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Astron. Astrophys. 330, 181-188 (1998)

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3. Analysis and results

3.1. Power density spectrum

The light curves show dramatic differences in the two spectral states when seen in 1 s time resolution (see Fig. 2). In the soft state there are large intensity variations at seconds to minutes times scales. To quantify these differences we have obtained the power density spectrum (PDS) for the two states. PDS are obtained for individual data segments and then co-added, using the XRONOS software package. The PDS are normalized to the squared fractional rms per unit frequency. In Fig. 3, the observed PDS is shown for the hard state (1996 May). The data were acquired in 1 s mode as well as 0.4 ms mode and the PDS were generated separately for each of the observations and co-added. As can be seen from the figure, the PDS at low frequencies (0.01 Hz to 0.3 Hz) is very flat with a power-law index of -0.09. At higher frequencies, the PDS steepens and it has a power-law index of -1.1, with a break frequency of 0.23 Hz. The rms variability at the break frequency is 26%. There are indications of additional structures at higher frequencies. A single power-law for frequencies above 0.23 Hz gives a [FORMULA] of 197 for 65 degrees of freedom and the [FORMULA] improves to 109 if we allow for one more steepening at higher frequencies. The fitted parameters are: additional break frequency at 2.5 Hz, power-law index between 0.23 and 2.5 Hz is -1.0, power-law index between 2.5 and 10 Hz is -1.9. In Fig. 4, the observed PDS for the 1996 July observations (the soft state) are shown. The power-law index for frequencies above 0.03 Hz is -0.39 and there is no indication of a break in the slope.

[FIGURE] Fig. 3. The power density spectrum (PDS) for the hard state of Cyg X-1 obtained during 1996 May from the PPC observations as well as from the ASM data.
[FIGURE] Fig. 4. Same as Fig. 3, but the data are for the 1996 July (soft state) observations.

The general trend of the observed PDS is similar to that obtained using the PCA data (Belloni et al. 1996; Cui et al. 1997a). In the soft state the PDS is a simple power-law. In the transition state, the PDS shows a break at around 0.3 - 0.7 Hz, and the power-law index is between -0.6 to -0.3 below this frequency. The PDS obtained by us in the hard state, though has a similar break frequency, is much flatter at lower frequencies (power-law index -0.09). It is interesting to note that the PDS shape obtained by us in the hard state agrees very well with that traditionally obtained in the hard state by other observers. Belloni & Hasinger (1990a) have analyzed about 30 EXOSAT observations on Cyg X-1 and found that the PDS is very flat below a break frequency, follows a power law with slope about -1 up to about 1 Hz, and then steepens to a slope of roughly -2. This is very similar to the behavior detected by us. The break frequency shows a negative correlation with the rms variability at the break frequency (Belloni & Hasinger 1990a) and the value of the break frequency (0.23 Hz) and the rms variability (26%) obtained by us, agrees very well with this correlation. It is interesting to note that the PDS in the hard state has a similar shape at all times, including immediately prior to a state transition, as observed by us. During the transition period, the PDS parameters show a trend of change such that the power at low frequencies increases and comes close to that seen in the soft state.

One of the major differences between the PDS in the soft and the hard states is the power-law index below 0.2 Hz. If the same shape extends to still lower frequencies, the variability characteristics should be substantially different in the two states at time scales of a few days. To investigate this, we have obtained the archival data from the ASM on-board the RXTE, in the hard and the soft states, centered around our observations. We have also obtained the contemporaneous ASM data on Crab, for calibration purpose. We find that in the hard state, Cyg X-1 has a average count rate of 35.9 s-1 (compared to 75.8 s-1 in Crab). The rms variability is 25%, compared to 3.1% for the Crab (which can be taken as due to observational errors). In the soft state Cyg X-1 shows 25% variability (with an average count rate of 76.1 s-1) compared to 3.5% for the crab (74.4 s-1 average count rate). Hence we can conclude that there is no significant difference in the variability over time scales of days in Cyg X-1 in the two spectral states. To compare this variability to the higher frequency range, we have obtained the PDS for the ASM data and shown these also in Figs. 3 and 4. It should be noted here that the ASM data are not contiguous and hence the PDS may not be an accurate representation, but the large rms value is reflected in the PDS also. The power-law index for frequencies between 10-5 Hz and 0.02 Hz is estimated to be -0.98 for the soft state and -1.02 for the hard state. Since the EXOSAT data (Belloni & Hasinger 1990a) showed that the hard state PDS is flat all the way down to 10-3 Hz, the very low frequency variations may be unrelated to the spectral states and may have a common source mechanisms for the two spectral states.

