Distribution of kilohertz QPO frequencies and their ratios in the atoll source 4U 1636-53
Gabriel Torok, Marek A. Abramowicz, Pavel Bakala, Michal Bursa, Jiri Horak, Wlodek Kluzniak, Paula Rebusco, Zdenek Stuchlik
aa r X i v : . [ a s t r o - ph ] A p r Distribution of kilohertz QPO frequencies and theirratios in the atoll source 4U 1636-53
Gabriel T¨or¨ok , Marek A. Abramowicz , , , Pavel Bakala , Michal Bursa , Jiˇr´ı Hor´ak ,Włodek Klu´zniak , , Paola Rebusco , , Zdenˇek Stuchl´ık Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezruˇcovo n´am.13,746-01 Opava, CZ e-mail: [email protected], pavel.bakala@ fpf.slu.cz, [email protected] Department of Physics, G¨oteborg University, S-412 96 G¨oteborg, SE e-mail: [email protected] Copernicus Astronomical Centre PAN, Bartycka 18, 00-716 Warsaw, PL e-mail: [email protected], [email protected] Astronomical Institute of the Academy of Sciences, Boˇcn´ı II 1401 / e-mail: [email protected], [email protected] Johannes Kepler Institute of Astronomy, Zielona Gora University, ul. Lubuska 2, 65-265 Zielona G´ora, PL MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue, 37, Cambridge, MA02139, US e-mail: [email protected] Max-Planck-Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching, D
Abstract.
A recently published study on long term evolution of the frequencies of the kilohertz quasi-periodic oscillations (QPOs) in the atoll source 4U 1636-53 concluded that there is no preferred frequencyratio in a distribution of twin QPOs that was inferred from the distribution of a single frequency alone. How-ever, we find that the distribution of the ratio of actually observed pairs of kHz QPO frequencies is peakedclose to the 3 / / ff ers from the distribution of frequency of the sameQPO in the subset of observations where both the kHz QPOs are detected. We conclude that detections ofindividual QPOs alone should not be used for calculation of the distribution of the frequency ratios. Keywords:
X-rays:binaries — Stars:neutron — Accretion, accretion disks
Klu´zniak & Abramowicz (2000) suggested that the kHz twin peak QPOs, observed in the Fourier powerspectra (PDS) from accreting neutron stars, originate in a non-linear resonance that is possible only in stronggravity. It was reported later (Abramowicz et al. 2003) that the ratio ν L /ν U of the upper and lower QPOfrequency in neutron stars usually clusters close to the rational ratio 2 /
3, with some frequency pairs possiblyclustering close to other ratios, such as 0.78.Belloni et al. (2005) re-examined the study of Abramowicz et al. (2003) for a larger set of detections ofa single kHz QPO and, on the assumption of a correlation between the observed QPO and the unobservedsecond QPO, confirmed that their (inverse) frequency ratio ν U /ν L would cluster most often close to the 3 / / / Left: Frequency histogram for the lower QPO. The subset of lower QPOs detected simultane-ously with the upper QPO is denoted by darker bars, which are labeled “lower in twin”. Right: Analogoushistograms for the upper QPO.A more recent study of Belloni et al. (2007) is based on a long term evolution of the QPO frequencies inthe atoll source 4U 1636 −
53 over an eighteen month period and on the results of their previous research. Theauthors now conclude that in fact there are no peaks in the frequency distribution of the lower kHz QPO inthis source. In keeping with their previous argument, they conclude that there are no peaks in the frequencyratio distribution either.While Abramowicz et al. (2003) examine the frequency ratios in pairs of observed frequencies, bothof the cited papers of Belloni et al. study primarily distributions of the frequencies and focus mainly on thelower QPO. Note that in most observations of Belloni et al. (2007) only single QPO frequencies have beendetected. − Using 4U 1636 −
53 data from the analysis of Barret et al. (2005b), we prepare a histogram of detections ofthe lower QPO over nine years (from 1996 till 2005) of monitoring by
RXTE (Figure 1, left), as well asof the upper QPO (Figure 1, right). We restrict our study to frequencies >
400 Hz for the lower QPO, and >
