Statistical analysis for the Q‐factor of twin kHz QPOs

Using the recently published data of twin kHz quasi‐period oscillations (QPOs) in neutron stars of low‐mass X‐ray binaries (LMXBs), we study the different profiles between bright Z sources and less luminous Atoll sources. The quality factors of upper kHz QPOs show a narrow distribution both for Z sources and Atoll sources, which concentrate at 7.98 and 9.75, respectively. The quality factors of lower kHz QPOs show a narrow distribution for Z sources and a broader distribution for Atoll sources, which concentrate at 5.25 and 86.22, respectively. In order to investigate the relation between the quality factor and the peak frequency of kHz QPOs, we fit the data with power‐law, linear, and exponential functions, respectively. There is an obvious trend that the quality factors increase with the peak frequencies both for upper and lower QPOs. The implications of our results are discussed (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The coherence of kilohertz quasi-periodic oscillations in the X-rays from accreting neutron stars

We study in a systematic way the quality factor of the lower and upper kilohertz quasi-periodic oscillations (kHz QPOs) in a sample of low-luminosity neutron star X-ray binaries, showing both QPOs varying over a wide frequency range. The sample includes 4U 1636−536, 4U 1608−522, 4U 1735−44, 4U 1728−34, 4U 1820−303 and 4U 0614+09. We find that all sources except 4U 0614+09 show evidence of a drop in the quality factor of their lower kHz QPOs at high frequency. For 4U 0614+09 only the rising part of the quality factor versus frequency curve has been sampled so far. At the same time, in all sources but 4U 1728−34, the quality factor of the upper kHz QPO increases all the way to the highest detectable frequencies. We show that the high-frequency behaviours of both the lower and the upper kHz QPO quality factors are consistent with what is expected if the drop is produced by the approach of an active oscillating region to the innermost stable circular orbit: the existence of which is a key feature of general relativity in the strong field regime. Within this interpretation, our results imply gravitational masses around 2 M_⊙ for the neutron stars in those systems.     

The correlations between the twin kHz quasi-periodic oscillation frequencies of low-mass X-ray binaries

We analyzed the recently published kHz quasi-period oscillaiton (QPO) data in the neutron star low-mass X-ray binaries (LMXBs), in order to investigate the different correlations of the twin-peak kHz QPOs in bright Z sources and in the less luminous Atoll sources. We find a power-law relation  ν₁∼ν ^b ₂  between the upper and the lower kHz QPOs with different indices: b ≃ 1.5 for the Atoll source 4U 1728-34 and b ≃ 1.9 for the Z source Sco X-1. The implications of our results for the theoretical models for kHz QPOs are discussed.     

On The Low Frequency Quasi Periodic Oscillations Of X-Ray Sources

Based on the interpretation of the twin kilohertz Quasi Periodic Oscillations (kHz QPOs) of X-ray spectra of Low Mass X-Ray Binaries (LMXBs) ascribed to the Keplerian and the periastron precession frequencies at the inner disk respectively, we ascribe the low frequency (0.1–10 Hz) Quasi Periodic Oscillations (LFQPO) and HBO (15–60 Hz QPO for Z sources or Atoll sources) to the periastron precession at some outer disk radius. It is assumed that both radii are correlated by a scaling factor of 0.4. The conclusions obtained include: All QPO frequencies increase with increasing accretion rate. The theoretical relations between HBO (LFQPO) frequency and the kHz QPO frequencies are similar to the measured empirical formula.     

kHz quasi-periodic oscillations in the low-mass X-ray binary 4U 0614+09

We report on a comprehensive analysis of the kilohertz (≥300 Hz) quasi-periodic oscillations (kHz QPOs) detected from the neutron star low-mass X-ray binary 4U 0614+09 with the Rossi X-ray Timing Explorer. With a much larger data set than previously analysed (all archival data from 1996 February up to 2007 October), we first investigate the reality of the 1330 Hz QPO reported by van-Straaten et al. This QPO would be of particular interest since it has the highest frequency reported for any source. A thorough analysis of the same observation fails to confirm the detection. On the other hand, over our extended data set, the highest QPO frequency we measure for the upper kHz QPO is at ∼1224 Hz; a value which is fully consistent with the maximum values observed in similar systems. Secondly, we demonstrate that the frequency dependence of the quality factor  ( Q =ν/Δν)  and amplitude of the lower and upper kHz QPOs follow the systematic trends seen in similar systems. In particular, 4U 0614+09 shows a drop of the quality factor of the lower kHz QPO above ∼700 Hz. If this is due to an approach to the innermost stable circular orbit, it implies a neutron star mass of  ∼1.9 M_⊙  . Finally, when analysing the data over fixed durations, we have found a gap in the frequency distribution of the upper QPO, associated with a local minimum of its amplitude. A similar gap is not present in the distribution of the lower QPO frequencies, suggesting some cautions when interpreting frequency ratio distributions, based on the occurrence of the lower QPO only.     

