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abstract: 'We present a dimensional analysis of two characteristic time scales in the boundary layer where the disk adjusts to the rotating neutron star (NS). The boundary layer is treated as a transition region between the NS surface and the first Keplerian orbit. The radial transport of the angular momentum in this layer is controlled by a viscous force defined by the Reynolds number, which in turn is related to the mass accretion rate. We show that the observed low- Lorentzian frequency is associated with radial oscillations in the boundary layer, where the observed break frequency is determined by the characteristic diffusion time of the inward motion of the matter in the accretion flow. Predictions of our model regarding relations between those two frequencies and frequencies of kHz QPO’s compare favorably with recent observations for the source 4U 1728-34. This Letter contains a theoretical classification of kHz QPO’s in NS binaries and the related low frequency features. Thus, results concerning the relationship of the low-Lorentzian frequency of viscous oscillations and the break frequency are presented in the framework of our model of kHz QPO’s viewed as Keplerian oscillations in a rotating frame of reference.'
author:
- Lev Titarchuk
- Vladimir Osherovich
title:  Correlations between kHz QPO and Low Frequency Features Attributed to Radial Oscillations and Diffusive Propagation in the Viscous Boundary Layer Around a Neutron Star
---

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Introduction
============

The discovery of kilohertz quasiperiodic oscillations (QPO’s) in the low mass X-ray neutron star (NS) binaries (Strohmayer 1996; Van der Klis 1996 and Zhang 1996) has stimulated both theoretical and observational studies of these sources. In the upper part of the spectrum (400- 1200 Hz) for most of these sources, two frequencies $\nu_k$ and $\nu_h$ have been seen. Initially, the fact that for some sources, the peak separation frequency $\Delta \nu=\nu_h-\nu_k$ does not change much led to the beat frequency interpretation (Strohmayer 1996; Van der Klis 1998) which was presented as a concept for the first time in the paper by Alpar & Shaham (1985). Beat-frequency models, where the peak separation is identified with the NS spin rate have been challenged by observations: for Sco X-1, $\Delta\nu$ varies by 40% (van der Klis 1997 hereafter VK97) and for source 4U 1608-52, $\Delta\nu$ varies by 26% (Mendez 1998). Mounting observational evidence that $\Delta\nu$ is not constant demands a new theoretical approach. For Sco X-1, in the lower part of the spectrum, VK97 identified two branches (presumably the first and second harmonics) with frequencies 45 and 90 Hz which slowly increase in frequency when $\nu_k$ and $\nu_h$ increase. Furthermore, in the spectra observed by Rossi X-ray Timing Explorer (RXTE) for 4U 1728-34, Ford and van der Klis (1998, herein FV98) found low frequency Lorentzian (LFL) oscillations with frequencies between 10 and 50 Hz. These frequencies as well as break frequency, $\nu_{break}$ of the power spectrum density (PSD) for the same source were shown to be correlated with $\nu_k$ and $\nu_h$. It is clear that the low and high parts of the PSD of the kHz QPO sources should be related within the framework of the same theory. Difficulties which the beat frequency model faces are amplified by the requirement of relating the observed low frequency features, described above, with $\nu_k$ and $\nu_h$.

Recently, a different approach to this problem has been suggested: kHz QPO’s in the NS binaries have been modeled by Osherovich & Titarchuk (1999) as Keplerian oscillations in a rotating frame of reference. In this new model the fundamental frequency is the Keplerian frequency $\nu_k$ (the lower frequency of two kHz QPO’s) $$\nu_k={{1}\over{2\pi}}\left({{GM}\over{R^3}}\right)^{1/2},$$ where G is the gravitational constant, M is the NS mass, and R is the radius of the corresponding Keplerian orbit. The high QPO frequency $\nu_h$ is interpreted as the upper hybrid frequency of the Keplerian oscillator under the influence of the Coriolis force $$\nu_h=[\nu_k^2+(\Omega/\pi)^2]^{1/2},$$ where $\Omega$ is the angular rotational frequency of the NS magnetosphere.

