Epicyclic oscillations of test particles near marginally stable circular orbits around charged Kiselev black holes

The marginally stable circular orbits (MSCOs) of test particles in the spacetime exterior to a charged Kiselev black hole are investigated for three characteristic values of the equation of state parameter ${\omega }_q$, namely (i) ${\omega }_q=-1/3$, (ii) ${\omega }_q=-1$, and (iii) ${\omega }_q=-2/3$, and for different values of the normalization factor $\alpha$ and electric charge $Q$ of the black hole. It is found that the presence of the quintessence field shifts outward the innermost stable circular orbits (ISCOs) around the Kiselev black hole, having the same charge parameter $Q$, as compared to the ISCOs around a Riessner-Nordstrom black hole, while the effect of the quintessence field on the outermost stable circular orbits (OSCOs) is just opposite to that on the ISCOs. Further, the radii of the photon circular orbits are also calculated for different ranges of the parameters $\alpha$ and $Q$. It is observed that the photon orbits are also shifted outward as the value of $\alpha$ increases. The radial and latitudinal epicyclic motion of test particles, which can be related to the quasiperiodic oscillations of test particles slightly above the MSCOs in the vicinity of the charged Kiselev black hole, is analyzed for the three different values of ${\omega }_q$. It is seen that the azimuthal and latitudinal frequencies coincide, and the radial epicyclic frequency is different in dependence on the spacetime parameters. In the case of ${\omega }_q=-1/3$, the azimuthal and latitudinal frequencies depend on the radial position $r$ of the particle, the charge $Q$, and the mass $M$ of the black hole, and do not depend on the factor $\alpha$. However, for ${\omega }_q=-2/3$ and ${\omega }_q=-1$, these two frequencies, along with the black hole parameters—i.e., $M$ and $Q$ and the radial position $r$—also depend on the factor $\alpha$. The radial epicyclic frequency for all the values of ${\omega }_q$ depends on $M$, $Q$, $r$, and also on the normalization factor $\alpha$. We also compare the epicyclic frequencies with that for an uncharged black hole. With the increase of electric charge, the ISCO becomes closer to the central object, and one can observe epicyclic frequencies closer to the central object, which makes the epicyclic frequencies larger. The ISCO gets larger as $\alpha$ increases, and thus the epicyclic frequencies can be observed away from the central object and would be smaller as compared to the case of a pure Riessner-Nordstrom black hole without quintessence. As the effect of the parameters $Q$ and $\alpha$ on the OSCOs is just opposite to that on the ISCOs, the epicyclic frequencies near the OSCOs behave the other way around.

Figure 1

Shaded region shows available range of the permitted values for the parameters $\alpha$ and $Q$ for $M=1$, allowing the existence of a black hole. Parameter $\alpha$ can take values in the interval $\alpha \in [0,1)$. The parameter $Q$ varies from 0 to $Q_{\max }$ given by the condition in Eq. (46) for the given value of the parameter $\alpha$. The maximal value of the positive charge $Q_{\max }$ is represented by the solid curve on the graph.

Figure 2

The left column shows plots of the effective potentials, the central column shows the specific energy, and the right column shows the specific angular momentum of the test particle for different values of the parameters $\alpha$ and $Q$. In each row, for a chosen value of the parameter $Q$, four graphs for four different values of $\alpha$ are presented. The red color corresponds to $Q=0$, while the green, blue, and violet colors correspond to $Q=0.6$, $Q=0.8$, and $Q=1$, respectively. The parameter $\alpha$ takes the values $\alpha =0$ for the first line, $\alpha =0.3$ for the second, $\alpha =0.6$ for the third, and $\alpha =0.8$ for the fourth line in the graphs presented in the left column. With increasing values of $\alpha$, the radii of the MSCOs also increase. The values of specific energy and the specific angular momentum for particles on ISCOs, with the radii of ISCOs, for the chosen parameters $Q$ and $\alpha$ are presented in Table 1.

Figure 3

Color map of the radius of the black hole event horizon for different values of the parameters $\alpha$ and $Q$, for $M=1$. Parameter $\alpha$ changes from 0 to 0.8, and parameter $Q$ changes from 0 to $Q_{\max }$ given by the condition in Eq. (46) for each value of $\alpha$. For higher values of the parameter $\alpha$ (shown by the shaded region on the graph), the qualitative behavior of the radius of the event horizon is the same—it increases with the parameter $\alpha$.

Figure 4

Plots of the square of the specific energy, $E^2$, for different values of the parameters $\alpha$ and $Q$, for $M=1$. Here $\alpha$ increases from red to violet colors and takes the values $\alpha =\{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines.

