On the origin of clustering of frequency ratios in the atoll source 4U 1636-53
Gabriel Torok, Michal Bursa, Jiri Horak, Marek A. Abramowicz, Pavel Bakala, Paola Rebusco, Zdenek Stuchlik
aa r X i v : . [ a s t r o - ph ] A p r On the origin of clustering of frequency ratios in theatoll source 4U 1636-53
Gabriel T¨or¨ok , Marek A. Abramowicz , , , Pavel Bakala , Michal Bursa , Jiˇr´ıHor´ak , 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 Department of Physics, G¨oteborg University, S-412 96 G¨oteborg, SE Copernicus Astronomical Centre PAN, Bartycka 18, 00-716 Warsaw, PL Astronomical Institute of the Academy of Sciences, Boˇcn´ı II 1401 / MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue, 37, Cambridge, MA02139, US e-mail: [email protected], [email protected], pavel.bakala@ fpf.slu.cz, [email protected],[email protected], [email protected], [email protected]
Abstract.
A long discussion has been devoted to the issue of clustering of the kHz QPO frequency ra-tios in the neutron star sources. While the distribution of ratios inferred from an occurrence of a single QPOseems to be consistent with a random walk, the distribution based on simultaneous detections of both peaksindicates a preference of ratios of small integers. Based on the public RXTE data we further investigate thisissue for the source 4U 1636 −
53. Quality factors and rms amplitudes of both the QPOs nearly equal to thepoints where the frequencies are commensurable, and where the twin QPO detections cluster. We discuss aconnection of the clustering with the varying properties of the two QPO modes. Assuming approximativerelations for the observed correlations of the QPO properties, we attempt to reproduce the frequency andratio distributions using a simple model of a random-walk evolution along the observed frequency-frequencycorrelation. We obtain results which are in qualitative agreement with the observed distributions.
Keywords:
X-rays: binaries — Stars: neutron — Accretion, accretion disks
Since the paper of Abramowicz et al. (2003), the issue of distribution of kHz QPOsin neutron-star low mass X-ray binaries has been discussed extensively. In their work,Abramowicz et al. examined simultaneous detections of the upper and lower QPOs inthe Z-source Sco X-1. The authors show that that the ratios of the lower and upperQPO frequencies cluster most often close to the value ν L /ν U = /
3. They find alsoevidence for the second peak in a distribution of frequency ratios at ν L /ν U ≈ .
78. Thisvalue is remarkably close to an another ratio of small integers, 4 / = .
8. In themost recent paper, T¨or¨ok et. al (2008) have examined occurrences of the twin QPOsin the atoll source 4U 1636 −
53 applying the same methodology as Abramowicz et al.(2003). They find that the distribution of the (inverse) frequency ratios ν U /ν L of twosimultaneously detected QPOs peaks near 3 / / −
53 (Belloni etal. 2005). They argue that such clustering does not provide any useful informationbecause frequencies of the two QPOs are correlated and the distribution of the ratio oftwo correlated quantities is completely determined by the distribution of one of them.1eeping this argument, a recent study of Belloni et al. (2007) based on a systematiclong term observation of 4U 1636 −
53 concludes that there is no preferred frequencyratio.The aparent disagreement in conclusions of the two groups comes from a confusionbetween the observed frequency distribution (the one which can be recovered fromobserved data) and the intrinsic distribution (the “invisible” one really produced bythe source). While Abramowicz et al. (2003) and T¨or¨ok et. al (2008) have examinedfrequency ratios of the actually observed QPO pairs (twin peaks) only, the analysisof Belloni et al. (2005, 2007) study primarily distributions of frequencies of a singleQPO and make implications for the distribution of the other, often invisible, QPO fromthe empirical correlation between frequencies.In this paper we show that the observed distributions are a ff ected by the way thesignal form a source is being detected and analyzed. We show that the observed clus-tering can be understood in terms of rms amplitude and quality factor correlations withQPO frequency. Taking these correlations into account, we simulate the ratio distribu-tion using a random walk model of QPO frequency evolution and we find that resultsof the simulation agree with empirical data. In the process of data reduction and searching for QPOs, an important quantity is thesignificance S of the peak in PDS, which measures the peak prominence. Shape of apeak in the PDS is most often fitted by a Lorentzian. Usually, S ≥ − ff ect the distribution of detections.The significance S is given by the relation