For three sources (Sco X-1, 4U 1608-52 and 4U 1702-429), we demonstrated that the solid body rotation ($\Omega=\Omega_0=const$) is a good first order approximation. Slow variation of $\Omega$ as a function of $\nu_k$ within the second order approximation is related to the differential rotation of the magnetosphere controlled by a frozen-in magnetic structure. This model allows us to address the relation between the high and low frequency features in the PSD of the neutron systems. We interpreted the $\sim 45$ and $90$ Hz oscillations as 1st and 2nd harmonics of the lower branch of the Keplerian oscillations in the rotating frame of reference: $$\nu_L=(\Omega/\pi)(\nu_k/\nu_h)\sin\delta,$$ where $\delta$ is the angle between ${\bf \Omega}$ and the vector normal to the plane of the Keplerian oscillations. For Sco X-1, we found that the angle $\delta=5.5^o$ fits the observations.

In this Letter we include the LFL oscillations and related break frequency phenomenon in our classification. We attribute LFL oscillations to radial oscillations in the viscous boundary layer surrounding a neutron star. According to the model of Shakura & Sunyaev (1973, hereafter SS73), the innermost part of the Keplerian disk adjusts itself to the rotating central object (i.e. neutron star). The recent modelling by Titarchuk, Lapidus & Muslimov (1998, hereafter TLM) led to the determination of the characteristic thickness of the viscous boundary layer $L$. In the following section, we present the extension of this work to relate the frequency of the viscous oscillations $\nu_v$ and $\nu_{break}$ with $\nu_k$. Comparison with the observations is carried out for 4U 1728-34. The last section of this Letter contains our theoretical classification of kHz QPO’s and related low frequency phenomena.

Radial Oscillations and Diffusion in the Viscous Boundary Layer
===============================================================

We define the boundary layer as a transition region confined between the NS surface and the first Keplerian orbit. The radial motion in the disk is controlled by the friction and the angular momentum exchange between adjacent layers resulting in the loss of the initial angular momentum by an accreting matter. The corresponding radial transport of the angular momentum in a disk is described by the equation (e.g. SS73): $$\dot M {d\over {dR}}(\omega R^2) =
2\pi {d\over {dR}} (W_{r\varphi}R^2),$$ where $\dot{M}$ is the accretion rate, and $ W_{r\varphi}$ is the component of a viscous stress tensor which is related to the gradient of the rotational frequency $\omega$, namely $$W_{r\varphi}=-2\eta HR{{d\omega}\over{dR}},$$ where $H$ is a half-thickness of a disk, and $\eta$ is the turbulent viscosity. The nondimensional parameter which is essential for equation (4) is the Reynolds number for the accretion flow $$\gamma={{\dot M}\over{4\pi\eta H}}={{3R v_r}\over {{\it v}_t{\it l}_t}},$$ which is the inverse $\alpha-$parameter in the SS73-model; $v_r$ is a characteristic velocity, $v_t$ and $l_t$ are a turbulent velocity and related turbulent scale respectively.