Evidence of a decrease of kHz quasi-periodic oscillation peak separation towards low frequencies in 4U 1728–34 (GX 354–0)

We have produced the colour–colour diagram of all the observations of 4U 1728–34 available in the Rossi X-ray Timing Explorer public archive (from 1996 to 2002) and found observations filling in a previously reported 'gap' between the island and the banana X-ray states. We have made timing analysis of these gap observations and found, in one observation, two simultaneous kHz quasi-periodic oscillations (QPOs). The timing parameters of these kHz QPOs fit in the overall trend of the source. The 'lower' kHz QPO has a centroid frequency of ∼308 Hz. This is the lowest 'lower' kHz QPO frequency ever observed in 4U 1728–34. The peak frequency separation between the 'upper' and the 'lower' kHz QPO is  Δν= 274 ± 11 Hz  , significantly smaller than the constant value of  Δν∼ 350 Hz  found when the 'lower' kHz QPO frequency is between ∼500 and 800 Hz. This is the first indication in this source for a significant decrease of kHz QPO peak separation towards low frequencies. We compare the result briefly to theoretical models for kHz QPO production.     

Statistical properties of twin kilohertz quasi‐periodic oscillations in neutron star low‐mass X‐ray binaries

We collect the data of twin kilohertz quasi‐periodic oscillations (kHz QPOs) published before 2012 from 26 neutron star (NS) low‐mass X‐ray binary (LMXB) sources, then we analyze the centroid frequency (ν) distribution of twin kHz QPOs (lower frequency ν₁ and upper frequency ν₂) both for Atoll and Z sources. For the data without shift‐and‐add, we find that Atoll and Z sources show different distributions of ν₁, ν₂ and ν₂/ν₁, but the same distribution of Δν (difference of twin kHz QPOs), which indicates that twin kHz QPOs may share the common properties of LXMBs and have the same physical origins. The distribution of Δν is quite different from a constant value, so is ν 2/ν₁ from a constant ratio. The weighted mean values and maxima of ν₁ and ν₂ in Atoll sources are slightly higher than those in Z sources. We also find that shift‐and‐add technique can reconstruct the distributions of ν₁ and Δν. The K‐S test results of ν₁ and Δν between Atoll and Z sources from data with shift‐and‐add are quite different from those without it, and we think that this may be caused by the selection biases of the sample. We also study the properties of the quality factor (Q) and the root‐meansquared (rms) amplitude of 4U 0614+09 with data from the two observational methods, but the errors are too big to make a robust conclusion. The NS spin frequency (ν_s) distribution of 28 NS‐LMXBs show a bigger mean value (∼408 Hz) than that (∼281 Hz) of the radio binary millisecond pulsars (MSPs), which may be due to the lack of the spin detections from Z sources (systematically lower than 281 Hz). Furthermore, on the relations between the kHz QPOs and NS spin frequency ν_s, we find the approximate correlations of the mean values of Δν with NS spin and its half, respectively. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The magnetohydrodynamics model of twin kilohertz quasi-periodic oscillations in low-mass X-ray binaries

We suggest an explanation for the twin kilohertz quasi-periodic oscillations (kHz QPOs) in low-mass X-ray binaries (LMXBs) based on magnetohydrodynamics (MHD) oscillation modes in neutron star magnetospheres. Including the effect of the neutron star spin, we derive several MHD wave modes by solving the dispersion equations, and propose that the coupling of the two resonant MHD modes may lead to the twin kHz QPOs. This model naturally relates the upper, lower kHz QPO frequencies with the spin frequencies of the neutron stars, and can well account for the measured data of six LMXBs.     

