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

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5. Summary and discussion

5.1. The LSSM[1] results - a generalization of the shot noise approach

We have shown that the short-term variability of the EXOSAT ME lightcurves of Cyg X-1 can be well described by a Linear State Space Model of first order, i.e. the dynamics of the system can be modeled as an autoregressive process with one temporal parameter - the relaxation timescale [FORMULA]. We find [FORMULA] to be (0.19 [FORMULA] 0.03) s for the ME energy-range. The relaxator is not found to be correlated either with time nor with the luminosity of the source, indicating that the physical process producing the emitted radiation is very stable and suggesting that the short-term variability is independent of the mass accretion rate of the system. The LSSM analysis in the time domain allows us to estimate the fraction of the countrate that is due to observation-noise. When analyzing the periodogram in the frequency domain this white noise component can only be represented by a constant whose extraction is not trivial (Belloni & Hasinger 1990a, Zhang et al. 1995). Furthermore, the problem of spectral leakage that has to be delt with in the frequency domain, is circumvented by fitting in the time domain. For these reasons our LSSM analysis delivers results with higher statistical significance than corresponding frequency domain fits (König & Timmer 1997).

These LSSM[1] results allow a much simpler description of hard state short-term variability than multi-timescale shot noise models. Although a detailed quantitative comparison is beyond the scope of this paper, the reproduction of the periodogram by the LSSM[1] (compared to its approximation by adding shot-profiles with different timescales, corrected for observation-noise and binning) as well as the greater sensitivity of the LSSM fitting procedure (compared to frequency domain fits, which are usually used to evaluate shot noise models) suggest that LSSM[1]s are better suited to describe the nature of the observed variability.

The LSSMs can model the different realizations of a stochastic relaxator [FORMULA], whereas shot noise models are generally restricted by the definition of special shot forms. Shot noise lightcurves therefore might be regarded as a subclass of LSSM lightcurves in the sense that a superposition of exponentially decaying shots can be interpreted as one possible realization of an intrinsic AR[1] process. The inspection of the measured lightcurves of Cyg X-1 implies that the source of the derived AR[1] process is indeed the stochastic superposition of individual shot events, corresponding to the basic idea of shot noise. We note that on larger timescales this kind of variability is also present in the X-ray emission of active galactic nuclei (AGN) (e.g. McHardy 1989, Mushotzky et al. 1993, König & Timmer 1997, König et al. 1997). The physical mechanism responsible for such a temporal behavior, however, is not yet understood.

5.2. The LSSM results in the light of time-dependent comptonization models

Recently, the discussion concerning accretion physics has begun to concentrate on the consideration of timing and spectral properties of the X-ray emission as two aspects of the same model (Kazanas et al. 1997, König et al. 1997, Wilms et al. 1997; and references therein). The spectrum of both, AGN and the hard state of galactic black hole candidates, is usually explained by inverse Comptonization, where soft X-ray photons, provided by a cold accretion disk, are upscattered by inverse Compton collisions in a hot plasma to produce the observed high energy power-law.

In this context it can be assumed that the observed X-ray temporal behavior is the result of the processing of short shots of seed photons within the accretion disk corona (Payne 1980). The initial photon pulse is hardened and temporally broadened while diffusing through the corona. As harder photons in the emerging spectrum have, on average, undergone more scattering events, they have stayed in the corona for a longer time than softer photons and the emerging pulse is comparatively broader (observed in AGN by König et al. 1997). In addition, variability structures in high energy lightcurves exhibit a characteristic time-lag with respect to those in low energy lightcurves (Fig. 7). Frequency-dependent time-lags have e.g. been observed in Ginga and RXTE observations of Cyg X-1 (Miyamoto et al. 1988, Miyamoto et al. 1992, Wilms et al. 1997). The question to what extent these lags are produced in a hot corona or wether they are intrinsic properties of the cold disk is still unresolved (Miyamoto et al. 1988, Miller 1995, Nowak & Vaughan 1996, Nowak et al. 1998). Nowak et al. 1998, however, show that the frequency-dependent time-lags of Cyg X-1 can be qualitatively reproduced using a simple propagation model (but they also point out that the longest observed time-lags ([FORMULA] 0.1 sec) can probably not be explained by Comptonization). Another challenge for combined spectral and temporal theories is given by modeling the coherence of variability structures in different energy bands which is observed to be near unity over a large frequency range in Cyg X-1 (Vaughan & Nowak 1997, Wilms et al. 1997).

[FIGURE] Fig. 7. Shot profiles for the energy-ranges 4-5 keV (dash-dotted line) and 12-23 keV (dashed line). The profiles have been computed for an accretion disk corona (ADC) model with an accretion geometry as described by Dove et al. 1997. This geometry is similar to an advection dominated flow (ADAF, Abramowicz et al. 1995, Narayan 1996). The ADC model parameters were chosen to be appropriate for Cyg X-1 (disk temperature: [FORMULA] =200 eV, coronal temperature: [FORMULA] keV, optical depth of the corona: [FORMULA] ; Dove et al. 1997).

Additional support for the time-dependent Comptonization model comes from the joint spectral and temporal analysis of AGN, which have similar properties to those of Cyg X-1. For a sample of AGN observations we were able to show that a linear correlation between the photon index [FORMULA] and the LSSM[1] timescale [FORMULA] exists, in the sense that harder spectra have a longer variability timescale (König et al. 1997). By comparing the observed [FORMULA] - [FORMULA] relation with Monte Carlo simulations of time-dependent Comptonization models, it is possible to scale the model geometry (König 1997). For the hard-state galactic black hole candidates a similar study cannot be performed since not enough objects are known. What can be done, however, is to check whether our best-fit LSSM models for Cyg X-1 could, at least roughly, be explained with a time-dependent Comptonization model.

We have computed shot profiles for several possible accretion geometries using a linear Monte Carlo code. The detailed results of these simulations will be discussed elsewhere. Since shot noise can be regarded as a special representation of an AR[1] process (see Sect. 5.1), we can use the computed shots to generate lightcurves based on these profiles, including an additional white noise component. In analogy to our treatment of the distribution of [FORMULA] seen in the observations, we derive a most probable value for [FORMULA] from the simulated lightcurve samples, [FORMULA]. We find [FORMULA], with [FORMULA] being the radius of the corona. The profile of a Compton-shot only depends on the relative size of the system, parameterized by the light travel time through one radius of the spherical corona, [FORMULA]. By identifying [FORMULA] with the measured value of [FORMULA] s, it is therefore possible to express the measured [FORMULA] in terms of the coronal radius. Our simulations give a first estimate of 320-640 Schwarzschild radii for the coronal radius (assuming a mass of 10 [FORMULA] for the black hole). Other authors estimate the size of the Comptonization cloud in the hard state to be [FORMULA] 100 Schwarzschild radii (Esin et al. 1997, ADAFs) or [FORMULA] 23 Schwarzschild radii (Nowak et al. 1998, from minimal time-lags). These values are only in rough agreement and a better understanding of the relation between spectral and temporal properties is needed to arrive at a consistent picture. We plan a more detailed study of the Compton-shot profiles as well as LSSM analyses of RXTE data to further constrain the accretion geometry of Cyg X-1.

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Online publication: May 12, 1998