Large-scale magnetic fields enabling fitting of the high-frequency QPOs observed around supermassive black holes

Stuchlík, Zdeněk; Kološ, Martin; Tursunov, Arman

doi:10.1093/pasj/psac066

Abstract

Recently, it has been argued that the high-frequency quasi-periodic oscillations (QPOs) observed in black hole systems of various scales in mass in cases of supermassive black holes (SMBH) are not consistent with any of the simple physical models, based on frequencies of the geodesic epicyclic motion (Smith et al. 2021, ApJ, 906, 92). We test if such a disease can be simply cured by geodesic models based on epicyclic frequencies modified by the effect of electromagnetic interaction of slightly charged orbiting matter, with large-scale magnetic fields with values observed around SMBHs in active nuclei. Inspired by GRAVITY/ESO observations, we assume a slightly charged hot spot, as the relativistic motion of a plasma in magnetic field leads to charge separation and non-negligible charge density in the orbiting plasma. Its electromagnetic interaction with the large-scale magnetic field around the black hole can be weak enough, allowing for nearly harmonic epicyclical oscillatory motion of the hot spot with frequencies given by modification of those applied in the geodesic model. Even the simplest epicyclic resonance variant of the geodesic model, modified by slight electromagnetic interaction admitted by observations, can fit the QPOs in the case of both stellar-mass and supermassive black holes. We have shown that even a tiny excess of charged particles in the quasi-neutral plasma of the radiating hot spot, allowed by observations, enable an explanation of QPOs observed in active galactic nuclei. We also estimate the effect of the electromagnetic interaction on the shift of the innermost stable circular orbits, implying the degeneracy in the measurements of spins of the black hole candidates.

1 Introduction

Many astrophysical systems containing black holes or neutron stars observed in X-ray clearly exhibit the presence of modulations of the X-ray flux in the form of quasi-periodic oscillations (QPOs). The frequencies of QPOs range from millihertz up to kilohertz, corresponding to the low- (LF) and high-frequency (HF) QPOs, respectively. Construction of a reliable physical model of QPO generation can shed light on the phenomena occurring around compact objects. Almost two decades of QPO observations suggest that the observed frequencies are in inverse proportion to the mass of the central object (Remillard & McClintock 2006) and the processes responsible for QPOs occur close to the black hole, likely on the scales of the innermost stable circular orbit (ISCO). On the other hand, the dependence of observed frequencies on the black hole spin is not that apparent and sometimes seems illusive (Török et al. 2005, 2011; Bursa 2011). The universal property of QPOs from both stellar and supermassive black holes is that they are usually observed with double peaks, having a frequency ratio close to 3 : 2 (Gierliński et al. 2008; Zhou et al. 2015; Kotrlová et al. 2020).

It is particularly interesting that the characteristic frequencies of HF QPOs are close to the values of the frequencies of test particle geodesic epicyclic oscillations in the regions near the innermost stable circular orbit (ISCO), which makes it reasonable to construct the model involving the frequencies of oscillations associated with the orbital motion around Kerr black holes. After the first detection of QPOs, there were many attempts to fit the observed QPOs, and different models have been proposed, such as the hot-spot models, disco-seismic models, warped disk model, and many versions of resonance models, developed in the framework of general relativity or alternative theories of gravity. However, until now, the exact physical mechanism of the generation of HF QPOs is not known, since none of the models can fit the observational data from different sources. An explanation of QPOs, i.e., fitting the observed frequencies with a single particular model, remains challenging, where none of the considered models based on frequencies of the epicyclic motion can explain the observed frequencies, especially in the case of supermassive black holes (Smith et al. 2021). Surprisingly, the situation changes considerably if one assumes the effect of an external magnetic field on the dynamics of slightly charged matter generating twin HF QPOs around a rotating black hole. The investigation of the influence of the magneto-gravitational interaction on the QPO phenomena is the main aim of the present paper. To demonstrate the possibility of the fitting of observed QPO frequencies, we consider the magnetic version of the simplest model of epicyclic resonances, which assumes coupling between the orbital and radial or vertical epicyclic oscillations during the motion of a QPO’s component around a black hole.

We have used the simple approximation of the geodesic models modified by the slight electromagnetic interaction for two reasons. First, the complex magnetohydrodynamics model (Tchekhovskoy 2015) and its modifications (Liska et al. 2018; Davis & Tchekhovskoy 2020) do not demonstrate systematic oscillatory motions that could be considered as an explanation of the observed QPOs in AGNs, implying therefore some doubts on the stability of the oscillatory models predicted by simplified models of oscillating magnetized disks (Fu & Lai 2009; Ortega-Rodríguez et al. 2015) that govern influence of the magnetic field on the oscillatory modes. Secondly, we are inspired by direct GRAVITY observations of flares corresponding to extended hot spots (Gravity Collaboration 2018). We use estimates of possible weak charge related to such hot spots, as presented in Tursunov et al. (2020b), assuming the zeroth approximation of the hot spot as a charged particle with corresponding specific charge. The charged particle representing the hot spot is interacting with the large-scale magnetic field extended near the black hole horizon, leading to near-harmonic oscillatory motion in radial and vertical directions. We assume the simplest approximation of the asymptotically uniform magnetic field centered around a Kerr black hole rotation axis that could be relevant especially for the near-harmonic oscillatory motion that is confined to fixed radius (Stuchlík et al. 2020). The character of the magnetic field near the black hole horizon, namely its variations in the vertical direction, could be considered as a possible source of non-linear effects leading to parametric resonance phenomena between the radial and vertical oscillatory motion. In such a situation we can directly apply the results of our previous works on the modified geodesic model as presented in Tursunov, Stuchlík, and Kološ (2016) and Kološ et al. (2017).

We begin in section 2 with a general discussion related to the magnetic field estimates in realistic black hole systems and their influence on a plasma revolving around black holes of difference mass scales. In section 3, we describe general properties of the harmonic oscillations of charged matter in the presence of an external magnetic field. In section 4, we introduce an epicyclic resonance model modified by the magnetic field influence and perform the fitting of all observed QPOs frequencies by a single model. We discuss the main results and give conclusions in section 5.

For general equations we use the geometrized system of units, in which c = 1 and G = 1. For the expressions with an astrophysical application and estimates we use the physical (Gaussian) units.

2 Magnetic field effect on a plasma rotating around a black hole

2.1 Magnetic field around black holes

It is natural to assume that a regular magnetic field is present around black holes, which is the case in nearly all celestial objects. Black holes are usually surrounded by accretion disks constituted by highly conducting plasma, the dynamics of which give rise to the generation of the regular magnetic field. One of the indications of the presence of a strong magnetic field around a black hole is the observed jet, which is supposed to be created due to the combined effect of magnetic and gravitational fields in the immediate vicinity of the black hole (Blandford & Znajek 1977). Collimation of jets likely indicates the existence of large-scale regular magnetic fields in the interstellar medium. Besides strong local magnetic fields, spiral galaxies (Beck & Wielebinski 2013), including our own, are likely to exhibit weak galactic magnetic fields, drastically increasing towards the galactic centre (Eatough et al. 2013).

