Quasiperiodic Oscillations and Dynamics of Test Particles around Regular-Kiselev Black Holes

Abstract

Testing gravity theories combining (massive and massless) scalar & electrodynamic fields become the most important issue in relativistic astrophysics using data from, black hole observations. In the present work, we first show a spherically symmetric black hole solution in general relativity coupling to generic-type nonlinear electrodynamics (NED) together with the quintessential field. We also obtain possible values for the parameters of the quintessential field and NED charge in the black hole environment for different values of degree of nonlinearity. Also, event horizon properties and scalar invariants of the black hole spacetime are studied. We investigate the equatorial motion of test particles around the regular-Kiselev black holes and study the combined effects of quintessential field and the NED charge of the black hole on particle angular momentum together with its energy at their circular orbits as well as their innermost circular stable orbits (ISCOs) and compared the obtained results with Reissner-Nordström black hole (RN BH) case. Moreover, we study particle oscillations along the orbits above than ISCO and applications to quasiperiodic oscillations (QPOs) where we obtain constrain values for the quintessential parameter and black hole mass charge parameters using observational QPO data from microquasars.

1. Introduction

In astrophysics, QPOs have been identified in the X-ray emissions of neutron stars and black holes, providing insights into the extreme gravitational environments near these objects. These quasiperiodic fluctuations in luminosity suggest the presence of underlying physical processes, such as oscillations of accretion disks or gravitational interactions, which could serve as probes to study the fundamental nature of spacetime.

The presence of QPOs in accretion disks around black holes suggests the existence of resonant oscillations that could be linked to the innermost stable circular orbits, shedding light on the strong gravity regime and testing Einstein’s theory of general relativity. Neutron star QPOs, on the other hand, provide a platform to study the behavior of ultra-dense matter and the nuclear matter properties under extreme density and gravity conditions.

The interpretation of QPOs involves interdisciplinary collaboration, integrating theoretical models with observational data. Spectral-timing analysis and numerical simulations are employed to decipher the intricate interplay between accretion flows, magnetic fields, and gravitational interactions.

Furthermore, the observation of twin-peak QPOs in some systems has sparked debates about their origins and potential connections to oscillatory modes within the accretion disk. These findings underscore the need for advanced theoretical frameworks and more precise observational data. After the first detection of QPO in the analyses of the spectrum and the flux from X-ray binaries [1] have been studied in numerous works together with the theoretical investigations of QPO and their nature. The most promising model of QPO is related to particle dynamics and corresponding oscillations of their trajectories. From this point of view, QPOs resulting in the collective motion of test charged particles containing accretion disk have been investigated in [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].

Regular black holes, an intriguing concept in astrophysics, challenge the traditional notion of singularities at the center of astrophysical black holes. Unlike classical black holes that exhibit infinite curvature and density at their centers, regular black holes are proposed to possess a core where the spacetime curvature remains finite [17,18,19,20,21,22,23]. This idea has potential implications for resolving the information paradox and harmonizing quantum mechanics with general relativity [24,25,26].

Regular black holes are posited to be formed via mechanisms that prevent the complete collapse of matter, avoiding the formation of singularities. Various theoretical frameworks, such as modifications to Einstein’s equations or the introduction of exotic forms of matter, have been proposed to describe these objects. One of the regularizations of black holes has been proposed within the Kiselev model [27].

Our aim of this work is to investigate test particle dynamics around regular Kiselev black holes and corresponding applications to describe QPOs. The paper is organized as follows: In Section 2 we review the generic regular black hole surrounded by a quintessential field. The Section 3 is devoted to studying the test particle dynamics around regular Kiselev black hole. We study the fundamental frequencies in spacetime around the regular Kiselev black hole in Section 4. In Section 5 we discuss the QPO models and possible applications to observed sources. We conclude our results in Section 7.

In this work, we use the geometrical system of units $G=c=1$ and run the Greek indices from 0 to 3, and Latin indices from 1 to 3.

