Following the accumulation of a number of TOAs, a surprisingly simple model is usually sufficient to account for the TOAs during the time span of the observations and to predict the arrival times of subsequent pulses. The model is a Taylor expansion of the rotational frequency about a model value at some reference epoch . The model pulse phase

where is the barycentric time and 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 in Figure 22. Ideally, the residuals should have a zero mean and be free from any systematic trends (see Panel a of Figure 22). To reach this point, however, 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 incorporated into the model.

From Equation (10), an error in the assumed results in a linear slope with time. A parabolic trend results from an error in (see Panel b of Figure 22). Additional effects will arise if the assumed position of the pulsar (the unit vector in Equation (9)) used in the barycentric time calculation is incorrect. A position error results in an annual sinusoid (see Panel c of Figure 22). A proper motion produces an annual sinusoid of linearly increasing magnitude (see Panel d of Figure 22).

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 with a precision which improves as the length of the data span increases. For example, timing observations of the original millisecond pulsar B1937+21, spanning almost 9 years (exactly 165,711,423,279 rotations!), measure a period of 1.5578064688197945±0.0000000000000004 ms [188, 184] defined at midnight UT on December 5, 1988! Measurements of other parameters are no less impressive, with astrometric errors of being presently possible for the bright millisecond pulsar J0437–4715 [389].

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