between the integral area of a Lorentzianin PDS and its error. For a particular detection, it depends on observational conditions,on the quality factor Q of the peak (defined as the QPO centroid frequency over thepeak full-width at its half-maximum) and on the fractional root-mean-squared ampli-tude r (a measure for the signal amplitude given as a fraction of the total source fluxthat is proportional to the root mean square of the peak power contribution to the totalpower spectrum), S = k r √ Q /ν , where the time-varying factor k ( t ) = I ( t ) √ T depedson the total length of observation T and the instantneous source intensity I , which at agiven time same for both upper and lower peak .Barret et al. (2005a,b,c, 2006) have shown that both quality factors and rms am-plitudes are determined by frequency and moreover that their profiles greatly di ff erbetween lower and upper QPO modes. The quality factor of the upper QPO is usu-ally small and tends to stay at an almost constant level around Q U ∼
10. In contrast, thelower QPO quality factor improves with frequency and can reach up to Q L ∼
200 beforea sharp drop of coherence at high frequencies. Amplitudes of upper QPOs generallydecrease with frequency, while the lower QPO amplitudes show first an increase andthen they start to decay too. The standard process of the QPO determining is in detail described in van der Klis (1989).
The quality factor (left), rms amplitude (middle) and inferred significance (right) behaviour inatoll source 4U 1636 −
53. Red points represent lower QPO data, blue points are for upper QPO data. Data infirst two panels comes from the study of Barret et al. (2005b) and cover large range of frequencies availablevia shift-add method through all segments of RXTE observations. Continuous curves are obtained frominterpolation by several exponentials (see, e.g., T¨or¨ok 2007). The prospected course of the QPO significancein the right panel is determined by the rms amplitude and quality factor profiles ( S ∝ r √ Q /ν ). Frequencyaxes are related using frequency correlation ( ν U = . ν L + Figure 1 shows the behaviour of amplitudes and quality factors of individual QPOmodes in 4U 1636 −
53 and how they change with frequencies. The displayed dataof Barret et al. (2005b) cover large frequency range available through the shift-addtechnique over all RXTE observations (see M´endez et al. 1998, 1999; Barret et al.2005a,b,c, for details).Note that both of the two properties are becoming similar as the frequency ap-proaches points corresponding to 3 / / I and observing time t constant (for simplicity). It is clearlyvisible that there is a similar equality of QPO significances close to points, where thefrequencies are close to the 3 / / Q and r at thosepoints), while they are much di ff erent elsewhere. We will hereafter call the points ofequal significances as the “3 /
2” and “5 /
4” points. We may also observe that the upperQPO mode is usually strong (much more significant) left from the 3 / / It is likely that if QPOs are produced in a source, they are always produced in pairs.Because the strength of oscillations is usually around the sensitivity threshold of mea-surements, often only one (the stronger) QPO is detected. Around the special points3 / /
4, where significances are comparable, there is a good chance that if onemode can be detected the other could be detected as well, because both peaks havenearly the same properties. Indeed, this agrees with what is observed and has beenlaboured or challenged many times (Abramowicz et al. 2003; Belloni et al. 2005; Bulik2005; Yin & Zhao 2007; Belloni et al. 2007) that pairs of QPOs cluster close to the 3 / In the figure we use a correlation ν U = . ν L + The distribution of observed frequency ratios. Left: The fraction of the number of observationswith simultaneous detections n UL to the number of observetions in which at least one QPO has been detected n UL + n U + n L (where n U and n L are respectively the numbers of observations with detections of the upper orlower QPO only). Middle: Simulated ratio distribution assuming a random-walk in frequency and variablecount rate (see text). The gray underlying histogram in the first two panels shows the actual observed ratiodistribution of twin QPO peaks (the data in both panels are those discussed in T¨or¨ok et. al 2008). Right:The individual distributions of lower and upper QPO frequencies from the random walk simulation. Black-shaded portions of bars represent simultaneous occurrences of both modes (twin QPOs) as shown in themiddle panel. and some other small rational number ratios.From time to time, the conditions at the source become such that both QPOs canbe detected simultaneously regardless of their