kHz QPOs Research Articles

SPECTRAL-TIMING ANALYSIS OF THE LOWER kHz QPO IN THE LOW-MASS X-RAY BINARY AQUILA X-1

ABSTRACT Spectral-timing products of kilohertz quasi-periodic oscillations (kHz QPOs) in low-mass X-ray binary (LMXB) systems, including energy- and frequency-dependent lags, have been analyzed previously in 4U 1608-52, 4U 1636-53, and 4U 1728-34. Here, we study the spectral-timing properties of the lower kHz QPO of the neutron star LMXB Aquila X-1 for the first time. We compute broadband energy lags as well as energy-dependent lags and the covariance spectrum using data from the Rossi X-ray Timing Explorer. We find characteristics similar to those of previously studied systems, including soft lags of ∼30 μs between the 3.0–8.0 keV and 8.0–20.0 keV energy bands at the average QPO frequency. We also find lags that show a nearly monotonic trend with energy, with the highest-energy photons arriving first. The covariance spectrum of the lower kHz QPO is well fit by a thermal Comptonization model, though we find a seed photon temperature higher than that of the mean spectrum, which was also seen in Peille et al. and indicates the possibility of a composite boundary layer emitting region. Lastly, we see in one set of observations an Fe K component in the covariance spectrum at 2.4-σ confidence, which may raise questions about the role of reverberation in the production of lags.

Probing the accretion disc structure by the twin kHz QPOs and spins of neutron stars in LMXBs

We analyze the relation between the emission radii of twin kilohertz quasi-periodic oscillations (kHz QPOs) and the co-rotation radii of the 12 neutron star low mass X-ray binaries (NS-LMXBs) which are simultaneously detected with the twin kHz QPOs and NS spins. We find that the average co-rotation radius of these sources is r_co about 32 km, and all the emission positions of twin kHz QPOs lie inside the corotation radii, indicating that the twin kHz QPOs are formed in the spin-up process. It is noticed that the upper frequency of twin kHz QPOs is higher than NS spin frequency by > 10%, which may account for a critical velocity difference between the Keplerian motion of accretion matter and NS spin that is corresponding to the production of twin kHz QPOs. In addition, we also find that about 83% of twin kHz QPOs cluster around the radius range of 15-20 km, which may be affected by the hard surface or the local strong magnetic field of NS. As a special case, SAX J1808.4-3658 shows the larger emission radii of twin kHz QPOs of r about 21-24 km, which may be due to its low accretion rate or small measured NS mass (< 1.4 solar mass).

Relation between spectral changes and the presence of the lower kHz QPO in the neutron star low-mass X-ray binary 4U 1636−53

We fitted the $3-180$-keV spectrum of all the observations of the neutron-star low-mass X-ray binary 4U 1636$-$53 taken with the {\it Rossi X-ray Timing Explorer} using a model that includes a thermal Comptonisation component. We found that in the low-hard state the power-law index of this component, $\Gamma$, gradually increases as the source moves in the colour-colour diagram. When the source undergoes a transition from the hard to the soft state $\Gamma$ drops abruptly; once the source is in the soft state $\Gamma$ increases again and then decreases gradually as the source spectrum softens further. The changes in $\Gamma$, together with changes of the electron temperature, reflect changes of the optical depth in the corona. The lower kilohertz quasi-periodic oscillation (kHz QPO) in this source appears only in observations during the transition from the hard to the soft state, when the optical depth of the corona is high and changes depends strongly upon the position of the source in the colour-colour diagram. Our results are consistent with a scenario in which the lower kHz QPO reflects a global mode in the system that results from the resonance between, the disc and/or the neutron-star surface, and the Comptonising corona.