2. Generic Regular Black Holes in Quintessential Fields

The generic case of the regular back solution is obtained in Ref. [23], in general, relativity coupling to nonlinear electrodynamics with the following Lagrangian:

$$\mathcal{L}\left(F\right)=\frac{4\mu }{\alpha }\frac{{\left(\alpha F\right)}^{\frac{\nu +3}{4}}}{{\left[1+{\left(\alpha F\right)}^{\frac{\nu }{4}}\right]}^{1+\frac{\mu }{\nu }}}$$

(1)

where dimensionless constants $\mu$ and $\nu$ are degree of nonlinearity, and can be $\mu >3\&\nu >0$, $\alpha$ dimension of length squared, and $F=\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$, with $F_{\mu \nu }=2{▿}_{[\mu }{A}_{\nu ]}$, where $F_{\mu \nu }$ denotes the Maxwell electromagnetic field tensor.

However, we assume the generalized action for this theory as:

$$\mathcal{S}=\frac{1}{16\pi }\int d^{4}x\sqrt{-\mathrm{g}}\left(R-\mathcal{L}\left(F\right)+{\mathcal{L}}_m\right),$$

(2)

where $\mathrm{g}$ is a determinant of the metric tensor, R is the Ricci scalar curvature, and ${\mathcal{L}}_m$ is Lagrangian for an anisotropic matter field.

Here we consider the field as a quintessential field. The components of the non-vanishing stress-energy tensor of the field derived by Kiselev [27].

$$\begin{array}{ccc}\hfill T_t^t& =& T_r^r={\rho }_q,\hfill \\ \hfill T_{\theta }^{\theta }& =& T_{\varphi }^{\varphi }=-\frac{1}{2}{\rho }_q(1+3{\omega }_q),\hfill \end{array}$$

(3)

where ${\omega }_q$ is the state parameter of the corresponding equation of state of the quintessential field $p_q={\omega }_q{\rho }_q$, and take values $-1<{\omega }_q<-1/3$. When ${\omega }_q=0$ corresponds to the Schwarzschild black hole case, ${\omega }_q=-1/3$ to RN BH case and the case ${\omega }_q=-1$ defines Schwarzschild-(anti)-de Sitter black hole spacetime. ${\rho }_q$ is the density which is always positive and ${\rho }_q=-3{\omega }_q(c/2)/r^{3(1+{\omega }_q)}$ and $c>0$ is a normalization factor.

The GR BH spacetime satisfying the weak energy condition can be obtained by coupling Einstein’s gravity to a nonlinear electrodynamic field [23,28]. This model is described by the metric given by

$$\begin{array}{c}\hfill d{s}^2=-f\left(r\right)d{t}^2+\frac{1}{f\left(r\right)}d{r}^2+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2),\end{array}$$

(4)

with

$$\begin{array}{c}\hfill f\left(r\right)=1-\frac{2M}{r}{\left(1+\frac{q^{\nu }}{r^{\nu }}\right)}^{-\frac{\mu }{\nu }}-\frac{\mathcal{C}}{r^{3{\omega }_q+1}}\end{array}$$

(5)

where M is the total mass of the BH, and q is the magnetic charge of a nonlinear self-gravitating monopole. When $q=\mathcal{C}=0$ the solution (5) turns to the Schwarzschild solution, while at $\mathcal{C}=0$, it becomes the generic regular black hole solution.

2.1. Horizon Structure

Here, we will analyze the radial profiles of the metric function $f\left(r\right)$.

In Figure 1, we analyze the zeros of the metric function (5) as a radial function by varying the non-linearity parameter $\nu$, the black hole charge q, and the quintessence field parameters $\mathcal{C}$, and ${\omega }_q$. Observed that when $\mathcal{C}=0$ an increase of $\nu$ causes a decrease (increase) in the radius of the event (Cauchy) horizon. However, at ${\omega }_q=-1/2$, the radius of the event horizon increases with increasing $\nu$ and $\mathcal{C}$. When, $\nu$ changes from 0 to 1, the event horizon shifts out sufficiently, while in charges from 2 to 3, the horizon shift is very small.

