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Comparison of the Hartle-Thorne Model of Neutron Stars with its Kerr Approximation by using Resonant Switch Model

Stuchlík Z., Klimovičová K., Schee J., Török G., Kotrlová A.

Acta Astronomica

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2021

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Vol. 71, No. 4

311--141

EN

The Hartle-Thorne (H-T) models of slowly rotating neutron or quark stars, characterized by the mass M, dimensionless spin a, and reduced quadrupole moment q, are constructed for the observationally given rotational frequency frot=580 Hz (290 Hz) of the compact star in the atoll source 4U 1636-53, and a wide range of equations of state (EoS) giving sequences of allowed states governed by the relations a(M), q(M). These sequences are used in the framework of the resonant switch (RS) model combining pairs of geodesic oscillation models to match the data of the twin high-frequency quasi-periodic oscillations observed in the 4U 1636-53 source. The results of the matching procedure using the H-T models are compared to those based on the Kerr approximation of the exterior of the neutron stars. The best H-T matches fix the only variant of the RS model combining particular modifications of the relativistic precession model, exclude the rotation frequency frot=290 Hz, restrict the considered EoS to six of them, excluding the strange quark stars, and significantly improve precision of the matches given by any single geodesic oscillations model. The Kerr matching allows two variants of the RS model, thus, giving false information, and only three EoS, thus, giving insufficient information. Our results demonstrate that in the matching procedure, the Kerr approximation can be used only for neutron stars governed by the H-T models with q<2, implying an important restriction on the applicability of the Kerr approximation for description of the oscillatory phenomena around neutron stars. On the other hand, the RS model is sufficiently discriminating for the spacetime metric to be largely determined by fitting to the data. The ranges of the external spacetime parameters of the neutron stars related to the best H-T matches are determined to be M≈2.10-2.13 Mo, a≈0.21-0.25, q≈1.8-2.3. Most compact neutron star is predicted by the Gandolfi EoS, when M≈2.10 Mo, a≈0.21, q≈1.8, with the equatorial radius R≈10.83 km and eccentricity ε=0.03.

2

Superspinning Quark Stars Limited by Twin High-Frequency Quasiperiodic Oscillations

Stuchlìk Z., Schee J., Šràmkovà E., Török G.

Acta Astronomica

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2017

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Vol. 67, No. 2

181--201

EN

We study properties of Keplerian disks and their high-frequency quasi-periodic oscillations (HF QPOs) in the field of quark stars with dimensionless spin a breaking the black-hole spin limit of a=1 up to a≈1.3. Using the external geometry of the superspinning quark stars approximated by the Kerr geometry, we show that the Keplerian disks have to touch the surface of such quark stars and their accretion efficiency η≈18% significantly exceeds the efficiency related to the Schwarzschild black holes. Using the geodesic oscillation models, we test possible existence of the superspinning quark stars in atoll sources demonstrating the twin HF QPOs with resonant frequency ratios 3:2, 4:3, 5:4. For explanation of the twin HF QPOs we consider the standard relativistic precession model and its modifications, the tidal distortion model, the resonance epicyclic and the warped disk model. In a given model, we assume occurrence of the twin oscillatory modes at a common resonant dimensionless radius x=r/M determined by the frequency ratio and the quark star spin a. The theoretical limit R>3M on the quark star surface radius puts strong restrictions on the relations between the resonant radii x and the quark star spin a. These restrictions imply that all the considered geodesic oscillation models can be excluded, except for one variant of the relativistic precession model, or alternatively the tidal distortion and warped disk models, that allow for appearance of the twin HF QPOs with frequency ratio 3:2 at radii slightly above the theoretical limit on the radius of the quark star surface, but exclude the smaller frequency ratios (4:3, 5:4).

3

Equations of State in the Hartle-Thorne Model of Neutron Stars Selecting Acceptable Variants of the Resonant Switch Model of Twin HF QPOs in the Atoll Source 4U 1636-53

Stuchlìk Z., Urbanec M., Kotrlovà A., Török G., Goluchovà K.

