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Astron. Astrophys. 334, 201-209 (1998)

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2. The short-term X-ray variability of Cyg X-1

2.1. The data

The EXOSAT raw data have been stored on Final Observation Tapes (FOTs) and are now available at the HEASARC archive. Table 1 lists the FOT observations of Cyg X-1 that we have analyzed with Linear State Space Models. We have chosen the ME datastreams provided by the primary timing telemetry modes HTR3 und HER6. These observations only contain events registered in the Argon counters (1-20 keV). The lightcurves are given as countrates normalized to one half of the detector array (i.e. four Argon counters). Using the Interactive Analysis (IA) software, we extracted lightcurves corrected for dead-time effects and collimator efficiency (for an overview of the IA see Parmar et al. 1995). For the purpose of this paper an explicit background subtraction is not necessary since the LSSM is implicitly modeling the measurement process (see Sect. 3.2, Eq. 7).


[TABLE]

Table 1. EXOSAT ME observations that have been analyzed with LSSMs.


All EXOSAT observations of Cyg X-1 have found the source in its usual hard state. An example for the characteristic hard state variability of Cyg X-1 on short timescales can be seen in Fig. 1.

[FIGURE] Fig. 1. a X-ray lightcurve of Cyg X-1, showing a part of the fifth observation on FOT Nr. EE8087 (No. 1 of Table 1, energy range=1-20 keV, bin time=62.4 ms, length=156 s). b Blow-up of the marked segment of subfigure a with a length of 3.9 s and the original bin time of 7.8 ms, showing a typical shot structure. Both lightcurves are plotted without error bars for reasons of clarity.

2.2. The periodogram of Cyg X-1

The periodogram [FORMULA] of a lightcurve [FORMULA] (where [FORMULA]) is providing the "strength" of harmonic variations with a certain frequency [FORMULA] in the lightcurve. [FORMULA] is defined as the squared modulus of the discrete Fourier transform of [FORMULA] and is calculated for the Fourier frequencies [FORMULA] with bin time [FORMULA] and [FORMULA], where M is the integer part of [FORMULA] (Scargle 1982):

[EQUATION]

Fig. 2 shows the logarithmically plotted sample periodogram (solid line) of the observation No. 1 (Table 1). It has been obtained by averaging over periodograms [FORMULA] from 48 different lightcurve segments (details see caption of Fig. 2). Since [FORMULA] is a [FORMULA] -distributed random variable, its standard deviation is equal to its mean (van der Klis 1989). Therefore, an individual periodogram exhibits large fluctuations (Fig. 2, dots). Calculating the sample periodogram significantly reduces the scatter and allows the possibility to estimate the theoretical spectrum of the process responsible for the observed variability.

[FIGURE] Fig. 2. Sample periodogram (solid line) of Cyg X-1, averaged over 48 individual periodograms, which have been obtained by dividing the original lightcurve (No. 1, Table 1) into 48 segments of 156 s each and calculating [FORMULA] for each segment (Eq. 1). The values of a typical individual periodogram, fluctuating wildly, are also displayed (dots).

The periodogram of the short-term variability of Cyg X-1 in its hard state is well known and exhibits the following distribution of timescales (Fig. 2):