Now, we will find the separate are in values of the quintessence parameter $\mathcal{C}$ and the black hole charge q which changes the event horizon oppositely, by two: black hole no black hole regions using the set of equations $f\left(r\right)=0$ and ${\partial }_{r}f\left(r\right)=0$. The black hole region means that when we choose the values for q and $\mathcal{C}$ the metric (5) should have zeros, where black hole horizons exist. In the black hole region, the metric (5) determines the spacetime of horizonless objects, such as a naked singularity.

It is clearly observed from Figure 2 that with an increase of $\nu$ the maximum value of $\mathcal{C}$ decreases, however, extreme charge q increases, and the black hole region expands.

2.2. Scalar Invariants

The curvature invariants of a spacetime are quantities that describe its curvature properties and are useful for understanding the geometry of spacetimes around black holes. Three well-known scalar curvature invariants are the Ricci scalar, the square of the Ricci tensor, and the Kretschmann scalar. By analyzing the behavior of the curvature invariants, we can learn about the geometry of spacetime. For example, the square of the Ricci tensor is defined as the square of the energy-momentum tensor of a field in the spacetime of a black hole as $\mathcal{R}=R_{\mu \nu }{R}^{\mu \nu }\equiv 1/\left(8\pi G\right)T_{\mu \nu }{T}^{\mu \nu }$. The Kretschmann scalar is responsible for the effective gravitational energy density of the spacetime ($\sqrt{\mathcal{K}}\sim {\rho }_M$). Now, we can calculate the Ricci scalar, the square of the Ricci tensor, and the Kretschmann scalar of spacetime (5). Due to the large form of their expressions, below, we plot to visualize the curvature properties of the spacetime and analyze them graphically.

Figure 3 shows the radial dependence of scalar invariants of spacetime (5), such as the Ricci scalar in the top left panel, the square of the Ricci tensor in the top right panel, and the Kretchman scalar (square of Riemann tensor) in the bottom panel. One can see from the figure that an increase in the values of $\nu$ and $\mathcal{C}$ causes an increase in the Ricci scalar and square of the Ricci tensor, while the Kretchman scalar decreases.

3. Test Particles in the Spacetime of the Regular-Kiselev Black Holes

In the present section, we consider the effects of NED and quintessential fields on particle dynamics around spherically symmetric black holes.

3.1. Deriving Equations of Electrically Particles Motion

The equations of motion for massive electrically neutral particles govern using time-like geodesics of the spacetime (5).

Taking into account integrals of motion, such as the particle’s specific energy $\mathcal{E}=E/m$ and its angular momentum $\mathcal{L}=L/m$, one may derive the equations in the form,

$$\begin{array}{ccc}\hfill {\dot{r}}^2& =& {\mathcal{E}}^2+g_{tt}\left(1+\frac{\mathcal{K}}{r^2}\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \dot{\theta }& =& \frac{1}{g_{\theta \theta }^2}\left(\mathcal{K}-\frac{l^2}{{sin}^2\theta }\right),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \dot{\varphi }& =& \frac{l}{g_{\varphi \varphi }},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \dot{t}& =& -\frac{\mathcal{E}}{g_{tt}},\hfill \end{array}$$

(9)

where $\mathcal{K}$ is the Carter constant which corresponds to the particle’s angular momentum.

Furthermore, one may restrict the dynamics of the particles to some constant plane as $\theta ={\theta }_0=\mathrm{const}$ and that implies the momentum along $\theta$ is zero ($\dot{\theta }=0$). Consequently, the Carter constant takes $\mathcal{K}={\mathcal{L}}^2/{sin}^2{\theta }_0$. Thus, the equation of motion in the constant plane takes in the following form

$${\dot{r}}^2={\mathcal{E}}^2-V_{\mathrm{eff}}\left(r\right),$$

(10)

where $V_{\mathrm{eff}}\left(r\right)$ is the effective potential corresponding to circular motion which has with the form,

$$V_{\mathrm{eff}}\left(r\right)=f\left(r\right)\left(1+\frac{{\mathcal{L}}^2}{r^2{sin}^2{\theta }_0}\right).$$

(11)

In order to observe the effects of the charge (q) sourced by the NED field and the quintessential field on the effective potential at the equatorial plane, we analyze it graphically for different values of the quintessential field and nonlinearity parameters in Figure 4.