Figure 5

Plot of the specific angular momentum, $L^2$, for different values of $\alpha$ and $Q$, for $M=1$. Here $\alpha$ increases from red to violet colors and takes the values $\alpha =\{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines.

Figure 6

The left graph shows how the radii of ISCOs depend on $Q$ for some different fixed values of $\alpha =\{0,0.1,\dots ,0.8\}$. The right graph shows how the radii of ISCOs depend on $\alpha$ for some different fixed values of $Q$ in the range from $Q=0$ to $Q=1$.

Figure 7

Color map demonstrates the radii of photon orbits for different values of the parameters $\alpha$ and $Q$, for $M=1$. Parameter $\alpha$ takes values from 0 to 0.8, and parameter $Q$ ranges from 0 to $Q_{\max }$ given by the condition in Eq. (46) for each value of the parameter $\alpha$.

Figure 8

Plots in the left column show the radial dependence of functions ${\alpha }_h(r)$, ${\alpha }_{ms}(r)$, and ${\alpha }_{rph}(r)$ [Eqs. (57), (59) and (58) in the text] for several values of the charge $Q$ with the fixed value of $M=1$. For better visualization, instead of the functions themselves on the plots, we show $A_h(r)=Sgn({\alpha }_h(r))|{\alpha }_h(r){|}^{1/4}$, $A_{ms}(r)=Sgn({\alpha }_{ms}(r))|{\alpha }_{ms}(r){|}^{1/4}$, and $A_{rph}(r)=Sgn({\alpha }_{rph}(r))|{\alpha }_{rph}(r){|}^{1/4}$. The thin black line is the graph of the function $A_h(r)$, which determines the positions of the event horizons. The blue dashed line is the graph of the function $A_{rph}(r)$, which determines the positions of the photon orbits. The thin green line is the graph of the function $A_{ms}(r)$, which determines the positions of the stable circular orbits. The gray dashed line defines the parameter ${\alpha }^{1/4}$, for which the functions $N(r)$, $f_{rph}(r)$, and $f_{rms}(r)$ are shown in the right column. In the right column, for some parameters $\alpha$ and $Q$, the following graphs are presented: The thin black line is a graph of the function $N(r)$ [Eq. (48) in the text]; zero values of this function determine the positions of the horizons for the selected values of the parameter $\alpha$. The blue dashed line is a graph of the function $f_{rph}(r)$, [Eq. (55) in the text]; zero values of this function determine the position of the photon orbit for the selected values of the parameter $\alpha$. The thin green line is a graph of function $f_{rms}(r)$ [Eq. (51) in the text]; zero values of this function determine the position of the ISCO for the selected values of the parameter $\alpha$. Parameters $\alpha$ and $Q$ take the values $\alpha =[0.05,0.1]$ and $Q=[0.9,1.1]$.

Figure 9

Color map of the radius of the event horizon for different values of the parameters $\alpha$ and $Q$, for $M=1$. The parameter $\alpha$ varies from 0 to ${\alpha }_{+}$ if $0\le Q<1$, and from ${\alpha }_{-}$ to ${\alpha }_{+}$ if $Q\ge 1$ (${\alpha }_{\max }=1/6$ and $Q_{\max }=3/4$). For the white regions, there is no event horizon in the spacetime, and they correspond to the naked singularity. The region where the event horizon exists is given by the condition in Eq. (62).

Figure 10

The gray shaded region shows the allowable range of the parameters $\alpha$ and $Q$ for which the event horizon exists. Parameter $\alpha$ takes values from 0 to ${\alpha }_{+}$ if $0\le Q<1$, and ${\alpha }_{-}\le \alpha <{\alpha }_{+}$ if $Q\ge 1$; the value of ${\alpha }_{\pm }$ is given by the condition in Eq. (62).

Figure 11

The red shaded region shows the allowable range of the parameters $\alpha$ and $Q$ for which the MSCOs exist. The parameter $\alpha$ takes values from 0 to ${\alpha }_{ms+}$ if $0\le Q<1$, and ${\alpha }_{-}\le \alpha <{\alpha }_{ms+}$ if $Q\ge 1$.

