Keplerian disks are here assumed to be corotating with the Kerr geometry.

Contrary to the Schwarzschild case with

$a=0$ where the canonical energy remains the same as the kinetic energy, for the magnetized rotating black holes there is a shift in the canonical energy governed by the non-zero

${A}_{t}$ component of the electromagnetic potential that allows escaping of the charged particle to infinity along the magnetic field lines – the acceleration due to the electric component of the electromagnetic field can be enormous [

33,

38]. For example, the escaping particles can undergo extremely efficient MPP reaching an ultra-high energy of the order of

${10}^{21}$ eV, if the process of acceleration occurs near a supermassive black hole having mass

$M\sim {10}^{10}{M}_{\odot}$ surrounded by magnetic field of the order of

${10}^{4}$ G [

10,

39,

81].

In the case of magnetized Schwarzschild black holes (

$a=0$) the situation is illustrated in

Figure 12. Here, and, in the following figures representing the disk fate, the uncharged test particles on innermost stable spherical orbit (the dashed circle) represent the inner edge of the Keplerian accretion disk; the fate is studied for a region close to the inner edge. For the disk that remains neutral, or for the vanishing magnetic field (

$\mathcal{B}=0$ case), all the orbits keep the original form and the inclined razor thin disk remains. For a weak electromagnetic interaction switched-on (

$\mathcal{B}=\pm 0.001$ cases), the charged particles enter epicyclic oscillatory motion around the circular orbit in both radial and latitudinal directions, only particles forming the innermost region are captured by the black hole in the case of magnetic attraction (

$\mathcal{B}=-0.001$); the accretion disk becomes slightly thicker due to the vertical epicyclic motion. For stronger electromagnetic interactions switched-on (

$\mathcal{B}=\pm 0.01,\pm 0.1$ cases), the charged particles enter a fully chaotic regime or they fall directly to the black hole; the accretion disk is destroyed, partly destroyed, or transformed into thick toroidal structure. The complete destruction of the inner region of the Keplerian disk occurs in the

$\mathcal{B}=-0.01$ case, when all the particles are captured by the black hole, a large part of the charged particles are captured for

$\mathcal{B}=-0.1$, the rest enter chaotic motion. If large electromagnetic interaction is switched-on (

$\left|\mathcal{B}\right|\ge 1$ cases), the Lorentz force dominates the particle motion. The charged particles can enter a special regime of the regular motion due to spiraling up and down along the magnetic field lines due to fast Larmor oscillations combined with slow motion around the black hole in the clockwise (

$\mathcal{B}>0$) or the counterclockwise (

$\mathcal{B}<0$) direction, demonstrating vertical oscillations—in this case, the thin Keplerian disk transforms into a special toroidal structure.

In the case of slowly rotating black holes (

$a=0.3$) illustrated in

Figure 13, the situation remains very similar to the Schwarzschild case for the weak and mediate values of the magnetic parameter of charged particles with an exception of the mediate repulsive magnetic parameter with

$\mathcal{B}=0.1$ when an increase of the particle canonical energy due to electric acceleration causes the overcoming of the repulsive centrifugal barrier and capturing of the innermost ionized particles by the black hole. For the large magnitudes of the magnetic parameter, the behavior of the ionized part of the disk becomes dramatically dependent on the sign of the magnetic parameter

$\mathcal{B}$. For the negative values (

$\mathcal{B}=-1$ case), the character of the motion remains similar to those corresponding to the Schwarzschild black holes, but, for positive values (

$\mathcal{B}=1$ case), the influence of the magnetic field is getting stronger with increasing radius of the original orbit causing increase of the amplitude of the vertical oscillatory motion.

For black holes with mediate spin (

$a=0.7$), the results are illustrated in

Figure 14. Now, the fate of the inner part of the disk is similar to the case of slow rotation, but the effects of black hole spin are amplified—namely, for the case of mediate positive values of the magnetic parameter,

$\mathcal{B}=0.1$ increase of the particle canonical energy due to electric acceleration causes entering of the chaotic motion creating winds in the innermost region of the disk. For the large positive values of the magnetic parameter (

$\mathcal{B}=1$ case), the combined influence of the black hole spin and the magnetic field implies escape of all the considered particles to infinity along the magnetic field lines with the exception of those originating almost at the ISSO that remain in oscillatory motion in the vertical direction combined with the Larmor orbital motion with very small radii.

For near-extreme rotating black holes (

$a=0.998$), the fate of the Keplerian disks is demonstrated in

Figure 15. The near-extreme spin now causes the decrease of charged particles from the innermost regions of the disk for both large and mediate negative values of the magnetic parameter (cases

$\mathcal{B}=-0.1,-1$), but the infall of the particles is stopped in the case of slightly negative magnetic parameters (case

$\mathcal{B}=-0.01$). For the large positive magnitudes of the magnetic parameter (case

$\mathcal{B}=1$), charged particles from whole the ionized region escape to infinity, those from the innermost part of the disk are executing a period of chaotic motion before the escape.

We can summarize that the increasing spin of the black hole for fixed magnetic parameters causes increasing influence of the electromagnetic field due to an increase of its electric component. For low values of the magnetic parameter, the increasing spin generally implies increasing chaotic character of the motion, but it can be transmuted to particle capture by the black hole for particles sufficiently close to the ISCO if the particle canonical energy overcomes the repulsive centrifugal barrier. The most interesting situation occurs for values of the magnetic parameter $\mathcal{B}\sim \pm 1$, as in this case around slowly rotating black holes with ${a}^{2}\sim 0$, for both negative and positive values of the magnetic parameter, an interesting regular type of the motion arises, demonstrating the Larmor rotation around the magnetic field lines mixed with a slow orbital motion combined with a quasi-geodesic “epicyclic” vertical and radial oscillatory motion. For $\mathcal{B}\sim -1$, this kind of motion survives with increasing spin. For $\mathcal{B}\sim 1$, increasing spin implies increasing amplitude of the vertical oscillatory motion and possible tendency of the particles to escape to infinity; moreover, for large values of the spin ($a>0.7$), the orbits in close vicinity of the black hole horizon demonstrate increasing degree of chaos. With magnetic parameter $\mathcal{B}$ increasing above the unit value, the particles escape to infinity and the region of chaotic motion starts to be more and more restricted. In the next section, we concentrate on the case of acceleration and motion of the escaping particles discussing in detail the MPP.