The strength of a magnetic field around a particular source is usually estimated from the intensity of the non-thermal synchrotron emission, by the measurements of the rotation angle of the plane of polarization, or by the Zeeman effect near stars. For stellar-mass black holes, magnetic field strength may range from a few up to 10⁸ G, while for SMBHs the characteristic magnetic field is of the order of 10⁴ G (Daly 2019). We expect the occurrence of these magnetic field intensity values due to both the environmental effect (a possible strong magnetic field present in the partner of the black hole in binary systems) and focusing of the magnetic field lines due to local spacetime curvature in the black hole vicinity. Energy densities of magnetic fields in that range are negligible for any sufficient contribution to the spacetime geometry (Gal’Tsov & Petukhov 1978). Therefore, the spacetime around realistic black hole candidate immersed into external magnetic field can be fully described by the Kerr metric.

Although the plasma surrounding a black hole is believed to be globally electro-neutral with the total charge being equal to zero, local charges can still be present in small regions along the disk. In that case, these charges can be influenced by the external magnetic fields (Cremaschini et al. 2013; Kovář et al. 2014; Stuchlík et al. 2020; Tursunov et al. 2020b). Moreover, in the case of electrons or ions the magnetic field plays a leading role due to the large values of the charge-to-mass ratio for elementary particles e/m, entering the dynamical equations. Below we show that it is also true in the case of possibly macroscopic QPO components due to the charge separation in a magnetized plasma revolving around black holes. This implies that the role of magnetic fields in the vicinity of astrophysical black holes cannot be neglected and could even be crucial.

2.2 Charge separation in a plasma

It is usually assumed that a plasma surrounding astrophysical black holes is electrically neutral due to the neutralization of charged plasma on relatively short timescales. Any oscillation of the net charge density in a plasma is supposed to disappear very quickly due to induction of the large electric field caused by charge imbalance. However, in the presence of an external magnetic field and when the plasma is moving at relativistic speeds, one can observe the charge separation effect in a magnetized plasma and consequently measure the net charge density. Applied to rotating neutron stars with magnetic field, this special relativistic effect of plasma charging is represented by the Goldreich–Julian (GJ) charge density (Goldreich & Julian 1969). In fact, the motion of a plasma induces an electric field, which in its co-moving frame should be neutralized, leading to the appearance of a net charge in the rest frame. The GJ charge density is usually referred to for pulsar magnetospheres, although it is applicable in more general cases, as we will show below. In fact, an analogous argument can be used to obtain the net-charge density of the plasma moving around a Kerr black hole and generating the QPOs.

For a black hole magnetosphere, the charging of the plasma was first described by Ruffini and Wilson (1975), who showed that the twisting of magnetic field lines due to the rotation of the black hole induces an electric charge in both the black hole and the surrounding magnetosphere. For a black hole in uniform magnetic field configuration, the charging mechanism was introduced by Wald (1974). Computing Maxwell equations inside the magnetosphere (see, for details, e.g., Ruffini & Wilson 1975), one can find that the total charge of the black hole magnetosphere should be equal to the charge of the black hole with the opposite sign, i.e., Q_mag = −Q_BH. This condition holds for the large class of accretion models. Dynamical effects of the QPO generation studied in this paper focus on the transient features of the hot spots during their orbital timescales. These timescales (which are different for stellar mass and supermassive black holes) can be estimated for a test particle moving at the ISCO as follows:

$$\begin{eqnarray} \tau _{\rm orb} (10^{9}\, M_{\odot }) &\approx & 126.6 \frac{M}{10^{9}\, M_{\odot }} \left(\frac{r}{r_{\rm ISCO}} \right)^{{3}/{2}}\:\mbox{hr}, \end{eqnarray}$$

(1)

$$\begin{eqnarray} \tau _{\rm orb} (10\, M_{\odot }) &\approx & 4.5 \frac{M}{10\, M_{\odot }} \left(\frac{r}{r_{\rm ISCO}} \right)^{{3}/{2}}\:\mbox{ms}. \end{eqnarray}$$

(2)

Therefore, we assume that the orbital timescales of the hot spots are shorter than the discharging timescale. To estimate this, the viscous timescale of electrons on similar scales can be used. Following Tursunov et al. (2020b) and Zajaček et al. (2018) the viscous timescale can be estimated to be of the order of

$$\begin{eqnarray} \tau _{\rm visc} (10^{9}\, M_{\odot }) \approx 8.5 \frac{M}{10^{9}\, M_{\odot }} \left( \frac{h}{r} \right)^{-2} \left(\frac{r}{r_{\rm ISCO}} \right)^{{3}/{2}} \left( \frac{\alpha _{\rm visc}}{0.1} \right)^{-1}\:\mbox{d},\\ \end{eqnarray}$$

(3)

$$\begin{eqnarray} \tau _{\rm orb} (10\, M_{\odot }) \approx 17.3 \frac{M}{10\, M_{\odot }} \left( \frac{h}{r} \right)^{-2} \left(\frac{r}{r_{\rm ISCO}} \right)^{{3}/{2}} \left(\frac{\alpha _{\rm visc}}{0.1} \right)^{-1}\:\mbox{ms},\\ \end{eqnarray}$$

(4)

where the height-to-radius ratio h/r ∼ 1 and the viscosity parameter α ∼ 0.1 (Tursunov et al. 2020b). Therefore, we neglect charge fluctuations (charging/discharging) in the orbital dynamics of the hot spots, which may affect their long-term behavior, i.e., being longer than the QPOs appearing timescales.

Following the works by Bardeen et al. (1972), Ruffini and Wilson (1975), Thorne and MacDonald (1982), Muslimov and Tsygan (1992), and Izzard et al. (2004), for our purposes, the most convenient way to describe the electrodynamics of the relativistic plasma around a black hole is to use the 3 + 1 splitting of spacetime introduced by Thorne and MacDonald (1982) and further developed in Izzard et al. (2004). In tensor form, the covariant Maxwell equations read

$$\begin{eqnarray} \nabla _\nu \, ^*\!F^{\mu \nu } = 0, \quad \nabla _\nu F^{\mu \nu } = J^\mu , \end{eqnarray}$$

(5)

where F^αβ and *F^αβ are the Maxwell and Faraday (dual) tensors, respectively, and J^μ is the four-current. Splitting these equations in to time and space components we get

$$\begin{eqnarray} \nabla \cdot {\bf B} = 0, \quad \nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t }, \end{eqnarray}$$

(6)

$$\begin{eqnarray} \nabla \cdot {\bf D} = 4\pi \rho , \quad \nabla \times {\bf H} = 4 \pi {\bf j} + \frac{\partial {\bf D}}{\partial t }. \end{eqnarray}$$

(7)