800 Hz for the upper QPO. The inaccuracies caused by the long-term frequency drift inside of continuousdata segments are not important for the purposes of our paper.The data of Barret et al. (2005b) have been obtained through a shift-add procedure carried out on indi-vidual continuous segments of observation. In this approach each continuous data segment (correspondingto few tens minutes of an e ff ective subset from 1.5 hour RXTE orbital period) is divided into N intervals, andsearched for a QPO. The shortest usable integration time is estimated such that the QPO is detected abovea certain significance in at least 80% of the N intervals; a linear interpolation is used to estimate the QPOfrequency in the remaining intervals. The N PDS are frequency-shifted to the mean QPO frequency over thesegment and averaged (M´endez et al. 1998). The resulting averaged PDS representing the complete continu-ous segment is then fitted with one or two Lorentzians plus a constant corresponding to the counting-statisticsnoise level (Barret et al. 2005b).When only one significant peak is detected, the QPO is identified as upper or lower from the parametersof the Lorentzian and we refer to such peaks as single QPOs. We stress that the value of the QPO frequencyitself is not used to distinguish between the upper and lower QPOs—in principle, a QPO of a given frequencycould be either the upper or the lower QPO. For instance, the quality factor for the lower kHz QPO is a welldetermined function of the frequency, and a di ff erent function of frequency for the upper QPO, and we canuse this and other relationships to identify a QPO of given frequency (see Barret et al. 2005a,b,c, 2006, fordetails). Of course, when two significant kHz QPOs are detected, the upper QPO is the one with the larger This way of QPO identification di ff ers from the method based on hardness diagram applied in Belloniet al. (2005, 2007). The two methods have a di ff erent range of applicability but give comparable results (see requency, by definition.Consequently, the frequency values are averaged through intervals of predetermined length ∼ ff erent lengths, resulting in a larger number of detections.Hence the histograms we use here are not comparable in details to those of Belloni et al., even for the sameRXTE data.We take into account only the detections of oscillations with quality factor (defined as the QPO centroidfrequency over the full-width of the peak at its half-maximum) Q ≥ S ≥ ff erent distributions Each of the histograms in the Figure 1 clearly reveals an accumulation of frequencies in the ν ∼
900 Hzvicinity of the power spectrum. However, for the lower kHz QPO this range ( ν L ∼ ν U are accumulated in thelow-frequency part of its own range of variation.If one is interested (for whatever reason) in the distribution of the frequency ratio ν U /ν L , then thoseobservations in which both QPO peaks are simultaneously detected should be considered. Accordingly, weapply our selection criteria to simultaneous significant detections of both QPO frequencies as well. Thehistograms of the upper QPO frequency in this sample (darker bars in Figure 1, right panel) are strikinglydi ff erent from the previous histogram of significant detections of the upper QPO (bars of lighter shade in thesame figure). A new cluster of frequencies appears, in the range ∼ ∼ ν U ≈ . ν L +
520 Hz , (1)very clearly the examined data do not support the assumption of Belloni et al. (2005, 2007) that the dis-tribution of the ratio of two linearly correlated frequencies is determined by the distribution of one of thefrequencies even when the second frequency remains undetected—there is apparently no direct link betweenthe histogram of all the lower QPO detections (Figure 1, left; lighter) and the histogram of the same QPOtaken from the subset of twin peak QPO detections (the same figure; darker). This result should have animpact on the theory of QPOs. Although a full discussion is beyond the scope of this paper, we note thatthe change in the frequency distribution when a second QPO is detected may be suggestive of a physicalmechanism, such as mode-coupling.To quantify this e ff ect we plot the cumulative distributions of the lower and of the upper QPO, which areshown in the Figure 2. Using the Kolmogorov-Smirnov (K-S) test we compare the frequency distributionsof each (the upper and the lower) QPO measured in all detections, with those measured for the same QPOwhen both the upper and the lower QPO are detected. We obtained the K-S probabilities p L , KS = . × − and p U , KS = . × − in the case of the lower and upper QPO respectively. Indeed, the two distributionsare di ff erent in both cases. We directly conclude that detections of individual QPOs alone cannot be used forcalculation of the distribution of the frequency ratios.It is interesting to note that the single upper QPOs are mostly detected at relatively low frequencies,while the single lower QPOs are detected at relatively high frequencies. Taking into account the linearcorrelation among QPO frequencies, the distributions of single lower and upper QPOs appear to be comple-mentary, in the following sense. The lowest-frequency detection of the single lower QPO is at ν L =
