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 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.
2.2. The periodogram of Cyg X-1
The periodogram of a lightcurve (where ) is providing the "strength" of harmonic variations with a certain frequency in the lightcurve. is defined as the squared modulus of the discrete Fourier transform of and is calculated for the Fourier frequencies with bin time and , where M is the integer part of (Scargle 1982):
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 from 48 different lightcurve segments (details see caption of Fig. 2). Since is a -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.
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):
2.3. The shot noise model
Here the are the times at which the shots occur, with the time intervals between the following a Poisson distribution.
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):
where is the average time interval between the shots. provides the flat top for but has a fixed logarithmic slope of for higher frequencies (Fig. 3 a). Since the shot noise process is continuous in t, whereas the observed lightcurve is the result of integrating the measured counts over a finite number of intervals , must be corrected for binning. Lochner et al. 1991have shown, that binning makes the shot noise power spectrum even steeper for high frequencies.
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 (Fig. 3 b).
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998