  • For frequencies above 10 Hz white noise dominates the periodogram which means that the variability of the lightcurves on all corresponding timescales is almost equally strong. These fluctuations (i.e. photon statistics, particle background) are introduced by the measurement procedure. It is common practice to subtract a constant (corrected for deadtime effects) which represents this noise component from the periodogram (Belloni & Hasinger 1990b, Zhang et al. 1995).
  • Towards lower frequencies the power of the variability increases within the frequency range of roughly 0.04-10 Hz. This behavior is called red noise. It is often modeled by a power law, [FORMULA]. Variability of the [FORMULA] -type is known from many X-ray binaries (van der Klis 1995) and also from active galactic nuclei (McHardy 1989). In the case of Cyg X-1, [FORMULA] is approximately 1 (Nolan et al. 1981, Lochner et al. 1991).
  • An important feature indicating the stationarity of the short-term variability process is the flattening of the power spectrum for frequencies below [FORMULA] 0.04 Hz. This flat top corresponds to the absence of additional long-term variations in the lightcurves. There are, however, some EXOSAT ME observations of Cyg X-1 which show low frequency noise in the form of increasing power towards lower frequencies below 0.001 Hz (Angelini et al. 1994). Since this component is not always present, it is most likely not produced in the same physical process as the short-term variability. Suggested causes for this low frequency variabilities are instabilities in the mass transfer process. Sometimes the low frequency noise is associated with absorption dips in the lightcurve (Kitamoto et al. 1984, Angelini et al. 1994).
  • Several authors reported a transient quasi-periodic oscillation (QPO) feature in the power spectrum with a central frequency of about 0.04 Hz (Kouveliotou et al. 1992, Angelini et al. 1994, Vikhlinin et al. 1994, Borkus et al. 1995), whereas other authors have not found any evidence for QPOs (Belloni & Hasinger 1990b, Miyamoto et al. 1992). Thus the significance of this feature is not yet clear. In Fig. 2 no QPO feature is present.

2.3. The shot noise model

Terrell 1972proposed that the structures observed in the lightcurves of Cyg X-1 might be due to a shot noise process [FORMULA], i.e. the superposition of randomly occuring shots with the shot profile [FORMULA]:

[EQUATION]

Here the [FORMULA] are the times at which the shots occur, with the time intervals between the [FORMULA] following a Poisson distribution.

The standard shot profile [FORMULA] is identical for all shots and consists of an instantaneous rise to height [FORMULA] and an exponential decay with the decay time [FORMULA]:

[EQUATION]

Shot noise models are often evaluated by comparing their theoretical power spectrum to the observed power spectrum (Belloni & Hasinger 1990b, Lochner et al. 1991). The theoretical power spectrum of the standard shot noise process is (Lehto 1989):

[EQUATION]

where [FORMULA] is the average time interval between the shots. [FORMULA] provides the flat top for [FORMULA] but has a fixed logarithmic slope of [FORMULA] for higher frequencies (Fig. 3 a). Since the shot noise process [FORMULA] is continuous in t, whereas the observed lightcurve is the result of integrating the measured counts over a finite number of intervals [FORMULA], [FORMULA] must be corrected for binning. Lochner et al. 1991have shown, that binning makes the shot noise power spectrum even steeper for high frequencies.

[FIGURE] Fig. 3. a Shot noise power spectra for three different values of [FORMULA], calculated according to Eq. 4([FORMULA] =1, [FORMULA] =1). b LSSM power spectra for the same values of [FORMULA] as in (a), calculated according to Eq. 8([FORMULA] =1). All power spectra are given in arbitrary units and [FORMULA] has been normalized to 1. In subfigure b, the variance of the observation-noise, [FORMULA], has been set to 0 to allow direct comparison of a and b.

Neither the observed slope nor the white noise floor of the Cyg X-1 periodogram (Fig. 2) can be reproduced by the standard shot noise process. In frequency domain fits the white noise level is usually treated as an additional constant parameter (cf. Sect. 2.2). To model the observed slope, different shot profiles with distributions of shot durations and shot amplitudes have been proposed. A number of those models, each having many degrees of freedom, are able to reproduce the observed periodogram and other second order statistics (Belloni & Hasinger 1990b, Lochner et al. 1991). Higher order statistics like the time skewness (Priedhorsky et al. 1979) or phase portraits (Lochner et al. 1991) were also studied but no special shot noise model could be singled out that allows for a homogeneous description of different observations. In Sect. 3.3 we discuss the theoretical power spectrum of a first order LSSM and show that it can reproduce all the features of the hard state periodogram of Cyg X-1, requiring only one temporal parameter [FORMULA] (Fig. 3 b).

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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