3. Analysis and results
The pulse period and the observed pulsed flux at 40 keV are plotted in Fig. 1 for TJD 8350 - 9650 (Fig. 1a and 1d, respectively). A linear fit to the pulse period gives an average value for of 2.12 s yr-1. A quadratic fit to the data gives a value of of 8.3 yr. Higher order polynomials do not improve the fit. To see the period variation in more detail, the residuals to the quadratic fit are shown in Fig. 1b. The pulsed X-ray flux is seen to increase by a factor of 4 for duration of 2 - 10 days compared to the average flux and by a factor of 2 for a duration of about 20 - 100 days. Since is obtained after fitting a OTTB spectrum, it is in fact a measure of the hard X-ray pulsed luminosity. Though there have been some indications of an anti-correlation between pulse fraction and total X-ray luminosity (Rao et al. 1994 ), the observed pulse fraction in the present spin-down era lies in a narrow range of 0.3 to 0.5. In fact, from a compilation of hard X-ray luminosity of GX 1+4 (Chitnis 1994 ) we find a positive correlation between X-ray luminosity and . Hence, in the following, we treat as a measure of the total X-ray luminosity.
To examine whether the luminosity is related to the pulse period variation, the instantaneous spin down rate is calculated for each of the data points by doing a linear fit to the neighboring 25 data points and this is shown in Fig. 1c. The similarity in the Figs. 1c and 1d led to further analysis of correlation and cross-correlation between and the pulsed flux .
3.1. Spin down rate and luminosity
To estimate the correlation of spin change rate and luminosity we choose only those values where the linear fit around that data point (for about 12 days) is acceptable (unlike Fig. 1c. where all the data points are included). As the regions of very high are of short durations, the determination of is not very reliable and hence we exclude those points from our analysis. The two quantities are positively correlated and we have calculated a correlation coefficient of 0.63 (for 102 data points) and the probability of no correlation in the given data set is estimated to be 10-12.
To investigate whether the pulse period variation is completely governed by the luminosity variation, we made an attempt to reproduce the pulse period history of GX 1+4 only from the luminosity history. The positive correlation seen between and is assumed to be the real torque transfer equation in the pulsar and the pulse period of the first data point is propagated with time depending on , using the linear relation obtained between and . For this purpose we have used those values even when the period determination is uncertain (Flag N in the archive). The resultant residuals in the period determination are shown in Fig. 2b. For comparison, we show in Fig. 2(a) the residuals to the period obtained by assuming a constant . The 100 - 200 days features in the upper plot is not present in the lower plot signifying that the pulse period changes are actually correlated to the luminosity. However, the reproduction of the pulse period for days later than TJD 9000 deviates from the observed one by up to about 0.5 s because of the lack of sufficient number of measurements. The rms deviation in the pulse period as estimated from only a constant (Fig. 2a) is 0.1 s and it improves to 0.04 s when pulse period is predicted from the - relation (Fig. 2b). The rms deviation reduced further to 0.03 s (which is the typical error in the period determination) for TJD 8370 to 9000 (where is well determined and well sampled). Hence, we can conclude that when is well sampled, all the variations in period can be explained correctly within the observational errors using a simple linear relation between and .
3.2. Time delay
The instantaneous spin-down rate and the pulsed flux were subjected to cross-correlation tests. For this purpose is calculated using two neighboring data points and the average value of is used. When the total data is taken we find a positive correlation between and at a confidence level of 99.4%. The reduced level of confidence is due to the fact that is calculated over 2 observations (unlike 12 data points used in the previous section). The correlation, however, was found to be delayed by a few days. To improve the confidence level, the total data are divided into several sets of 128 data points and the derived cross-correlation values are co-added. The resultant profile is shown in Fig. 3. The central part of the figure is shown in an expanded form in the inset to the figure. As can be seen from the figure, there is a clear asymmetry near 0. A Gaussian fit to the profile near 0 gives a of 20 for 35 degrees of freedom (dof) and the derived value of delay is (4.8 1.0) 105 s (5.6 1.2 days). The errors are calculated by the criterion of +2.3 (1 error for two free parameters). A constant fit to the profile gives 75 for 36 dof showing the existence of correlation at a confidence level of 99.99%. This confidence level improves further if the value of the constant is kept fixed at 0 (i.e., there is no correlation instead of constant correlation). A Gaussian fit with the centroid frozen at zero gives 54 signifying the existence of a delay at a very high confidence level (the value of being 34 for one additional parameter). Hence the co-adding method resulted in the detection of a delay at a high confidence level, and could be the reason for the lack of detection of any such delay by other workers (Chakrabarty 1996). The delay between and is seen for the first time in an X-ray pulsar.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998