X-ray binary system consists of a compact object, usually a neutron star (NS) or a black hole, and an optical companion star. According to the ratio of X-ray luminosity to optical wavelength and the mass of companion star, X-ray binaries fall into three categories [1] - [3] , i.e., high-mass X-ray binaries, intermediate-mass X-ray binaries and low-mass X-ray binaries (LMXBs).

In NS/LMXBs, the companions of NSs are low-mass ( ⊙) stars, such as late-type main-sequence stars, white dwarfs, or subgiant stars with F-G type spectra [4] [5] . Neutron stars accrete material via Roche lobe overflow through the inner Lagranigian point, emitting soft X-ray, with typical luminosity of 10 ³⁶ - 10 ³⁸ ergs/s. Their galactic distribution is concentrated towards the galactic center, which characterizes a relatively old stellar population, with ages in the range of (0.5 - 1.5) × 10 ¹⁰ yrs. They possess relatively weak surface magnetic fields of 10 ⁸ - 10 ⁹ G and short spin periods of a few milliseconds [6] .

Z sources and atoll sources are two classes of NSs in LMXBs [7] [8] , which are characterized by Z-like and atoll-like shapes on X-ray color-color diagrams (CCDs) or hardness-intensity diagrams (HIDs), respectively. Z sources accrete matter at an appreciable fraction of the Eddington critical rate, 0.5 - 1 L_Edd. The Z-like track consists of three branches (see panel (c) in Figure 1), which are referred to as horizontal branch (HB), normal branch (NB) and flaring branch (FB) from top left to bottom right, differing somewhat between sources [7] [9] and showing slow drifts [10] [11] . Atoll sources cover a much wider range in luminosity, from 0.001 L_Edd all the way up to the range of Z sources. At high luminosity, atoll sources describe a curved banana branch in CCD/ HIDs (Figure 1(a) and Figure 1(b)), along which sources move back and forth on timescales of hours to a day or so [7] [12] . The banana branch is further subdivided into the upper banana (UB), the lower banana (LB) and the lower left banana (LLB). The spectra become harder in CCD/HIDs at lower luminosity, and the CCD motion is often much slower here, from days to weeks, which causes isolated patches to form. These patterns are called island state (IS, see Figure 1(a)), and a hardest, lowest luminosity extreme island state (EIS [13] and references therein).

Quasi-periodic oscillations (QPOs) in time series stands out in the power density spectrum (PDS) as a peak with finite width (e.g. Figure 2), whose profile usually can be described by a Lorentzian function

where is the peak frequency (or centroid frequency); w denotes the full width at half maximum (FWHM); and A₀ represents the amplitude of the signal. A QPO signal is characterized by three characteristic quantities,

i.e. centroid frequency, quality factor, and the fractional root-mean-squared (rms)

amplitude. The centroid frequency tells us the position of the signal. While FWHM w that is related to the

coherence time is often reported in terms of quality factor Q, which is a measure of coherence. Con-

ventionally, signals with Q > 2 are called QPOs and those with Q < 2 are peaked noises. The rms, as a function of the source flux, represents a measure of strength, which is proportional to the square root of the integrated

power of its contribution to the PDS, and is often expressed in percent.

Neutron stars in LMXBs show considerable QPO components, and their frequencies display some correlations.

Low-frequency QPOs―Low-frequency QPOs are observed in the range of 5 - 60 Hz, with Q ≥ 2, and amplitudes of 1% - 10% [14] , see Figure 2. They come in different types, which were named horizontal branch oscillations (HBOs), normal branch oscillations (NBOs), and flaring branch oscillations (FBOs), and correlate with the spectral states in CCDs of NS/LMXBs. The frequencies of HBOs come in a characteristic range of about 15 - 55 Hz and display a strong positive correlation with X-ray intensity [15] , for which the magnetospheric beat-frequency model was proposed [16] . In a range of about 5 - 7 Hz, a signal with approximately constant frequency is often observed in the middle of NB, with a similar coherence and strength as HBO. Such NBOs have no obvious strong trend of correlation with the X-ray intensity. FBOs, as an intensity-dependent peak, have been reported in frequency range of about 10 - 20 Hz, which merge smoothly with NBOs. The frequency becomes strongly correlated to X-ray intensity as soon as the source rounds the corner from NB into FB. The peak gets broader as the source moves up FB and finally dissolves into a very broad excess noise component. QPOs belonging to different branches could sometimes occur simultaneously in the same source.

