Quasi-periodic Oscillations Frequency Research Articles

A Possible X-ray Quasi-periodic Oscillation in the Narrow-line Seyfert 1 Galaxy Mrk 142

A possible quasi-periodic oscillation (QPO) at frequency 7.045 × 10−5 Hz is found in the narrow-line Seyfert 1 galaxy Mrk 142 in the data of XMM-Newton collected on 2020 April 11. We find that the QPO signal is statistically significantly larger than the 5σ level and highly coherent with quality factor Q > 5 at the 0.3–10 keV band by using the method of the Lomb–Scargle Periodogram, the Weighted Wavelet Z-transform and the REDFIT. We analyze the data in 0.3–0.6 keV, 0.6–1 keV, 1–3 keV and 3–10 keV energy bands, and find obvious QPO signals at 0.3–0.6 keV and 1–3 keV bands. We then analyze the time-average spectra and time variability at the QPO frequency of 7.045 × 10−5 Hz, and use a model to fit them. We find that the QPO signal mainly comes from the X-ray hot corona.

Discovery of evolving low-frequency QPOs in hard X-rays (∼100 keV) observed in black hole Swift J1727.8−1613 with AstroSat

ABSTRACT We report the first detection of evolving low-frequency quasi-periodic oscillation (LFQPO) frequencies in hard X-rays upto 100 keV with AstroSat/LAXPC during ‘unusual’ outburst phase of Swift J1727.8−1613 in hard intermediate state (HIMS). The observed LFQPO in 20–100 keV has a centroid $\nu _{_{\rm QPO}}=1.43$ Hz, a coherence factor Q = 7.14 and an amplitude ${\rm rms_{_{\rm QPO}}} = 10.95{{\ \rm per\ cent}}$ with significance σ = 5.46. Type-C QPOs (1.09–2.6 Hz) are found to evolve monotonically during HIMS of the outburst with clear detection in hard X-rays (80−100 keV), where ${\rm rms_{_{\rm QPO}}}$ decreases ($\sim 12\!-\!3{{\ \rm per\ cent}}$) with energy. Further, $\nu _{_{\rm QPO}}$ is seen to correlate (anticorrelate) with low- (high-) energy flux in 2–20 keV (15–50 keV). Wide-band (0.7−40 keV) energy spectrum of NICER/XTI and AstroSat/LAXPC is satisfactorily described by the ‘dominant’ thermal Comptonization contribution (∼88 per cent) in presence of a ‘weak’ signature of disc emissions (kTin ∼ 0.36 keV) indicating the harder spectral distribution. Considering source mass $M_{\rm BH}=10\, \mathrm{M}_\odot$ and distance 1.5 < d (kpc) < 5, the unabsorbed bolometric luminosity is estimated as $\sim 0.03\!-\!0.92{{\ \rm per\ cent}}\, L_{\rm Edd}$. Finally, we discuss the implications of our findings in the context of accretion dynamics around black hole X-ray binaries.

Timing and spectral analysis of the black hole X-ray binary MAXI J1803–298 with Insight-HXMT data

Abstract We present a comprehensive analysis of the 2021 outburst of MAXI J1803–298 utilizing observations of the Insight-Hard X-ray Modulation Telescope (Insight-HXMT) spanning from the low hard state to the high soft state. Within the Insight-HXMT dataset, compared to the previous work, we identify a more prolonged presence of type-C quasi-periodic oscillations (QPOs) with centroid frequencies ranging from ∼0.16 to 6.3 Hz, which present correlations with the hardness ratio and the photon index of the Comptonized component. For QPO frequencies less than ∼2 Hz, the QPO phase lags are hard (photons of 10–19 keV arrive later than those of 1–4 keV), while at higher frequencies, lags transition to soft lags after reaching ∼4 Hz. Furthermore, the spectra in all Insight-HXMT observations consist of a multi-color blackbody component and a Comptonized component, as commonly observed in classical black-hole X-ray binaries. We analyze state transitions and the evolution of accretion geometry in this work. The fitted inner disk radius increases abnormally during the low hard state, hypothesized to result from the corona condensing onto the inner disk. Additionally, two significant drops in flux are observed during the soft intermediate state, maybe implying changes in the corona/jet and the disk, respectively.