It should be noted that D and H coincide with E and B measured by a zero angular momentum observer (ZAMO), whose four-velocity in axially symmetric spacetime is defined by

$$\begin{eqnarray} n^{\mu } = (n^t,0,0,n^\phi ), \end{eqnarray}$$

(8)

where

$$\begin{eqnarray} (n^t)^2=\frac{g_{\phi \phi }}{g_{t\phi }^2 -g_{tt}g_{\phi \phi }}, \quad n^\phi = -\frac{g_{t\phi }}{g_{\phi \phi }}\, n^t. \end{eqnarray}$$

(9)

Applying the covariant derivative ∇ to equation (7) we get the charge conservation law

$$\begin{eqnarray} \partial _{t}\rho + \nabla \cdot {{\bf J}}= 0. \end{eqnarray}$$

(10)

Assuming that the magnetosphere of a black hole shares its background symmetry, i.e., applying stationarity and axial symmetry, we get the effective charge density in the form (Muslimov & Tsygan 1992)

$$\begin{eqnarray} && \rho = - \frac{1}{4 \pi } \nabla \cdot \left[ \frac{1}{\alpha } \left(1- \frac{k }{\eta _{\rm r}^3} \right) {\bf v} \times {\bf B} \right], \end{eqnarray}$$

(11)

$$\begin{eqnarray} && {\bf v} = {\bf \Omega } \times {\bf r}, \quad k = \frac{R_{\rm g} \beta }{ a }, \quad \eta _{\rm r} = \frac{a}{r}, \quad \alpha = \sqrt{g_{tt}}, \end{eqnarray}$$

(12)

where a is the black hole’s spin parameter, Ω is the angular velocity of orbiting matter, β is the moment of inertia of the plasma rotating around a black hole, and α is the lapse function.

It was argued by Komissarov (2004) that an electric field measured by ZAMO drives the electric current along the magnetic field lines, resulting in the separation of charges and the drop in the electrostatic potential at least within the ergosphere. In this scenario, the black hole can act as the unipolar generator (Blandford & Znajek 1977) similar to the classical Faraday disk, which is based on the use of electromotive force qv × B, resulting in the charge separation due to a drop in voltage between the edge of the disk and its center. Further analysis led to the conclusion that any rotating compact object, like a neutron star or black hole immersed into an external magnetic field and surrounded by plasma or an accretion disk, generates a rotationally induced electric field at the object as well as in the surrounding magnetosphere.

The orientation of electric field lines, and consequently, the sign of the rotationally induced electric charge, depends on the relative orientation of the black hole spin, external magnetic field, and the direction of the angular velocity of accreting matter. This orientation does not alter the values of acquired charge. In total, one can distinguish four types of orbits, leading to the four distinguishable scenarios (see figure 1 in Tursunov et al. 2016). It has been argued in Zajaček et al. (2018) that in the case of Sgr A* the induced charge of the black hole is likely positive, for various reasons (more positive charge is in the range of the gravitational influence of the SMBH than negative charge, and expectation of alignment between the magnetic field lines and the SMBH spin). Later, based on the analysis of the flare dynamics observed in the close vicinity of the Galactic centre SMBH (Tursunov et al. 2020b), it was shown that the flare components (as well as surrounding plasma matter) are likely to have a small excess of negatively charged particles, which is in accordance with the expectations derived long ago by Ruffini and Wilson (1975). In the current work, however, we consider all possible situations, without restricting ourselves on the particular orientation of orbits and lines of electromagnetic fields.

In order to estimate the value of charge density caused by the charge separation in a plasma, one can use the simplified special-relativistic version of equation (11) in the following form, which is somewhat similar to the GJ formula (Goldreich & Julian 1969):

$$\begin{eqnarray} \rho _q \sim \frac{1}{2 \pi c} \frac{\nu _{\phi } \, B_{\perp }}{ |e|} , \end{eqnarray}$$

(13)

where ν_φ is the orbital angular velocity of a charged hot spot and B_⊥ is the strength of the magnetic field orthogonal to the orbital plane. The orbital velocity of the geodesic motion of matter at the distance of the ISCO ∼ 6GM/c² is |$\nu _{\phi }^{\rm BH} \sim 220\:$|s⁻¹ for typical stellar mass of 10 |${M_{\odot}}$| black hole and |$\nu _{\phi }^{\rm SMBH} \sim 10^{-6}\:$|s⁻¹ for a supermassive black hole of 10⁹ |${M_{\odot}}$| mass. Applying typical strengths of magnetic fields around black holes as 10⁴ G, one can estimate the number densities of extra charged particles in a unit volume as

$$\begin{eqnarray} && \rho _q^{\rm SMBH} \sim 10^{-4} \left(\frac{B}{10^4 \:\mbox{G}} \right) \left( \frac{M}{10^9\, M_{\odot }}\right)^{-1}\:\mbox{cm}^{-3}, \end{eqnarray}$$

(14)

$$\begin{eqnarray} && \rho _q^{\rm BH} \sim 10^{4} \left(\frac{B}{10^4 \:\mbox{G}} \right) \left( \frac{M}{10\, M_{\odot }}\right)^{-1}\:\mbox{cm}^{-3}. \end{eqnarray}$$

(15)

In a complex magnetic field (Contopoulos et al. 2015), we have to expect fully chaotic particle motion and no observed QPOs. Therefore, an ordered magnetic field has to be expected, where the observed QPOs arise. The simplest case is the uniform field assumed here. General relativistic magnetohydrodynamics (GRMHD) simulations (Tchekhovskoy 2015) indicate in all initial states of accreting tori for vicinity of rotation axis an ordered parabolic magnetic field (Nakamura et al. 2018; Kološ & Janiuk 2020). In such realistic cases we can simply consider the parameter B in equations (14) and (15) as a function of radius given by the formula of parabolic field (Nakamura et al. 2018). Such a modification does not influence our results for epicyclic oscillations that are governed by B locally, at given radius.

These values are less than the typical total number density in a plasma by several orders of magnitude (Blandford 2001). For example, the particle number density of a plasma surrounding an SMBH of mass 10⁹|${M_{\odot}}$| to support a magnetic field of the order of 10⁴ G is ∼10¹² cm⁻³ (see, e.g., Blandford 2001), while the plasma density around Sgr A* at the Galactic centre is of the order of 10⁶–10⁸ cm⁻³ (Eckart et al. 2012, 2017). For a characteristic spherical volume of the radius of r ∼ R_g = GM/c², we get a net charge excess of the order of

$$\begin{eqnarray} && |q|^{\rm SMBH} \sim 10^{21} \left(\frac{B}{10^4 \:\mbox{G}} \right) \left(\frac{M}{10^9\, M_{\odot }}\right)^3 {\rm C}, \end{eqnarray}$$

(16)

$$\begin{eqnarray} && |q|^{\rm BH} \sim 10^{5} \left(\frac{B}{10^4 \:\mbox{G}} \right) \left(\frac{M}{10\, M_{\odot }}\right)^3 {\rm C}. \end{eqnarray}$$

(17)

One can see that the charge separation in a relativistic magnetized plasma can lead to the presence of a sufficient net charge in the chosen characteristic volume that can have considerable effect on the dynamics.