Hectohertz QPOs―The hectohertz (hHz) QPO [17] is a peaked noise, sometimes coherent enough to be called a QPO, with frequency in the range of 100 - 200 Hz (see panel (b) in Figure 3 for example) that is seen in most states of atoll sources and the millisecond pulsar SAX J1808.4-3658 when the break frequency ≥ 1 Hz. They may also occur in Z sources. It stands out from all other NS components by its approximately constant frequency. The hHz QPO usually has rather low quality factor but occasionally peaks up to Q > 2 and has a rms amplitude between 2% and 20%, which becomes weaker and more coherent as the break frequency rises.

Kilohertz QPOs―The fastest variability components in X-ray binaries are the kilohertz (kHz) QPOs, which were firstly detected by the Rossi X-ray Timing Explorer (RXTE) [19] in Scorpius X-1 [20] and 4U 1728-34 [21] , respectively, see [6] [22] for a historical account. Sometimes only one peak can be detected, but in some sources the kHz QPOs occur in pairs (see panel (a) of Figure 3 and Figure 4 for examples). The twin peaks move up and down in frequencies together in the range from 200 Hz to 1300 Hz, in correlation with source states [6] . The peak with higher frequency in PDS is called the upper kHz QPO, and the signal with

lower frequency is assigned to be the lower kHz QPO. Towards the edges of their observed frequency range peaks also occur alone. In all Z sources and in 4U 1728-34 the kHz QPOs are seen down to the lowest inferred accretion rate levels (HB and upper NB) these sources reach. The QPOs always become undetectable at the highest levels. In most atoll sources, QPOs are seen in the part of LLB closest to the islands, i.e. near the lower left corner of UB, and they are often not detected in IS that may be related to low sensitivity at the low count rates there [23] .

Frequency correlations of QPOs in NS/LMXBs―The frequencies between twin kHz QPOs and other components display some correlations [13] , which are consistent in Z and atoll sources. The frequencies of HBOs turn out to correlate well with those of kHz QPOs [18] [26] - [30] , and the same is true for NBO in Scorpius X-1 [20] . A systematical study for 0.1 - 1200 Hz QPOs and broad noise components [31] of non-pulsing NS indicates that the frequencies between lower kHz QPOs and low frequency QPOs follow a tight correlation (e.g. see Figure 5), which can be described by an empirical model that is based on a superposition of multiple Lorentzians [32] . It was proposed that the origin of these correlations is a physical dependence of the frequencies on one another [33] , and a combination of the sonic-point and magnetospheric beat-frequency models can explain these correlations [34] .

The brightest persistent NS/LMXB Scorpius X-1 traces a “Z” track in CCD, whose typical Fourier PDS are composed of noise components (broad structure) and QPOs in most of these branches. There are three distinct types of QPOs, i.e. the NBO at about 6 Hz, the HBO at about 45 Hz with a harmonic at about 90 Hz, and the twin kHz QPOs of 800 - 1000 Hz [18] [20] , see Figure 6.

About 6 Hz NBO in Scorpius X-1, which occurs on the mid- and lower NB, show some correlations with variability of about 45 Hz HBO and the twin kHz QPOs [20] [37] , and therefore imply some kind of coupling between the three types of QPOs [20] [38] .