The First Polarimetric View on Quasiperiodic Oscillations in a Black Hole X-Ray Binary

We present the first polarimetric analysis of quasiperiodic oscillations (QPOs) in a black hole binary utilizing IXPE data. Our study focuses on Swift J1727.8–1613, which experienced a massive outburst that was observed by various telescopes across different wavelengths. The IXPE observation we studied was conducted during the hard-intermediate state. The polarization degree (PD) and polarization angle (PA) were measured at 4.28% ± 0.20% and 1.°9 ± 1.°4, respectively. Remarkably, significant QPO signals were detected during this observation, with a QPO frequency of approximately 1.34 Hz and a fractional rms amplitude of about 12.3%. Furthermore, we conducted a phase-resolved analysis of the QPO using the Hilbert–Huang transform technique. The photon index showed a strong modulation with respect to the QPO phase. In contrast, the PD and PA exhibit no modulations in relation to the QPO phase, which is inconsistent with the expectation of the Lense–Thirring precession of the inner flow. Further theoretical studies are needed to conform with the observational results.

First Observational Evidence for an Interconnected Evolution between Time Lag and QPO Frequency among AGNs

Quasiperiodic oscillations (QPOs) have been widely observed in black hole X-ray binaries (BHBs), which often exhibit significant X-ray variations. Extensive research has explored the long-term evolution of the properties of QPOs in BHBs. In contrast, such evolution in active galactic nuclei (AGNs) has remained largely unexplored due to limited observational data. By using the 10 new XMM-Newton observations for the narrow-line Seyfert 1 galaxy RE J1034 + 396 from publicly available data, we analyze the characteristics of its X-ray QPOs and examine their long-term evolution. The hard-band (1–4 keV) QPOs are found in all 10 observations and the frequency of these QPOs evolves ranging at (2.47–2.83) × 10−4 Hz. Furthermore, QPO signals in the soft (0.3–1 keV) and hard bands exhibit strong coherence, although, at times, the variations in the soft band lead those in the hard band (the hard-lag mode), while at other times, it is the reverse (the soft-lag mode). The observations presented here serendipitously captured two ongoing lag reversals between these two modes within about two weeks, which are first seen in RE J1034 + 396 and also among all AGNs. A transition in QPO frequency also takes place within a two-week timeframe, two weeks prior to its corresponding lag reversal, indicating a possible coherence between the transitions of QPO frequency and lag mode with delay. The diagram of time lag versus QPO frequency clearly evidences this interconnected evolution with hysteresis, which is, for the first time, observed among AGNs.

The ongoing spin-down episode of 4U 1626-67

Abstract We report the X-ray characteristics of the persistent X-ray pulsar 4U 1626-67 using simultaneous NuSTAR and NICER observations. The X-ray pulsar 4U 1626-67 has just encountered a torque reversal in 2023 and is presently in the spin-down state. We have examined the temporal and spectral characteristics of the source during its ongoing spin-down episode. The pulse profiles of the source are characterized by multiple substructures at lower energies and a wide asymmetric single-peaked structure at higher energies. The pulse fraction follows an overall increasing trend with energy. We confirm the existence of mHz quasi-periodic oscillation (QPO) exclusively during the current spin-down phase in all the observations. The source is spinning down at 0.00045(4) s yr−1. The broadband spectrum during this phase is described by empirical NPEX model and a soft blackbody component with kT ∼ 0.25 keV. In addition to the iron emission line, we also confirm the presence of cyclotron line at ∼ 36 keV. The source flux continues to decrease during the current spin-down phase, and the corresponding luminosity ∼ (3.3–4.9) × 1036 ergs s−1 lies in the intermediate range of accreting X-ray pulsars that may be associated with a hybrid accretion geometry. The magnetic field strengths estimated using the cyclotron line measurements and QPO frequency are consistent. The evolution of the spectral parameters relative to the pulsed phase is examined using phase-resolved spectroscopy.

