ࡱ> egd#` Rbjbjmm޷'<T0    L!T0 C\"r%:%%%)) )UCWCWCWCWCWCWC$Eh@Hp{C90) )00{C%%pC6660%8%UC60UC66V=@=%P" \~ R2BY= !B4C0Ce=0I3g=Ilg$=)V>+62,,))){C{C|4&)))C00000 0 0  4 0 0 0 40 0 0   MACROBUTTON MTEditEquationSection2 Equation Chapter 1 Section 1 SEQ MTEqn \r \h \* MERGEFORMAT  SEQ MTSec \r 1 \h \* MERGEFORMAT  SEQ MTChap \r 1 \h \* MERGEFORMAT Hard X-ray lags of Cygnus X-1 on short timescales JIN Jing1, LU Yu2,3, QU JinLu2 & Li TiPei1,2 1 Department of Engineering Physics & Center for Astrophysics, Tsinghua University, Beijing 100084, China; 2 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; 3 Graduate School of Chinese Academy of Sciences, Beijing 100049, China Calculating the time lags over different timescales using the cross-correlation technique may lead to a biased estimate at small timescales. Given a timescale for lightcurve binning, we propose to subtract the local average instead of the global average during the cross-correlation, in order to filter variations on timescales larger than the bin size. The new means allows us to make an unbiased estimate of the time lags using RXTE/PCA data on timescales as small as ~5 ms, where the Fourier technique becomes invalid. We calculate the time lag spectra of Cygnus X-1 at different spectral states with the new means, and find that the source appears to have similar X-ray lags on small timescales independent of its spectral states. Spectral lags, Cygnus X-1, data analysis 1 Introduction The time lag spectrum, which is the distribution of time lags over timescales between two lightcurves in different energy bands, is useful to reveal the X-ray emission mechanism from black hole binaries. The most widely used method to obtain such a lag spectrum is based on the Fourier analysis in the frequency domain. However, with RXTE/PCA data of Cygnus X-1, it is difficult to obtain reliable hard X-ray lags from the Fourier analysis at frequencies higher than ~30 Hz, where the lag information may be important[1]. A time domain method has been developed to get lag spectra using the modified cross-correlation function (MCCF) [2][3][4]. With MCCF, one can calculate lags on timescales of milliseconds using the same data of Cygnus X-1. However, the timescales are not independent with each other in the calculation of MCCF, leading to a biased estimate of the time lags especially on short timescales. In order to disentangle the lag information on small timescales from the large ones, we propose to use local average instead of the usually adopted global average in MCCF. In this paper we make simulations to confirm the improvement on the time lag estimate by using the local average, and apply it to the timing analysis of Cygnus X-1 at four spectral states. 2 Method and simulations Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 be two simultaneously observed counting series in two different energy bands with an original time resolution of EMBED Equation.DSMT4 . In order to calculate the time lag at a given timescale EMBED Equation.DSMT4  ( EMBED Equation.DSMT4  is an integer), the original counting series should first be re-binned to the timescale  EMBED Equation.DSMT4  as follows,  EMBED Equation.DSMT4  (1)  EMBED Equation.DSMT4  (2) The zero-mean counting series of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are given by  EMBED Equation.DSMT4  (3) and  EMBED Equation.DSMT4  (4) Corresponding author (email: jinj02@mails.tsinghua.edu.cn) Supported by National Basic Research Program of China (Grant No. 2009CB824800) and the National Natural Science Foundation of China (Grant No. 10533020) where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are the average counts of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , respectively. The MCCF at lag  EMBED Equation.DSMT4  is defined as[4]  EMBED Equation.DSMT4  (5) where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  are the root mean square of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , respectively. The value  EMBED Equation.DSMT4  which satisfies the condition  EMBED Equation.DSMT4  (6) can thus be found, and used to define the time lag of band 2 relative to band 1 on timescale  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4  (7) With this definition, the lag distribution over different timescales can be calculated. However, it is unavoidable that the variations on large timescales will influence the measurement of the lag on small timescales. This is due to the fact that the lightcurve binned with a small timescale necessarily contains variations on large timescales. For example, a sinusoidal signal with a period of 100 ms will show variation in a counting series with a time bin of 1 ms. In order to reduce the crosstalk on different timescales and make an unbiased estimate of time lags on a given timescale, we use a local average instead of the global average in Eq. (3) and (4).  