3 Pulse to pulse flux variations

The last section discussed the flux variations of Crab pulsar over time scales of hours and days. The current section discusses flux variations from period to period.

3.1 Spectrum of flux variations

Figure 3 shows the so called X-ray fluctuation spectrum of Crab pulsar. At each of the 33 samples (phases) of the integrated profile of Fig. 1, a time series was formed comprising of the X-ray flux as a function of the period number in the data file. This was Fourier transformed in arrays of length 128 $\times$ 1024 periods. The data were centered in the array and zero padded, and then a Hamming window was applied. To remove long term variations, the data in blocks of 32768 periods were normalized with the mean value of this block (see Ritchings 1976 and Vivekanand & Joshi 1997 for details). Fourteen data files were chosen that had at least 75000 periods each, totalling to 1347028 periods. Figure 3 shows the power spectrum averaged over the 33 spectra, after normalizing each spectrum with the variance of its time series. A polynomial of the form

y = a₀ + a₁ x + a₂ x² + a₃ x³ (1)

was fit to the power spectrum in Fig. 3. the coefficients are $a_0 = 0.999 \pm 0.001$, $a_1 = 0.003 \pm 0.013$, $a_2 = -0.003 \pm 0.059$ and $a_3 = -0.004 \pm 0.077$. The standard deviation of the power spectrum with respect to the above fitted curve is 0.046, which is mainly determined by photon noise. It is clear that none of the coefficients are significant except the first.

Figure 3: Average power spectrum of pulse to pulse X-ray flux variations of Crab pulsar. The time series, consisting of X-ray photons as a function of the period number, was Fourier transformed at each of the 33 phases of the integrated profile; the modulus squared of the 33 transforms was averaged. The abscissa in is units of inverse periods, extending up to 0.5 due to the Nyquist criterion. A small range of frequencies has been removed around the sampling spectral feature ($\approx$1/10 periods), and two of its harmonics.

The upper limit to any possible weak and broad spectral feature that might be hidden in the data can be computed to be $\approx$0.06% of the total power in the spectrum. Therefore Fig. 3 is consistent with the Crab pulsar having no spectral feature in its pulse to pulse X-ray flux variations.

3.2 Modulation index

The radio flux of rotation powered pulsars varies significantly from pulse to pulse; this could be due to intrinsic flux variation of the sub pulse, as well as random position of the sub pulse within the on-pulse window (here one is ignoring the flux variations due to propagation in the interstellar medium). This is characterized by the so called modulation index $\mu$, defined as

$$\mu = \frac{\sigma_I}{\langle I \rangle}$$ (2)

where $\langle I \rangle$ and $\sigma_I$ are the mean and the standard deviation of the pulsar flux (see Manchester & Taylor 1977); $\mu$ represents the fractional flux variation of the pulsar, and is usually greater than 1.0 at radio wavelengths.

Figure 4 shows the square of $\mu$, which is the natural quantity to average, for all 1868112 periods. First, the mean $\langle I \rangle$ and the variance $\sigma^2$ are computed at each sample in the integrated profile ( $\langle I \rangle$ is plotted in Fig. 1 and the top panel of Fig. 4). Now $\sigma^2$ has contribution from two sources:

$$\begin{array}{ll} \sigma^2 & = \sigma_K^2 + \sigma_I^2 \\ & = \langle I \rangle + \sigma_I^2 \\ \end{array}$$ (3)

where $\sigma_K^2$ is the variance due to Poisson statistics of photons, and equals the mean number of photons $\langle I \rangle$, while $\sigma_I^2$ represents the fluctuation of the average intensity of the pulsar; the two are referred to as photon noise and wave noise, respectively (Goodman 1985). The modulation index squared in Fig. 4 was estimated by subtracting the mean intensity $\langle I \rangle$ from the estimated variance $\sigma^2$, and then dividing by $\langle I \rangle^2$ at each sample of the integrated profile. Because of the PCA dead time correction to the data, the photon rate of Crab pulsar in any time sample is about 6% higher than the corresponding integer value of photons. This correction was estimated self-consistently by averaging $\sigma^2 - \langle I \rangle$ over all 33 samples of the integrated profile, for each file. The average value of the correction for all 23 files is 1.062 with standard deviation of 0.003. The mean flux $\langle I \rangle$ at each sample was scaled by this constant (for that data file) before subtracting from $\sigma^2$. This constant is not dependent upon the average flux $\langle I \rangle$ at each sample, so the above procedure is unlikely to introduce artifacts in the $\mu ^2$ of Fig. 4.

