The uncertainty and the significance of the pulse period derived using the Z2 statistics in each of the XRT observations reported in Sect. 2.2, could be reliably estimated only by performing Monte Carlo simulations, as the individual determinations obtained by scanning frequencies from a single event file are not independent. Furthermore, we verified that the formal significance of the Z2 peak is not a good estimate of the reliability of the measurement for the Swift observations. The latter comprise several relatively short (≲ 1 ks) snapshots containing only a few hundreds photons each and, as a result, the corresponding periodogram has the shape of a wide-peak (the diffraction figure of the window function) with evident interference fringes produced by the fragmentation of the observations (see Fig. A.1).
Periodograms obtained from the Z2 statistics applied to the XRT observations reported in Sect. 2.2. The OBSIDs are indicated by the two last digits in each panel.
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Spin periods derived from the simulated event files for each of the XRT observation reported in Sect. 2.2. The OBSIDs are indicated with the two last digits for each panel.
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To perform the Monte Carlo simulations, we determined in each observation the best pulse period and approximated the corresponding pulse profile by using its first two Fourier components (a check was performed a posteriori to verify that this description is accurate in each case by inspecting the single pulse profiles; see Fig. 2). Event files with an average event rate equal to that measured from the source were then simulated in the good time intervals (GTIs) of the observations by introducing two sinusoidal modulations at the estimated spin period and its first harmonic. The amplitudes of the modulations were assumed to be the same as those measured from the Fourier decomposition of the real source pulse profiles. In each of the simulated event files, the pulse period was then measured with the Z2-statistics method and recorded. We report the results of this analysis in Fig. A.2.
The spin periods determined from the simulated files cluster around the real value measured in the corresponding observations and follow roughly a Gaussian distribution. The shape of the Gaussian could in principle be used to estimate in each case the uncertainty in the spin-period determination, but this procedure is complicated in all cases by a certain number of “outliers”. The number of these points, whose “wrong” value of the spin period is due to the sporadic displacement of the peak in the Z2-statistics periodogram at the moment of the best period determination, is larger for shorter observations and observations characterized by a smaller number of source events. In Bozzo et al. (2011), we argued that the number of outliers with respect to the total number of measurements could give an estimate of the reliability (significance) of the spin-period detection in an observation.
To identify the outliers in the present case, we: (i) collected all the spin periods determined from the 90% of the simulations that gave periods closer to those measured from the real data (“central spin periods”), (ii) estimated the average spin period and the variance σ2 of the corresponding Gaussian distribution, and (iii) identified all the remaining realizations that gave a spin period differing by more that 2.6σ from the averaged one. We rejected as “not-reliable” the XRT observations in which the number of outliers were found to significantly exceed (99% c.l., taking into account also the intrinsic uncertainty in σ and the Poissonian nature of the key variables; see e.g., Sheskin 2007) the value expected from the Gaussian distribution of the simulated central spin periods.
As a final remark, we note that, in all cases, the timing analysis was performed on XRT event files that were uncorrected for pile-up (see Sect. 2), as the latter is known to affect
mostly the spectral energy distribution of photons recorded from the source4. We checked a posteriori that an analysis similar to that described above performed on the pile-up corrected XRT event files of the reliable spin-period determinations in observations 00032293001 and 00032293002 would give values fully in agreement (to within the uncertainties) with those reported in Table 4. For the observations 00032293004 and 00032293008, the lower flux and/or number of counts led, in contrast to an unreliable spin-period determination once the corresponding data are corrected for pile-up. As the effect of pile-up on these observations cannot be checked further owing to the limited statistics, the corresponding values of the spin periods reported in Table 4 should be interpreted with caution.
© ESO, 2012