Acta Astronomica

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2015

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Vol. 65, No. 2

169--195

EN

The Resonant Switch (RS) model of twin high-frequency quasi-periodic oscillations (HF QPOs) observed in neutron star binary systems, based on switch of the twin oscillations at a resonant point, has been applied to the atoll source 4U 1636-53 under assumption that the neutron star exterior can be approximated by the Kerr geometry. Strong restrictions of the neutron star parameters M (mass) and a (spin) arise due to fitting the frequency pairs admitted by the RS model to the observed data in the regions related to the resonant points. The most precise variants of the RS model are those combining the relativistic precession frequency relations with their modifications. Here, the neutron star mass and spin estimates given by the RS model are confronted with a variety of equations of state (EoS) governing structure of neutron stars in the framework of the Hartle-Thorne theory of rotating neutron stars applied for the observationally given rotation frequency frot≈580 Hz (or alternatively frot≈290 Hz) of the neutron star in 4U 1636-53. It is shown that only two variants of the RS model based on the Kerr approximation are compatible with two EoS applied in the Hartle-Thorne theory for frot≈580 Hz, while no variant of the RS model is compatible for frot≈290 Hz. The two compatible variants of the RS model are those giving the best fits of the observational data. However, a self-consistency test by fitting the observational data to the RS model with oscillation frequencies governed by the Hartle-Thorne geometry described by three spacetime parameters M,a and (quadrupole moment) q related by the two available EoS puts strong restrictions. The test admits only one variant of the RS model of twin HF QPOs for the Hartle-Thorne theory with the EoS predicting the parameters of the neutron star M≈2.10 M⊙, a≈0.208, and q/a2≈1.77.

4

Test of the Resonant Switch Model by Fitting the Data of Twin-Peak HF QPOs in the Atoll Source 4U 1636-53

Stuchlìk Z., Kotrlovà A., Török G., Goluchovà K.

Acta Astronomica

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2014

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Vol. 64, No. 1

45--64

EN

Resonant Switch (RS) model has recently been proposed as an alternative to the standard models of twin-peak high-frequency quasi-periodic oscillations (HF QPOs) observed in low-mass X-ray binaries containing a neutron star. The model assumes switch of twin oscillations at a resonant point, where frequencies of the upper and lower oscillations νU and νL become commensurable and one pair of the oscillating modes (corresponding to a specific model of HF QPOs) changes to some other pair due to non-linear resonant phenomena. We test the RS model for the atoll source 4U 1636-53, where we assume two resonant points observed at frequency ratios νU:νL=3:2, 5:4, by fitting the pairs of the oscillatory modes to the observed data in the regions related to the resonant points. Among acceptable variants of the RS model the most promising are those combining the relativistic precession (RP) and the total precession (TP) frequency relations or their modifications. The precision of the fits is shown to be strongly increased in comparison to fits realized by individual pairs along the whole data range. We demonstrate that the χ2 test is significantly improved. Fitting of the HF QPO data in the source 4U 1636-53 by the RP1-RP variant of the RS model gives the best results and implies that the neutron star mass and dimensionless spin are M≈2.2 Msun and a≈0.27.

5

Multi-Resonance Orbital Model Applied to High-Frequency Quasi-Periodic Oscillations Observed in Sgr A*

Kotrlová A., Stuchlík Z., Török G.

Acta Astronomica

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2013

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Vol. 63, No. 2

275--292

EN

The multi-resonance orbital model of high-frequency quasi-periodic oscillations (HF QPOs) enables precise determination of the black hole dimensionless spin a if observed set of oscillations demonstrates three (or more) commensurable frequencies. The black hole spin a is related to the frequency ratio only, while its mass M is related to the frequency magnitude. The model is applied to the triple frequency set of HF QPOs observed in Sgr A* source with frequency ratio 3:2:1. Acceptable versions of the multi-resonance model are determined by the restrictions on the Sgr A* supermassive black hole mass. The version of strong resonances related to the black hole "magic" spin a=0.983 is acceptable but the version demonstrating the best agreement with the mass restrictions predicts spin a=0.980.

6

Resonant Switch Model of Twin Peak HF QPOs Applied to the Source 4U 1636-53

Stuchlík Z., Kotrlová A., Török G.

Acta Astronomica

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2012

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Vol. 62, No. 4

389--407

EN

Resonant Switch (RS) model of twin peak high-frequency quasi-periodic oscillations (HF QPOs) assumes switch of twin oscillations at a resonant point where frequencies of the upper and lower oscillations νU and νL become commensurable and the twin oscillations change from one pair of the oscillating modes (corresponding to a specific model of HF QPOs) to some other pair due to non-linear resonant phenomena. The RS model is used to determine range of allowed values of spin a and mass M of the neutron star located in the atoll source 4U 1636-53 where two resonant points are observed at frequency ratios νU:νL=3:2, 5:4. We consider the standard specific models of the twin oscillations based on the orbital and epicyclic geodetical frequencies. The resonant points are determined by the energy switch effect exhibited by the vanishing of the amplitude difference of the upper and lower oscillations. The predicted ranges of the neutron star parameters are strongly dependent on the twin modes applied in the RS model. We demonstrate that for some of the oscillatory modes used in the RS model the predicted parameters of the neutron star are unacceptable. Among acceptable RS models the most promising are those combining the Relativistic Precession and the Total Precession frequency relations or their modifications.

7

On the Origin of Clustering of Frequency Ratios in the Atoll Source 4U 1636-53

Török G., Abramowicz M. A., Bakala P., Bursa M., Horák J., Rebusco P., Stuchlík Z.