It is observed from Figure 4 that the presence of NED charge causes an increase in the maximum of the effective potential and the orbit where the effective potential takes maximum shifts towards the black hole. While the maximum decrease is due to the presence of the quintessential field. Also, the effective potential at larger distances decreases at smaller values of the ${\omega }_q$ parameter.

3.2. Circular Orbits

The circularity of the orbits of test particles orbiting a central black hole is described by the following conditions.

$$\begin{array}{c}\hfill V_{\mathrm{eff}}=\mathcal{E},V_{\mathrm{eff}}^{\prime }=0.\end{array}$$

(12)

In circular orbits, no radial motion means that radial forces compensate each other at the corresponding values of angular momentum of the particles. Expressions of the particle energy and its angular momentum along circular orbits in the equatorial plane using the conditions (12) in the form

$$\begin{array}{ccc}\hfill {\mathcal{L}}^2& =& \frac{r^3\left(2Mr\left(2q^{\nu }-r^{\nu }\right)+\mathcal{C}{\left(q^{\nu }+r^{\nu }\right)}^{\frac{\nu +3}{\nu }}\right)}{6M{r}^{\nu +2}+(\mathcal{C}r-2){\left(q^{\nu }+r^{\nu }\right)}^{\frac{\nu +3}{\nu }}},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\mathcal{E}}^2& =& -\frac{2{\left(q^{\nu }+r^{\nu }\right)}^{\frac{\nu -3}{\nu }}{\left[2M{r}^2+(\mathcal{C}r-1){\left(q^{\nu }+r^{\nu }\right)}^{\frac{3}{\nu }}\right]}^2}{6M{r}^{\nu +2}+(\mathcal{C}r-2){\left(q^{\nu }+r^{\nu }\right)}^{\frac{\nu +3}{\nu }}}.\hfill \end{array}$$

(14)

Figure 5 presents radial profiles of specific energy and angular momentum of the particles for fixed values of the parameters $\mu =3$, $C=0.25$ and ${\omega }_q=-1/3$ and different values of the parameter $\nu$. Moreover, to show the effects of the quintessential medium and the feature of the nonlinear electrodynamics, in the figures, we have also provided the plots for the energy and angular momentum of test particles around RN BH. It is shown that the existence of the quintessential medium with the parameter ${\omega }_q=-1/3$ decreases the asymptotics of the energy values. Vice versa, an increase in $\nu$ also causes to increase in it. However, the effect of the NED field is more sensitive to the angular momentum compared to the energy and causes an increase over that of the RN BH case.

3.3. ISCO Studies

The ISCO is an important concept in the study of black holes and accretion disks around them, and it is a region of spacetime where the gravitational field of the black hole is extremely strong and also where the physics of accretion disks is most complex. The material in the accretion disk will eventually spiral inwards and cross the ISCO. Once particles cross the ISCO, they will inevitably fall into the central black hole. In fact, the circular orbits can be stable when the effective potential satisfies the condition ${\partial }_{rr}{V}_{\mathrm{eff}}\left(r_{\mathrm{ISCO}}\right)>0$, while ISCO satisfies ${\partial }_{rr}{V}_{\mathrm{eff}}\left(r_{\mathrm{ISCO}}\right)=0$, mostly, together with the conditions (12). One may get the following equation using the ISCO conditions,

$$\begin{array}{c}\hfill 2\mathcal{C}(3-\mathcal{C}r){\left(q^{\nu }+r^{\nu }\right)}^{\frac{6}{\nu }+2}+24M^2{r}^{\nu +3}\left((\nu -2)q^{\nu }+r^{\nu }\right)-4Mr{\left(q^{\nu }+r^{\nu }\right)}^{\frac{2}{\nu }}\\ \hfill \times \left[q^{2\nu }(3\mathcal{C}r-8)+q^{\nu }{r}^{\nu }(3\nu -3\mathcal{C}\nu r+2)+(6\mathcal{C}r+1)r^{2\nu }\right]=0.\end{array}$$

(15)

Equation (15) is impossible to solve concerning r, thus we obtain contour plots of the equation for different NED fields and quintessential field parameters.