Disc corona radii and QPO frequencies in black hole binaries: testing Lense– Thirring precession origin

ABSTRACT Stellar-mass black hole binary systems in the luminous X-ray states show a strong quasi-periodic oscillation (QPO) in their Comptonized emission. The frequency of this feature correlates with the ratio of a disc to Comptonized emission rather than with total luminosity. Hence, it changes dramatically during spectral transitions between the hard and soft states. Its amplitude is also strongest in these intermediate states, making them an important test of QPO models. However, these have complex spectra which generally require a disc and two separate Comptonization components, making it difficult to uniquely derive the spectral parameters. We build a new energy-conserving model of the accretion flow, SSsed model, which assumes a fixed radial emissivity but with a changing emission mechanism. This is similar to the agnsed model in xspec but tuned to be more suitable for stellar mass black holes. It uses a combination of the disc luminosity and temperature to constrain the inner radius of the (colour temperature corrected) blackbody disc, separating this from the more complex Comptonization spectra emitted inwards of this radius. We show a pilot study of this model fit to hundreds of RXTE spectra of the black hole binary XTE J1550 − 564. We show that the derived disc radius tightly anticorrelates with the central frequencies of the low-frequency QPO detected in the same observations. The relation is consistent with the quantitative predictions of Lense–Thirring precession of the entire inner Comptonization regions for the assumed system parameters. This supports the scenario that low-frequency QPOs are caused by Lense–Thirring precession.

Evidence for a dynamic corona in the short-term time lags of black hole X-ray binary MAXI J1820+070

ABSTRACT In X-ray observations of hard state black hole X-ray binaries (BHXRBs), rapid variations in accretion disc and coronal power-law emission are correlated and show Fourier-frequency-dependent time lags. On short ($\sim$0.1 s) time-scales, these lags are thought to be due to reverberation and therefore may depend strongly on the geometry of the corona. Low-frequency quasi-periodic oscillations (QPOs) are variations in X-ray flux that have been suggested to arise because of geometric changes in the corona, possibly due to general relativistic Lense–Thirring precession. Therefore, one might expect the short-term time lags to vary on the QPO time-scale. We performed novel spectral-timing analyses on Neutron Star Interior Composition ExploreR observations of the BHXRB MAXI J1820+070 during the hard state of its outburst in 2018 to investigate how the short-term time lags between a disc-dominated and a coronal power-law-dominated energy band vary on different time-scales. Our method can distinguish between variability due to the QPO and broad-band noise, and we find a linear correlation between the power-law flux and lag amplitude that is strongest at the QPO frequency. We also introduce a new method to resolve the QPO signal and determine the QPO phase dependence of the flux and lag variations, finding that both are very similar. Our results are consistent with a geometric origin of QPOs, but also provide evidence for a dynamic corona with a geometry varying in a similar way over a broad range of time-scales, not just the QPO time-scale.

A spectral study of GRS 1915+105 during its March 2017 NuSTAR observations

ABSTRACT The Nuclear Spectroscopic Telescope Array Mission (NuSTAR) observed the enigmatic microquasar GRS 1915+105 twice, on 2017 March 26 and 28. Using these NuSTAR observations, we performed a time-resolved spectral analysis. We detected a type C quasi-periodic oscillation (QPO) along with the harmonic and the sub-harmonic in both epochs. From our spectral modelling, we found a dual reflection component, where one of the components comes from the hot inner flow and the second component originates from the exterior region of the accretion disc. Additionally, we found strong, highly ionized accretion disc winds as indicated by Fe xxv (on 2017 March 26) and Fe xxvi (on 2017 March 28) absorption lines, likely ultrafast outflows. Finally, at the end of the second observation, we detected variability class, which marked the onset of heartbeat oscillations in the X-ray light curve. We perform time-resolved spectral analysis to comprehend the physical conditions responsible for the onset of the heartbeat. We found that the accretion disc became highly ionized (log ξ > 4), and the high-energy cut-off showed some signs of increase, approaching >100 keV before the onset of the heartbeat class. We have not detected any signature of accretion disc instabilities thought to cause the heartbeat oscillations. We also found a strong positive correlation between the QPO frequency and the power-law index.

Unveiling hidden variability components in accreting X-ray binaries using both the Fourier power and cross-spectra

ABSTRACT We present a novel method for measuring the lags of (weak) variability components in neutron-star and black-hole low-mass X-ray binaries (LMXBs). For this we assume that the power and cross-spectra of these sources consists of a number of components that are coherent in different energy bands, but are incoherent with one another. The technique is based on fitting simultaneously the power spectrum (PS) and the Real and Imaginary parts of the cross-spectrum (CS) with a combination of Lorentzian functions. We show that, because the PS of LMXBs is insensitive to signals with a large Imaginary part and a small Real part in the CS, this approach allows us to uncover new variability components that are only detected in the CS. We also demonstrate that, contrary to earlier claims, the frequency of the type-C quasi-periodic oscillation (QPO) in the black-hole binary GRS 1915+105 does not depend on energy. Rather, the apparent energy dependence of the QPO frequency can be explained by the presence of a separate QPO component with a slightly higher frequency than that of the QPO, whose rms amplitude increases faster with energy than the rms amplitude of the QPO. From all the above we conclude that, as in the case of the PS, the CS of black-hole and neutron-star binaries can be fitted by a combination of Lorentzian components. Our findings provide evidence that the frequency-dependent part of the transfer function of these systems can be described by a combination of responses, each of them acting over relatively well-defined time-scales. This conclusion challenges models that assume that the main contribution to the lags comes from a global, broadband, transfer function of the accreting system.