A NEW CORRELATION WITH LOWER KILOHERTZ QUASI-PERIODIC OSCILLATION FREQUENCY IN THE ENSEMBLE OF LOW-MASS X-RAY BINARIES

ABSTRACT We study the dependence of kilohertz quasi-periodic oscillation (kHz QPO) frequency on accretion-related parameters in the ensemble of neutron-star low-mass X-ray binaries. Based on the mass accretion rate, , and the magnetic field strength, B, on the surface of the neutron star, we find a correlation between the lower kHz QPO frequency and . The correlation holds in the current ensemble of Z and atoll sources and therefore can explain the lack of correlation between the kHz QPO frequency and X-ray luminosity in the same ensemble. The average run of lower kHz QPO frequencies throughout the correlation can be described by a power-law fit to source data. The simple power law, however, cannot describe the frequency distribution in an individual source. The model function fit to frequency data, on the other hand, can account for the observed distribution of lower kHz QPO frequencies in the case of individual sources as well as the ensemble of sources. The model function depends on the basic length scales, such as the magnetospheric radius and the radial width of the boundary region, both of which are expected to vary with to determine the QPO frequencies. In addition to modifying the length scales, and hence the QPO frequencies, the variation in , being sufficiently large, may also lead to distinct accretion regimes, which would be characterized by Z and atoll phases.

ARE THE kHz QPO LAGS IN NEUTRON STAR 4U 1608–52 DUE TO REVERBERATION?

X-ray reverberation lags have recently been discovered in both active galactic nuclei (AGN) and black hole X-ray binaries. A recent study of the neutron star low-mass X-ray binary 4U 1608-52 has also shown significant lags, whose properties hint at a reverberation origin. Here, we adapt general relativistic ray tracing impulse response functions used to model X-ray reverberation in AGN for neutron star low-mass X-ray binaries. Assuming relativistic reflection forms the broad iron line and associated reflection continuum, we use reflection fits to the energy spectrum along with the impulse response functions to calculate the expected lags as a function of energy over the range of observed kHz QPO frequencies in 4U 1608-52. We find that the lag energy spectrum is expected to increase with increasing energy above 8 keV, while the observed lags in 4U 1608-52 show the opposite behavior. This demonstrates that the lags in the lower kHz QPO of 4U 1608-52 are not solely due to reverberation. We do note, however, that the models appear to be more consistent with the much flatter lag energy spectrum observed in the upper kHz QPO of several neutron star low-mass X-ray binaries, suggesting that lower and upper kHz QPOs may have different origins.

A Monte Carlo estimate for the fraction of thermal Comptonized photons that impinge back on the soft source in neutron star LMXBs

In earlier works, it was shown that the energy dependent soft time lags observed in kHz QPOs of neutron star low mass X-ray binaries (LMXBs) can be explained as being due to Comptonization lags provided a significant fraction ($\eta \sim 0.2$ - $0.8$) of the Comptonized photons impinge back into the soft photon source. Here we use a Monte Carlo scheme to verify if such a fraction is viable or not. In particular we consider three different Comptonizing medium geometries: (i) a spherical shell, (ii) a boundary layer like torus and (iii) a corona on top of an accretion disk. Two set of spectral parameters corresponding to the 'hot' and 'cold' seed photon models were explored. The general result of the study is that for a wide range of sizes, the fraction lies within $\eta \sim 0.3$ - $0.7$, and hence compatible with the range required to explain the soft time lags. Since there is a large uncertainty in the range, we cannot concretely rule out any of the geometries or spectral models, but the analysis suggests that a boundary layer type geometry with a 'cold' seed spectral model is favoured over an accretion corona model. Better quality data will allow one to constrain the geometry more rigorously. Our results emphasise that there is significant heating of the soft photon source by the Comptonized photons and hence this effect needs to be taken into account for any detailed study of these sources.

Constraining the size of the Comptonizing medium by modelling the energy-dependent time lags of kHz QPOs of neutron star system