frequency, only because of their actualhigh brightness (as the observational sensitivity is relatively low). These events allowsus not only to see QPO pairs close to the critical points, but sporadically also all theway along the frequency-frequency correlation line, even far from 3 / / ff ectedby the behaviour of rms amplitudes and quality factors and namely by the fact that thesequantities become equal close to that frequency ratio. This is demonstrated in Figure 2(left), where we show a fraction of number of observations, in which both QPOs havebeen detected simultaneously, to a number of those, in which at least one QPO has beendetected. The figure is based on data used in T¨or¨ok et. al (2008). Clearly, the positionsof maxima remarkably well correlate with points, where the two significances equal.Moreover, these positions coincide with peaks in the distribution of frequency ratiosfound in T¨or¨ok et. al (2008) which justify a hypothesis that there is a link betweenQPO properties and the ratio clustering. As previously noticed in several works and further suggested by Belloni et al. (2005),the observed time evolution of QPO frequency appears consistent with a series of ran-dom walks.This has been later critised by Bulik (2005) who pointed out that contrary to thedistributions of QPOs that appear qualitativelly similar at di ff erent times, distributionsarising from random walk di ff er significantly among di ff erent realisations (with di ff er-ent seeds). Nevertheless, using a simple model of random walk we attempt to at leastroughly reproduce the frequency ratio distribution.Starting with ν L = ± ν L is assigned to each step. This setup roughly corresponds to thedocumented frequency drifting through 32 sec integration intervals (Barret et al. 2004;Paltani et al. 2004) and each segment then mimic 1.6 kiloseconds of QPO evolution.The QPO frequencies are averaged over each segment, and the linear correlation ν U = . ν L + k =
1. For each point we calculate its significance based on ob-served profiles of Q and rms, which are based merely on datapoints corresponding totwin peak QPO observations. Only such points are considered in the simulation, whereboth upper and lower QPOs have significance above 3 σ level. The resulting histogramof frequency ratios shows strong clustering around 3:2 ratio, however, it does not re-produce the second peak around 5 /
4, which indicates that the assumption of constantcount rate my not be su ffi cient.As a second step, we adopt an additional (still very simplifying) assumption to thesimulation that countrate is varying with frequency. The motivation here comes froma known fact that for a given source there is not a global correlation between sourceluminosity and QPO frequency, but the two quantities stay correlated during individual(temporary) observational events (so-called parallel-track phenomenon, e.g., M´endezet al. 1999). In the case of 4U 1636 −
53, the maximal countrates related to the highestobserved lower QPO frequencies (up to 950Hz) are 2–3 times higher than the highestcountrates at ν L ∼ ν L ∼ ν L ∼ ∼ / /
4. Whilethe presence of the 3 / / ν U ∼ ff limit on significance orconsider lower countrates, we would loose the 5 / Focused on the atoll source 4U 1636-53 we demonstrate that at frequencies, where theboth QPO modes have comparable properties, there is a high probability of detectingboth peaks of a twin pair simultaneously. We have found a precise match comparing theobserved twin QPO distribution with our simulation based on the observed correlations5etween QPO frequencies and their properties. The simulation not only reproduces theobserved clustering, but it also shows the “complementarity” between upper and lowerQPO distributions that has been noticed by T¨or¨ok et. al (2008). This suggests that theratio clustering may origin in the exchange of dominance between the two modes whenone mode fades in and the other one fades out.Even if the intrinsic distributions of both the mode frequencies were uniform, therewould be a non-trivial profile of the observed distributions and clustering of the twinpeak detections around certain points (narrow regions) prominent due to behaviour ofthe QPO amplitudes and coherence times determined by the QPO mechanism. It willrequire a further detailed analysis to investigate whether the above influence of theQPO properties can explain the ratio clustering observed in 4U 1636-53 completely.For a further understanding of the ratio clustering mechanism (and importance) it isalso highly needed to perfom a similar analysis for the other sources. For instance, avery recent study of the atoll source 4U 1820-30 (Barret & Boutelier 2008) found thata point close to the 4 / . Acknowledgements.