Let us now estimate the limits on the characteristic ratio of the Lorentz and gravitational forces acting on charged matter in a selected spherical volume with estimated charge [equation (16)] moving around a black hole at the ISCO radius. If the magnetic field lines are orthogonal to the orbital plane of matter in this volume, following Tursunov et al. (2020b) we get the limits

$$\begin{eqnarray} 10^{-5} < \frac{F_{\rm Lor}}{F_{\rm grav}} < 10. \end{eqnarray}$$

(18)

We will give tighter constraints to this ratio by fitting the observed QPOs by the magnetized epicyclic model. We shall further parametrize the above ratio by the dimensionless parameter |${\cal B}$| in equation (25) and compare the oscillatory frequencies of matter with those observed in QPOs.

3 Influence of electromagnetic interaction on the harmonic epicyclic oscillations

3.1 Spacetime geometry

The presence of an external magnetic field does not influence the metric of the black hole spacetime, which is assumed to be a rotating Kerr metric;

$$\begin{eqnarray} d s^2 = g_{tt}d{t}^2 + 2g_{t\phi }d{t}d\phi + g_{\phi \phi }\mathrm{d}\phi ^2 + g_{rr}d{r}^2 + g_{\theta \theta }d\theta ^2 , \end{eqnarray}$$

(19)

where non-zero components of the metric tensor g_μν, in the standard Boyer–Lindquist coordinates, take the form

$$\begin{eqnarray} && g_{tt} = - \left( 1- \frac{2Mr}{\Sigma } \right), \quad g_{t\phi } = - \frac{2Mra \sin ^2\theta }{\Sigma } , \\ && g_{\phi \phi } = \left( r^2 +a^2 + \frac{2Mra^2}{\Sigma } \sin ^2\theta \right) \sin ^2\theta , \\ && g_{r r} = \frac{\Sigma }{\Delta }, \quad g_{\theta \theta } = \Sigma , \end{eqnarray}$$

(20)

where

$$\begin{eqnarray} \Sigma = r^2 + a^2 \cos ^2\theta , \quad \Delta = r^2 - 2Mr + a^2. \end{eqnarray}$$

(21)

Here, M and a are the mass and spin parameters of the black hole. The physical singularity is placed at the ring with r = 0, θ = π/2. The outer horizon is placed at

$$\begin{eqnarray} r_{+} = M + (M^2-a^2)^{1/2}. \end{eqnarray}$$

(22)

3.2 External magnetic field

Large-scale magnetic fields can have complicated global structures; however, in a small limited region of space, their configuration can be assumed as locally approximately homogeneous. We concentrate our attention on this simplified scenario introduced by Wald (1974) for black holes immersed in asymptotically homogeneous magnetic fields with field lines parallel to the black hole symmetry axis. Thus, the electromagnetic four-potential corresponding to an asymptotically homogeneous magnetic field around a black hole takes the following simple form

$$\begin{eqnarray} A^{\alpha } = \frac{B}{2} \xi _{(\phi )}^{\alpha }, \end{eqnarray}$$

(23)

where |$\xi _{(\phi )}^{\alpha }$| is the axial Killing vector and B is the magnetic field intensity. One of the important features of the Wald solution is that equation (23) corresponds to the case of a black hole with an induced electric charge Q_W that exists due to the frame-dragging effect of the twisting of magnetic field lines and whose value is Q_W = 2aMB (for more details, see Wald 1974; Aliev & Özdemir 2002; Tursunov et al. 2016; Stuchlík et al. 2020). The gravitational effect of the induced charge is tiny in realistic situations, typically being 12 orders of magnitude less than the maximum Reissner–Nordstrom charge (Kološ et al. 2017). However, its electrostatic effect can play an important role in the mechanisms of the particle acceleration around realistic black hole candidates (see, e.g., Stuchlík & Kološ 2016c; Zajaček et al. 2018; Tursunov & Dadhich 2019; Tursunov et al. 2020a, 2020b, 2021; Stuchlík et al. 2020, 2021; Kološ et al. 2021). The electromagnetic tensor, F_μν = A_ν,μ − A_μ,ν, for the potential (23), and in the equatorial plane θ = π/2, has the following non-vanishing components:

$$\begin{eqnarray} F_{r \phi } = \frac{B (r^3 - a^2)}{r^2}, \quad F_{r t} = \frac{a B}{r^2}. \end{eqnarray}$$

(24)

The components of the F_μν tensor in the equatorial plane will be used for the calculation of the frequencies of harmonic epicyclic oscillations related to the observed HF QPOs.

In order to parametrize the dynamical equations, it is useful to introduce the dimensionless parameter

$$\begin{eqnarray} \quad {\cal B}= \frac{q B G M}{2m c^4}, \end{eqnarray}$$

(25)

where q and m are the charge and mass of the QPO matter component, and G and c are the constants. Hereafter, we refer to |${\cal B}$| as the magnetic parameter (Kološ et al. 2015; Tursunov et al. 2016). For qBa > 0 or |${\cal B}a>0$|⁠, the magnetic field lines are aligned with the black hole spin, while for qBa < 0 or |${\cal B}a<0$|⁠, the magnetic field lines and black hole spin are directed in the opposite directions.

3.3 Dynamical equations for circular orbits

We give a short summary of the geodesic epicyclic model modified by the electromagnetic interaction introduced in Tursunov, Stuchlík, and Kološ (2016).

The charged particle moving around a Kerr black hole in the presence of a magnetic field obeys the Lorentz equation

$$\begin{eqnarray} \frac{{d}u^\alpha }{{d}\tau } + \Gamma _{\mu \nu }^{\alpha } u^{\mu } u^{\nu } = \frac{q}{m} F^{\alpha }_{\, \, \, \beta } u^{\beta }, \end{eqnarray}$$

(26)

where u^μ = dx^μ/dτ is the four-velocity of the particle. The motion in the presence of a magnetic field sharing the axial symmetry of the background spacetime is always radially bounded. In particular, we are interested in the circular orbits in the equatorial plane θ = π/2, corresponding to the minimum of the effective potential (for details, see Tursunov et al. 2016). In this case, u^μ = {u^t, 0, 0, u^φ}, so that the radial component of equation (26) takes the form

$$\begin{eqnarray} (a^2 - r^3) (u^{\phi })^2 - 2 {\cal B}(r^3 - a^2) u^{\phi } - 2 a (u^{\phi } + {\cal B}) u^{t} + (u^{t})^2 = 0.\\ \end{eqnarray}$$

(27)

The velocity is normalized by the condition g_μνu^μu^ν = −1, which gives

$$\begin{eqnarray} [a^2 (2 + r) + r^3] (u^{\phi })^2 - 4 a u^{\phi } u^{t} - (r-2) (u^{t})^2 + r = 0. \end{eqnarray}$$

(28)

Then, equations (27) and (28) can be solved for two unknown quantities u^t and u^φ, thus determining the circular orbits for charged particles around the magnetized black holes. Detailed discussion of these circular orbits can be found in Kološ, Stuchlík, and Tursunov (2015, 2017), Tursunov, Stuchlík, and Kološ (2016), and Stuchlík et al. (2020).

