Study on Correlations between the Twin Kilohertz Quasi-periodic
  Oscillations in Low-mass X-ray Binaries
Study on the Correlations between the Twin
Kilohertz Quasi-periodic Oscillations in
Low-mass X-ray Binaries
H. X. Yin1∗, Y. H. Zhao1
1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China
Email:yhx@lamost.org
Abstract
The recently updated data of the twin kilohertz quasi-periodic oscillations (kHz
QPOs) in the neutron star low-mass X-ray binaries are analyzed. The power-law
fitting ν1 = a(ν2/1000)
b and linear fitting ν2 = Aν1 + B are applied, individually,
to the data points of four Z sources (GX 17+2, GX 340+0, GX 5-1 and Sco X-1)
and four Atoll sources (4U 0614+09, 4U 1608-52, 4U 1636-53 and 4U 1728-34). The
χ2-tests show that the power-law correlation and linear correlation both can fit data
well. Moreover, the comparisons between the data and the theoretical models for
kHz QPOs are discussed.
Key words: QPOs; Accretion disks; Neutron stars; X-rays binaries.
1 Introduction
The kilohertz quasi-periodic oscillations (kHz QPOs) were firstly discov-
ered in Sco X-1, a luminous Z source in neutron star (NS) low-mass X-ray
binaries (LMXBs) (e.g. van der Klis et al. 1996), and now they have been
detected in twenty more sources (e.g. van der Klis 2000, 2006, for reviews).
Usually, these kHz QPOs appear in pairs, the upper kHz QPO frequency (ν2,
hereafter the upper-frequency) and the lower kHz QPO frequency (ν1, here-
after the lower-frequency), which are discovered in three classes of sources,
i.e. accretion powered millisecond pulsars, bright Z sources and less luminous
Atoll sources (e.g., Hasinger & van der Klis 1989).
The kHz QPO peak separation, ∆ν = ν2 − ν1, in a given source generally
decreases with frequency, except the recently detected kHz QPOs in Cir X-
1, in which the peak separation increases with frequency (Boutloukos et al.
Accepted for publication in Advances of Space Research
http://arxiv.org/abs/0704.0689v1
2006). In addition, the variable peak separations are not equal to the NS spin
frequencies. However, the averaged peak separation is found to be either close
to the spin frequency or to half of it (e.g., van der Klis 2006; Linares et al.
2005).
The above observations offer strong evidence against the simple beat-
frequency model, in which the lower-frequency is the beat between the upper-
frequency ν2 and the NS spin frequency νs (e.g. Strohmayer et al. 1996; Zhang
et al. 1997; Miller et al. 1998), i.e. ν1 = ν2 − νs. Furthermore, with the dis-
covery of pairs of 30–450 Hz QPOs from a few black-hole candidates with
the frequency ratios 3:2 (e.g., van der Klis 2006), Abramowicz et al. (2003)
reported that the ratios of twin kHz QPOs in Sco X-1 tend to cluster around
a value about 3:2, and they argued this fact to be a promising link with the
black hole high-frequency QPOs (e.g. van der Klis 2006).
For the all Z and Atoll sources, the data plots of the upper-frequency
versus the lower-frequency can be fitted by a power law function (e.g., Zhang
et al. 2006a), and also roughly fitted by a linear function (Belloni et al. 2005).
However, for the individual kHz QPO source, for instance Sco X-1, its kHz
QPOs can be well fitted by a power law function (e.g. Psaltis et al. 1998; Yin
et al. 2005).
In this paper, to investigate the twin kHz QPO correlation for the individ-
ual Z or Atoll source, we fitted the data with a power-law and a linear function
for four typical Z sources and four typical Atoll sources, and a comparison of
both fittings by χ2-tests is discussed in section 2, where comparisons with the
models are discussed. The conclusions and consequences are given in section
2 Correlations between twin kHz QPOs
Until now, twin kHz QPOs have been detected in 21 LMXBs, including 2
accretion powered millisecond X-ray pulsars, 8 Z sources and 11 Atoll sources,
as listed in Tab. 1. In Fig. 1 and Fig. 2, we plotted twin kHz QPO data for
the Z sources and Atoll sources, showing the correlations of ν1 vs. ν2, ∆ν vs.
ν2 and ν2/ν1 vs. ν2, where the power-law and linear fitting lines for the eight
Z and Atoll sources are presented. The results of the fittings and χ2 -tests are
listed in Tab. 2.