In Figure 6, the dependence of ISCO of a neutral particle orbiting regular generic BH in the quintessential field on the fixed values of parameters $\mu =3$, $C=0.01$ and ${\omega }_q=-2/3$ and different values of $\nu$ parameter. The value of $\nu$ increases (decreases), and the minimum value of the radius of ISCO also increases (decreases).

We have obtained that the ISCO radius decreases faster as $\nu =1$ compared with $\nu =2,3$ cases as well as the case of magnetically Reissner-Nordström black hole case. That means in the $\nu =1$ case, the gravitational effect of the charge q is much stronger than in the other cases.

4. Fundamental Frequencies

Here we consider the fundamental frequencies in oscillations of test particles orbiting a regular black hole in the quintessential field. In particular, the orbital frequencies of the particles in their Keplerian orbits and the radial and vertical oscillation frequencies along the orbits are higher than ISCOs, and we apply them to QPO analyses.

4.1. Keplerian Frequency

The angular velocity of the particles in circular orbits ${\mathsf{\Omega }}_K=\dot{\varphi }/\dot{t}$ around regular-Kiselev black holes take th form,

$${\mathsf{\Omega }}_K^2=\frac{M\left(r^{\nu }-2q^{\nu }\right)}{{\left(q^{\nu }+r^{\nu }\right)}^{\frac{\nu +3}{\nu }}}-\frac{\mathcal{C}}{2r}.$$

Furthermore, we express all the frequencies in the unit of Hz, multiplying by $c/\left(2\pi GM\right)\mathsf{\Omega }$. We take the speed of light as $c=3\times {10}^8 \mathrm{m}/\sec$, and the gravitational Newtonian constant as $G=6.67\times {10}^{-11} {\mathrm{m}}^3/({\mathrm{kg}}^2\times \sec )$.

4.2. The Radial and Vertical Oscillation Frequencies

The test particles orbiting a black hole along stable circular orbits may oscillate in the directions of the radial and vertical axes, due to small displacement $r_0+\delta r$ and $\pi /2+\delta \theta$. Harmonic oscillator equations The frequencies of radial and vertical oscillations can be evaluated by the following [29]:

$$\begin{array}{c}\hfill \frac{d^2\delta r}{d{t}^2}+{\mathsf{\Omega }}_r^2\delta r=0,\frac{d^2\delta \theta }{d{t}^2}+{\mathsf{\Omega }}_{\theta }^2\delta \theta =0,\end{array}$$

(16)

where

$$\begin{array}{ccc}& & {\mathsf{\Omega }}_r^2=-\frac{1}{2g_{rr}{\left(u^t\right)}^2}{\partial }_r^2{V}_{\mathrm{eff}}(r,\theta ){|}_{\theta =\pi /2},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & {\mathsf{\Omega }}_{\theta }^2=-\frac{1}{2g_{\theta \theta }{\left(u^t\right)}^2}{\partial }_{\theta }^2{V}_{\mathrm{eff}}(r,\theta ){|}_{\theta =\pi /2},\hfill \end{array}$$

(18)

are the frequencies of radial and vertical oscillations, respectively.

5. QPO Models

In this section, we theoretically apply the fundamental frequencies to obtain high-frequency upper and lower frequencies (${\nu }_U$ and ${\nu }_L$) in twin QPOs. Several QPO models explain the QPO frequencies as combinations of Keplerian, radial, and vertical frequencies of test particles, i.e., the peak frequencies are described in terms of frequencies of radial and vertical oscillations and Keplerian frequency. For example, according to the RP model, ${\nu }_U$ is interpreted as orbital frequency (${\nu }_U={\nu }_{\varphi }$) while the lower frequency is the difference between Keplerian and radial frequencies in stable orbits ${\nu }_L={\nu }_{\varphi }-{\nu }_r$. Now, we analyze relationships between ${\nu }_U$ and ${\nu }_L$ to QPOs generated around regular black holes surrounded by quintessential fields by test particles.