In earlier works, we had shown that the observed soft lags and r.m.s versus energy of the lower kHz QPO of neutron star binaries can be explained in the framework of a thermal Comptonization model. It was also shown that such an interpretation can provide estimates of the size and geometry of the Comptonizing medium. Here we study the dependence of these estimates on the time-averaged spectral model assumed and on the frequency of the QPO. We use the high quality time lag and r.m.s obtained during March 3rd 1996 observation of 4U 1608-52 by RXTE as well as other observations of the source at different QPO frequencies where a single time-lag between two broad energy bands have been reported. We compare the results obtained when assuming that the time-averaged spectra are represented by the spectrally degenerate "hot" and "cold" seed photon spectral models. We find that for the "hot" seed photon model the medium size is in the range of 0.3-2.0 kms and the size decreases with increasing QPO frequency. On the other hand for the "cold" seed photon model the range for the sizes are much larger 0.5-20 kms and hence perhaps show no variation with QPO frequency. Our results emphasis the need for broad band spectral information combined with high frequency timing to lift this degeneracy. We further show that the r.m.s as a function of energy for the upper kHz QPO is similar to the lower one and indeed we find that the driver for this QPO should be temperature variations of the corona identical to the lower kHz QPO. However, the time lag reported for the upper kHz QPO is hard, which if confirmed, would challenge the simple Comptonization model presented here. It would perhaps imply that reverberation lags are also important and/or the dominate spectral component is not a single temperature medium but a multi-temperature complex one.

Application of the relativistic precession model to the accreting milisecond X‐ray pulsar IGR J17511–3057

AbstractThe observation of a pair of simultaneous twin kHz QPOs in the power density spectrum of a neutron star or a black hole allows its mass‐angular‐momentum relation to be constrained. Situations in which the observed simultaneous pairs are more than one allow the different models of the kHz QPOs to be falsified. Discrepancy between the estimates coming from the different pairs would call the used model into question. In the current paper, the relativistic precession model is applied to the twin kHz QPOs that appear in the light curves of three groups of observations of the accreting millisecond X‐ray pulsar IGR J17511–3057. It was found that the predictions of one of the groups are practically in conflict with the other two. Another interesting result is that the region in which the kHz QPOs have been born is rather broad and extends quite far from the ISCO. (© 2016 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Investigation of the emission radii of kHz QPOs for the accreting millisecond X-Ray pulsars, Atoll and Z sources

We infer the emission positions of twin kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low mass X-ray binaries (NS-LMXBs) based on the Alfven wave oscillation model (AWOM). For most sources, the emission radii of kHz QPOs cluster around a region of 16-19 km with the assumed NS radii of 15 km. Cir X-1 has the larger emission radii of 23-38 km than those of the other sources, which may be ascribed to its large magnetosphere-disk radius or strong NS surface magnetic field. SAX J1808.4-3658 is also a particular source with the relative large emission radii of kHz QPOs of 20 - 23 km, which may be due to its large inferred NS radius of 18 - 19 km. The emission radii of kHz QPOs for all the sources are larger than the NS radii, and the possible explanations of which are presented. The similarity of the emission radii of kHz QPOs (16-19 km) for both the low/high luminosity Atoll/Z sources is found, which indicates that both sources share the similar magnetosphere- disk radii.

THE SPECTRAL-TIMING PROPERTIES OF UPPER AND LOWER kHz QPOs

Soft lags from the emission of the lower kilohertz quasi-periodic oscillations (kHz QPOs) of neutron star low mass X-ray binaries have been reported from 4U1608-522 and 4U1636-536. Those lags hold prospects for constraining the origin of the QPO emission. In this paper, we investigate the spectral-timing properties of both the lower and upper kHz QPOs from the neutron star binary 4U1728-34, using the entire Rossi X-ray Timing Explorer archive on this source. We show that the lag-energy spectra of the two QPOs are systematically different: while the lower kHz QPO shows soft lags, the upper kHz QPO shows either a flat lag-energy spectrum or hard variations lagging softer variations. This suggests two different QPO-generation mechanisms. We also performed the first spectral deconvolution of the covariance spectra of both kHz QPOs. The QPO spectra are consistent with Comptonized blackbody emission, similar to the one found in the time-averaged spectrum, but with a higher seed-photon temperature, suggesting that a more compact inner region of the Comptonization layer (boundary/spreading layer, corona) is responsible for the QPO emission. Considering our results together with other recent findings, this leads us to the hypothesis that the lower kHz QPO signal is generated by coherent oscillations of the compact boundary layer region itself. The upper kHz QPO signal may then be linked to less-coherent accretion-rate variations produced in the inner accretion disk, being detected when they reach the boundary layer.