651 Hz,which in the linear correlation corresponds to ν U =
976 Hz while the highest-frequency detection of thesingle upper QPO is at ν U =
961 Hz, which corresponds to ν L =
628 Hz. In other words, if one assumed thateach of the single upper QPOs is accompanied by a lower QPO of frequency determined from eq. (1), theresulting points would all fall to the left of the 3:2 line in Fig. 4 (left panel), and if the same procedure wereapplied to the single lower QPOs, the resulting points would fall to the right of the 3:2 line in the same figure.This is illustrated in Fig. 3. Given this fact, it is not surprising that the distribution of actually detected upper(or lower) kHz QPOs is completely di ff erent from the distribution that would be predicted on eq. (1) fromthe distribution of the other kHz QPO, when detections of single QPOs dominate the data set (Fig. 2).e.g., Barret et al. 2005b,c). The cumulative distributions for the kHz QPOs corresponding to Fig. 1. Left: Solid curvescorrespond to the detected lower QPOs (see left panel of Fig. 1), the dotted line labeled “inferred lower”indicates the lower QPO frequency calculated from eq. (1) using all detections of the upper QPO. The dashedvertical line shows the greatest di ff erence D max = .
515 between the distributions of “all lower” and “lowerin twin”. Right: Analogous lines for the upper QPO. The dashed vertical line on right panel corresponds tothe maximal di ff erence D max = . Figure 3:
Distribution of single QPOs. Frequency axes are aligned according to the correlation of eq. (1).The shadow denotes a 50 Hz scatter about the lower QPO frequency of 650 Hz, corresponding to a 3:2 ratio.
The left panel of Figure 4 depicts the mutual dependence of frequencies of the lower and upper QPO whenboth were significantly detected. It also display a corresponding histogram of the frequency ratio. As forSco X-1 (Abramowicz et al. 2003), this histogram is peaked close to the 3 / p ( r ) = f λ /π ( r − r ) + λ + (1 − f ) λ /π ( r − r ) + λ , (2)where r = ν U /ν L is the frequency ratio and r , r , λ , λ and f are free parameters. Their values obtainedby the maximum likelihood method are r = . r = . λ = . λ = . f = . p , KS = . r = .
50 and λ = . p , KS = . Left: The frequencies of detected twin QPOs. The inset shows a corresponding histogram of thefrequency ratio. Right: Cumulative distribution of the frequency ratios, the thick solid line denotes the bestfit by a sum of two Lorentzians. Fit by a single Lorenzian is marked by dotted line.
We have demonstrated for a set of uniform data (Barret et al. 2005b) that the frequency distribution of asingle kHz QPO is not equivalent to the distribution of the corresponding frequency when a pair of kHzQPOs have been detected.We stress that if there is a one-to-one correspondence between the frequencies and their ratio, as is thecase for linear functions with a non-vanishing intercept, the question whether to consider the QPO frequencydistribution or the ratio distribution as fundamental is one of theoretical assumptions, as the two distributionsare mathematically equivalent. However, the distribution of a single kHz QPO frequency is not predictiveof the distribution of two frequencies detected simultaneously, nor of the distribution of their ratio, even ifthese frequencies are correlated when both are actually detected. Thus, the study of Belloni et al. (2007),who conclude that “ there is no preferred frequency or frequency ratio in 4U 1636 − ” is based on an invalidassumption, and cannot be accepted as applying to the distribution of ratios, as long as it is based on thedetection of a single frequency.The finding that the frequency distribution of a QPO depends on whether or not a second QPO canbe detected as well should restrict models of the physical origin of the QPO and of X-ray flux modulation,regardless of whether or not the value of the frequency ratio is clustered about the specific value of 3 / Acknowledgements.
We thank Didier Barret for providing the data and software on which this paper buildsand for several discussions. We have also benefited from helpful comments by Tomek Bulik. We thank thereferee for very useful suggestions. The authors are supported by the Czech grants MSM 4781305903 andLC06014, by the Polish grants KBN N203 009 31 / References
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