In Figure 7 we demonstrate the relations between the upper and lower frequencies of the twin-peaked QPOs generated around the generic regular BH for various values of the nonlinearity parameter $\nu$, and the quintessential field parameter. Here, the black hole mass is chosen as $10M_{\odot }$, the NED field parameter $\mu =3$. In the top left panel, we present the effects of the NED charge of the black hole and nonlinearity parameters on the upper and lower frequencies in the absence of the quintessential field. The frequency ratio at $\nu =1$ is observed to be more significant than in the case $\nu =2$ in the range of ${\nu }_U<200$ Hz. Furthermore, low-frequency QPOs below 20 Hz disappear in the presence of a quintessential field with $\mathcal{C}=0.001$, and the variation of ${\omega }_q$ does not give much change in QPO frequencies.

QPO Orbits

For the first time, R. Sunyaev has suggested that rotating and nonrotating black holes can be distinguished by signals generated by oscillated particles in orbits close to ISCO, called QPOs. According to his idea, one may measure ISCOs around black holes by studying QPO orbits. Our previous papers [my papers] showed that twin-peak QPOs are generated in stable orbits close to the ISCO in the framework of various gravity theories. As the frequency ratio tends to 1, the QPO orbit approaches ISCO. In this subsection, we consider the radius of QPO orbits around regular black holes in the quintessential field in the RP. One may obtain the dependence of the QPO radius from the black hole charge by solving the equation with respect to r,

$$\begin{array}{c}\hfill 3{\nu }_L(r;q,\nu ,\mu ,\mathcal{C},{\omega }_q)=2{\nu }_U(r;q,\nu ,\mu ,\mathcal{C},{\omega }_q),\end{array}$$

(19)

Now we are interested in how the QPO orbits depend on the black hole, NED, and quintessential field parameters. Figure 8 shows the dependence of the radius of QPO orbits for different values of nonlinearity and quintessential field parameters. Our performed analyses have shown that an increase in the black hole charge causes a decrease in the radius. It decreases faster at $\nu =1$ case compared to $\nu =2,3$ cases which means the gravitational effect of the black hole charge becomes weaker at higher values of the nonlinearity parameter $\nu$ weak. However, the orbit shifts slightly out due to the presence of a quintessential field. Also, an increase of the ${\omega }_q$ parameter slightly increases the radius (see Figure 9).

6. Constraining Black Hole Mass and Its Magnetic Charge Using QPO Data

The ${\chi }^2$ method is a powerful tool for constraining the parameters of black hole models. It has been used to constrain the parameters of black holes in gravity theories in a variety of astrophysical systems, including active galactic nuclei, (low-mass) X-ray binaries, and gravitational wave sources. Here we obtain the constraints for mass and charge of the regular black hole in a quintessential field using QPOs data observed in the microquasars GRS 1915+105 and GRO J1655-40 performing ${\chi }^2$-method analyses [30]

$$\begin{array}{ccc}\hfill {\chi }^2(M,B,r_1,r_2)& =& \frac{{({\nu }_{1\varphi }-{\nu }_{1\mathrm{U}})}^2}{{\sigma }_{1\mathrm{U}}^2}+\frac{{({\nu }_{1\mathrm{per}}-{\nu }_{1\mathrm{L}})}^2}{{\sigma }_{1\mathrm{L}}^2}\hfill \\ & +& \frac{{({\nu }_{1\mathrm{nod}}-{\nu }_{1\mathrm{C}})}^2}{{\sigma }_{1\mathrm{C}}^2}+\frac{{({\nu }_{2\varphi }-{\nu }_{2\mathrm{U}})}^2}{{\sigma }_{2\mathrm{U}}^2}+\frac{{({\nu }_{2\mathrm{nod}}-{\nu }_{2\mathrm{C}})}^2}{{\sigma }_{2\mathrm{C}}^2}.\hfill \end{array}$$

(20)