THE APERIODIC X-RAY VARIABILITY OF THE ACCRETING MILLISECOND PULSAR SAX J1808.4−3658

We have studied the aperiodic variability of the 401 Hz accreting millisecond X-ray pulsar SAX J1808.4-3658 using the complete data set collected with the Rossi X-ray Timing Explorer over 14 years of observation. The source shows a number of exceptional aperiodic timing phenomena that are observed against a backdrop of timing properties that show consistent trends in all five observed outbursts and closely resemble those of other atoll sources. We performed a detailed study of the enigmatic ~410 Hz QPO, which has only been observed in SAX J1808.4-3658. We find that it appears only when the upper kHz QPO frequency is less than the 401 Hz spin frequency. The difference between the ~410 Hz QPO frequency and the spin frequency follows a similar frequency correlation as the low frequency power spectral components, suggesting that the ~410 Hz QPO is a retrograde beat against the spin frequency of a rotational phenomenon in the 9 Hz range. Comparing this 9 Hz beat feature with the Low-Frequency QPO in SAX J1808.4-3658 and other neutron star sources, we conclude that these two might be part of the same basic phenomenon. We suggest that they might be caused by retrograde precession due to a combination of relativistic, classical and magnetic torques. Additionally we present two new measurements of the lower kHz QPO, allowing us, for the first time, to measure the frequency evolution of the twin kHz QPOs in this source. The twin kHz QPOs are seen to move together over 150 Hz, maintaining a centroid frequency separation of $(0.446 \pm 0.009) \nu_{spin}$.

LOW-FREQUENCY QPOS AND POSSIBLE CHANGES IN THE ACCRETION GEOMETRY DURING THE OUTBURSTS OF AQUILA X−1

We have studied the evolution of the Low Frequency Quasi-Periodic Oscillations (LFQPOs) during the rising phase of seven outbursts of the neutron star Soft X-ray Transient (SXT) Aql X$-$1 observed with the \textit{Rossi X-ray Timing Explorer (RXTE)}. The frequency correlation between the low frequency break and the LFQPO sampled on the time scale of $\sim$2 days was seen. Except for the peculiar 2001 outburst, the frequency of the LFQPOs increased with time before the hard-to-soft state transition up to a maximum $\nu_{max}$ at $\sim$31 Hz, a factor of $\sim$5 higher than those seen in black hole transients such as GX 339$-$4, making the maximum QPO frequency a likely indicator of the mass of the central compact object. The characteristic frequencies increased by around ten percent per day in the early rising phase and accelerated to nearly one hundred percent per day since $\sim$2 days before the hard-to-soft state transition. We examined the dependence of the frequency $\nu_{LF}$ on the source flux $f$ and found an anti-correlation between the maximum frequency of the LFQPOs and the corresponding X-ray luminosity of the hard-to-soft transition (or outburst peak luminosity) among the outbursts. We suggest that X-ray evaporation process can not be the only mechanism that drives the variation of the inner disk radius if either of the twin kHz QPO corresponds to the Keplerian frequency at the truncation radius.

THE MAGNETOHYDRODYNAMICAL MODEL OF KILOHERTZ QUASI-PERIODIC OSCILLATIONS IN NEUTRON STAR LOW-MASS X-RAY BINARIES (II)

We study the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low mass X-ray binaries (LMXBs) with a new magnetohydrodynamics (MHD) model, in which the compressed magnetosphere is considered. The previous MHD model (Shi \& Li 2009) is re-examined and the relation between the frequencies of the kHz QPOs and the accretion rate in LMXBs is obtained. Our result agrees with the observations of six sources (4U 0614+09, 4U 1636--53, 4U 1608--52, 4U 1915--15, 4U 1728--34, XTE 1807--294) with measured spins. In this model the kHz QPOs originate from the MHD waves in the compressed magnetosphere. The single kHz QPOs and twin kHz QPOs are produced in two different parts of the accretion disk and the boundary is close to the corotation radius. The lower QPO frequency in a frequency-accretion rate diagram is cut off at low accretion rate and the twin kHz QPOs encounter a top ceiling at high accretion rate due to the restriction of innermost stable circular orbit.