We have to stress that the assumption of small (linear) perturbative motion around circular orbits is relevant in Keplerian disks (Novikov & Thorne 1973). However, non-circularity (strongly “elliptical” motion) can be relevant for matter objects totally destroyed in the vicinity of the black hole, creating so-called flares, as observed in Sgr A* (Gravity Collaboration 2020).

3.4 Harmonic epicyclic oscillations

Small perturbations of the equations of motion around stable circular orbits lead to the appearance of the quasi-harmonic epicyclic oscillations in the radial and latitudinal directions, which we later compare with the QPOs data. Following Aliev and Galtsov (1981), let us introduce the deviation vector |$\xi ^{\mu } (\tau )=x^{\mu } (\tau )-x^{\mu }_{0} (\tau )$|⁠. Substituting ξ^μ into equation (26) and using the first-order expansion, one can get the following equation for ξ^μ:

$$\begin{eqnarray} \frac{d^2\xi ^{s}}{dt}+\Omega ^2_s\xi ^s=0, \quad s=\lbrace r,\theta \rbrace , \end{eqnarray}$$

(29)

where Ω_r and Ω_θ are the epicyclic radial and vertical oscillatory frequencies, measured by an observer at rest at infinity.¹ These frequencies for a Kerr black hole’s spacetime in the presence of a magnetic field can be written explicitly (Kološ et al. 2017; Tursunov et al. 2016):

$$\begin{eqnarray} \Omega _ \theta ^2 &=&\frac{\alpha u^\phi (u^\phi +2 {\cal B})+\beta u^t(u^\phi +{\cal B})+2a^2 (u^t)^2}{r^5(u^t)^2} , \end{eqnarray}$$

(30)

$$\begin{eqnarray} \Omega _ r^2 &=& \frac{\gamma u^\phi (u^\phi +2{\cal B})+\mu u^t(u^\phi +{\cal B})+\rho (u^t)^2+\sigma }{r^5(u^t)^2}, \end{eqnarray}$$

(31)

where

$$\begin{eqnarray} && \alpha = r^5 + a^2 r^2 (4 + r) + 2 a^4, \\ && \beta = -4 a (r^2 + a^2), \\ && \gamma = r^4 (-8 + 3 r) + a^2 r (2 - 10 r + r^2) - 4 a^4, \\ && \mu = 4 a (2 a^2 - r + 3 r^2), \\ && \rho = 2 (-2 a^2 + r - r^2), \\ && \sigma = 4 {\cal B}^2 r [(-2 + r) r^3 - a^2 (1 + 2 r)], \end{eqnarray}$$

(32)

and u^φ and u^t are taken from the solutions of equations (27) and (28). The azimuthal Keplerian frequency Ω_K, and the Larmor angular frequency Ω_L associated with the external magnetic field, are given by (Kološ et al. 2017; Tursunov et al. 2016)

$$\begin{eqnarray} \Omega _{\rm K}\equiv \Omega _{\phi } = \frac{d \phi }{d t} = \frac{u^\phi }{u^{t}}, \qquad \Omega _{\rm L} \equiv \frac{q B}{m u^t} = \frac{2{\cal B}}{u^t}. \end{eqnarray}$$

(33)

In Gaussian units, the corresponding frequencies [equations (30), (31), and (33)] are modified by the factor c³/GM:

$$\begin{eqnarray} \nu _{i} = \frac{1}{2\pi } \frac{c^3}{GM} \, \Omega _{i} [\mbox{Hz}], \quad i = \lbrace r,\theta ,\phi \rbrace . \end{eqnarray}$$

(34)

In general, the circular orbits of a charged body moving around a rotating Kerr black hole in an external uniform magnetic field can be divided into four qualitatively different classes, namely co- and counter-rotating orbits for each parallel and anti-parallel orientation of the magnetic field lines with respect to the black hole spin (for details, see Tursunov et al. 2016). The characteristic radial profiles of the frequencies ν_θ, ν_r, and ν_φ are shown in figure 1 for the four different classes of orbits, corresponding to criteria u^φ > 0 and u^φ < 0 combined with |${\cal B}>0$| and |${\cal B}<0$|⁠. The electromagnetic interaction significantly enriches the properties of the radial profiles of frequencies ν_r, ν_θ, ν_K, and their relations, thus strongly extending the abilities of the models based on epicyclic frequencies to fit HF QPO data. We demonstrate later that this is crucial for fitting data generated around supermassive black holes.

$Radial profiles of the frequencies of epicyclic oscillations νθ, νr, and νφ given by equation (34), for a charged matter revolving around a Kerr black hole with mass M = 100 ${M_{\odot}}$ and spin a = 0.7 for different values of the magnetic parameter ${\cal B}$ and the orbital direction. The first row represents the frequencies in the absence of electromagnetic interaction, i.e., ${\cal B}=0$ corresponding to a co-rotating uφ > 0 and counter-rotating uφ < 0 test body orbiting a Kerr black hole. The second and third rows show the effect of positive and negative magnetic parameter ${\cal B}$. At the same time, the first and the second columns of the figure correspond to the case with uφ > 0, while third and the fourth columns are for the case with uφ < 0. The symmetries of the orbital parameters in relation to the direction of the magnetic field, black hole spin, and orbital motion are discussed in Tursunov, Stuchlík, and Kološ (2016).$

Fig. 1.

Radial profiles of the frequencies of epicyclic oscillations ν_θ, ν_r, and ν_φ given by equation (34), for a charged matter revolving around a Kerr black hole with mass M = 100 |${M_{\odot}}$| and spin a = 0.7 for different values of the magnetic parameter |${\cal B}$| and the orbital direction. The first row represents the frequencies in the absence of electromagnetic interaction, i.e., |${\cal B}=0$| corresponding to a co-rotating u^φ > 0 and counter-rotating u^φ < 0 test body orbiting a Kerr black hole. The second and third rows show the effect of positive and negative magnetic parameter |${\cal B}$|⁠. At the same time, the first and the second columns of the figure correspond to the case with u^φ > 0, while third and the fourth columns are for the case with u^φ < 0. The symmetries of the orbital parameters in relation to the direction of the magnetic field, black hole spin, and orbital motion are discussed in Tursunov, Stuchlík, and Kološ (2016).

