1 Introduction

The intrinsic temporal flux variations of pulsars are probably an important clue to understanding their emission mechanism at any wavelength. In fact, the flux variations at high energies (optical, X-ray, $\gamma$-ray) might provide important constraints to the emission mechanism at radio wavelengths also (Cheng et al. 1986b; Kawai et al. 1991; Lundgren et al. 1995; Moffet & Hankins 1996; Patt et al. 1999). Patt et al. (1999) searched for period to period flux variations in about 105000 periods of Crab pulsar data at X-ray energies (in the range 1 to 10 kilo electron volts (KeV)), with a time resolution of 100 microseconds ($\mu$s), obtained by the PCA detector aboard the RXTE X-ray observatory. This work reports the results of analyzing 1868112 periods of Crab pulsar from the same instrument, with a time resolution of 3.815 $\mu$s, in the energy range 13.3 to 58.4 KeV.

The RXTE data archive was searched using the XDF tool, for public data acquired by the Proportional Counter Array (PCA). A uniform set of 23 data files were found observed during August/September 1996, their ObsId numbers ranging from 10203-01-01-00 to 10203-01-03-01. They were obtained in the EVENT mode (XTE_SE), combining events from all five Proportional Counter Units (D[0~4]), and also from both halves of all three Xenon anode layers of each PCU (X1L^X1R^X2L^X2R^X3L^X3R). Channels 50 to 249 of the PCA were also combined, which corresponds to the energy range 13.3 to 58.4 KeV.

The first phase of data analysis used the FTOOLS software. First, the Good Time Intervals (GTI) were obtained for each data file by using the MAKETIME tool on the corresponding XTE filter file; the selection criteria were (a) pointing OFFSET less than 0.02 $\hbox{$^\circ$ }$, (b) elevation (ELV) greater than 10 $\hbox{$^\circ$ }$, and (c) all five PCUs to be switched on. Next, the GTI extension of each data file was edited to insert the above GTI values. Then the FSELECT tool was run to filter out data outside these time ranges. Next the SEFILTER tool was run with the M[1]{1} option (without bypassing the FSELECT tool) to retain only the valid photon events. Then the FXBARY tool was run using the orbit file for that day, to convert the arrival times of photons from the Terrestrial Time system (TT) to the solar system barycenter system (TDB). Penultimately, the SEEXTRCT tool was used to obtain the light curve for each file, in time intervals of 1.010895 millisecond (ms), which is 265 times the basic time resolution of the data. Finally, the PCADTLC tool was used to correct the light curve for dead time of the PCA; before this the corresponding Standard 1 files were also converted to the TDB system for consistency.

Time samples having incomplete exposure were deleted. These occurred naturally at the beginning and end of each light curve, and also whenever the RXTE observatory shut off some PCUs, for technical reasons. Each light curve was then converted into the ASCII format for further processing.

The second phase of data analysis used self-developed software. First the power spectrum of each light curve was computed to obtain the period of Crab pulsar in that file. This was used to separate the light curve into individual periods (also called single pulses). Each period has 33 time samples (also called bins), giving a synthesized sampling interval that is different from file to file, but is $\approx$1.013967 ms. The above separation was done such that the photon counts in each original sampling interval were not split across more than one synthesized sampling intervals; otherwise the Poisson statistics of the data would be distorted, and would cause problems for some studies as discussed later on. The separation into individual periods for radio data is much simpler, since standard resampling techniques can be used.

Figure 1: Integrated profile of Crab pulsar after summing 1868112 periods from 23 files. The abscissa is time (also called phase) within the period (in ms), while the ordinate is the average number of photons obtained in one synthesized time sample (1.013967 ms). The mis-alignment of the profiles from file to file does not exceed half a synthesized time sample. The dotted curves represent the two peaks modeled as Gaussian.

The appropriate period for each data file was ascertained more accurately by checking for "drift'' of the integrated profile between the first and second halves of the light curve; this part was done iteratively (see Vivekanand et al. 1998 for details). A straight line fit, to the starting epoch (TDB system) of each data file versus the Crab pulsar period in that file, gives a period derivative of 4.208 ($\pm$0.005 $) \times 10^{-13}$ s/s, which compares excellently with the actual value for Crab pulsar. The standard deviation of the periods about the fitted straight line is $\approx$2 nano seconds (ns), which is consistent with the expected value; the data files contain typically 50000 to 98000 periods, and one can recognize a relative shift of a small fraction of the time sample between the integrated profiles of the first and second halves of a data file. However the periods differ systematically by $\approx$8 ns from those obtained using Crab pulsar's ephemeris. It is not clear to this author why this systematic difference should occur, but this does not affect the rest of the analysis.

Figure 1 shows the integrated profile of Crab pulsar for 1868112 periods. The integrated profiles of all files are aligned, correct to half a time sample, so that one can analyze them as if all 23 files have been obtained "in phase''. However it does smear the integrated profile to a maximum of half a synthesized time sample. Samples 5 to 30 are considered to represent the on-pulse window, and the rest of the seven samples the off-pulse window, although the Crab pulsar might emit X-rays all through its period.

Details of the analysis in the coming sections can be found in Vivekanand & Joshi (1997), Vivekanand et al. (1998), and Vivekanand (2000); they will be described only briefly in this article.