Probing X-ray burst – accretion disk interaction in low mass X-ray binaries through kilohertz quasiperiodic oscillations

The intense radiation flux of Type I X-ray bursts is expected to interact with the accretion flow around neutron stars. High frequency quasiperiodic oscillations (kHz QPOs), observed at frequencies matching orbital frequencies at tens of gravitational radii, offer a unique probe of the innermost disk regions. In this paper, we follow the lower kHz QPOs, in response to Type I X-ray bursts, in two prototypical QPO sources, namely 4U 1636-536 and 4U 1608-522, as observed by the Proportional Counter Array of the Rossi X-ray Timing Explorer. We have selected a sample of 15 bursts for which the kHz QPO frequency can be tracked on timescales commensurable with the burst durations (tens of seconds). We find evidence that the QPOs are affected for over ~200 s during one exceptionally long burst and ~100 s during two others (although at a less significant level), while the burst emission has already decayed to a level that would enable the pre-burst QPO to be detected. On the other hand, for most of our burst-kHz QPO sample, we show that the QPO is detected as soon as the statistics allow and in the best cases, we are able to set an upper limit of ~20 s on the recovery time of the QPO. This diversity of behavior cannot be related to differences in burst peak luminosity. We discuss these results in the framework of recent findings that accretion onto the neutron star may be enhanced during Type I X-ray bursts. The subsequent disk depletion could explain the disappearance of the QPO for ~100 s, as possibly observed in two events. However, alternative scenarios would have to be invoked for explaining the short recovery timescales inferred from most bursts. Heating of the innermost disk regions would be a possibility, although we cannot exclude that the burst does not affect the QPO emission at all. Clearly the combination of fast timing and spectral information of Type I X-ray bursts holds great potential in the study of the dynamics of the inner accretion flow around neutron stars. However, as we show, breakthrough observations will require a timing instrument providing at least ten times the effective area of the RXTE/PCA.

Broad iron emission line and kilohertz quasi-periodic oscillations in the neutron star system 4U 1636–53

Both the broad iron (Fe) line and the frequency of the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low-mass X-ray binaries (LMXBs) can potentially provide independent measures of the inner radius of the accretion disc. We use XMM-Newton and simultaneous Rossi X-ray Timing Explorer observations of the LMXB 4U 1636-53 to test this hypothesis. We study the properties of the Fe-K emission line as a function of the spectral state of the source and the frequency of the kHz QPOs. We find that the inner radius of the accretion disc deduced from the frequency of the upper kHz QPO varies as a function of the position of the source in the colour-colour diagram, in accordance with previous work and with the standard scenario of accretion disc geometry. On the contrary, the inner disc radius deduced from the profile of the Fe line is not correlated with the spectral state of the source. The values of the inner radius inferred from kHz QPOs and Fe lines, in four observations, do not lead to a consistent value of the neutron star mass, regardless of the model used to fit the Fe line. Our results suggest that either the kHz QPO or the standard relativistic Fe line interpretation does not apply for this system. Furthermore, the simultaneous detection of kHz QPOs and broad Fe lines is difficult to reconcile with models in which the broadening of the Fe line is due to the reprocessing of photons in an outflowing wind.

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

AbstractWe 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 (νs) 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 νs, we find the approximate correlations of the mean values of Δν with NS spin and its half, respectively. (© 2014 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Time lags of the kilohertz quasi-periodic oscillations in the low-mass X-ray binaries 4U 1608−52 and 4U 1636−53