A systematic study of the high-frequency bump in the black-hole low-mass X-ray binary GX 339 − 4

ABSTRACT The high-frequency bump, characterized by a frequency exceeding ∼30 Hz, represents a seldom-explored time-variability feature in the power density spectrum (PDS) of black-hole X-ray binaries. In the 2002, 2004, 2007, and 2010 outbursts of GX 339 − 4, the bump has been occasionally observed in conjunction with type-C quasi-periodic oscillations (QPOs). We systematically study the properties of the bump during these four outbursts observed by Rossi X-ray Timing Explorer (RXTE) in the 2–60 keV bands and detect the bump in 39 observations. While the frequencies of the type-C QPOs are in the range of ∼0.1–9 Hz, the root-mean-square (rms) amplitude of the bump shows an evolution in the hardness ratio versus the type-C QPO frequency plot. By comparing the rms amplitude of the bump with the corona temperature and simultaneous radio jet flux of the source, as previously studied in GRS 1915 + 105, we establish that in the hard state of GX 339 − 4, the bump is always strong, with the measurements of the rms amplitude in the range of 4–10 per cent. At the same time, the corona temperature is high and the radio flux is low. These findings indicate that, using the bump as a proxy, the majority of the accretion energy is directed towards the hot corona rather than being channeled into the radio jet. We discuss this phenomenon in terms of an inefficient energy transfer mechanism between the corona and jet in GX 339 − 4.

Fast-varying time lags in the quasi-periodic oscillation in GRS 1915 + 105

ABSTRACT The properties of subsecond time variability of the X-ray emission of the black hole binary GRS 1915 + 105 are very complex and strictly connected to its patterns of variability observed on long time-scales. A key aspect for determining the geometry of the accretion flow is the study of time lags between emission at different energies, as they are associated to key time-scales of the system. In particular, it is important to examine the lags associated to the strong low-frequency quasi-periodic oscillations (QPOs), as the QPOs provide unambiguous special frequencies to sample the variability. We have analysed data from an observation with the AstroSat satellite, in which the frequency of the low-frequency QPO varies smoothly between 2.5 and 6.6 Hz on a time-scale of ∼10 h. The derived phase lags show the same properties and evolution of those observed on time-scales of a few hundred days, indicating that changes in the system geometry can take place on times below one day. We fit selected energy spectra of the source and rms and phase-lag spectra of the QPO with a time-variable Comptonization model, as done previously to RossiXTE data of the same source, and find that indeed the derived parameters match those obtained for variations on much longer time-scales.

Gamma-ray burst precursors from tidally resonant neutron star oceans: potential implications for GRB 211211A

ABSTRACT Precursors have been observed seconds to minutes before some short gamma-ray bursts. While the precursor origins remain unknown, one explanation relies on the resonance of neutron star pulsational modes with the tidal forces during the inspiral phase of a compact binary merger. In this paper, we present a model for short gamma-ray burst precursors that relies on tidally resonant neutron star oceans. In this scenario, the onset of tidal resonance in the crust–ocean interface mode ignites the precursor flare, possibly through the interaction between the excited neutron star ocean and the surface magnetic fields. From just the precursor total energy, the time before the main event, and a detected quasi-periodic oscillation frequency, we may constrain the binary parameters and neutron star ocean properties. Our model can immediately distinguish neutron star–black hole mergers from binary neutron star mergers without gravitational wave detection. We apply our model to GRB 211211A, the recently detected long duration short gamma-ray burst with a quasi-periodic precursor, and explore the parameters of this system. The precursor of GRB 211211A is consistent with a tidally resonant neutron star ocean explanation that requires an extreme mass ratio neutron star–black hole merger and a high-mass neutron star. While difficult to reconcile with the main gamma-ray burst and associated kilonova, our results constrain the possible precursor mechanisms in this system. A systematic study of short gamma-ray burst precursors with the model presented here can test precursor origin and probe the possible connection between gamma-ray bursts and neutron star–black hole mergers.