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The electromagnetic interaction increases the values of the radial frequency ν_r, which shifts towards the black hole and vanishes at the location of the ISCO, ν_r(r_ISCO) = 0. A detailed analysis of the ISCO position under the influence of electromagnetic interaction can be found in Tursunov, Stuchlík, and Kološ (2016). Below the ISCO, the particle spirals down towards the event horizon. The behaviour of the azimuthal frequency ν_φ is different for different types of orbits. When the Lorentz force is directed outwards from the black hole, i.e., |${\cal B}u^\phi >0$|⁠, ν_φ decreases with increasing |$|{\cal B}|$|⁠. When the Lorentz force is directed towards the black hole, i.e., |${\cal B}u^\phi < 0$|⁠, ν_φ increases with increasing |$|{\cal B}|$|⁠. Since we choose the uniform magnetic field configuration, the Lorentz force does not affect the motion in the vertical direction. A change of the black hole mass changes the scales of the plots and does not affect their shapes. This is due to the inverse dependence on mass; see equation (34). We also remind the reader that the features of the frequencies ν_i described here appear only due to the combined effects of the gravitational and magnetic fields. In the absence of electromagnetic interaction, the intersections of the radial profiles of the fundamental frequencies do not appear; they coincide at an asymptotically flat infinity. In the Newtonian gravity, the fundamental frequencies ν_i are equal to each other: ν_r ≡ ν_θ ≡ ν_φ = c³/(2πGMr^3/2). The existence of intersection points in the frequency profiles close to the black hole, as shown in figure 1, can increase the probability of the appearance of resonances, whence the QPOs phenomena are believed to originate.

3.5 Charged particle oscillations and resonances

Charged particle dynamics in the field of a black hole with a uniform magnetic field is generally chaotic (Kopáček et al. 2010; Pánis et al. 2019), but a small perturbation of a test particle around a stable circular orbit leads to regular motion–harmonic oscillation with frequencies derived in equations (30), (31), and (33); see the first row of plots in figure 2. Such regularity will be lost when the particle’s displacement from its stable position is large—here the chaotic regime will fully manifest; see the second row of plots in figure 2. Due to the non-linear bound between the particle’s vertical and radial oscillations, one can observe the appearance of resonance where the particle oscillatory frequencies are locked in a small integer ratio. We demonstrate this effect on the last row of plots in figure 2, where the charged particle trajectory with the frequency ratio Ω_r : Ω_θ = 2 : 1 is given. Due to the mass difference between protons (or ions) and electrons, the considered effects of electromagnetic interaction are relevant predominantly for electrons in a fixed electromagnetic field (Stuchlík et al. 2021; Tursunov et al. 2018).

Fig. 2.

Charged particle orbits around a black hole with uniform magnetic field. The first three subfigures are different views of a full 3D particle trajectory, with the gray disk representing the black hole horizon. The Fourier image or radial r and vertical θ coordinate are given in the last subfigure along with the main peak frequency. The first row of figures shows particle trajectory with a small perturbation from a stable circular orbit; main peak frequencies are given by fundamental harmonic frequencies [equations (30)–(31)]. The second row of figures shows a chaotic trajectory with large displacement forming a circular orbit. The third row of figures show resonant particle trajectory, where the main peak frequencies of radial and vertical oscillations are locked in 2 : 1 resonance.

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We focus our attention on the problem of the fitting data around SMBHs using the standard approach of linear analysis, corresponding to harmonic oscillations of epicyclic characters connected to the radial and vertical epicyclic oscillations around a circular orbit, which is of course general 3D motion with a small deviation from the circular orbits. Equivalently, we can consider the radial and vertical oscillations of a slender, slightly charged torus (Stuchlík & Kološ 2016a).

A similar resonant effect cannot exist for neutral particle geodesic motion around a Kerr black hole. In this case, the motion is fully separable and the trajectory can be expressed analytically as the function of the evolution parameter for arbitrary initial conditions. If one would like to consider models for geodesic particle oscillations (Rana & Mangalam 2019, 2020), one should add some binding between orbital modes to see the resonance.

3.6 Modifications of spin estimates due to electromagnetic interactions

The masses of black holes in microquasars and active galactic nuclei in many cases are based on averaged measurements using various methods, and the influence of the magnetic field on the mass estimate seems unlikely. However, estimates of spins of given sources are mainly based on the continuum spectrum fittings, which do not include the possible influence of an electromagnetic interaction on the motion of slightly charged plasma near the inner edge of the thin disk. Nearly all spin estimation techniques, including spectral continuum models, are quite sensitive on the position of the inner edge of an accretion disk of the source, which is believed to coincide with the ISCO position. However, the ISCO radius for charged matter is strongly dependent on the strength and orientation of the magnetic field (Tursunov et al. 2016). Moreover, electromagnetic interaction can shift the ISCO location down to the event horizon of the black hole, thus strongly mimicking the black hole spin. Therefore, taking into account an electromagnetic interaction in the spectral continuum fitting models could, in principle, lead to the corrections of current estimates of the black hole spins. Most likely, the electromagnetic interaction would lower the spin value. In figure 3, we plot the dependence of the ISCO radii on the magnetic parameter |${\cal B}$| for different values of the black hole spin. As one can see, in the chosen uniform magnetic field configuration the ISCO for a charged matter can be shifted towards the event horizon, down to the static radius r_ISCO = 2 M if the Lorentz force is directed outwards from the black hole, and up to r_ISCO = 4.3 M if the Lorenz force is directed towards the black hole, thus mimicking the black hole spin up to the values of a = 0.943 M and a = 0.483 M, respectively.

$ISCO radii given in relation to the magnetic field parameter ${\cal B}$ for typical values of the black hole spin a = 0, a = 0.7, and a = 0.998. The curves represent four classes of the circular orbits differing in the orientation of the orbits with respect to the black hole spin and magnetic field.$

Fig. 3.

ISCO radii given in relation to the magnetic field parameter |${\cal B}$| for typical values of the black hole spin a = 0, a = 0.7, and a = 0.998. The curves represent four classes of the circular orbits differing in the orientation of the orbits with respect to the black hole spin and magnetic field.

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4 Observations and fitting

Following the work by Smith, Tandon, and Wagoner (2021), we have chosen five black hole microquasars and 12 SMBH candidate systems, for which the HF QPOs have been observed. The frequency data from observations of considered sources are listed in the tables of the paper by Smith, Tandon, and Wagoner (2021). For completeness and comparison we added the twin HF QPOs observed with frequency ratio 3 : 2 in Sgr A* (Aschenbach 2004; Török 2005). All sources under consideration are summarized in figure 4. For more details on the features of the presented frequencies, we refer to the original work by Smith, Tandon, and Wagoner (2021). Below, we only show the fitting results using the magnetic version of the epicyclic resonance model.

Fig. 4.

Scaled mass-to-frequency product versus black hole spin for observed uppermost frequency of the HF QPOs (in cases when more than one frequency is observed) from stellar-mass black holes (left), SMBHs (middle), and joined data for stellar mass and SMBHs (right). Each particular source in the left-hand and middle plots is indicated by color and accompanied by the source’s name. We aimed to fit both types of sources (stellar and SMBHs) simultaneously by an epicyclic resonance model with an electromagnetic interaction. The data related to frequencies observed in Sgr A* with ratio 3 : 2 are included in the quasar case.