2.1 A power law fitting
The power-law function is chosen as
ν1 = a
1000 Hz
Hz (1)
to fit twin kHz QPO data points of all Atoll (Z) sources, as well as 4 individual
Atoll (Z) sources, separately. It is noted that a same function was applied to
the fitting of kHz QPOs of Sco X-1 by Psaltis et al. (1998) with a smaller set
of kHz QPO data points. The fitting results of the normalization coefficient a,
the power-law index b and χ2/d.o.f. for various cases are listed in Tab. 2, which
correspond to the fitting curves as presented in Fig. 1. We find that the power-
law index for the fitting of all Z sources (see Tab. 2) is 1.87, obviously bigger
than that of the fitting for all Atoll sources (1.61). Then, for the individual
case, the power-law index for Z source is generally bigger than that in Atoll
source, except GX 17+2.
2.2 A linear fitting
For the same data sets, the linear fitting function is chosen as,
ν2 = Aν1 +B Hz , (2)
which was exploited by Belloni et al. (2005) to discuss the kHz QPO fitting in
Sco X-1, 4U 1608-52, 4U 1636-53, 4U 1728-34 and 4U 1820-30. By means of the
χ2 -tests, as shown in Tab. 2, we find that the linear fitting concordes with the
data well in some cases, and there is no much systematic difference between
the linear slope parameters of the Atoll sources and those of Z sources.
2.3 Comparison between the power-law and the linear correlation
As a comparison between models and the data, it is remarked that the
relativistic precession model (e.g. Stella & Vietri 1999) and the Alfvén wave
oscillation model (e.g. Zhang 2004; Li & Zhang 2005) both can lead to power-
law relations approximately, and then the beat-frequency model (e.g. Miller et
al. 1998) and the 3:2 resonance model (e.g. Abramowicz et al. 2003, this model
is successfully applied to black hole candidates) predicted the linear relations
between twin kHz QPO frequencies in the lowest approximation (Abramowicz
et al. 2005). In Tab. 2, we can see that the χ2/d.o.f. of the power-law relation
is usually less than the linear one for the same source, except the two Atoll
sources 4U 0614+09 and 4U 1636-53. And a linear function cannot give a
firstly increasing and then decreasing tendency of all Z data as shown in Fig.
2b. But a power-law one would fit it well as shown in Fig. 1b. So, these maybe
mean that a power-law correlation is better than a linear one.
2.4 Testing the constant peak separation ∆ν = 300 Hz
Since the discovery of kHz QPOs, it is known that the peak separation
for Sco X-1 (van der Klis et al 1997; Méndez & van der Klis 2000) is a not
constant, and the same is true for the other Z sources, e.g., GX 17+2 (Homan
et al. 2002) and Cir X-1 (Boutloukos 2006). As for the Atoll sources, the peak
separation of 4U 1728–34 (Migliari, van der Klis, & Fender 2003; Méndez
& van der Klis 1999) is always significantly lower than the burst oscillation
frequency, and the peak separation of 4U 1636–53 (Jonker, Méndez, & van der
Klis 2002b; Méndez, van der Klis, & van Paradijs 1998) is varying between
being lower and higher than half the spin frequency. In addition, 4U 1608-52
( Méndez et al. 1998) and 4U 1735-44 (Ford et al. 1998) are found to share
the varied peak separations.
In Fig. 1b or Fig. 2b, we show that the peak separations in all Z sources
decrease (increase) systematically with the upper frequency if the upper fre-
quency is larger (less) than ∼700 Hz (e.g. van der Klis 2000, 2006; Boutloukos
et al. 2006). But this firstly increasing and then decreasing with frequency is
not clearly found for the kHz QPO data of all Atoll sources, as shown in Fig.
1e and Fig. 2e, which perhaps is on account of the less amount of data in the
low kHz QPO frequencies in Atoll sources. From Fig. 1 and Fig. 2, we find
that the peak separations are scattered in a wide range of frequency for each
source. Therefore, a constant peak separation, i.e. ∆ν = 300 Hz, cannot fit
for these data.
Fig. 3 shows the results of χ2-tests against a general constant peak sep-
aration of the twin kHz QPOs in the 8 individual sources. The minimum
χ2/d.o.f. are all with values >> 1 , which means that any constant peak
separation model cannot fit for these data anymore.
2.5 Testing the constant peak ratio ν2/ν1 = 3/2
From Fig. 1c (Fig. 1f) or Fig. 2c (Fig. 2f), we find that twin frequency
ratios distribute in a wide range from 1.2 to 4.2, with the averaged value 1.73
(1.50) for all known Z (Atoll) data. Obviously, a constant ratio ν2/ν1 = 3/2,
which can be applicable to some black hole QPO sources, is not consistent with
the observed NS/LMXB data. In the ν2/ν1 vs. ν2 plots of Fig. 1 and Fig. 2, the
frequency ratios systematically decrease with the upper-frequency for both Z
and Atoll sources. In detail, the incompatible 3:2 ratio peak distribution has
been also studied by Belloni et al. (2005) in several sources, who showed that
the distribution of QPO frequencies in Sco X-1, 4U 1608–52, 4U 1636–53, 4U
1728–34, and 4U 1820–30 is multi-peaked, with the peaks occurring at the
different ν2/ν1 ratios.