Type-A quasi-periodic oscillation in the black hole transient MAXI J1348−630

ABSTRACT We present a detailed analysis of the spectral and timing characteristics of a 7-Hz type-A quasi-periodic oscillation (QPO) detected in NICER observations of the black hole X-ray binary MAXI J1348−630 during its high-soft state. The QPO is broad and weak, with an integrated fractional rms amplitude of 0.9 per cent in the 0.5–10 keV band. Thanks to the large effective area of NICER, combined with the high flux of the source and a relatively long accumulative exposure time, we construct the first rms and phase-lag spectra for a type-A QPO. Our analysis reveals that the fractional rms amplitude of the QPO increases with energy from below 1 per cent at 1 keV to ∼3 per cent at 6 keV. The shape of the QPO spectrum is similar to that of the Comptonized component, suggesting that the Comptonized region is driving the variability. The phase lags at the QPO frequency are always soft taking the lowest energy as reference. By jointly fitting the time-averaged spectrum of the source and the rms and phase-lag spectra of the QPO with the time-dependent Comptonization model vkompthdk, we find that the radiative properties of the type-A QPO can be explained by a vertically extended Comptonized region with a size of ∼2300 km.

Test Particles and Quasiperiodic Oscillations around Gravitational Aether Black Holes

This paper is devoted to the analysis of the dynamics of test particles in the vicinity of a black hole within the framework of a gravitational aether model. First, we explored the structure of spacetime by analyzing the curvature scalars. Then, we studied particle dynamics around a black hole using the Hamilton–Jacobi equation.The influence of the aether on the effective potential of the radial motion of test particles around the black hole has been investigated. The dependence of the innermost stable circular orbits (ISCO) on the aether parameter has also been investigated. We also considered particle collision near the black hole in the presence of aether, and studied the fundamental frequencies of the orbital motion of the test particles around the black hole in the presence of aether. Further, we applied the obtained results to the analysis of the upper and lower frequencies of twin-peaked quasiperiodic oscillations (QPOs) occurring near black holes. Finally, we use theoretical and numerical results to obtain constraints on model parameters using observation data in QPO.

EOS-dependent millihertz quasi-periodic oscillation in low-mass X-ray binary

ABSTRACT We studied the frequency and critical mass accretion rate of millihertz quasi-periodic oscillations (mHz QPOs) using a one-zone X-ray burst model. The surface gravity is specified by two kinds of equation of states: neutron star (NS) and strange star (SS). The base flux, Qb, is set in the range of 0–2 MeV nucleon−1. It is found that the frequency of mHz QPO is positively correlated to the surface gravity but negatively to the base heating. The helium mass fraction has a significant influence on the oscillation frequency and luminosity. The observed 7–9 mHz QPOs can be either explained by a heavy NS/light SS with a small base flux or a heavy SS with a large base flux. As base flux increases, the critical mass accretion rate for marginally stable burning is found to be lower. Meanwhile, the impact of metallicity on the properties of mHz QPOs was investigated using one-zone model. It shows that both the frequency and critical mass accretion rate decrease as metallicity increases. An accreted NS/SS with a higher base flux and metallicity, combined with a lower surface gravity and helium mass fraction, could be responsible for the observed critical mass accretion rate ($\dot{m}\simeq 0.3\dot{m}_{\rm Edd}$). The accreted fuel would be in stable burning if base flux is over than ∼2 MeV nucleon−1. This finding suggests that the accreting NSs/SSs in low-mass X-ray binaries showing no type I X-ray bursts possibly have a strong base heating.