The NBO frequency remains approximately constant at about 6 Hz, in general. However, the timing properties along the Z track show that the NBO frequency extends to at most about 21 Hz and its position moves

smoothly into the lower part of the FB [38] [39] . This smooth transition indicates that NBO and FBO are physically related to each other [38] [40] [41] . Properties of both twin kHz QPO and 45 Hz HBO depend on the NBO flux. The upper kHz QPO frequency is anti-correlated with the NBO flux, and the lower kHz QPO becomes stronger when the NBO flux is low [42] . Significant HBOs are detected during the NBO phase of high flux, while the HBO disappears at the low flux of NBO phase [37] . The coupling between the properties of kHz QPO and HBOs and the phase of the NBO makes the NBO a unique phenomenon.

The characteristic quantities of NBO display energy dependence on the oscillation photons [43] . The centroid frequency of NBO changes with energy of photons from a negative correlation to a positive one, and the turning points are located at 6 - 8 keV (see left panel of Figure 7). The rms amplitude increases significantly with the photon energy below 13 keV (see middle panel of Figure 7), which means that the strength of the NBO signal increases with the photon energy. However, the increase of rms amplitude with photon energy seems to stop

when the photon energy is higher than around 13 keV, the quality factors have no obvious relation with the photon energy (see the right panel of Figure 7).

The non-monotonic energy dependence of centroid frequency implies that the emission zone of NBO experiences a radial variation during the accretion process. The dependence of rms amplitude on photon energy may indicate that the NBO signals reach the strongest strength at 13 - 20 keV. Higher energy photons are often emitted from the inner region of an accretion disk. Consequently, the most likely region responsible for such physics can be referred to the transition zone between the inner boundary of accretion disk and the magnetosphere, in which the transition of the properties for accretion flow occurs [44] - [46] . NBO occurs at high accretion rates, i.e. near the Eddington rate. The high-mass flux deposits increasingly more matter in the transition zone, leading to expansion and radial scale change of this region. However, the pile of more deposits with greater viscosity can contribute to the radiation enhancement, as well as to the emission of photons with higher energies. As a result, the radial scale change of the transition zone is responsible for the observed non-monotonic energy dependence of the centroid frequency. In addition, because of the limited region of the transition zone, more deposits enhance the viscosity and thus suppress the further monotonically increasing strength of NBO, which explain the energy dependence of rms amplitude at high energy. We therefore suggest that NBO is a type of oscillation in the transition zone. This kind of oscillation may originate in transition layer due to the viscosity of clumps [47] . On the other hand, this oscillation may also carry the information of frequency oscillation modes in the accretion disk and manifest itself as modulation of the accretion rate during the course when disk matter penetrates the transition layer [48] - [50] .

The kHz QPOs was discovered in February 1996, just two months after RXTE was launched, which have the timescales of the same order as the dynamical timescales in the inner region of accretion disk. Therefore, it may open a window that allows us to see down to the very bottoms of the potential wells of NSs in LMXBs, to investigate the motion of clumps of matter orbiting in strong field and of hot spots corotating on NS surfaces, and thereby to map out curved space-time near accreting compact objects, and directly to measure the compactness of NSs and study the dense matter.

NSs are rotating objects, and the rotation of NS has influence on the behavior of accretion flow in its vicinity. On the one hand, the rotational gravitational mass, in analogy with electromagnetism described by Maxwell’s equations, gives rise to Gravitoelectro-magnetism (GEM) [97] - [99] . The GEM of rotating NS contributes to a

gravitational Larmor frequency (is the angular velocity of NS) of the orbiting

plasma, leading to orbital split in vertical direction of disk. On the other hand, the rotation of NS will break the spherical symmetry and produce axisymmetric spacetime. As a result, the plasma obtain a local angular velocity

that makes the orbits of particles deform the original circular orbits. In general, the

motion of accretion flow follows helical trajectories, i.e. open circular orbits at each radius, and spiral in towards the NS. If and only if the gravitational Larmor frequency and angular velocity of accreted plasma satisfy [100]

where n and l are integers with n, l ≥ 1, the vertical split along disk and deformation in the direction of rotation of NS can be harmonious [101] . Consequently, the orbital motion of accreted matter is holonomy invariant, and the physical orbits for accreted particles will be closed. The radii of closure circular orbits are given by

