The timing model

4.2 The timing model

To model the rotational behaviour of the neutron star we ideally 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 solar system centre-of-mass (barycentre) can be regarded as an inertial frame. It is now standard practice [155 ] to transform the observed TOAs to this frame using a planetary ephemeris such as the JPL DE405 [346 ]. The transformation between barycentric ( $𝒯$ ) and observed ( $t$ ) takes the form

Here $r --$ is the position of the observatory with respect to the barycentre, $ˆs --$ is a unit vector in the direction of the pulsar at a distance $d$ and $c$ is the speed of light. The first term on the right hand side of Equation (9

) is the light travel time from the observatory to the solar system barycentre. Incoming pulses from all but the nearest pulsars can be approximated by plane wavefronts. The second term, which represents the delay due to spherical wavefronts, yields the parallax and hence $d$ . This has so far only been measured for five nearby millisecond pulsars [322 , 380

, 216 , 336 , 218 ]. The term $Δtrel$ represents the Einstein and Shapiro corrections due to general relativistic time delays in the solar system [18

]. Since measurements can be carried out at different observing frequencies with different dispersive delays, TOAs are generally referred to the equivalent time that would be observed at infinite frequency. This transformation is the term $ΔtDM$ (see also Equation (1

)).

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 $Ω = 2π∕P$ about a model value $Ω0$ at some reference epoch $𝒯0$ . The model pulse phase

where $𝒯$ is the barycentric time and $ϕ0$ is the pulse phase at $𝒯0$ . 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.

Figure 22: Timing model residuals for PSR B1133+16. Panel a: Residuals obtained from the best-fitting model which includes period, period derivative, position and proper motion. Panel b: Residuals obtained when the period derivative term is set to zero. Panel c: Residuals showing the effect of a 1-arcmin position error. Panel d: Residuals obtained neglecting the proper motion. The lines in Panels b–d show the expected behaviour in the timing residuals for each effect. Data provided by Andrew Lyne.

From Equation (10 ), an error in the assumed $Ω0$ results in a linear slope with time. A parabolic trend results from an error in $Ω˙0$ (see Panel b of Figure 22 ). Additional effects will arise if the assumed position of the pulsar (the unit vector $ˆs-$ 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 $∼ 3 μarcsec$ being presently possible for the bright millisecond pulsar J0437–4715 [389 ].