Bondi-Hoyle-Lyttleton accretion onto a rotating black hole with ultralight scalar hair

We present a numerical study of relativistic Bondi-Hoyle-Lyttleton (BHL) accretion onto an asymptotically flat black hole with synchronized hair. The hair is sourced by an ultralight, complex scalar field, minimally coupled to Einstein's gravity. Our simulations consider a supersonic flow parametrized by the asymptotic values of the fluid quantities and a sample of hairy black holes with different masses, angular momenta, and amount of scalar hair. For all models, steady-state BHL accretion solutions are attained that are characterized by the presence of a shock-cone and a stagnation point downstream. For the models of the sample with the largest component of scalar field, the shock-cone envelops fully the black hole, transitioning into a bow-shock, and the stagnation points move further away downstream. Analytical expressions for the mass accretion rates are obtained after fitting the numerical results, which can be used to analyze black-hole formation scenarios in the presence of ultralight scalar fields. The formation of a shock-cone leads to regions where sound waves can be trapped and resonant oscillations excited. We measure the frequencies of such quasi-periodic oscillations and point out a possible association with quasi-periodic oscillations in the X-ray light curve of Sgr A* and microquasars.

Testing the dynamic origin of Quasi-periodic Oscillations in MAXI J1535−571 and H 1743−322

ABSTRACT We investigate spectro-temporal properties for two black hole X-ray binary sources, MAXI J1535−571 and H 1743−322, during their hard and hard-intermediate states. For MAXI J1535−571, we analyse swift/XRT, NuSTAR, and nicer observations, specifically focusing on the occurrence of type-C Quasi-periodic Oscillations (QPOs). Regarding H 1743−322, we analyse multi-epoch observations of nicer and AstroSat, identifying a type-C QPO with centroid frequency ranging from 0.1–0.6 Hz. In both sources, we fit the spectra with a relativistic truncated disc and a power-law component. In MAXI J1535−571, we also observe an additional relativistically smeared iron-line. Through temporal and spectral analysis, we estimate the QPO centroid frequency and spectral parameters, such as the accretion rate and inner disc radii. We test the origin of type-C QPOs as relativistic precession frequency, and dynamic frequency (i.e. the inverse of the sound crossing time $\frac{r}{c_\mathrm{ s}(r)}$). The dependence of QPO frequency on both the accretion rate and inner disc radii favours the QPO origin as dynamic frequency. We discuss the implications of these results.

Correlations between QPO frequencies and spectral parameters of GRS 1915+105 using AstroSat observations

Abstract In this work, we study the correlation between Quasi-periodic Oscillation (QPO) frequency and the spectral parameters during various X-ray states in the black hole binary GRS 1915+105 which matches well with the predicted relativistic dynamic frequency (i.e. the inverse of the sound crossing time) at the truncated radii. We have used broadband data of LAXPC and SXT instruments onboard AstroSat. Spectral fitting shows that the accretion rate varies from ∼0.1 to ∼5.0 × 1018 gm s−1 and the truncated radius changing from the last stable orbit of an almost maximally spinning black hole, ∼1.2 to ∼19 Gravitational radii. For this wide range, the frequencies of the C-type QPO (2–6 Hz) follow the trend predicted by the relativistic dynamical frequency model and interestingly, the high-frequency QPO at ∼70 Hz also follows the same trend, suggesting they originate from the innermost stable circular orbit with the same mechanism as the more commonly observed C-type QPO. While the qualitative trend is as predicted, there are quantitative deviations between the data and the theory, and the possible reasons for these deviations are discussed.

Quasiperiodic oscillations around hairy black holes in Horndeski gravity

Testing gravity theories and their parameters using observations is an important issue in relativistic astrophysics. In this context, we investigate the motion of test particles and their harmonic oscillations in the spacetime of non-rotating hairy black holes (BHs) in Hordeski gravity, together with astrophysical applications of quasiperiodic oscillations (QPOs). We show possible values of upper and lower frequencies of twin-peak QPOs which may occur in the orbits from innermost stable circular orbits to infinity for various values of the Horndeski parameter q in relativistic precession, warped disk models, and three different sub-models of the epicyclic resonant model. We also study the behaviour of the QPO orbits and their position relative to innermost stable circular orbits (ISCOs) with respect to different values of the parameter q. It is obtained that at a critical value of the Horndeski parameter ISCO radius takes 6M which has been in the pure Schwarzschild case. Finally, we obtain mass constraints of the central BH of microquasars GRS 1915+105 and XTE 1550-564 at the GR limit and the possible value of the Horndeski parameter in the frame of the above-mentioned QPO models. The analysis of orbits of twin peak QPOs with the ratio of upper and lower frequencies 3:2, around the BHs in the frame of relativistic precession (RP) and epicyclic resonance (ER4) QPO models have shown that the orbits locate close to the ISCO. It is obtained that the distance between QPO orbits and ISCO is less than the error of the observations.