3 Conclusions
In this paper, the updated data sets of twin kHz QPO frequencies simul-
taneously detected in NS LMXBs are analyzed, and the power-law and linear
fittings are studied for the individual Z/Atoll and all Z/Atoll sources, respec-
tively. Our main conclusions are presented as follows. (1)In Fig.1 and Fig.2,
we can notice that a simple constant peak separation model, such as the beat-
frequency model (e.g. Strohmayer et al. 1996; Zhang et al. 1997; Miller et
al. 1998), or a constant peak ratio assumption, as in a naive extrapolation
of the observed resonant frequency ratio from the black hole sources to the
neutron stars (e.g. Abramowicz et al. 2003), cannot fit the observed data.
Namely, any simple constant peak separation and constant peak ratio models
are generally inconsistent with the data. The peak separations in all Z sources
tend to increase (decrease) with the upper-frequency if the upper-frequency
is less (larger) than ∼700 Hz. But this tendency does not appear in all Atoll
sources because of less amount of data at low kHz QPO frequency. Statis-
tically, the twin frequency ratios tend to decrease with the upper-frequency
in both Z and Atoll sources. (2)Our results show that the index of the fitted
power-law relation of Z source is generally bigger than that of Atoll source,
except GX 17+2. On the consideration of model, this different index value in
Z and Atoll sources might be related to the diversity in their luminosity or
magnetic field. However the linear correlations do not show any systematic
differences between Z and Atoll sources. (3)The power-law fitting is somewhat
better than the linear one for most of the sources, because the χ2/d.o.f. value
of the power-law correlation is generally less than that of linear one, and a lin-
ear correlation cannot give the firstly increasing and then decreasing tendency
of peak separations in all Z data. As a comparison with model’s prediction,
we mention the Relativistic Precession model (e.g. Stella & Vietri 1999) and
the Alfvén Wave Oscillation model (Zhang 2004), since both models can give
an approximated power-law correlation between the twin kHz QPOs, however
none of them can distinguish the influences of the luminosity of Z and Atoll
sources.
As a summary, if the future data still support the conclusions obtained
in the paper, they will pose the meaningful constraints on the models for
explaining kHz QPOs.
4 Acknowledgements
We are grateful for T. Belloni, M. Méndez, D. Psaltis and J. Homan for
providing the QPO data, and thank C.M. Zhang for discussions. We highly
appreciate the anonymous reviewers for their helpful comments.
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0 200 400 600 800 1000 1200 1400
 (Hz)
 GX 5-1
 GX 17+2
 GX 340+0
 Sco X-1
 Other Z data
 Sco X-1
 GX 5-1
 GX 340+0
 GX 17+2
 All Z data
 = 300 Hz
0 200 400 600 800 1000 1200 1400
 4U 1728-34
 4U 0614+09
 4U 1608-52
 4U 1636-53
 Other data
 4U 1728-34
 4U 0614+09
 4U 1608-52
 4U 1636-53
 All Atoll data
 =300Hz
 =3/2
 (Hz)
Fig. 1. Plots of a and d ν1 vs. ν2, b and e ∆ν vs. ν2 and c and f ν2/ν1 vs.
ν2 for Z sources and Atoll sources. Power-law fitting lines and the two
reference lines (ν2/ν1 = 3/2, and ∆ν = 300 Hz) are presented also.
0 200 400 600 800 1000 1200 1400
 (Hz)
 GX 5-1
 GX 17+2
 GX 340+0
 Sco X-1
 Other Z data
 Sco X-1
 GX 5-1
 GX 17+2
 GX 340+0
 All Z data
 =300 Hz
 =3/2
0 200 400 600 800 1000 1200 1400
 4U 1728-34
 4U 0614+09
 4U 1608-52
 4U 1636-53
 Other Atoll data
 =3/2
 =300Hz
 4U 1728-34
 4U 0614+09
 4U 1608-52
 4U 1636-53
 All Atoll data
 (Hz)
Fig. 2. Plots of a and d ν1 vs. ν2, b and e ∆ν vs. ν2 and c and f ν2/ν1 vs. ν2
for Z sources and Atoll sources. Linear fitting lines and the two reference
lines (ν2/ν1 = 3/2, and ∆ν = 300 Hz) are presented also.
200 300 400
10000
 GX 5-1
 GX 17+2
 GX 340+0
 Sco X-1
 4U 1728-34
 4U 1636-53
 4U 1608-52
 4U 0614+09
 (Hz)
Fig. 3. χ2−tests for a constant peak separation of the 8 individual sources
list in Tab. 2.
Table 1
List of LMXBs with the simultaneously detected twin kHz QPOs.
