Astron. Astrophys. 321, 776-790 (1997)

3. Analysis

For the spectral analysis we made CDs and HIDs. The choice of the bands for the CDs and HIDs was dictated by three concerns: (i) we want approximately the same number of counts in each band to optimize the statistics, (ii) all energy boundaries should coincide with the energy channels of the MPC3 mode, since we do not want to interpolate in such broad channels, (iii) the highest energy bands should be high enough to be sensitive to a change in the power law tail in the spectrum, since this may be a good diagnostic tool for the analysis of BHCs. We find that a good balance between these considerations could be obtained with the following choice of the energy channels: 2.3-4.6 keV, 4.6-6.9 keV, 6.9-11.5 keV, and 11.5-23.0 keV.

For the timing analysis we made FFT's of data segments with a length of 256 seconds of data with a highest time resolution of 1 ms (PC data, other data have lower time resolution). This results in power spectra which have a range from 0.004 to 512 Hz. For data with a lower time resolution (MPC3 and MPC2 data) the Nyquist frequency is accordingly lower. The lowest Nyquist-frequency we used was 8 Hz (for the MPC2 data in high bit rate mode). All power spectra were normalized to obtain the power density in units of (fractional rms)² /Hz or (rms/mean)² /Hz (see Van der Klis 1994c for a recipe how to obtain such a normalization, see also Belloni and Hasinger 1990, and Miyamoto et al. 1991). For the background count rate we used 68 counts/second (Turner et al. 1989). Variations in the background result in errors in the normalization of the order of 5%.

We fitted the power spectra with a model consisting of three components: (i) A Lorentzian component which is centered at zero frequency and which has a full width at half maximum (FWHM) of 0.1 Hz; (ii) A Lorentzian component with a zero central frequency and a FWHM of a 2-3 Hz and (iii) for the spectra which have a Nyquist frequency of 512 Hz, a Lorentzian component centered around 30 Hz. This model gives a satisfactory description of the shape of the power spectra (Fig. 8 and 9) and follows the 'wiggles' in the power spectrum. However, the values which are found for the are formally too high for the fits to be acceptable ( for 81 d.o.f., and for 45 d.o.f.). We note that a similar can also be obtained with other fit models (e.g. a power law with different slopes in the various frequency intervals), but almost always more free parameters are needed. This is probably caused by the fact that the "bump" around 2-3 Hz is most easily fitted with a Lorentzian. To obtain acceptable fits to power spectra which are obtained in the first 10 days of the outburst we found it necessary to include a power-law component in the fit-model. This power law is needed to obtain a good fit to the lowest frequency points and has indices of 2.

The Poisson level was subtracted from each power spectrum before we made fits to it. The method we used to determine the Poisson level takes into account dead time, and has been described by Mitsuda & Dotani (1989). We find that the dead time calculation is sufficiently accurate to predict the Poisson level for the power spectra at the observed count rates.

We have checked that power introduced by changes in pointing of the instrument is negligible. We did this by making a power spectrum of the raw uncorrected data and a power spectrum of the data which had been corrected for aspect (and dead time and background). We found that the difference is very small (less than 0.001 ((rms/mean)² /Hz) at 0.005 Hz; this approximately corresponds to an rms fractional variation of less than 0.3% for frequencies below 0.01 Hz).

We also made FFT's of 256 second long data segments of the energy resolved data (PC, MPC2, and MPC3 data). We used these FFT's of data obtained at different energies to study the coherence and phase delays of the variability between different energy bands (see Vaughan , 1994). We use the technique ( -technique) in which all FFT's at all energies are used simultaneously to extract the phase delays between these energy bands. These FFT's were also used to construct power spectra in different energy bands. From these power spectra we constructed rms spectra, which give the fractional variability of the various spectra components as a function of energy.

We found that in some data sets occasional spikes were present which greatly increased the variability in one energy channel (or a few non-neighbouring channels) only. These spikes can not be astrophysical in origin, since they occur in a narrower energy band than the detector energy resolution ( 20% at 6 keV). We also find that these spikes are only present above 10 keV and occur more often at higher energies. When we made the final rms spectra (and phase delay spectra) we filtered the data which were used to make the power spectra. This was done by calculating the local mean of a few points (for the MPC2 data with 61.5 msec resolution we used 9 points, i.e. an average over 0.5 s), and then calculated the level which has a very low probability to be exceeded (we took 1.10^-10) based on Poisson statistics. All data points above this level were substituted with the mean level. Per FFT only a few (less than 10) spikes were generally found, and due to filtering them out the power was only slightly diminished (normally less than 1 percent); we concluded this from a comparison to the neighbouring energy channels. From a comparison of rms-spectra made with the original (unfiltered) data and rms-spectra made from the filtered data, we conclude that spikes are removed very well, while real source variations are not or only very slightly affected.

Another analysis we performed on the data is the calculation of high-to-low flux X-ray spectral ratios. In order to do this we calculated the running mean of the dead time and background corrected data (over 64 s) and accepted the spectra when the instantaeous flux was above a certain flux level (we find that 1.1 times the running mean works well) to make PHA-spectra of the high-intensity data. This was done with data obtained in the MPC2 and MPC1 mode, resulting in 48 channels spectral resolution. We also made PHA-spectra of the low intensity data by taking the instantaneous count rates below 0.9 times the running mean. In this way we obtained dead time and background corrected PHA-spectra at high count rates, which contained about 20-30% of the total data, and PHA-spectra at low count rates, which contained about the same amount of data. The PHA-ratio spectrum was then calculated as the ratio of the spectrum obtained at the peaks to the spectrum obtained at the valleys. These PHA-ratios give us an additional way to look at the energy dependence of the flux variations.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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