## 4.2 The timing model

Ideally, in order to model the rotational behaviour of the neutron star, we require TOAs measured by an inertial observer. An observatory located on Earth experiences accelerations with respect to the neutron star due to the Earth's rotation and orbital motion around the Sun and is therefore not in an inertial frame. To a very good approximation, the centre-of-mass of the solar system, the solar system barycentre, can be regarded as an inertial frame. It is standard practice [103] to transform the observed TOAs to this frame using a planetary ephemeris such as the JPL DE200 [224]. The transformation is summarised as the difference between barycentric () and observed (t) TOAs:

Here is the position of the Earth with respect to the barycentre, is a unit vector in the direction towards the pulsar at a distance d, and c is the speed of light. The first term on the right hand side of this expression is the light travel time from the Earth to the solar system barycentre. For all but the nearest pulsars, the incoming pulses can be approximated by plane wavefronts. The second term, which represents the delay due to spherical wavefronts and which yields the trigonometric parallax and hence d, is presently only measurable for four nearby millisecond pulsars [117, 46, 208]. The term represents the Einstein and Shapiro corrections due to general relativistic effects within the solar system [16]. Since measurements are often carried out at different observing frequencies with different dispersive delays, the TOAs are generally referred to the equivalent time that would be observed at infinite frequency. This transformation corresponds to the term and may be calculated from Equation (1).

Following the accumulation of about ten to twenty barycentric TOAs from observations spaced over at least several months, a surprisingly simple model can be applied to the TOAs and optimised so that it is sufficient to account for the arrival time of any pulse emitted during the time span of the observations and predict the arrival times of subsequent pulses. The model is based on a Taylor expansion of the angular rotational frequency about a model value at some reference epoch . The model pulse phase as a function of barycentric time is thus given by:

where is the pulse phase at . Based on this simple model, and using initial estimates of the position, dispersion measure and pulse period, a ``timing residual'' is calculated for each TOA as the difference between the observed and predicted pulse phases.

A set of timing residuals for the nearby pulsar B1133+16 spanning almost 10 years is shown for illustrative purposes in Fig.  19 . Ideally, the residuals should have a zero mean and be free from any systematic trends (Fig.  19 a). Inevitably, however, due to our a-priori ignorance of the rotational parameters, the model needs to be refined in a bootstrap fashion. Early sets of residuals will exhibit a number of trends indicating a systematic error in one or more of the model parameters, or a parameter not initially incorporated into the model.

Figure 19: Timing model residuals versus date for PSR B1133+16. Case (a) shows the residuals obtained from the best fitting model which includes period, the period derivative, position and proper motion. Case  (b) is the result of setting the period derivative term to zero in this model. Case (c) shows the effect of a 1 arcmin error in the assumed declination. Case (d) shows the residuals obtained assuming zero proper motion. The lines in (b)-(d) show the expected behaviour in the timing residuals for each effect (see text).

From Equation (9), an error in the assumed results in a linear slope with time. A parabolic trend results from an error in (Fig.  19 b). Additional effects will arise if the assumed position of the pulsar (the unit vector in equation (8)) used in the barycentric time calculation is incorrect. A position error of just one arcsecond results in an annual sinusoid (Fig.  19 c) with a peak-to-peak amplitude of about 5 ms for a pulsar on the ecliptic; this is easily measurable for typical TOA uncertainties of order one milliperiod or better. A proper motion produces an annual sinusoid of linearly increasing magnitude (Fig.  19 d).

After a number of iterations, and with the benefit of a modicum of experience, it is possible to identify and account for each of these various effects to produce a ``timing solution'' which is phase coherent over the whole data span. The resulting model parameters provide spin and astrometric information about the neutron star to a precision which improves as the length of the data span increases. Timing observations of the original millisecond pulsar B1937+21, spanning almost 9 years (exactly 165,711,423,279 rotations!), measure a period of ms [117, 114] defined at midnight UT on December 5 1988! Astrometric measurements based on these data are no less impressive, with position errors of arcsec being presently possible.

 Binary and Millisecond Pulsars at the New Millennium Duncan R. Lorimer http://www.livingreviews.org/lrr-2001-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de