Moving on these closure circular orbits, the accreted material is in a stable state, with first derivative of angular momentum larger than or equal zero, which correspond to a minimum of an effective potential. With a slight perturbation, the test particle will oscillate around the minimum, manifesting as drift of the orbital frequency. If the perturbation is strong enough to transfer sufficient angular momentum outwardly and drive the particles to leave the stable state, the plasma will continue to follow the original helical track and spirals in. In a turbulent and viscous accretion disk, the dissipative processes, i.e., viscosity, collisions of elements, shocks, and so on, are responsible for the perturbation.

From astrophysical accretion point of view, the sudden change of physical environment may give rise to particular phenomenon. In the quasi-quantized structure of inner accretion disk, the motion of accretion flows changes from helical infall to closure circular orbital motion at some preferred radii, which accordingly contribute to special emission. On a closure orbit at radius r, the plasma moves at Keplerian velocity, radiating X-ray

photons with an observed flux modulated at the Keplerian frequency. However, a slight perturbation due to dissipative processes in the turbulent disk will lead to an oscillation around r. As a result, the original Keplerian frequency at r goes with some drift, instead of the exact Keplerian frequency.

Such phenomenon can manifest as some width of frequency in the Fourier PDS, which is referred to as signals of QPOs [2] [13] .

We review the properties and theoretical explanations of QPOs in NS/LMXBs. The fastest X-ray variability, with timescales on the same order as the dynamical timescale of region under strong field and general relativity in accreting low-magnetic-field NSs, comes closer towards the strong-field physics. Consequently, these components, especially the kHz variability, can be the diagnostics of accretion flow in strong magnetic field and general relativity of rotating NSs.

Theoretical framework for the fastest variability has been emerging, which falls into two concerns: 1) the interaction between magnetosphere and accretion flow at the innermost boundary of accretion disk and 2) relativistic effects near the NS. Most interpretations identified the mechanism of upper kHz QPO as the orbital frequency at a preferred radius in the innermost region of disk. With more and more released data, the beat-fre- quency mechanism has been excluded due to a predicted constant frequency separation that conflicts with observations. The disk mode that involves different oscillation modes should have, in nature, similar origin to the relativistic models. The relativistic precession model [33] [77] seems fascinating, but remains questionable. The modulation of X-ray flux at the predicted frequencies, the reasons for variation of with luminosity, and the correlation with burst oscillations are still open questions. The relativistic resonance model [102] based on a purely mathematical scheme, offers exact description of the intrinsic oscillatory properties of QPOs, without developments of detailed physical interpretations. Considering the relatively weak gravity of NSs but strong magnetic field and fast rotation, we investigate the rotational effects of NSs in the framework of general relativity, which produce some closure circular orbits in the inner region of accretion disk [100] . We expect that the radial oscillation of plasma, with different fixed frequencies on closure orbits, e.g. epicyclical frequencies, may result in resonance or certain frequency correlations.

The photons from different regions of accretion disk carry different energies and present distinct physical properties. Therefore, the QPOs detected in different energy band may have varying characteristics, which can be an entry to investigate the nature of QPOs. It was found that about 6 Hz NBO in Scorpius X-1 displayed some photon energy dependence [43] . However, to study the energy dependence of twin kHz QPOs, we require high quality data with both good time resolution and enough spectral information in future observations.

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 161gpy49) at Sun Yat-sen University.

Jing Wang, (2016) A Brief Review for Quasi-Periodic Oscillations in Neutron-Star Low-Mass X-Ray Binaries. International Journal of Astronomy and Astrophysics,06,82-98. doi: 10.4236/ijaa.2016.61006