3.2. Shot statistics

Traditionally, the time variability in Cyg X-1 was sought to be explained by models invoking random occurrences of shots. The hard state PDS were qualitatively explained by Belloni & Hasinger (1990a) by a shot noise model with a distribution in shot times. The observed distribution of shots in Cyg X-1 has also been explained under the premises of self-organized criticality (SOC) model (Mineshige et al. 1994a). In this model it is assumed that the inner portions of accretion disk are composed of numerous small reservoirs. If a critical mass density is reached at some reservoir, an instability gets established. This model predicts a power-law distribution of shot energy and duration (Mineshige et al. 1994b). The observed distribution of shots are, however, exponential (Negoro et al. 1995), and it was reconciled with the SOC model by assuming a gradual mass diffusion in the reservoir. To investigate whether the shot distribution obtained by Negoro et al. (1995) has a similar shape in both the spectral states, we have subjected our data to a shot distribution analysis. We have taken the total energy in the shots and the shot width rather than the shot peak intensity. For this purpose, the light curves are taken with 1 s binning. Background subtracted counts are compared in each bin with a running average of length 20 times the bin width. When the observed counts in a bin exceed the average value, a shot is deemed to have started. All the successive bins are considered to be a part of the same shot if all of them have count rates in excess of the running average. The duration and the excess counts in each shot are calculated and histograms are generated. The excess counts are normalized to the average count rate such that the excess counts are expressed in equivalent width in seconds.

The number distribution for the shot equivalent width is shown in Fig. 5 for the hard state observations. The distribution for Poissonian noise of equivalent average count rate is much lower than the observed distribution. We find that the distribution can be described by an exponential function with an e-folding constant of 0.35 s. We find that other simple functional forms like power-law etc do not fit the observations.

[FIGURE] Fig. 5. The shot statistics of Cyg X-1 for the hard state (1996 May) observations. The total number of shots of a given total excess counts (normalized to the local mean) are plotted against the excess counts (differential plot). The units of the excess counts are equivalent width in seconds. The best fit exponential function is shown as a straight line, and the derived time constant is 0.35 s. Time resolution of the light curve used for generating shot statistics is 1s.

A similar shot statistics is shown in Fig. 6 for the soft state observations. The distribution is much steeper with an e-folding constant of 0.2 s. Apart from this, there are a large number of high intensity shots over and above that described by the exponential function. We have also generated the shot distribution for the soft state when the count rates are taken in 0.1 s bins (Fig. 7). The distribution is steep with an e-folding constant of 0.04 s. We also find a shot with large excess counts (with an equivalent width of 0.92 s). The formal probability that this shot occurred at random assuming a shot occurrence rate described by an exponential function is 2 [FORMULA] 10-7. This shot was detected by PPC-1 on 1996 July 5. The light curve for this shot is shown in Fig. 8 for the total counts (top panel), 2-6 keV counts (middle panel) and 6-18 keV counts (bottom panel). The shot can be described by exponential rise and decays with a time constant of 0.4 s.

[FIGURE] Fig. 6. Same as Fig. 5, but the data are for the 1996 July (soft state) observations. The exponential fit has a time constant of 0.2 s.

[FIGURE] Fig. 7. Same as Fig. 6, but the time resolution for the flare statistics generation is 0.1 s. The exponential fit has a time constant of 0.04 s.

[FIGURE] Fig. 8. The details of the largest shot observed during the soft state (1996 July). The observed counts per bin (for a bin size of 0.1 s) are shown for PPC-1 total counts (top panel), 2 keV - 6 keV counts (middle panel) and 6 keV - 18 keV counts (bottom panel). The time is given in seconds starting from 1996 July 5 14:21 UT.

The shot width distribution also shows an exponential distribution. We find that in the hard state it has a time constant of 1.4 s, and the distribution becomes flatter in the soft state with a time constant of 2.2 s. We also find that the distribution cannot be described by other simple functional forms like power-law etc.

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998