Acta Astronomica

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2008

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Vol. 58, No. 2

113--119

EN

A long discussion has been devoted to the issue of clustering of the kHz quasi periodic oscillation (QPO) frequency ratios in neutron star sources. While the distribution of ratios inferred from an occurrence of a single QPO seems to be consistent with a random walk, the distribution based on simultaneous detections of both peaks indicates a preference of ratios of small integers. Based on the public RXTE data we further investigate this issue for the source 4U 1636-53. Quality factors and rms amplitudes of both the QPOs nearly equal at the points where the frequencies are commensurable, and where the twin QPO detections cluster. We discuss a connection of the clustering with the varying properties of the two QPO modes. Assuming approximate relations for the observed correlations of the QPO properties, we attempt to reproduce the frequency and ratio distributions using a simple model of a random-walk evolution along the observed frequency-frequency correlation. We obtain results which are in qualitative agreement with the observed distributions.

8

Modeling the Twin Peak QPO Distribution in the Atoll Source 4U 1636-53

Török G., Bakala P., Stuchlík Z., Čech P.

Acta Astronomica

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2008

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Vol. 58, No. 1

1--14

EN

Relation between the lower and upper frequency mode of the twin peak quasi-periodic oscillations observed in the neutron star X-ray binaries is qualitatively well fitted by the frequency relation following from the relativistic precession model. Assuming this model with no preferred radius and the probability of an observable twin QPO excitation being uniform across the inner edge of an accretion disk we compare the expected and observed twin peak QPO distribution in the case of atoll source 4U 1636-53. We find these two distributions highly incompatible. We argue that the observed distribution roughly corresponds to the expected one if an additional consideration of preferred resonant orbits is included. We notice that our findings are relevant for some disk-oscillation QPO models as well.

9

Distribution of Kilohertz QPO Frequencies and Their Ratios in the Atoll Source 4U 1636-53

Török G., Abramowicz M. A., Bakala P., Bursa M., Horák J., Kluźniak W., Rebusco P., Stuchlík Z.

Acta Astronomica

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2008

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Vol. 58, No. 1

15--21

EN

A recently published study on long term evolution of the frequencies of the kilohertz quasi-periodic oscillations (QPOs) in the atoll source 4U 1636-53 concluded that there is no preferred frequency ratio in a distribution of twin QPOs that was inferred from the distribution of a single frequency alone. However, we find that the distribution of the ratio of actually observed pairs of kHz QPO frequencies is peaked close to the 3/2 value, and possibly also close to the 5/4 ratio. To resolve the apparent contradiction between the two studies, we examine in detail the frequency distributions of the lower kHz QPO and the upper kHz QPO detected in our data set. We demonstrate that for each of the two kHz QPOs (the lower or the upper), the frequency distribution in all detections of a QPO differs from the distribution of frequency of the same QPO in the subset of observations where both the kHz QPOs are detected. We conclude that detections of individual QPOs alone should not be used for calculation of the distribution of the frequency ratios.

10

Black Holes Admitting Strong Resonant Phenomena

Stuchlík Z., Kotrlová A., Török G.

Acta Astronomica

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2008

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Vol. 58, No. 4

441--461

EN

High-frequency twin peak quasiperiodic oscillations (QPOs) are observed in four microquasars, i.e., Galactic black hole binary systems, with frequency ratio very close to 3:2. In the microquasar GRS 1915+105 the structure of QPOs exhibits additional frequencies and more than two frequencies are observed in the Galaxy nuclei Sgr A* or in some extragalactic sources (NGC 4051, MCG-6-30-15 and NGC 5408 X-1). The observed QPOs can be explained by a variety of the orbital resonance model versions assuming resonance of oscillations with the Keplerian frequency νK or the vertical epicyclic frequency νθ, and the radial epicyclic frequency νr, or some combinations of these frequencies. Generally, different resonances could arise at different radii of an accretion disk. However, we have shown that for special values of dimensionless black hole spin a strong resonant phenomena could occur when different resonances can be excited at the same radius, as cooperative phenomena between the resonances may work in such situations. The special values of a are determined for triple frequency ratio sets νK:νθ:νr=s:t:u with s,t,u being small integers. The most promising example of such a special situation arises for black holes with extraordinary resonant spin a=0.983 at the radius r=2.395 M, where νK:νθ:νr=3:2:1. We also predict that when combinations of the orbital frequencies are allowed, QPOs with four frequency ratio set 4:3:2:1 could be observed in the field of black holes with a=0.866,0.882 and 0.962. Assuming the extraordinary resonant spin a=0.983 in Sgr A*, its QPOs with observed frequency ratio ≈3:2:1 imply the black hole mass in the interval 4.3×106 Mʘ< M < 5.4×106 Mʘ, in agreement with estimates given by other, independent, observations.