Figure 4: Square of the estimated modulation index ($\mu ^2$) of the Crab pulsar X-ray flux, at each sample of the integrated profile. The top panel is the same as Fig. 1. The vertical bars represent two standard deviation error bars. Since $\mu ^2$ depends upon the small difference between two much larger quantities (Eq. (3)), it can be negative also depending upon measurement errors. The distribution of these negative values about the value 0.0 is consistent with their rms errors, as expected.

The average value of $\mu ^2$ in Fig. 4 is $-0.0002 \pm 0.0040$, which is is consistent with zero as expected. The $\chi^2$ of $\mu ^2$ of Fig. 4 with respect to the expected value of 0.0 is 76.3 for 33 degrees of freedom. Removing sample number 10 reduces the $\chi^2$ to 57.0, and further removing sample number 11 reduces it to 48.7, which is just 2.25 standard deviations away from the expected value of 31.0. Therefore to the zeroth order of accuracy it is reasonable to assume that $\mu ^2$ is the same (i.e., 0.0) for all samples in the integrated profile. Its standard deviation is 0.0024 in the on-pulse window; then the standard deviation of $\mu$ can be estimated as $\sqrt{0.0024 / 2} \approx 3.5$% (by simple algebra). Thus the rms X-ray flux variation at any phase in the on-pulse window of the integrated profile of Crab pulsar is $\approx$3.5% of its mean value. Then the rms variation of the total on-pulse X-ray flux of the Crab pulsar will be $\approx$3.5/ $\sqrt{26} \approx 0.7$%. A similar calculation for the entire integrated profile gives $\sqrt{0.0023 / 2} / \sqrt{33} \approx 0.6$%, which is essentially the same result. This is a much tighter limit than the $\approx$7% quoted by Patt et al. (1999). These authors used totally 105000 periods and a different method of analysis, on account of which their result might be dominated by photon noise. The result of this section should ideally reflect the actual X-ray intensity variations of Crab pulsar (wave noise), that contain information about the physics of the X-ray emission mechanism.

To the next order of accuracy $\mu ^2$ appears to be correlated with the shape of the integrated profile; both the lower panel of Fig. 4 as well as the $\chi^2$ discussed above point to this. The $\mu ^2$ at the phase of the second peak of the integrated profile also appears to be enhanced. The on-pulse and off-pulse $\chi^2$ are 69.0 and 7.3, for 26 and 7 degrees of freedom respectively; the former is 6 standard deviations away from the expected value, which is quite high. However more data, or better analysis, is needed to confirm this with good statistical significance.

3.3 Giant pulses

Comparison of photon counts in the off- and on-pulse windows of Fig. 1 shows that the Crab pulsar does not emit "giant pulses'' at X-ray energies. The mean on-pulse window photon rate is 17.57 photons in 26 time samples, while the maximum is about 31 photons. This implies that at X-ray energies the Crab pulsar emission occasionally increases by $\approx$31.0/ $17.57 \approx 1.76$ times, at the most, whereas at radio wavelengths the giant pulse energy is about 10 to 100 times its mean value (Lundgren et al. 1995). Further, the on-pulse photon distribution fits a Poisson distribution very well, and there is no discernible excess probability at higher photon rates. Finally, following the method of Ritchings (1976) and Vivekanand (1995), a deconvolved photon distribution was obtained that represents the true on-pulse photon distribution of the Crab pulsar; it also does not indicate the presence of an excess probability at higher photon rates.