## 3. Analysis and resultsThe 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
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 luminosityTo 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 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 delayThe 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) 10
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |