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 ${\ensuremath{\omega}}_{q}$, namely (i) ${\ensuremath{\omega}}_{q}=\ensuremath{-}1/3$, (ii) ${\ensuremath{\omega}}_{q}=\ensuremath{-}1$, and (iii) ${\ensuremath{\omega}}_{q}=\ensuremath{-}2/3$, and for different values of the normalization factor $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$ and $Q$. It is observed that the photon orbits are also shifted outward as the value of $\ensuremath{\alpha}$ 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 ${\ensuremath{\omega}}_{q}$. 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 ${\ensuremath{\omega}}_{q}=\ensuremath{-}1/3$, 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 $\ensuremath{\alpha}$. However, for ${\ensuremath{\omega}}_{q}=\ensuremath{-}2/3$ and ${\ensuremath{\omega}}_{q}=\ensuremath{-}1$, these two frequencies, along with the black hole parameters---i.e., $M$ and $Q$ and the radial position $r$---also depend on the factor $\ensuremath{\alpha}$. The radial epicyclic frequency for all the values of ${\ensuremath{\omega}}_{q}$ depends on $M$, $Q$, $r$, and also on the normalization factor $\ensuremath{\alpha}$. 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 $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$ on the OSCOs is just opposite to that on the ISCOs, the epicyclic frequencies near the OSCOs behave the other way around.