Figure 12

The left column shows plots of the effective potentials, the central column shows the specific energy, and the right column shows the specific angular momentum of the test particle for different values of the parameters $\alpha$ and $Q$. In each row, for a chosen value of the parameter $Q$, eight graphs for four different values of $\alpha$ are presented. The red color corresponds to $Q=0$, while the green, blue, and violet colors correspond to $Q=0.6$, $Q=0.8$, and $Q=0.999$, respectively. The parameter $\alpha$ takes the values $\alpha =0.001$ for the first line, $\alpha =0.002$ for the second, $\alpha =0.003$ for the third, and $\alpha =0.005$ for the fourth line in the graphs presented in the left column. In the first column, solid lines represent the effective potential of the particles in the ISCOs, and dashed lines represent the effective potential of the particles in the OSCOs. With increasing values of $\alpha$, the radii of the ISCOs also increase, and the radii of the OSCOs decrease. The values of specific energy and the specific angular momentum for particles on MSCOs, with the radii of MSCOs, for chosen parameters $Q$ and $\alpha$ are presented in Tables 2 and 3.

Figure 13

Plot of $E^2$ for different values of the parameters $\alpha$ and $Q$. Here $\alpha$ increases from the red color to the violet one and takes the values $\alpha =\{0,0.001,0.002,0.003,0.004,0.006,0.008,0.01\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines.

Figure 14

Plot of $L^2$ for different values of $\alpha$ and $Q$. Here $\alpha$ increases from the red color to the violet color and takes the values $\alpha =\{0,0.001,0.002,0.003,0.004,0.006,0.008,0.01\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines.

Figure 15

In the left plot, the dependence of the radii of the ISCO and OSCO for the whole range of $Q^2$ is shown; each graph is for different values of $\alpha$, from $\alpha =0$ to $\alpha =0.01$. In the right plot, the dependence of the radii of the ISCO and OSCO for the whole range of $\alpha$ is shown; each graph corresponds to different values of $Q^2$, from $Q^2=0$ to $Q^2=1.0$.

Figure 16

Color map shows the radii of photon orbits for different values of the parameters $\alpha$ and $Q$. The parameter $\alpha$ takes values from 0 to ${\alpha }_{+}$ if $0\le Q<1$, and ${\alpha }_{-}\le \alpha <{\alpha }_{+}$ if $Q\ge 1$; the value of ${\alpha }_{\pm }$ is given by the condition in Eq. (62).

Figure 17

Plots in the left column show the radial dependences of the functions ${\alpha }_h(r)$, ${\alpha }_{rst}(r)$, ${\alpha }_{ms}(r)$, and ${\alpha }_{rph}(r)$ [Eqs. (70), (73), (72), and (71) in the text] for several values of the parameter $Q$ with fixed $M=1$. For better visualization, instead of the functions themselves in the plots, we show $A_h(r)=Sgn({\alpha }_h(r))|{\alpha }_h(r){|}^{1/4}$, $A_{rst}(r)=Sgn({\alpha }_{rst}(r))|{\alpha }_{rst}(r){|}^{1/4}$, $A_{ms}(r)=Sgn({\alpha }_{ms}(r))|{\alpha }_{ms}(r){|}^{1/4}$, and $A_{rph}(r)=Sgn({\alpha }_{rph}(r))|{\alpha }_{rph}(r){|}^{1/4}$. The thin black line is the graph of the function $A_h(r)$, which determines the positions of the event horizons. The blue dashed line is the graph of the function $A_{rph}(r)$, which determines the position of the photon orbit. The thin green line is the graph of the function $A_{ms}(r)$, which determines the positions of the stable circular orbits. The red dotted line is the graph of the function $A_{rst}(r)$, which determines the position of the static radius. The gray dashed line defines the parameter ${\alpha }^{1/4}$, for which the functions $N(r)$, $f_{rph}(r)$, $f_{rst}(r)$, and $f_{rms}(r)$ are shown in the right column. In the right column, for some values of the parameters $\alpha$ and $Q$, the following graphs are presented: The thin black line is the graph of the function $N(r)$ [Eq. (60)]; zero values of this function determine the positions of the horizons for the selected values of the parameter $\alpha$. The blue dashed line is the graph of the function $f_{rph}(r)$ [Eq. (69)]; zero values of this function determine the position of the photon orbit for the selected values of the parameter $\alpha$. The thin green line is the graph of the function $f_{rms}(r)$ [Eq. (68)]; zero values of this function determine the position of the ISCO for the selected values of the parameter $\alpha$. The red dotted line is the graph of the function $f_{rst}(r)$ [Eq. (67)]; zero values of the function determine the position of the static radius for the selected values of the parameter $\alpha$. Parameters $\alpha$ and $Q$ take the values $\alpha =[0.05,0.001,0.005]$ and $Q=[0.9,1.001]$.

Figure 18

Color map of the radius of the event horizon for different values of the parameters $\alpha$ and $Q$. Parameter $Q$ changes from 0 to $Q_{\max }$, and parameter $\alpha$ takes values from 0 to ${\alpha }_{h(\max )}$ if $0\le Q<1$, and ${\alpha }_{h(\min )}\le Q<{\alpha }_{h(\max )}$ if $Q\ge 1$. The functions ${\alpha }_{h(\min )}$ and ${\alpha }_{h(\max )}$ are given by the conditions in Eqs. (77) and (76) for each value of $Q$.

Figure 19

The black line limits the interval of the variation of the parameters $\alpha$ and $Q$ in which the existence of black hole spacetime is allowed. The parameters $\alpha$ and $Q$ lying outside the region confined by the black line are consistent with naked singularities. Parameter $\alpha$ takes values $0\le \alpha <{\alpha }_{h(\max )}$ if $Q<1$ and ${\alpha }_{h(\min )}\le \alpha <{\alpha }_{h(\max )}$ if $Q\ge 1$. The functions ${\alpha }_{h(\min )}(Q)$ and ${\alpha }_{h(\max )}(Q)$ are given by the conditions in Eqs. (76) and (77).

Figure 20

The red line limits the range of parameters $\alpha$ and $Q$ for which stable circular orbits exist. The values of the parameter $\alpha$ in the case of $Q>1$ are also bounded from below by the minimum value of the parameter ${\alpha }_{h(\min )}$ for which there is a black hole event horizon.

Figure 21

The left column shows plots of effective potentials, the central column shows the specific energy, and the right column shows the specific angular momentum of the test particle for different values of parameters $\alpha$ and $Q$. In each row, for chosen values of the parameter $Q$, eight graphs are presented for four different values of $\alpha$. The red color corresponds to $Q=0$, while the green, blue, and violet colors correspond to $Q=0.6$, $Q=0.8$, and $Q=0.999$, respectively. Parameter $\alpha$ takes the values $\alpha =0.00005$ for the first line, $\alpha =0.0001$ for the second, $\alpha =0.00015$ for the third, and $\alpha =0.0002$ for the fourth line in the graphs presented in the left column. In the first column, solid lines represent the effective potential of the particles on the ISCOs, and dashed lines represent the effective potential of the particles on the OSCOs. With increasing values of $\alpha$, the radii of ISCOs also increase, but the radii of OSCOs decrease. The values of specific energy and the specific angular momentum for particles in MSCOs, with the radii of MSCOs for chosen parameters $Q$ and $\alpha$, are presented in Tables 4 and 5.

Figure 22

Plot of the specific energy $E^2$ for different values of parameters $Q$ and $\alpha$. Here $\alpha$ increases from the red color to the violet color and takes the values $\alpha =\{0,0.00005,0.0001,0.00015,0.0002,0.0003,0.0004,0.0006\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines. Parameter $M$ is fixed and equals 1.

Figure 23

Plot of the specific angular momentum $L^2$ for different values of $\alpha$ and $Q$. Here $\alpha$ increases from the red color to the violet one and takes the values $\alpha =\{0,0.00005,0.0001,0.00015,0.0002,0.0003,0.0004,0.0006\}$, respectively. For each value of $\alpha$, $Q$ takes two characteristic values: $Q=0$ for dashed lines and $Q=1$ for solid lines. Parameter $M$ is fixed and equals 1.

Figure 24

Color map demonstrates the radii of photon orbits for different values of the parameters $\alpha$ and $Q$, for fixed $M=1$. The radius of the photon orbit does not depend on $\alpha$, but the parameter $\alpha$ defines the region where the existence of the black hole spacetime is allowed. Parameter $Q$ changes from 0 to $Q_{\max }$, and parameter $\alpha$ takes values from 0 to ${\alpha }_{h(\max )}$ if $0\le Q<1$, and ${\alpha }_{h(\min )}\le Q<{\alpha }_{h(\max )}$ if $Q\ge 1$. The functions ${\alpha }_{h(\min )}$ and ${\alpha }_{h(\max )}$ are given by the conditions in Eqs. (77) and (76) for each value of parameter $Q$.

Figure 25

In the left plot, the dependence of the radii of ISCO and OSCO for the whole range of $Q^2$ is shown; each graph is for different values of $\alpha$, from $\alpha =0$ to $\alpha =0.0006$. In the right plot, the dependence of the radii of ISCO and OSCO for the whole range of $\alpha$ is shown; each graph corresponds to different values of $Q^2$, from $Q^2=0$ to $Q^2=1.0$.