6.1. QPOs from GRO J1655-40

The microquasar GRO J1655-40 is a black hole candidate binary system located in the constellation Scorpius. It is one of the most well-studied black hole candidates, and it has been used to constrain the parameters of black holes and to test theories of gravity. This subsection is devoted to obtaining constraint values of the black hole mass and its NED charge surrounding a quintessential field using observed QPO frequencies in the microquasar GRO J1655-40 [11],

$$\begin{array}{cc}\hfill {\nu }_{1\mathrm{U}}=441\mathrm{Hz},& {\sigma }_{1\mathrm{U}}=2\mathrm{Hz},\\ \hfill {\nu }_{1\mathrm{L}}=298\mathrm{Hz},& {\sigma }_{1\mathrm{L}}=4\mathrm{Hz},\\ \hfill {\nu }_{1\mathrm{C}}=17.3\mathrm{Hz},& {\sigma }_{1\mathrm{C}}=0.1\mathrm{Hz},\end{array}$$

(21)

and

$$\begin{array}{cc}\hfill {\nu }_{2\mathrm{U}}=451\mathrm{Hz},& {\sigma }_{2\mathrm{U}}=5\mathrm{Hz},\\ \hfill {\nu }_{2\mathrm{C}}=18.3\mathrm{Hz},& {\sigma }_{2\mathrm{C}}=0.1\mathrm{Hz}.\end{array}$$

(22)

Our ${\chi }^2$-analyses have shown that the error reaches its minimum at $r\simeq 4.60986M$, $q\simeq -0.9695M_{\odot }$, $M\simeq 6.60771M_{\odot }$, and $\mathcal{C}=0.00199784/\sqrt{M}$ when ${\omega }_q=-\frac{2}{3}$ and equals to ${\chi }_{min}^2=2.35409\times {10}^{-11}$. While for the case ${\omega }_q=-\frac{1}{2}$, ${\chi }_{min}^2=5.99518\times {10}^{-11}$, at $r\simeq 4.68708M$, $q\simeq 0.481337M_{\odot }$, $M\simeq 3.21581M_{\odot }$ and $\mathcal{C}=0.297692/M$.

6.2. QPOs from GRS 1915+105

The QPOs in GRS 1915+105 have also been used to test theories of gravity. For example, the frequency of the type-C QPO has been used to constrain the mass and spin of the black hole in GRS 1915+105. This information has been used to test different models of black holes, such as the Kerr metric. Here, we get black hole mass and charge constraints by performing similar analyses done in the above subsection, but for QPO data from the GRS 1915+105 microquasar [31],

$$\begin{array}{cc}\hfill {\nu }_{1\mathrm{U}}=184.10\mathrm{Hz},& {\sigma }_{1\mathrm{U}}=1.84\mathrm{Hz},\\ \hfill {\nu }_{2\mathrm{U}}=142.98\mathrm{Hz},& {\sigma }_{2\mathrm{U}}=3.48\mathrm{Hz},\end{array}$$

(23)

and

$$\begin{array}{cc}\hfill {\nu }_{1\mathrm{L}}=67.40\mathrm{Hz},& {\sigma }_{1\mathrm{L}}=0.60\mathrm{Hz},\\ \hfill {\nu }_{2\mathrm{L}}=65.89\mathrm{Hz},& {\sigma }_{2\mathrm{L}}=0.52\mathrm{Hz},\\ \hfill {\nu }_{3\mathrm{L}}=69.58\mathrm{Hz},& {\sigma }_{3\mathrm{L}}=0.49\mathrm{Hz}.\end{array}$$

(24)

Here, we also perform similar numerical analyses for the cases ${\omega }_q=-\frac{2}{3}$ and ${\omega }_q=-\frac{1}{2}$. It is obtained that in the case ${\omega }_q=-\frac{2}{3}$, the minimum of ${\chi }_{min}^2=3.8563\times {10}^{-11}$ at the values of the QPO orbit $r\simeq 10.5034M$, the black hole charge $q\simeq 0.518973M_{\odot }$, quintessential field parameter $\mathcal{C}\to 0.00357793/\sqrt{M}$, and the black hole mass $M\to 5.02719M_{\odot }$.

And when ${\omega }_q=-\frac{1}{2}$, the minimum is about ${\chi }_{min}^2=4.7845\times {10}^{-10}$, at $r\simeq 13.9982M$, $q\simeq -1.18079M_{\odot }$, $\mathcal{C}\simeq 0.0237965/M$, and $M\simeq 2.97625M_{\odot }$.

