Pulsar Timing Studies

Pulsar timing studies involve accurate measurements of the time of arrival of the pulses, followed by appropriate modelling of the observed arrival times to study and understand various phenomena that can effect the arrival times.

The first step of accurate estimation of arrival times is achieved as follows. First, at each epoch of observation, data from the pulsar is acquired with sufficient resolution in time and frequency and over a long enough stretch so that a reliable estimate of the average profile can be obtained. The effective time resolution should be about one-thousandth of the period. Second, the absolute time for at least one well defined point in the observation interval is measured with the best possible accuracy. Traditionally, atomic clocks have been used for this purpose. With the advent of the Global Positioning System (GPS), absolute time (UTC) tagging with an accuracy of $\sim$ 100 nanosec is possible using commercially available GPS receivers. Third, the fractional phase offset with respect to a reference epoch is calculated for the data at each epoch. This is generally best achieved by cross-correlating the average profile at the epoch with a template profile and estimating the shift of the peak of the cross-correlation function. This shift, in units of time, is added to the arrival time measurement to reference the arrival times to the same phase of the pulse. Fourth, the arrival times measured at the observatory on the Earth are referred to a standard inertial point, which is taken as the barycenter of the solar system. These corrections include effects due to the rotation and revolution of the Earth, the effect of the Earth-Moon system on the position of the Earth and the effect of all the planets in the Solar System. Relativistic corrections for the clock on the Earth are also included, as are corrections for dispersion delay at the doppler corrected frequency of observation. Last, a pulse number, relative to the pulse at the reference epoch, is attached to the arrival time for each epoch. This can be a tricky affair, since to start with the pulsar period may not be known accurately, and it is possible to err in integer number of pulses when computing the pulse number. To avoid this danger, a boot-strapping technique is used where the initial epochs of observations are close enough so that, given the accuracy of the period, the phase error can not exceed one cycle between two successive epochs. As the period gets determined with better accuracy by modelling the initial epochs, the spacing between successive epochs can be increased. The net result of the above exercise is a series of data pairs containing time of arrival and pulse number, both relative to the same starting point.

The second step in the analysis is the modelling of the above data points. This is usually done by expressing the pulse phase at any given time in terms of the pulsar rotation frequency and its derivatives as follows

$$\phi_{i} ~~=~~ \phi_{0} \, +\, \nu_{0}t_{i} \, +\, \dot{\nu_0} {t_{i}}^{2}/2 \, +\, \ldots ~~~,$$ (17.7.5)

where $\nu_{0} \,=\, 1/P$. Least squared fits for $\phi_{0},\,\nu_{0},\,\dot{\nu_{0}}$ etc., can be obtained from such a model. In addition, by examining the residuals between the model and the data, other parameters that effect the pulsar timing can be estimated. These include errors in the positional estimate of the pulsar, its proper motion, perturbations to the pulsar's motion due to the presence of companions, sudden changes in the pulsar's rotation rate etc. In fact, good quality timing observations can be used to extract a wealth of information, including stability of pulsars vis-a-vis the best terrestial clocks!