A&A 471, 381-384 (2007)
DOI: 10.1051/0004-6361:20077069

The correlations between the spin frequencies and kHz QPOs of neutron stars in LMXBs
(Research Note)

H. X. Yin1 - C. M. Zhang1 - Y. H. Zhao1 - Y. J. Lei2 - J. L. Qu2 - L. M. Song2 - F. Zhang2


1 - National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, PR China
2 - Astronomical Institute, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, PR China

Received 9 January 2007 / Accepted 7 May 2007

Abstract
Aims. We studied the correlations between spin frequencies and kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low-mass X-ray binaries.
Methods. The updated data on kHz QPOs and spin frequencies are statistically analyzed.
Results. We find that when two simultaneous kHz QPOs are present in the power spectrum, the minimum frequency of upper kHz QPO is at least 1.3 times higher than the spin frequency, i.e. $\nu_{\rm s}<\nu_{\rm 2~min}/1.3$. We also find that the average kHz QPO peak separation in 6 Atoll sources anti-correlates with the spin frequency in the form $\langle\Delta\nu\rangle=-(0.19\pm0.05)\mbox{$\nu_{\rm s}$ }+(389.40\pm21.67)$Hz. If we shift this correlation in the direction of the peak separation by a factor of 1.5, this correlation matches the data points of the two accretion-powered millisecond X-ray pulsars, SAX J1808.4-3658 and XTE J1807-294.

Key words: X-rays: binaries - accretion, accretion disks - stars: neutron

1 Introduction

Since the launch of the Rossi X-Ray Timing Explorer (RXTE) ten years ago, kilohertz quasi-periodic oscillations (kHz QPOs) have been detected in about thirty neutron star low-mass X-ray binaries (NS/LMXBs; see van der Klis 2006, for a recent review). The kHz QPOs often occur  in pairs, the upper-frequency ( $\mbox{$\nu_{2}$ }$) and the lower-frequency ( $\mbox{$\nu_{1}$ }$). These kHz QPOs appear in four categories of NS/LMXBs, i.e. the bright Z sources, the less luminous Atoll sources (see Hasinger & van der Klis 1989, for the definition of Atoll and Z classes), accretion-powered millisecond X-ray pulsars (AMXPs), and other unidentified sources (see e.g., van der Klis 2006, and references therein). The kHz QPOs and other observed characteristic frequencies in these sources follow tight correlations among each other (e.g., Psaltis et al. 1998, 1999a,b; Stella et al. 1999; Belloni et al. 2002, 2005, 2007; Zhang et al. 2006a).

A 401 Hz coherent pulsation and a near 401 Hz X-ray burst oscillation frequency are found in SAX J1808.4-3658 (Chakrabarty et al. 2003; Wijnands et al. 2003), suggesting that the burst frequency is equal to the spin frequency ( $\nu _{\rm s}$) in this object (e.g., Strohmayer & Bildsten 2003; Wijnands et al. 2003; Muno 2004). In some sources showing both twin kHz QPOs and spin frequencies, the peak separation ( $\Delta\nu=\mbox{$\nu_{2}$ }-\mbox{$\nu_{1}$ }$) is generally inconsistent with being equal to the spin frequency (e.g. Méndez & van der Klis 1999; Jonker et al. 2002a). But the ratio between the peak separation and spin frequency clusters at around $\sim$1 or $\sim$0.5 (e.g., Wijnands et al. 2003; Wijnands 2005; Linares et al. 2005; Zhang et al. 2006b).

In this research note, we study the relation between kHz QPOs and spin frequencies.

2 Correlations between spins and kHz QPOs

From 35 NS/LMXBs with the kHz QPOs and/or spin frequencies, 21(6) sources show twin (single) kHz QPOs, and 22 sources show spin or/and burst frequencies (7 spin and 17 burst sources; see Table 1).

Table 1: List of LMXBs with the simultaneously detected twin kHz QPOs or spin frequencies.

2.1 Distribution of spin frequencies in LMXBs

In Fig. 1 we plot the distribution of the 22 spin frequencies with an average value of 440.8 Hz. For the 8 sources with both twin kHz QPOs and spin frequencies, the ratio between the minimum upper-frequency and the spin frequency is $\nu_{\rm 2~min}/\nu_{\rm s}>1.3$. If the upper-frequency is interpreted as the Keplerian frequency at the inner edge of the accretion disk (Miller et al. 1998; van der Klis 2000, 2006), this lower limit means that the inner edge of disc penetrates inside the corotation radius where the Keplerian frequency equals the spin frequency. If this applies to the other kHz QPO sources, we can use the relation $\nu_{\rm s}<\nu_{\rm 2~min}/1.3$ to constrain their spin frequencies. For example, we could obtain upper limits of spin frequencies for the 8 Z sources and 5 Atoll sources with simultaneously detected twin kHz QPOs but with unknown spin frequencies (see the inferred upper limits of spin frequencies for these sources in Table 1).


  \begin{figure} \par\includegraphics[width=8.5cm,clip]{7069fig1.eps}\end{figure} Figure 1: Distribution of the spin frequencies of the 22 neutron stars in LMXBs.
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Table 2: List of the sources with spin frequencies and peak separations.

We also notice that, when only a single kHz QPO is detected as in 4U 1608-52 (van Straaten et al. 2003) and 4U 1728-34 (van Straaten et al. 2002), these QPOs do not satisfy the relation $\nu_{\rm s}<\nu_{\rm 2~min}/1.3$. We argue that this relation only holds when two simultaneous kHz QPOs are detected, Our proposal would be ruled out if a pair of kHz QPOs were found and $\nu_{\rm s}>\nu_{2}/1.3$.

2.2 Correlations between $\mbox{$\nu_{\rm s}$ }$ and $\Delta \nu $

The sonic-point beat-frequency model (Miller et al. 1998) predicted a constant $\Delta \nu $ equal to the stellar spin frequency, whereas the sonic-point and spin-resonance model by Lamb & Miller (2003) predicts that the kHz QPO peak separation should be approximately equal to one or half the spin frequency considering that the disk flow at the spin-resonant radius is smooth or clumped. To confirm the above conjecture, we average the value of peak separations and plot $\langle\Delta\nu\rangle- \mbox{$\nu_{\rm s}$ }$ diagram for the six Atoll sources and two AMXPs in Fig. 2. We notice that for the sources (XTE J1807-294, 4U 1702-43, 4U 1728-34, and 4U 1915-05) with $\mbox{$\nu_{\rm s}$ }<400$ Hz $\langle\Delta\nu\rangle/\mbox{$\nu_{\rm s}$ }\sim 1$, whereas for those (SAX J1808.4-3658, 4U 1608-52, 4U 1636-53, and 4U 1731-28) with $\mbox{$\nu_{\rm s}$ }>400$ Hz $\langle\Delta\nu\rangle/\mbox{$\nu_{\rm s}$ }\sim 0.5$. Alternatively, the relation between the averaged peak separation and spin frequency of the six Atoll sources ( $\langle\Delta\nu\rangle$ and $\mbox{$\nu_{\rm s}$ }$) can be fitted by a linear relation $\langle\Delta\nu\rangle=-(0.19\pm0.05)\mbox{$\nu_{\rm s}$ }+(389.40\pm21.67)$ Hz (Fig. 2). The relation that crosses the points of the two AMXPs is the same relation as for the Atoll sources divided by 1.5 along the y direction.

If the above anti-correlation between $\langle\Delta\nu\rangle$ and $\mbox{$\nu_{\rm s}$ }$ is real, we can use it to infer the averaged kHz QPO peak separations of sources, such as EXO 0748-676 ( $\nu_{\rm s}=45$ Hz), XB 1254-690 ( $\nu_{\rm s}=95$ Hz), and XTE J1739-285 ( $\nu_{\rm s}=1122$ Hz) to be around 380 Hz, 370 Hz, and 160 Hz, respectively. However, this anti-correlation is still a conjecture since it is based on data from only six sources. But if this result were confirmed, it means that the spin frequency would play a role in the mechanism that produces the kHz QPOs, but a different one from the one so far proposed. Further measurements of kHz QPOs in the accretion powered millisecond X-ray pulsars are required to uncover the role of the spin of the neutron star in the mechanism that produces the kHz QPOs.


  \begin{figure} \includegraphics[width=8.5cm,clip]{7069fig2.eps}\end{figure} Figure 2: Plot of $\langle\Delta\nu\rangle/\nu_{\rm s}$ vs. $\nu _{\rm s}$. The solid curve stands for $\langle\Delta\nu\rangle=-(0.19\pm0.05)\mbox{$\nu_{\rm s}$ }+(389.40\pm21.67)$ Hz, and the dashed curve is the result of shifting the solid curve down by a factor of 1.5 along the direction of the peak separation.
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3 Conclusion

Our main conclusions are the following.

(1)
We find that for the 8 sources for which twin kHz QPOs and spins are known, the minimum upper-frequency is at least 1.3 times higher than the spin frequency, i.e. $\nu_{\rm 2~min}/\mbox{$\nu_{\rm s}$ }>1.3$. This relation might be used to estimate the spin frequencies of sources with twin kHz QPOs.

(2)
In 6 Atoll sources, the averaged peak separation anti-correlates with the spin frequency as $\langle\Delta\nu\rangle=-(0.19\pm0.05)\mbox{$\nu_{\rm s}$ }+(389.40\pm21.67)$ Hz, although their ratios cluster roughly around either 1 or 0.5 as reported (van der Klis 2006). This correlation would also apply to the two AMXPs (SAX J1808.4-3658 and XTE J1807-294) if in this cases the peak separation is divided by a factor of 1.5 (see Fig. 2). It is noted that this kind of shifting of about 1.5 is required to reconcile the frequency-frequency correlation of the AMXPs and the Atoll and Z sources (see van Straaten et al. 2005; Linares et al. 2005). This factor 1.5 remains unexplained, but it could reflect a different stellar magnetic field strength or magnetic angle between the magnetic polar axis and rotational axis between these types of sources.
If the above correlations between the spin frequency and kHz QPOs were confirmed in the future, it implies that the kHz QPOs are related to the spin frequencies of neutron stars in some manner. Thus, the spin frequency would play a role in the twin kHz QPO production (indirectly perhaps), and any successful model of kHz QPOs should have to take these relations into account. Usually, we consider the production of kHz QPO to be related to the magnetosphere radius defined by the instantaneous accretion rate, then the spin frequency should be involved with the magnetosphere radius defined by the long-term accretion rate (matched with the different magnetic B-field when considering Atoll and Z), which is almost stable in a short observational time. The instantaneous accretion rate varies around the long-term accretion rate, which accounts for the variation of kHz QPOs, thus the "averaged'' QPO variation would be related to the spin frequency. Therefore, in the sense of the average treatment, the kHz QPO has a relation to the spin frequency.

Acknowledgements
We thank T. Belloni, M. Méndez, D. Psaltis, S. Boutloukos, and J. Homan for providing the QPO data. Discussions with J. Petri, P. Rebusco, J. Horák, V. Karas, S. Boutloukos, T.P. Li, and S.N. Zhang are highly appreciated. This research has been supported by the innovative project of CAS of China. We are very grateful for the comments from M. Méndez.

References

 
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