Equations $\rm \omega=\omega_0~{\rm~at}~R=R_0$ (NS radius) and ${\rm \omega=\omega_K~at~R=R_{out}}$ (radius where the boundary layer adjusts to the Keplerian motion), and $\rm {{d\omega}\over{dr}}={{d\omega_k}\over{dr}}~
at~\rm R=R_{out}$ were assumed by TLM as boundary conditions. Thus the profile $\omega(R)$ and the outer radius of the viscous boundary layer $R_{out}$ are uniquely determined by these boundary conditions. Presenting $\omega(R)$ in terms of dimensionless variables: namely angular velocity $\theta=\omega/\omega_0$, radius $ r=R/R_0$ ($ R_0=x_0R_s$, $ R_s=2GM/c^2$ is the Schwarzschild radius), and mass $ m=M/M_{\odot}$, we express Keplerian angular velocity as $$\theta_K={{6}/(a_K r^{3/2}}),$$ where $a_K=m(x_0/3)^{3/2}(\nu_0/363~{\rm Hz})$ and the NS rotational frequency $\nu_0$ has a particular value for each star. The particular coefficient, 6, presented in formula (7) is obtained for the frequency of nearly coherent (burst) oscillations for 4U 1728-34, i.e. for $\nu_0=363$ Hz. The solution of equations (4-5 ) satisfying the above boundary conditions is $$\theta(r)=D_1 r^{-\gamma} + (1-D_1) r^{-2},$$ where $D_1=(\theta_{out}-r_{out}^{-2})/(r_{out}^{-\gamma}
-r_{out}^{-2})$ and $\theta_{out}=\theta_K(r_{out})$. Equation $\theta^{\prime}(r_{out})=\theta_K^{\prime}(r_{out})$ determines $r_{out}$: $${3\over2} \theta_{out}=D_1 \gamma r_{out}^{-\gamma}+2(1-D_1)r_{out}^{-2}.$$

The solution of equations (4-5) subject to the inner sub-Keplerian boundary condition has a regime corresponding to the super-Keplerian rotation (TLM). For such a regime matter piles up in the vertical direction thus disturbing the hydrostatic equilibrium. The vertical component of the gravitational force prevents this matter from further accumulation in a vertical direction and drives relaxation oscillations. The radiation drag force, which is proportional to the vertical velocity, determines the characteristic decay time of the vertical oscillations (TLM). The characteristic time $t_r$, over which the matter moves inward through this region, bounded between the innermost disk and relaxation oscillations zone is $$t_r\sim {L\over v_r},$$ where $L=R_{out}-R_0$ is the characteristic thickness of this region. Even though the specific mechanism providing the modulation of the observed X-ray flux over this timescale needs to be understood, this timescale apparently “controls” the supply of accreting matter into the innermost region of the accretion disk. Any local perturbation in the transition region would propagate diffusively outward over a timescale $$t_{diff}\sim \left({L\over {l_{fp}}}\right)^2 {{l_{fp}}\over v_r},$$ where $l_{fp}$ is the mean free path of a particle.

Note, that the $\gamma-$parameter is proportional to the accretion rate (see Eq. 6), and therefore $v_r\propto \gamma$. Using this relationship, we can exclude $v_r$ from the above equations and get the relations for the corresponding inverse timescales (frequencies). For the frequency of viscous oscillations $$\nu_{v}\propto {\gamma\over {r_{out}-r_0}},$$ and for the break frequency, related to the diffusion $$\nu_{break}\propto{\gamma\over {(r_{out}-r_0)^2}}.$$

In the following section, we compare the predictions of this model with the observations and also establish the theoretical relation between $\nu_v$ and $\nu_{break}$.

Comparisons with Observations
=============================

The results of FV98 for the low frequency Lorentzian in the X-ray binary 4U 1728-34 are presented in Figure 1 and for the break frequency $\nu_{break}$ in Figure 2. In Figure 1, crosses represent the frequencies (with the appropriate error bars) observed during four days. Data collected on February 16 (open circles) are situated apart from the rest of observations and they are not included in the empirical power law fit which is suggested by FV98. In the work discussed above, the authors plotted the observed low frequencies versus high-frequency QPO which for all days, except February 16 was $\nu_k$ and apparently for February 16 it was $\nu_h$. Our theoretical curve for $\nu_v$ versus $\nu_k$ is based on equation (12). The $\chi^2$ dependence on this parameter is rather strong: the parabola $\chi^2=38024-73076\cdot a_k+35732\cdot a_k^2$ has a minimum at $a_k=1.03$, which determines the best fit. Using $\Omega/2\pi=340$ Hz in the upper hybrid relation (2), we calculate $\nu_k$ for the points observed on February 16 and show that they belong to the set of frequencies modeled by our theoretical curve for the viscous radial oscillations (closed circles).

