## 2. The short-term X-ray variability of Cyg X-1## 2.1. The dataThe 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).
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-1The 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 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): - 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, . Variability of the -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, 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 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 modelTerrell 1972proposed that the structures observed in the lightcurves of Cyg X-1 might be due to a shot noise process , i.e. the superposition of randomly occuring shots with the shot profile : Here the are the times at which the shots occur, with the time intervals between the following a Poisson distribution. The standard shot profile is identical for all shots and consists of an instantaneous rise to height and an exponential decay with the decay time : 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
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 |