Figure 26

Plots in the left column show the radial dependences of the functions ${\alpha }_h(r)$, ${\alpha }_{rst}(r)$, ${\alpha }_{ms}(r)$ and $r_{ph}(Q)$ [Eqs. (87), (89), (88), and (85) in the text; $r_{ph}(Q)$ does not depend on $\alpha$ in the case $w_q=-1$] for several values of the parameter $Q$ with fixed $M=1$. For better visualization, instead of the functions themselves in the plots, we show $A_h(r)=Sgn({\alpha }_h(r))|{\alpha }_h(r){|}^{1/4}$, $A_{rst}(r)=Sgn({\alpha }_{rst}(r))|{\alpha }_{rst}(r){|}^{1/4}$, and $A_{ms}(r)=Sgn({\alpha }_{ms}(r))|{\alpha }_{ms}(r){|}^{1/4}$. The thin black line is the graph of the function $A_h(r)$, which determines the positions of the event horizons. The blue dashed line is the graph of the function $r_{ph}(Q)$, which determines the position of the photon orbit. The thin green line is the graph of the function $A_{ms}(r)$, which determines the positions of the MSCOs. The red dotted line is the graph of the function $A_{rst}(r)$, which determines the position of the static radius. The gray dashed horizontal line represents the chosen value of the parameter ${\alpha }^{1/4}$; intersections with this line determine the positions of the horizons, photon circular orbit, MSCOs, and static radius. For the value of the parameter $\alpha$ represented by the gray dashed horizontal line, in the right column, the functions $N(r)$, $f_{rph}(r)$, $f_{rms}(r)$, and $f_{rst}(r)$ are shown; the zero values of these functions determine the radii of the corresponding quantities (event horizons, photon orbit, ISCO and OSCO, static radius). In the right column, for some values of the parameters $\alpha$ and $Q$, the following graphs are presented: The thin black line is the graph of the function $N(r)$ [Eq. (74)]; zero values of this function determine the positions of the horizons for the selected values of the parameter $\alpha$. The blue dashed line is the graph of the function $f_{rph}(r)$ [Eq. (84)]; zero values of this function determine the position of the photon orbit for the selected values of the parameter $\alpha$. The thin green line is the graph of the function $f_{rms}(r)$ [Eq. (86)]; zero values of this function determine the position of the ISCO for the selected values of the parameter $\alpha$. The red dotted line is the graph of the function $f_{rst}(r)$ [Eq. (80)]; zero values of the function determine the position of the static radius for the selected values of the parameter $\alpha$. Parameters $\alpha$ and $Q$ take the values $\alpha =[0.0005,0.001]$ and $Q=[0.9,1.0001]$.

Figure 27

Fundamental frequencies of a test particle for ${\omega }_q=-1/3$, and for the different values of parameters $\alpha$ and $Q$. $\alpha =0.05,0.1,0.2$ in the first, second, and third rows, respectively; $Q=0,0.5,0.9$ in the first, second, and third columns, respectively.

Figure 28

Fundamental frequencies of a test particle for ${\omega }_q=-2/3$, and for the different values of parameters $\alpha$ and $Q$. $\alpha =0.0005,0.001,0.002$ in the first, second, and third rows, respectively; $Q=0,0.5,0.9$ in the first, second, and third columns, respectively.

Figure 29

Fundamental frequencies of a test particle for ${\omega }_q=-1$, and for the different values of the parameters $\alpha$ and $Q$. $\alpha =0.00005,0.0001,0.0002$ in the first, second, and third rows, respectively; $Q=0,0.5,0.9$ in the first, second, and third columns, respectively.

Figure 30

The plots show a comparison of the acceleration acting on a particle located in the gravitational field of a Kiselev black hole with mass $M={10}^6{M}_{\odot }$ for three values of the parameter ${\omega }_q$, with a Schwarzschild black hole in the case of zero charge $Q$ (top two graphs), as well as with a Riessner-Nordstrom black hole for a charge $Q={10}^6$ C (lower two graphs). For nonzero values of the parameter $\alpha$, the graphs diverge at the horizons.

Figure 31

The plots show a comparison of the acceleration acting on a particle located in the gravitational field of a Kiselev black hole with mass $M={10}^6{M}_{\odot }$ for three values of the parameter ${\omega }_q$, with a Riessner-Nordstrom black hole (top two graphs), as well as with a Schwarzschild black hole (lower two graphs), with a charge $Q={10}^{26}$ C for both cases. For nonzero values of the parameter $\alpha$, the graphs diverge at the horizons.