Sources ν
(3) ν2/ν
1 References
(Hz) (Hz) (Hz)
Millisecond pulsar (2)
XTE J1807-294 127-360 353-587 179-247 1.51-2.78 1,2,
SAX J1808.4-3658 499 694 195 1.39 3
Z source (8)
Cir X-1 56-226 229-505 173-340 2.23-4.19 4
Sco X-1 544-852 844-1086 223-312 1.26-1.57 B,M,K
GX 340+0 197-565 535-840 275-413 1.49-2.72 B,K,P,5
XTE J1701-462 620 909 289 1.47 6
GX 349+2 712-715 978-985 266-270 1.37-1.38 B,K,7
GX 5-1 156-634 478-880 232-363 1.38-3.06 B,K,P,8
GX 17+2 475-830 759-1078 233-308 1.28-1.60 B,K,P,9
Cyg X-2 532 856.6 324 1.61 B,K,P
Atoll source (11)
4U 0614+09 153-823 449-1162 238-382 1.38-2.93 B,K,P,10,11
4U 1608-52 476-876 802-1099 224-327 1.26-1.69 M,B,K,12
4U 1636-53 644-921 971-1192 217-329 1.24-1.51 B,K,P,13,14
4U 1702-43 722 1055 333 1.46 K,P,15
4U 1705-44 776 1074 298 1.38 B,K,P
4U 1728-34 308-894 582-1183 271-359 1.31-1.89 B,K,P,11,16
KS 1731-260 903 1169 266 1.29 B,K,P
4U 1735-44 640-728 982-1026 296-341 1.41-1.53 B,K,P
4U 1820-30 790 1064 273 1.35 B,K,P
4U 1915-05 224-707 514-1055 290-353 1.49-2.3 B,K,P
XTEJ2123-058 849-871 1110-1140 261-270 1.31-1.31 B,K,P
(1):the range of ν1;
(2): the range of ν2;
(3): the range of ∆ν; (4): the range of
ν2/ν1. K: van der Klis 2000, van der Klis 2006; M: Méndez et al. 1998ab,
Méndez & van der Klis 1999, 2000; B: Belloni et al. 2002, Belloni et al.
2005; P: Psaltis et al. 1999ab. 1: Linares 2005; 2: Zhang et al. 2006b; 3:
Wijnands et al. 2003; 4: Boutloukos et al. 2006; 5: Jonker et al. 2000; 6:
Homan 2006 (personal communication); 7: O’Neill et al. 2002; 8: Jonker
et al. 2002a; 9: Homan et al. 2002; 10: van Straaten et al. 2002; 11: van
Straaten et al. 2000; 12: van Straaten et al. 2003; 13: Di Salvo et al.
2003; 14: Jonker et al. 2002b; 15: Markwardt et al. 1999; 16: Migliari et
al. 2003.
Table 2
List of the results of fittings and χ2-tests.
ν1 = a(ν2/(1000 Hz))
b Hz ν2 = Aν1+B Hz
Source∗
a b χ2/d.o.f. A B χ2/d.o.f.
Z source
Sco X-1 721.95 ± 0.69 1.85 ± 0.01 33.9/87 0.765 ± 0.007 445.84 ± 4.73 54.8/87
GX 340+0 763.85 ±38.03 2.12 ± 0.15 21.3/17 0.884 ± 0.067 374.96 ±24.41 27.2/17
GX 5-1 833.02 ±26.08 2.26 ± 0.10 29.7/27 0.828 ± 0.039 379.54 ±15.10 43.7/27
GX 17+2 723.40 ± 5.53 1.56 ± 0.06 11.2/19 0.906 ± 0.038 341.13 ±24.27 12.0/19
All Z 725.38 ± 2.50 1.87 ± 0.02 205.8/169 0.924 ± 0.012 336.22 ± 6.81 380.1/169
Atoll source
4U 0614+09 673.74 ± 4.77 1.49 ± 0.06 80.1/40 1.002 ± 0.033 320.75 ±20.37 68.4/40
4U 1608-52 717.36 ± 4.89 1.81 ± 0.08 7.8/16 0.781 ± 0.036 436.60 ±25.12 9.1/16
4U 1636-53 685.16 ±15.44 1.72 ± 0.16 10.7/11 0.737 ± 0.064 505.18 ±53.51 10.1/11
4U 1728-34 667.86 ± 5.59 1.51 ± 0.07 20.9/23 0.997 ± 0.046 329.33 ±32.09 25.5/23
All Atoll 683.48 ± 3.01 1.61 ± 0.04 293.3/109 0.912 ± 0.020 371.08 ±13.91 308.7/109
	Introduction
	Correlations between twin kHz QPOs
	A power law fitting
	A linear fitting
	Comparison between the power-law and the linear correlation
	Testing the constant peak separation =300 Hz
	Testing the constant peak ratio 2/1=3/2
	Conclusions
	Acknowledgements
	References