(abridged) We studied the energy and frequency dependence of the Fourier time lags and intrinsic coherence of the kHz QPOs in the NS LMXBs 4U 1608-52 and 4U 1636-53 using RXTE data. In both sources we confirmed energy-dependent soft lags of 10-100 \mu s for the lower kHz QPO. We also found that the time lags of the upper kHz QPO are independent of energy and inconsistent with the soft lags of the lower kHz QPO. The intrinsic coherence of the lower kHz QPO remains constant at 0.6 from 5 to 12 keV, and then drops to zero, while for the upper kHz QPO the intrinsic coherence is consistent with zero across the full energy range. The intrinsic coherence of the upper kHz QPO is consistent with zero over the full frequency range of the QPO, except in 4U 1636-53 at ~780 Hz where it increases to 0.13. In 4U 1636-53, for the lower kHz QPO the 4-12 keV photons lag the 12-20 keV ones by 25 \mu s in the QPO frequency range 500-850 Hz, with the lags decreasing to 15 \mu s at higher frequencies. In 4U 1608-52 the soft lags of the lower kHz QPO remain constant at 40 \mu s. In 4U 1636-53, for the upper kHz QPO the 12-20 keV photons lag the 4-12 keV ones by 11 +/- 3 \mu s, independent of QPO frequency; we found consistent results for the time lags of the upper kHz QPO in 4U 1608-52. The intrinsic coherence of the lower kHz QPO increases from ~0-0.4 at 600 Hz to 1 and 0.6 at 800 Hz in 4U 1636-53 and 4U 1608-52, respectively. In 4U 1636-53 it decreases to 0.5 at 920 Hz, while in 4U 1608-52 we do not have data above 800 Hz. We discuss our results in the context of scenarios in which the soft lags are either due to reflection off the accretion disc or up-/down-scattering in a hot medium close to the neutron star. We finally explore the connection between, on one hand the time lags and the intrinsic coherence of the kHz QPOs, and on the other the QPOs' amplitude and quality factor in these two sources.

TESTING THE TWIN kHz QPO MODELS OF NEUTRON STAR

We take the data of twin kilohertz quasi-periodic oscillations (kHz QPOs) in neutron stars (NSs) published before 2011 as the samples, which include 15 Atoll sources and 8 Z sources. Then we test the correlation between the twin peak kHz QPOs, and compare with the samples. We find the power law relation between the lower-frequency (ν1) and the upper-frequency (ν2) of kHz QPOs can fit the data much better than the other models. For the Atoll (Z) sources, the best fitting coefficient and index of power law are a = 687 (720) and b =1.50 (1.83) respectively. The Chi-square per degree of freedom (χ2/d.o.f.) are 1.93 and 1.49 respectively.

STATISTICS OF TWIN kHz QPOs IN NEUTRON STARS

We collect the recently published data of twin kilohertz quasi-periodic oscillations (kHz QPOs) in neutron stars (NSs), and analyze two kinds of samples for five Atoll sources and five Z sources. We study the the relation between the quality factor and the peak frequency of upper frequency. We take three possible models, i.e. linear, power-law and exponential models to fit the data and find that the power-law and the exponential models can reflect the relation better than the linear model.

An Attempt to Describe Frequency-Correlations among kHz QPOs and HBOs by Two-Armed Vertical p-Mode Oscillations: Case of No Magnetic Field

The trapping of two-armed ($ m$$ =$ 2) vertical p-mode oscillations in relativistic disks is examined. The disks are assumed to be isothermal in the vertical direction, but are truncated at a certain height by the presence of corona. The same issues were examined in a previous paper (Kato 2012a). In this paper, unlike the previous paper, however, we do not use the approximation that the oscillations are nearly vertical, but limit to a simpler case of no magnetic field. As in the previous paper, the results suggest that the two basic oscillation modes [both are the fundamental ($ n$$ =$ 1) in the vertical direction, but in the horizontal direction one is the fundamental ($ n_{\rm r}$$ =$ 0) and the other the first overtone ($ n_{\rm r}$$ =$ 1)] correspond to the twin kHz QPOs. Second, the oscillation mode, which is the first overtone ($ n$$ =$ 2) in the vertical direction and the fundamental in the horizontal direction ($ n_{\rm r}$$ =$ 0), will correspond to the horizontal branch oscillation (HBO) of Z-sources. The latter suggests that the horizontal branch of Z-sources is a sequence of temperature change in disks whose vertical thickness is strongly terminated. The temperature increases leftward along the sequence from the apex between the normal and horizontal branches.