We thank M. M´endez for several discussions on the subject and, especially, we arethankful to D. Barret for ideas, comments and for providing the data and software on which this paper builds.We are grateful to W. Klu´zniak and Tomek Bulik for several comments and advice. We thank the refereefor all suggestions. We also thank the Yukawa Institute for Theoretical Physics at Kyoto University, wherethis work was initiated during the YITP-W-07-14 on ”Quasi-Periodic Oscillations and Time Variabilities ofAccretion Flows”. The authors are supported by the Czech grants MSM 478130590384 and LC06014, andby the Polish grants KBN N203 009 31 / References
M. A. Abramowicz, et al., 2005, AN 326, 864M. A. Abramowicz, T. Bulik, T., M. Bursa, W. Klu´zniak, 2003, A& A, 404, L21M. A. Abramowicz, W. Klu´zniak, A&A, v.374, p.L19–L20 (2001)Barret, W. Klu´zniak, J. F. Olive, S. Paltani, G. K. Skinner, in Proc. of the Annualmeeting of the French Astronomical Society (SF2A)Barret, W. Kluzniak, J. F. Olive, S. Paltani, G. K. Skinner, MNRAS, 357, 4, 1288–1294Barret D., J. F. Olive, M. C. Miller, 2005, MNRAS, 361, 3, 855Barret D., J. F. Olive, M. C. Miller, 2005, AN, 326, 9, 808Barret D., J. F. Olive, M. C. Miller, 2006, MNRAS, 370, 3, 1140Barret, D., Boutelier, M., in the Proceedings of 2007 Jean-Pierre Lasota Conference,2008 Note also that in contrary to the case of 4U 1636 they reported a lack of the twin QPO detections closeto the 3 /
6. Belloni, M. M´endez, J. Homan, 2005, A&A 437, 209T. Belloni, M. M´endez, J. Homan, 2007, MNRAS, 379, 1, 247T. Bulik, 2005, AN 325, 861W. Klu´zniak, M.A. Abramowicz, 2000, astro-ph / astro-ph/0306213 S. Paltani, D. Barret, J.F. Olive, G. K. Skinner, in Proceedings of the Annual meetingof the French Astronomical Society (SF2A)M. M´endez, MNRAS, 371, 4, 1925M. M´endez, M. van der Klis, R. Wijnands, E. C. Ford, J. van Paradijs, B. A. Vaughan,1998, ApJ, 505, L23M. M´endez, M. van der Klis, R. Wijnands, E. C. Ford, J. van Paradijs, J., 1999, ApJ,511, L49T¨or¨ok, G., A&A, 2007, submittedG. T¨or¨ok, M. A. Abramowicz, P. Bakala, M. Bursa, J. Hor´ak, W. Klu´zniak, P. Rebusco,Z. Stuchl´ık, Acta Astronomica, 58 /
1, 2008, pp. 1–14M. van der Klis, NATO Advanced Study Institute on Timing Neutron Stars, p. 27–69H. X. Yin, Y. H. Zhao, 2007, AdSpR 40, 1522C.M. Zhang,H. X. Yin, Y. H. Zhao, L.M. Song, F. Zhang, MNRAS, 2006, 366, 1373,astro-ph //