Identification of the observed $\nu_{break}$ with the inverse diffusion time (formulas 11 and 13) is illustrated by theoretical curves in Figure 2. It is worth noting that these two correlations with kHz frequencies are fit by two theoretical curves using [*only one parameter $a_k$*]{}. The $\chi^2-$ dependence on $a_k$ is obtained with inclusion of all data points for the break and low frequency correlations (75 data points). The theoretical dependences of $\nu_k$ and $r_{out}$ on $\gamma-$parameter are calculated numerically using equations (7) and (9) and employed here for calculations of the theoretical curves in Figures 2 and 3 using equations (12) and (13).

We were unable to interpret data for $\nu_{break}$ collected in February 16 (open circles). Neither $\nu_v$ nor $\nu_{break}$ in our theory have a power law relation with $\nu_k$. However, the theoretical relation between $\nu_{break}$ and $\nu_v$, shown in Figure 3 by a solid curve, is close to the straight line (in log-log diagram), suggesting an approximate power law $$\nu_{break}=0.041\nu_v^{1.61}.$$ This relation is derived from the theoretical dependence for the best fit parameter $a_k=1.03$. Observations of FV98 (except February 16) are also presented in the Figure 3.

**4. Discussion and Conclusions**

We present a model for the radial oscillations and diffusion in the viscous boundary layer surrounding the neutron star. Our dimensional analysis has identified the corresponding frequencies $\nu_v$ and $\nu_{break}$ which are consistent with the low Lorentzian and break frequencies for 4U 1728-34. and predicted values for $\nu_{break}$ related to the diffusion in the boundary layer are consistent with the break frequency observed for the same source. Both oscillations (Keplerian and radial) and diffusion in the viscous boundary layer are controlled by the same parameter - Reynolds number $\gamma$ which in turn is related to the accretion rate. It is shown in TLM that $\nu_k$ is a monotonic function of $\gamma$. Therefore, the observed range of $\nu_k$, (350-900 Hz) corresponds to the range $1<\gamma<5$ (or $0.2<\alpha<1$).

The results in this Letter extend the classification of kHz QPO’s and the related low frequency phenomena suggested by Osherovich & Titarchuk 1999. Figure 4 summarizes the new classification. Solid lines represent our theoretical curves and open circles observations for Sco X-1 (from VK97). As one can see, formulas (2) and (3), for the Keplerian oscillator under the influence of the Coriolis force, reproduce the observations well. Indeed, $\Delta\nu=\nu_h-\nu_k$ is not constant, as observed (see OT99 for details of comparisons of the data with the theory). Effectively, the main viscous frequency $\nu_v$ and the diffusive $\nu_{break}$ introduce the second oscillator with two new branches in the lower part of the spectra. The unifying characteristic of spectra for both oscillators is the strong dependence on $\nu_k$. This common dependence on $\nu_k$ can be viewed as a result of the interaction between Keplerian oscillator and the viscous oscillator which share the common boundary at the outer edge of the viscous transition layer.

Our parametric study indicates that the power law index 1.6 in Eq. (14) should be the same for different neutron stars. We expect a similar relation for black holes but with a distinctly different index.

The found value of $a_k$ leads ultimately to independent constraints in the determination of mass and radius for the neutron star (Haberl & Titarchuk 1995).

LT thanks NASA for support under grants NAS-5-32484 and RXTE Guest Observing Program. The authors acknowledge discussions with Alex Muslimov, Jean Swank, Lorella Angelini, Will Zhang, Joe Fainberg and fruitful suggestions by the referee. Particularly, we are grateful to Eric Ford, and Michiel van der Klis, for the data which enable us to make comparisons with the data in detail.

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