In Figure 10 and Figure 11 we have shown constraint values for the electric charge and mass of the black hole located at the microquasars GRO J1655-40 and GRS 1915-105, respectively, for different nonlinearity and quintessential field parameters in the $1\sigma$, $2\sigma$, and $3\sigma$ error bars.

6.3. Comparisons of the Black Hole Mass Constraints with Other Independent Findings

Here, in order to compare our results, we provide black hole mass constraints in the above-mentioned microquasars obtained using independent observations such as the X-ray timing techniques; their estimations on the mass of the central black hole in the microquasar GRO J1655-40 has mass $M/M_{\odot }=5.31\pm 0.07$ [32].

The constraints on the black hole mass using QPOs data considering the central black hole as a rotating Kerr black hole in the RP model have been obtained as $M/M_{\odot }=5.3\pm 0.1$ in Ref. [33].

Constraints of the black hole mass in the microquasar GRS 1915+105 obtained using the infrared spectroscopy technique as $M/M_{\odot }=12\pm 1.4$ [34], using QPOs data and considering the black hole as a Kerr black hole as $M/M_{\odot }=13.1\pm 0.2$ [35].

In fact, constraints on black hole parameters using astrophysical observations strongly depend on the type of observation data, its accuracy, and the gravity model of the black hole spacetime, as well as the theoretical model of the study. Therefore, obtained constraint values may differ from each other for different black holes in different observational data. Unfortunately, it is quite hard to know which one is more trustworthy. It is seen from the above-mentioned constraints from independent measurement that our results differ from them. In our theoretical model, the test particles feel combined gravitational effects of the black hole mass, NED field, and quintessential field on the spacetime around black holes. Consequently, our obtained results show not only constraints on the black hole mass and charge but also the value of the quintessential parameter for the models of the equation of state of the quintessential field ${\omega }_q=-1/2$ and ${\omega }_q=-2/3$.

7. Conclusions

In fact, testing gravity theories combined with different including massive and massless scalar and linear/nonlinear electrodynamics fields using observational data from black holes and neutron stars is the most important and actual issue in modern relativistic astrophysics. In this paper, we have obtained a new spherically symmetric black hole solution in general relativity coupling to nonlinear electrodynamics (NED) and quintessential fields. Also, we have analyzed the possible values for the quintessential field parameter and NED charge of the black hole for various values of the nonlinearity parameter $\nu$ in the ${\omega }_q=-\frac{2}{3}$ case. Event horizon properties and scalar invariants of the black hole spacetime are studied. We have investigated the motion of test particles around the regular Kiselev black holes in the equatorial plane and studied the combined effects of the quintessential field and the NED charge of the black hole on the angular momentum and energy of particles corresponding to circular orbits together with their ISCOs. Moreover, we calculated the oscillation frequencies of the particles along stable circular orbits in radial and vertical directions together with the Keplerian frequency. Then, we applied the frequencies to study QPOs in the RP model. It is shown that low-frequency QPOs below 20 Hz disappear in the presence of quintessential field with $\mathcal{C}=0.001$, and the variation of ${\omega }_q$ does not give much change in QPO frequencies. Also, we have obtained the dependence of the radius of orbits where the QPOs with the frequency ratio is 3:2, on the black hole charge for different nonlinearity and quintessential field parameters. Our performed analyses have shown that an increase in the black hole charge causes a decrease in the radius. It decreases faster at $\nu =1$ case compared to $\nu =2,3$ cases which means the gravitational effect of the black hole charge becomes weaker at higher values of the nonlinearity parameter $\nu$ weak. However, the orbit shifts slightly out due to the presence of a quintessential field. Also, an increase of the ${\omega }_q$ parameter slightly increases the radius.

Finally, we have obtained constrain values for the quintessential parameter together with the black hole mass and its charge using observational data from QPOs observed in the microquasars GRO J1655-40 and GRS 1915+105 performing ${\chi }^2$-method analyses and compared the obtained results with the mass constraints obtained in the independent astronomical observations.