The applications of the MHD Alfvén wave oscillation model for kHz quasi‐periodic oscillations

In this paper, we improve the previous work on the MHD Alfvén wave oscillation model for the neutron star (NS) kHz quasi‐periodic oscillations (QPOs), and compare the model with the updated twin kHz QPO data. For the 17 NS X‐ray sources with the simultaneously detected twin kHz QPO frequencies, the stellar mass M and radius R constraints are given by means of the derived parameter A in the model, which is associated with the averaged mass density of the star as 〈ρ 〉 = 3M /(4πR³) ≃ 2.4 × 10¹⁴ (A /0.7)² g/cm³, and we also compare the M ‐R constraints with the stellar equations of state. Moreover, we also discuss the theoretical maximum kHz QPO frequency and maximum twin peak separation, and some expectations on SAX J1808.4–3658 are mentioned, such as its highest kHz QPO frequency ∼ 870 Hz, which is about 1.4–1.5 times less than those of the other known kHz QPO sources. The estimated magnetic fields for both Z sources (about Eddington accretion rate ) and Atoll sources (∼ 1% ) are approximately ∼10⁹ G and ∼10⁸ G, respectively. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Is there a link between the neutron-star spin and the frequency of the kilohertz quasi-periodic oscillations?

There is a general consensus that the frequencies of the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron-star low-mass X-ray binaries are directly linked to the spin of the neutron star. The root of this idea is the apparent clustering of the ratio of the frequency difference of the kHz QPOs, and the neutron-star spin frequency,  Δν/ν_s  , at around 0.5 and 1 in 10 systems for which these two quantities have been measured. Here, we re-examine all available data of sources for which there exist measurements of two simultaneous kHz QPOs and spin frequencies, and we advance the possibility that Δν and  ν_s  are not related to each other. We discuss ways in which this possibility could be tested with current and future observations.     

Sidebands to the lower kilohertz quasi-periodic oscillation in 4U 1636–53

In this paper we report on further observations of the third and fourth kilohertz quasi-periodic oscillations (QPOs) in the power spectrum of the low-mass X-ray binary (LMXB) 4U 1636−53. These kilohertz QPOs are sidebands to the lower kilohertz QPO. The upper sideband has a frequency  55.5 ± 1.7 Hz  larger than that of the contemporaneously measured lower kilohertz QPO. Such a sideband has now been measured at a significance  >6σ  in the power spectra of three neutron-star LMXBs (4U 1636−53, 1728−34 and 1608−52). We also confirm the presence of a sideband at a frequency ∼55 Hz less than the frequency of the lower kilohertz QPO. The lower sideband is detected at a 3.5σ level only when the lower kilohertz QPO frequency is between 800 and 850 Hz. In that frequency interval, the sidebands are consistent with being symmetric around the lower kilohertz QPO frequency. The upper limit to the rms amplitude of the lower sideband is significantly lower than that of the upper sideband for lower kilohertz QPO frequencies >850 Hz. Symmetric sidebands are unique to 4U 1636−53. This might be explained by the fact that lower kilohertz QPO frequencies as high as 800–850 Hz are rare for 4U 1728−34 and 1608−52. Finally, we also measured a low-frequency QPO at a frequency of ∼43 Hz when the lower kilohertz QPO frequency is between 700 and 850 Hz. A similar low-frequency QPO is present in the power spectra of the other two systems for which a sideband has been observed. We briefly discuss the possibility that the sideband is caused by Lense–Thirring precession.     

Frequency ratio of twin kHz quasi-periodic oscillations: The case of 4U1820-303

By analyzing all archival Rossi X-ray timing explorer (RXTE) data of the neutron star low mass X-ray binary 4U1820-303, we investigate the detectability of simultaneous twin kHz quasi-periodic oscillations (QPOs) as a function of their frequency, width and root mean squared (RMS) amplitudes. In a blind search over the whole data set (spanning over about 10 years), we show that in continuous time intervals (2000–3000 s), twin QPOs are preferentially detected over narrow range of frequencies (100 Hz wide), leading to a clustering in the distribution of frequency ratios, and a gap around the point (600, 900) Hz in the correlation line that links the upper QPO frequency to the lower QPO frequency. A deficit of lower QPOs around 600 Hz had already been noticed by Belloni et al. [Belloni, T., Méndez, M., Homan, J., 2005. A&A 437, 209], it is now confirmed using a much larger data set. We show that the lack of twin QPOs within the gap is not due to a lack of sensitivity for QPO detection, if the parameters of the QPOs (RMS and width) can be interpolated within the gap, using values measured before and after. Since as previously noticed, the gap cannot be attributed to an incomplete sampling of the source states, it thus implies a sudden change of the QPO properties within the gap, either a loss of coherence or an amplitude decrease, or alternatively, that there may be forbidden frequencies in this system.     