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are re-defined as the average of the nearby five bins of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , respectively, as follows,  EMBED Equation.DSMT4  (8)  EMBED Equation.DSMT4  (9) With this modification, the influence from larger timescales to smaller ones on the estimate of the time lag is expected to be greatly reduced. Simulations are carried out to compare the MCCF with the global and local average. Two counting series of length 1000 s with independent random shot components in band 1 are simulated. The original time step  EMBED Equation.DSMT4  is 1 ms. Each series consists of random shots with a profile of  EMBED Equation.DSMT4  (i= 1 for series 1 and i = 2 for series 2), where a is randomly sampled from a uniform distribution between zero and the maximum. The separation of two successive shots obeys the exponential distribution with the average value EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  is the characteristic timescale of the random variation for each series, with  EMBED Equation.DSMT4  ms for series 1 and  EMBED Equation.DSMT4  ms for series 2. The counting rates for series 1 and 2 are 200 cts/s and 600 cts/s, respectively. For each series in band 1, a corresponding series in band 2 is generated with the same events in band 1, but each delayed EMBED Equation.DSMT4 , with  EMBED Equation.DSMT4  ms for series 1 and  EMBED Equation.DSMT4  ms for series 2. By summing up the two series for each band, two light curves are obtained, with one in band 1 and the other in band 2. By adding independent white noises at a level of 50 cts/s to each light curve, two final light curves are produced. It is expected that the lag between these two light curves has two components, a 6 ms component on the characteristic timescale of 10 ms and a 30 ms component on 80 ms. Both the MCCF with global average and with the local average are used to calculate the time lag on the timescale of 10 ms between the two light curves. The results are shown in Figure 1. The MCCF result with the global average has two peaks, caused by the lags on timescales of 10 ms and 80 ms, respectively. The estimated time lag is 8 ms, larger than the intrinsic lag of 6 ms on the timescale of 10 ms. Thus, it is obvious that the estimate of the time lag on the timescale of 10 ms has been influenced by the variation on the timescale of 80 ms. In contrary, the measured lag given by the MCCF with the local average is exactly 6 ms with a single peak in the MCCF, indicative of an unbiased estimate.  Figure 1. The MCCF vs. time lag of the two light curves with given time lags. Left panel: MCCF with the global average. Right panel: MCCF with the local average. 3 Hard X-ray lags of Cygnus X-1 The re-defined MCCF with the local average is used to study the time lag of the variation in the 13-60 keV band with respect to 2-5 keV of Cygnus X-1 at four states, using the data available in the archive of RXTE. The observations used are listed in Table 1. The time lag spectra of Cygnus X-1 are shown in Figure 2. Table 1. The log of observations used. Observation IDObservation dateSource state10241-01-01-00023-24/10/96hard10512-01-08-0017/06/96soft10412-01-01-0022/05/96hard to soft10412-01-06-0011/08/96Soft to hard Figure 2. The time lag spectra of Cygnus X-1 at the four states. The left panel is a semilog plot and the right panel is a loglog plot. The circles: the hard state. The stars: the soft state. The downward-pointing triangles: the hard to soft state. The upward-pointing triangles: the soft to hard state. The unbiased lag spectra for Cygnus X-1 in the four states at timescales down to ~5 ms are shown in Figure 2. Reliable results at frequencies below ~30 Hz that are unavailable from the Fourier analysis can be obtained. On the large timescales, it is obviously shown that the transition states have the largest time lags and the soft sate has the smallest time lags, which is consistent with the results by Fourier analysis[5]. However, the lags at different states get very close to each other on the small timescales. It has to be mentioned that, below 5 ms, one cannot make reliable measurement of the lag due to the dead time of RXTE/PCA, which causes a pit centered at the time lag of zero in the MCCF. This pit will affect the profile of MCCF and make the position of the intrinsic peak hard to find. On the timescales larger than ~5 ms, this effect can be neglected. Nowak M A, Wilms J, Vaughan B A. Rossi X-ray timing explorer observation of Cygnus X-1. a!. Timing analysis. Astrophys. J., 1999, 510: 874-891 Li T P, Feng Y X, Chen L. Temporal and spectral correlations of Cygnus X-1. Astrophys. J., 1999, 521: 789-797 Li T P. Timing in the time domain: Cygnus X-1. Chin. J. Astron. Astrophys., 2001, 1: 313-332 Li T P, Qu J L, Feng H, et al. Timescale analysis of spectral lags. Chin. J. Astron. Astrophys., 2004, 4: 583-598 Cui W, Zhang S N, Focke W, et al. Temporal properties of Cygnus X-1 during the spectral transitions. Astrophys. J., 1997, 484: 383-393 Pottschmidt K, Wilms J, Nowak M A, et al. Temporal evolution of X-ray lags in Cygnus X-1. Astron. 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