Astron. Astrophys. 363, 1013-1018 (2000)

2. Observations, data reduction and deadtime corrections

For our analysis we used the data from the Proportional Counter Array (PCA) aboard RXTE. To study the source variability in the hard spectral state, we chose a compact group of observations of Cyg X-1 performed on 15-17 Dec. 1996 (proposal P10236) with total exposure time of 80 ksec and 56 available observations of Cyg X-1 from the proposal P30157 performed between Dec. 1997 and Dec. 1998 with total exposure time of 110 ksec. The soft spectral state data were accumulated on 4-18 June 1996 data (proposal P10512, see also Cui et al. 1997) with total exposure time of 10 ksec. All observations were performed with 5 PCUs switched on.

The power density spectra were calculated from the light curves with the bin duration equal to the intrinsic time resolution of the data (ss in proposals P10236 and P10512 and ss in proposal P30157). Some deviations from the noise level predicted by the PCA dead time models are observed in the power spectra constructed in the PCA energy sub-bands with the amplitude of (rms/mean)²/Hz. We therefore followed recommendations of the RXTE GOF and analyzed the light curves in the total PCA energy band, where these deviations seem to be considerably less significant (William Zhang, private communication).

The power spectra calculated for individual time series were normalized to units of squared fractional rms (Miyamoto et al. 1991)

and averaged. In the above formula is the Fourier amplitude at the frequency ; - is total observed number of counts in the time series; - averaged observed count rate for the time series. Note that the ideal Poissonian noise component is already subtracted, therefore for a Poissonian distribution of counts in the time series.

In a real detector various effects could lead to modification of the Poissonian noise level by an amount . Two major effects contribute to in the case of PCA detector:

where is a modification of the noise level due to counts dead time, - additional noise component resulting from vetoing of the PCA detectors by the Very Large Events (VLE).

With an accuracy sufficient for our purpose the PCA dead time caused by the incident photons can be described as a non-paralizable process ¹ (Jahoda et al. 1996), for which modification of the noise level was derived by Vikhlinin et al. 1994 (see also Zhang et al. 1995 for the expression accounting for finite length of the time series). We shall use below Eq. (A4) from Vikhlinin et al. 1994 transformed to the units of squared fractional rms:

where

Here R is the observed count rate, is the photon dead time, is the bin duration, f is the frequency. For the practical purposes of the order of 10 is sufficient. This equation (for ) is valid for an infinitely long time sequence. One can use more complicated expression (see Eq. (44) from Zhang et al. 1995) if the analyzed time sequence contains only small number of bins.

To the first approximation the Very Large Events lead to appearance of an additional (positive) component (Zhang et al. 1996)

where - the duration of the VLE window, - the VLE count rate in one PCU. Note that the above expression does not include the binning and sampling effects and is valid in the limit .

Finally, the total noise component model in the first approximation would be:

where - the total observed count rate in one PCU, - number of PCUs. The and are given by the Eqs. (1) and (2) respectively. The factor accounts for the fact that the count streams from independent units were merged together. The factor accounts for reduction of the observed count rate due to Very Large Events. This formula is valid under the following conditions: , and . It is important to stress out that in the adopted dead time model is the the total observed count rate of the events causing the dead time (as opposite to the observed count rate in the analyzed time series). In the case of PCA detector it should include total Good Xenon rate, Propane and Remaining counts. We also note that for sufficiently small count rates, 3-5 kcnts/s/PCU dependence of and, therefore, of upon R vanishes. The model has been tested on a series of Monte-Carlo simulations and has proven to be sufficiently accurate in the parameters range of interest.

Note that we did not include in our model an additional background term (Jahoda 1998 or Jernigan et al. 2000). For bright sources (1 kcnts/s/PCA) its contribution can be neglected.

To verify our model for the noise component we compared it with the data of PCA observations of an extremely bright source Sco X-1 (100 000 cnts/s/PCA) for which all flavors of the dead time distortions are much more prominent than for Cygnus X-1. We used 3 different observations of Sco X-1- 10059-01-01-00 (Feb. 14, 1996; hereafter #1), 10059-01-03-00 (Feb. 19, 1996; #2) and 10056-01-02-02 (May 26, 1996; #3), having different value of the VLE window and different time resolution. Ideally, once the values of the dead time and VLE window are calibrated, the model should reproduce the noise level with count rates and set to the measured values. However, as we discuss below, this is not strictly the case. We therefore followed the approach adopted by Jernigan et al. 2000 and fitted the power spectra at high frequencies, Hz, with the model for the noise component plus the model for the source component leaving the noise model parameters free. The source contribution in this particular case was modeled as a superposition of two Lorentzians representing kHz QPOs. This approach has a disadvantage that it requires a priory assumptions about the shape of the power spectrum of the source. In particular, if the source has very weak and very flat power spectrum component it might be not detected by our procedure.

In Fig. 1, Fig. 2 and Fig. 3 we present the power spectra of Sco X-1 in observations #1 #2 and #3 along with the best fit noise models and residuals. The best fit parameters and their expected values are given in Table 1. For observation #3 we fixed parameters and because the limited frequency range of the power spectrum (4096 Hz) does not allow us to constrain their values from the fit. From Fig. 1, Fig. 2 and Fig. 3 one can see that with appropriate tuning of the parameters the adopted noise model is capable to reproduce the observed shape of the noise component.

Fig. 1. The power spectrum of Sco X-1 from the observation #1 performed with medium VLE window. The solid line shows the best fit model consisting of the noise level model and two kHz QPOs (see text). The dashed line shows the noise level component due to photon dead time. The lower panel shows the residuals data-model.

Fig. 2. The same as Fig. 1 but for observation #2, performed with short VLE window.

Fig. 3. The same as Fig. 1 but for observation #3, performed with short VLE window and lower time resolution, s.

Table 1. The expected and best fit model parameters of the deadtime correction for 3 observations of Sco X-1.
Notes:
a) e.g. Jahoda et al. 1996, Jahoda et al. 1997, Jahoda 2000
b) measured total count rate (Good Xenon, Propane Counts and Remaining Counts) per PCU
c) measured Good Xenon count rate per PCU
d) preflight values for all PCA, see Giles 1995
e) measured count rate (HouseKeeping data)

In general the best fit parameters of the noise level model (Table 1) look reasonable and are close to the expected values. The best fit value of , in observation #2 is by higher than the preflight value. It is possible, however, that the duration of the VLE window differs from the preflight values (Alan Smale, private communication). The most apparent difference is a large, by a factor of , discrepancy between the best fit and observed total count rate. But we should note that the observed GoodXenon count rate is much closer to the best fit value (see Table 1). The reason of this discrepancy is not clear, it might be due to unaccounted processes in the PCA detectors at high count rates. We note however, that dependence of the noise component (in units of squared fractional rms) on the total count rate is a specific feature of a non-paralyzable dead time process at sufficiently high count rates. At lower count rate (3000-5000 cnts/s) and in particular in the case of Cyg X-1 (1000-1500 cnts/s) this dependence vanishes.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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