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The HF QPOs are usually observed in a pair of two peaks in the timing spectra with ratio of an upper f_U to lower f_L frequency being close to the ratio 3 : 2. This property of QPO is explained by the occurrence of resonances between two oscillatory modes (Kluzniak & Abramowicz 2001; Abramowicz & Kluźniak 2001; Abramowicz et al. 2004). In that case, the observed frequencies can be associated with the fundamental frequencies ν_r, ν_θ, and ν_φ, or their linear combinations. In general, the frequencies are the functions of black hole mass M, spin a, magnetic parameter |${\cal B}$|⁠, and the resonance position r, so that

$$\begin{eqnarray} \nu _{\mathrm{U}} = \nu _{\mathrm{U}}(r,M,a,{\cal B}), \quad \nu _{\mathrm{L}} = \nu _{\mathrm{L}}(r,M,a,{\cal B}). \end{eqnarray}$$

(35)

In order to treat the fitting data due to resonant phenomena at the frequency ratio 3 : 2 efficiently, we apply for the magnetized resonance model the framework developed for the purely geodesic motion around Kerr black holes in Stuchlík, Kotrlová, and Török (2013). In order to fit the observed frequencies, we calculate the so-called resonant radii, r_3:2 and r_2:3, so that

$$\begin{eqnarray} \nu _{\mathrm{U}}(r_{3:2}):\nu _{\mathrm{L}}(r_{3:2})=3:2, \end{eqnarray}$$

(36)

$$\begin{eqnarray} \nu _{\mathrm{U}}(r_{2:3}):\nu _{\mathrm{L}}(r_{2:3})=2:3. \end{eqnarray}$$

(37)

The radii r_3:2 and r_2:3 occur when there is a crossing point of the radial profiles, where ν_U(r_cross) = ν_L(r_cross). Note that the appearance of the resonant radius r_2:3 is a result of purely electromagnetic interaction. In the absence of a magnetic field when ν_K(r) > ν_θ(r) > ν_r(r), the only resonant radius is r_3:2. The resonant radii r_3:2 and r_2:3 are found from the numerical solution of the higher-order polynomial in r, for a given spin a and magnetic parameter |${\cal B}$|⁠. Substituting the obtained resonant radii into equation (35), we get the frequencies ν_U and ν_L in terms of the black hole mass M, spin a, and magnetic parameter |${\cal B}$| (for details, see Kološ et al. 2017).

4.1 Magnetized epicyclic resonances

There is a large variety of models of HF QPOs based on the frequencies of the orbital epicyclic geodesic motion that assume the resonances of some of the oscillatory modes (see, e.g., Stuchlík et al. 2013; Stuchlík & Kološ 2016a, 2016b). These geodesic models can be generalized in a straightforward manner to the magnetized versions related to the orbital epicyclic motion of slightly charged matter around magnetized black holes (Kološ et al. 2017; Tursunov et al. 2016, 2020b). In the following, we apply just one version of these magnetized models, namely the simple epicyclic resonance models.

Original epicyclic resonance models (Abramowicz & Kluźniak 2001; Török et al. 2005) consider the resonances of axially symmetric oscillatory modes of accretion disks that are believed to increase at the frequency ratio 3 : 2 (Török et al. 2005; Stuchlík et al. 2013). In this case, the two resonant frequency modes are identified with the radial and vertical epicyclic frequencies (Török et al. 2005), i.e.,

$$\begin{eqnarray} \nu _{\rm U}=\nu _{\theta }, \quad \nu _{\rm L}=\nu _{ r} \end{eqnarray}$$

(38)

In the geodesic motion in a pure Kerr metric (i.e., in the absence of electromagnetic interaction), the condition ν_θ(r) ≥ ν_r(r) is always true for an arbitrary value of the spin a and the resonant radius r. The equality sign holds at infinity, r → ∞, where the two frequencies coincide. In the presence of electromagnetic interaction, a new region of parameters appears, where the radial frequency ν_r(r) can be greater or lower than the vertical epicyclic frequency ν_θ(r) in the vicinity of the black hole horizon.

Fits of HF QPOs observed from black hole sources, both stellar-mass and supermassive, are given in figures 5 and 6, for the resonant radii r_3:2 and r_2:3, respectively. For the resonance models there is a large region of parameters a and |${\cal B}$| where the crossing ν_θ(r_c) = ν_r(r_c) occurs and the double solution with r_3:2(ν_θ : ν_r = 3 : 2) and r_2:3(ν_θ : ν_r = 2 : 3) exists. For a fixed magnetic parameter, we have r_3:2 > r_2:3. In each figure we consider four qualitatively different orbits: co- and counter-rotating orbits for a magnetic field that is parallel and anti-parallel with respect to the black hole spin. These four cases are distinguished by the combinations of signs of the magnetic parameter |${\cal B}$| and angular velocity u^φ, as shown inside each plot.

$Fitting the HF QPOs sources including electromagnetic interaction at resonant radius r3:2. The colored regions correspond to figure 4 (right) with four stellar-mass and 12 supermassive black hole candidates in the scaled mass-to-frequency product–black hole spin relations. Thin lines are predictions of the model for various values of the magnetic parameter ${\cal B}$. The pure Kerr case with ${\cal B}=0$ is indicated by thick lines. The figure shows the four qualitatively different types of bounded orbits corresponding to four combinations of the signs of ${\cal B}$ and uφ. While stellar-mass black holes are fitted for each type of orbit, SMBH sources at r3:2 can be fitted with magnetic parameter ${\cal B}$ only for uφ < 0. The data is taken from Smith, Tandon, and Wagoner (2021) and references therein.$

Fig. 5.

Fitting the HF QPOs sources including electromagnetic interaction at resonant radius r_3:2. The colored regions correspond to figure 4 (right) with four stellar-mass and 12 supermassive black hole candidates in the scaled mass-to-frequency product–black hole spin relations. Thin lines are predictions of the model for various values of the magnetic parameter |${\cal B}$|⁠. The pure Kerr case with |${\cal B}=0$| is indicated by thick lines. The figure shows the four qualitatively different types of bounded orbits corresponding to four combinations of the signs of |${\cal B}$| and u^φ. While stellar-mass black holes are fitted for each type of orbit, SMBH sources at r_3:2 can be fitted with magnetic parameter |${\cal B}$| only for u^φ < 0. The data is taken from Smith, Tandon, and Wagoner (2021) and references therein.

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$Fitting the HF QPOs sources including electromagnetic interaction at resonant radius r2:3. The colored regions correspond to figure 4 (right) with four stellar-mass and 12 supermassive black hole candidates in the scaled mass-to-frequency product–black hole spin relations. Thin lines are predictions of the model for various values of the magnetic parameter ${\cal B}$. The pure Kerr case with ${\cal B}=0$ is indicated by thick lines. The figure shows the four qualitatively different types of bounded orbits corresponding to four combinations of the signs of ${\cal B}$ and uφ. Fitting of all sources is possible for any type of orbit, independently from the orientation of magnetic field lines and the direction of the motion with respect to the black hole spin. The data is taken from Smith, Tandon, and Wagoner (2021) and references therein.$

Fig. 6.