Studies of accretion flows around rotating black holes – III. Shock oscillations and an estimation of the spin parameter from QPO frequencies

In the present communication of our series of papers dealing with the accretion flows in the pseudo-Kerr geometry, we discuss the effects of viscosity on the accretion flow around a rotating black hole. We find the solution topologies and give special attention to the solutions containing shocks. We draw the parameter space where standing shocks are possible and where the shocks could be oscillating and could produce quasi-periodic oscillations (QPOs) of X-rays observed from black hole candidates. In this model, the extreme locations of the shocks give the upper limits of the QPO frequencies  (ν_QPO)  which could be observed. We show that both the viscosity of the flow and the spin of the black hole a increase the QPO frequency while, as expected, the black hole mass reduces the QPO frequencies. Our major conclusion is that the highest observed frequency gives a strict lower limit of the spin. For instance, a black hole exhibiting  ν_QPO∼ 400  and  700 Hz  must have the spin parameters of   a > 0.25  and  >0.75  , respectively, provided viscosity of the flow is small. We discuss the implications of our results in the light of observations of QPOs from black hole candidates.     

Are there three peaks in the power spectra of GX 339−4 and Cyg X-1?

Among the variability behaviours exhibited by neutron star systems are the so-called 'horizontal branch oscillations' (HBO, with frequencies ≈50 Hz), the 'lower-frequency kHz quasi-periodic oscillation' (QPO) and the 'upper-frequency kHz QPO', with the latter two features being separated in frequency by an amount comparable to, but varying slightly from, the suspected spin-frequency of the neutron star. Recently, Psaltis, Belloni & van der Klis have suggested that there exists a correlation between these three frequencies that, when certain identifications of variability features are made, even encompasses black hole sources. We consider this hypothesis by reanalysing a set of GX 339−4 observations. The power spectral density (PSD) constructed from a composite of seven separate, but very similar, observations shows evidence for three broad peaks in the PSD. If the peak frequencies of these features are identified with QPO, then their frequencies approximately fit the correlations suggested by Psaltis, Belloni, & van der Klis. We also reanalyse a Cyg X-1 observation and show that the suggested QPO correlation may also hold, but that complications arise when the QPOs (which, in reality, are fairly broad features) are considered as a function of energy band. These fits suggest the existence of at least three separate, independent physical processes in the accretion flow, a hypothesis that is also supported by consideration of the Fourier frequency-dependent time lags and coherence function between variability in different energy bands. If these variability features have a common origin in neutron star and black hole systems, then 'beat frequency models' of kHz QPO in neutron star systems are called into question.     

On the maximum amplitude and coherence of the kilohertz quasi-periodic oscillations in low-mass X-ray binaries

I study the behaviour of the maximum rms fractional amplitude, r _max, and the maximum coherence, Q _max, of the kilohertz quasi-periodic oscillations (kHz QPOs) in a dozen low-mass X-ray binaries. I find that (i) the maximum rms amplitudes of the lower- and upper-kHz QPOs,   r ^ℓ_max  and   r ^u_max  , respectively, decrease more or less exponentially with increasing luminosity of the source; (ii) the maximum coherence of the lower-kHz QPO,   Q ^ℓ_max  , first increases and then decreases exponentially with luminosity, at a faster rate than both   r ^ℓ_max  and   r ^u_max  ; (iii) the maximum coherence of the upper-kHz QPO,   Q ^u_max  , is more or less independent of luminosity; and (iv) r _max and Q _max show the opposite behaviour with hardness of the source, consistent with the fact that there is a general anticorrelation between luminosity and spectral hardness in these sources. Both r _max and Q _max in the sample of sources, and the rms amplitude and coherence of the kHz QPOs in individual sources show a similar behaviour with hardness. This similarity argues against the interpretation that the drop of coherence and rms amplitude of the lower-kHz QPO at high QPO frequencies in individual sources is a signature of the innermost stable circular orbit around a neutron star. I discuss possible interpretations of these results in terms of the modulation mechanisms that may be responsible for the observed variability.     

Rossi XTE monitoring of 4U 1636–53 – I. Long-term evolution and kHz quasi-periodic oscillations

We have monitored the atoll-type neutron star low-mass X-ray binary 4U 1636−53 with the Rossi X-ray Timing Explorer ( RXTE ) for more than 1.5 yr. Our campaign consisted of short (∼2 ks) pointings separated by 2 d, regularly monitoring the spectral and timing properties of the source. During the campaign we observed a clear long-term oscillation with a period of ∼30–40 d, already seen in the light curves from the RXTE All-Sky Monitor, which corresponded to regular transitions between the hard (island) and soft (banana) states. We detected kilohertz (kHz) quasi-periodic oscillations (QPOs) in about a third of the observations, most of which were in the soft (banana) state. The distribution of the frequencies of the peak identified as the lower kHz QPO is found to be different from that previously observed in an independent data set. This suggests that the kHz QPOs in the system shows no intrinsically preferred frequency.     

A Study on the Correlations between the Twin kHz QPO Frequencies in Sco X-1

For the bright neutron star low-mass X-ray binary Sco X-1, we analyzed all updated frequencies of the twin kilohertz quasi-periodic oscillations (kHz QPOs), their correlations and distributions. We found that the frequency separation of the kHz QPO peaks appears not to be a constant, rather, it decreases with increasing inferred mass accretion rate. We show that the currently available data of Sco X-1 by Rossi X-ray Timing Explorer are inconsistent with the proposals of the beat model that the frequency separation is a constant. Our conclusions are consistent with those of some previous researchers and we discuss further implications for the kilohertz QPO models.     

The correlations and anticorrelations in QPO data

Double peak kHz QPO frequencies in neutron star sources varies in time by a factor of hundreds Hz while in microquasar sources the frequencies are fixed and located at the line ν ₂ = 1.5ν ₁ in the frequency‐frequency plot. The crucial question in the theory of twin HFQPOs is whether or not those observed in neutron‐star systems are essentially different from those observed in black holes. In black hole systems the twin HFQPOs are known to be in a 3:2 ratio for each source. At first sight, this seems not to be the case for neutron stars. For each individual neutron star, the upper and lower kHz QPO frequencies, ν ₂ and ν ₁, are linearly correlated, ν ₂ = Aν ₁ + B , with the slope A < 1.5, i.e., the frequencies definitely are not in a 1.5 ratio. In this contribution we show that when considered jointly on a frequency‐frequency plot, the data for the twin kHz QPO frequencies in several (as opposed to one) neutron stars uniquely pick out a certain preferred frequency ratio that is equal to 1.5 for the six sources examined so far. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

QPOs in microquasars and Sgr A.: measuring the black hole spin

In all four microquasars which show double peak kHz QPOs, the ratio of the two frequencies is 3:2. This strongly supports the suggestion that twin peak kHz QPOs are due to a resonance between some modes of accretion disk oscillations. Here, we stress that fits to observations of the hypothetical resonances between vertical and radial epicyclic frequencies (particularly of the parametric resonance) give an accurate estimate of the spin for the three microquasars with known mass. Measurement of double peak QPOs frequencies in the Galaxy centre seems also to be consistent with the 3:2 ratio established by previous observations in microquasars, however the SgrA* data are rather difficult for the same exact analysis. If confirmed, the 3:2 ratio of double peak QPOs in SgrA* would be of a fundamental importance for the black hole accretion theory and the precise measurement could help to solve the question of QPOs nature. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of the Coronal Sources of the Fast Solar Wind

Starting from the observation that kilohertz quasi-periodic oscillations (kHz QPOs) occur in a very narrow range of X-ray luminosities in neutron star low-mass X-ray binaries, we try to link the kHz QPO observability to variations of the neutron star magnetospheric radius, in response to changing mass inflow rate. At low luminosities, the drop-off of kHz QPO activity may be explained by the onset of the centrifugal barrier, when the magnetospheric radius reaches the corotation radius. At the opposite side, at higher luminosities, the magnetospheric radius may reach the neutron star and the vanishing of the magnetosphere may lead to the stopping of the kHz QPO activity. If we apply these constraints, the magnetic fields of atoll [B approximately 0.3-1x108 G for Aql X-1] and Z [B approximately 1-8x108 G for Cyg X-2] sources can be derived. These limits naturally apply in the framework of beat-frequency models but can also work in the case of general relativistic models.