Fitting the HF QPOs sources including electromagnetic interaction at resonant radius r_2:3. The colored regions correspond to figure 4 (right) with four stellar-mass and 12 supermassive black hole candidates in the scaled mass-to-frequency product–black hole spin relations. Thin lines are predictions of the model for various values of the magnetic parameter |${\cal B}$|⁠. The pure Kerr case with |${\cal B}=0$| is indicated by thick lines. The figure shows the four qualitatively different types of bounded orbits corresponding to four combinations of the signs of |${\cal B}$| and u^φ. Fitting of all sources is possible for any type of orbit, independently from the orientation of magnetic field lines and the direction of the motion with respect to the black hole spin. The data is taken from Smith, Tandon, and Wagoner (2021) and references therein.

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As one can see from figures 5 and 6, inclusion of the magnetic parameter |${\cal B}$| into the consideration allows one to fit all observed frequencies for both stellar and supermassive black holes simultaneously by using the epicyclic resonance model, if we vary the magnetic parameter |${\cal B}$| along with the black hole spin appropriately. It is important to note that while increasing the parameter |${\cal B}$|⁠, the frequencies become more independent from the black hole spin a, i.e., the electromagnetic interaction prevails over the spin influence and becomes dominant.

5 Conclusion

It is well established that the rotation of plasma around a black hole in the presence of an external magnetic field leads to the induction of an electric field in both the black hole and the surrounding magnetosphere, which can be associated with the electric charge of the black hole and the magnetosphere, respectively. We have shown that by taking into account the effect of electromagnetic interaction of a matter moving around a black hole in the presence of large-scale magnetic field one can potentially explain the HF QPOs from supermassive black holes, as well as simultaneously explaining the HF QPOs in the microquasars containing stellar-mass black holes using a simple, appropriate modification of the orbital and epicyclic frequencies caused by electromagnetic interaction limited by observation. Therefore, no more complex model of plasma effects is necessary for an explanation of HF QPOs observed in active galactic nuclei.

To demonstrate the importance of electromagnetic interaction in the explanation of the QPO’s phenomena we have considered the simplest model of epicyclic resonances modified by an influence of the magnetic field and characterized by the dimensionless parameter |${\cal B}$|⁠, corresponding to the relative ratio of the Lorentz and gravitational forces.

We have shown that most of the QPO sources can be fitted well by the proposed model with the values of the parameter |${\cal B}$| of the order of 10⁻². In the case of Sgr A* the parameter |${\cal B}$| is shifted to the values of the order of 10⁻¹ sufficient to fit the QPOs. |${\cal B}\sim 10^{-2}$| also means that the ratio of the number density of extra charged particle to the total particle’s number density in the plasma around a black hole of mass 10⁸|${M_{\odot}}$| in the presence of a magnetic field of the order of 10⁴ G is just of the order of ρ_q/ρ_N ∼ ×10⁻¹³. This implies that one would expect an extra electron (or proton) per ∼10¹³ of pairs of protons and electrons in the plasma, which fits well with the general limits on this ratio in the quasi-neutral plasma from the estimates based on the charge separation mechanism and induction in the relativistic magnetized plasma.

Assuming the sharing symmetries of the magnetic field and background Kerr metric, one can distinguish four qualitatively different orbital types corresponding to the co-rotating and counter-rotating orbits with parallel and anti-parallel magnetic fields. The frequencies of epicyclic oscillations in all four cases have been investigated. It is important to note that the ISCOs in all cases shift towards the black hole event in the non-rotating black hole case; i.e., in the chosen asymptotically homogeneous magnetic field configuration, the electromagnetic interaction leading to the shifts of ISCO can mimic the black hole rotation up to the values of a = 0.943 M. As one can see from figures 5 and 6, the electromagnetic interaction suppresses the effect of the black hole rotation on the shifts of frequencies of the epicyclic oscillations, which allows simultaneous fitting of the QPO frequencies from various sources by the proposed model. For the values of the parameter |${\cal B}\sim 10^{-2}$| that fit the observed frequencies, there are no considerable shifts of the ISCO radii from that of the non-magnetized case.

One should emphasize that the present paper aims to show the potential importance of electromagnetic interaction in the explanation of the QPO phenomena, which is done by considering only the simplest model of the epicyclic resonances. All presented estimates for typical stellar mass and supermassive black holes are given to show the consistency of the idea of the weakly charged black hole magnetosphere with the QPO phenomena. More precise constraints on the charges of the black hole magnetospheres need to be obtained for each of the QPO sources separately, which however requires observational estimates of the magnetic fields and matter densities around these sources. We would like to stress that the fitting of data from the supermassive black hole is possible due to the existence of the resonance radius r_2:3 existing in all four variants of the orbiting charged matter. If we use the simplest epicyclic model, the fitting is not possible for the resonant radii r_3:2; however, it should be tested, if some of the other variants of the magnetically modified geodesic models of twin peak HF QPOs allow for possibility of r_3:2 resonances in the fitting.

On the other hand, we have to mention that all the observational data taken around the supermassive black holes in AGNs have to be considered with great carefulness, as there are some doubts on their correctness (Bao & Li 2022). Nevertheless, we could expect their correctness in at least at some of the considered sources. Our results then clearly demonstrate a simple explanation of these data by using a very simple model considering electromagnetic interactions of slightly charged matter localized to the vicinity of a marginally stable orbit of the Kerr spacetime with an observed large-scale magnetic field.

Acknowledgements

The authors would like to acknowledge the Research Centre for Theoretical Physics and Astrophysics and Institute of Physics of Silesian University in Opava for institutional support. Z. S. acknowledges the support of the Grant No. 19-03950S of Czech Science Foundation (GAČR).

Footnotes

1

For a different way to derive the epicyclic frequencies based on the effective potential of the motion, see, e.g., Kološ, Stuchlík, and Tursunov (2015) and Stuchlík et al. (2020).

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Article Contents

Large-scale magnetic fields enabling fitting of the high-frequency QPOs observed around supermassive black holes

Abstract

1 Introduction

2 Magnetic field effect on a plasma rotating around a black hole

2.1 Magnetic field around black holes

2.2 Charge separation in a plasma

3 Influence of electromagnetic interaction on the harmonic epicyclic oscillations

3.1 Spacetime geometry

3.2 External magnetic field

3.3 Dynamical equations for circular orbits

3.4 Harmonic epicyclic oscillations

3.5 Charged particle oscillations and resonances

3.6 Modifications of spin estimates due to electromagnetic interactions

4 Observations and fitting

4.1 Magnetized epicyclic resonances

5 Conclusion

Acknowledgements

Footnotes

References

Citations

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Article Contents

Large-scale magnetic fields enabling fitting of the high-frequency QPOs observed around supermassive black holes

Abstract

1 Introduction

2 Magnetic field effect on a plasma rotating around a black hole

2.1 Magnetic field around black holes

2.2 Charge separation in a plasma

3 Influence of electromagnetic interaction on the harmonic epicyclic oscillations

3.1 Spacetime geometry

3.2 External magnetic field

3.3 Dynamical equations for circular orbits

3.4 Harmonic epicyclic oscillations

3.5 Charged particle oscillations and resonances

3.6 Modifications of spin estimates due to electromagnetic interactions

4 Observations and fitting

4.1 Magnetized epicyclic resonances

5 Conclusion

Acknowledgements

Footnotes

References

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

Most Read

Most Cited

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