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Astron. Astrophys. 322, 846-856 (1997)

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4. Data analysis

Except for the observations at 2.25 GHz and 8.5 GHz, our measurements were not made simultaneously at different frequencies. For this reason, we have to apply timing models to the data in order to keep track of the pulse phase between the observations. A time delay given by Eq. (11) will then appear as a deviation of the observed arrival time from the expected one, determined by the timing model.

4.1. Timing models

Regular timing observations of long period pulsars are made by the Jodrell Bank pulsar group, using the 76m-Lovell telescope (Martin & Lyne in prep.). We consider the timing models obtained from these observations as the best available and use the Jodrell Bank models for the reduction of our data. However, the timing data of many pulsars exhibit deviations from the simple spin-down model including only parameters for the period and period derivative, i.e.


where [FORMULA] is the rotational phase at time T (measured in the pulsar frame), [FORMULA] the phase at [FORMULA], [FORMULA] the pulsar spin frequency and [FORMULA] its first derivative. These rotational variations known as timing noise can arise from different kind of processes like discrete jumps in [FORMULA] or its derivatives, a systematic value of [FORMULA] related to a real physical process, "random walks" in one of the spin parameters or even orbital motions (see e.g. Cordes & Downs 1985, hereafter CD85).

From an analysis of many pulsars, it has been shown that the amplitude of the timing noise is strongly correlated with the period derivative, [FORMULA], (e.g. CD85, Arzoumanian et al. 1994). Depending on the time interval analyzed, timing residuals may appear as regular variations with fairly large amplitudes (for an impressive collection see CD85). In general one can try to account for such large variations if the second derivative of the spin frequency, [FORMULA], is included in the spin down model (Eq.  13). Although the resulting value, noticeable by a non-zero [FORMULA], might not have any physical meaning (i.e. it might not be related to a systematic physical process), it is desirable to do so if one tries to phase align pulse profiles observed at widely spaced epochs. Although the Jodrell timing models include such fits for the second period derivative, we took nevertheless care that our analyzed observations were done within a short time interval. However, depending on the known timing noise activity (cf. CD85 and discussion below), for some pulsars we will include data observed at different epochs if necessary.

4.2. TOA measurements

The topocentric TOA of a pulse profile was determined by cross-correlating the profile with a high signal-to-noise ratio (S/N) template. In order to calculate the maximum of the cross-correlation function (CCF), we applied a least-square fit of a polynomial to the CCF around its maximal peak (Press et al. 1986). The location of the maximum and its corresponding uncertainty, combined with the recorded "time tag" which corresponds to the midpoint of the integration, yielded finally the TOA and its formal error (e.g. Ryba 1991). The actual value of the TOA is affected by the choice of a pulse longitude that is identified with pulse phase zero, i.e. by the choice of a fiducial point. Since we want to time profiles measured at different frequencies, it is in general critical that the fiducial point marks the location of the fiducial plane. The fiducial plane is the one which contains simultaneously the rotation and magnetic axis of the pulsar as well as the line-of-sight towards the observer. A TOA of a pulse emitted at this longitude will only suffer the delay given by Eq. (11). The choice of any other longitude, on the other hand, would introduce a frequency dependent offset due to a combined effect of a RFM and spreading of the magnetic field lines.

Craft (1970) showed that the midpoint of pulse profiles is such a fiducial point. Phillips & Wolszczan (1992, hereafter PW92) and P92, respectively, investigated dispersion measures and emission altitudes of data observed between 26 MHz and 4.80 GHz. As a selection criterion, PW92 restricted their sample of pulsars to those with simple or double component profiles, so that a determination of the profile midpoint was possible with high accuracy. Our sample, however, is determined by other aspects, i.e. the ability to detect the sources at the highest possible frequencies. Therefore, our sources do generally not exhibit such regular pulse shapes as would be desirable for a straightforward choice of the fiducial point. Consequently, we used a special technique reported by Kramer et al. (1994) in order to determine the profile midpoint. We first separated the profile into individual components by fitting Gaussians curves to the pulse shape. Averaging the results of several such fits applied to different data, we obtained the mean properties of individual components. These mean components were finally combined again to form a typical average waveform. Since this average waveform represents the sum of several pure Gaussians curves, it is completely free of noise and, therefore, the ideal template for timing.

This approach has three major advantages. Besides the fact that the obtained profile template has an infinitely large signal-to-noise ratio, S/N, the determination of the profile midpoint, even in cases of rather complicated pulse shapes, is fairly accurate since the template is a composition of well defined functions. For the same reason, the template is also independent of any profile resolution since it can be scaled to match any possible gridding of the real data. This ensures, that regardless of the actual resolution chosen during the observations, the same template with the same determined fiducial point can always be used for the analysis.

For observations made at 1.41 GHz, 4.75 GHz and 10.55 GHz, respectively, we constructed the template out of fit results presented by Kramer (1994), since they reflect the results of a large number of independent measurements. For observations at 2.25 GHz and 8.5 GHz, fewer independent measurements were typically available, so that resulting TOAs have larger uncertainties. The 32-GHz profiles were of much lower S/N and of generally simple waveform (cf. Kramer et al. 1996), so that in most of the cases, we thus chose the profile peak as the fiducial point.

4.3. Profile alignment

In order to align pulse profiles, corresponding topocentric TOAs were converted into arrival times at the solar system barycentre using Jodrell Bank timing ephemeredes (Martin & Lyne in prep.), the pulsar timing software TEMPO (Taylor & Weisberg 1989) and the DE200 ephemeris of the Jet Propulsion Laboratories (Standish et al. 1992). The resulting barycentric arrival times were finally corrected for dispersion delays by transforming them into arrival times at a formal infinite radio frequency using Eq. (4) and [FORMULA] -values determined from the Jodrell observations. The resulting TOA residuals, i.e. the phase differences of the observed arrival times to those expected from the timing model, were used to shift the corresponding pulse profiles with respect to each other. From each timing residual we additionally subtracted the phase offset for the profile measured at the lowest frequency, so that the TOA of this profile is identified with phase zero.

In contrast to most analyses of timing data (e.g. Manchester & Taylor 1977), we adjusted only the global phase offset [FORMULA] (cf. Eq.  13) to minimize the TOA residuals. Most importantly, we did not adjust the dispersion measure, [FORMULA]! This is in contrast to all previous, low frequency studies (e.g. PW92), since at low frequencies the dispersion measure is actually the crucial parameter. The use of a [FORMULA] -value which deviates only slightly from the true value can lead to dramatic results which might be even interpreted as new physical phenomena. A few authors (Shitov & Malofeev 1985; Kuzmin 1986; Shitov et al. 1988), for instance, have reported that low frequency data lead to systematic larger dispersion measures than those obtained at high frequencies, referring to it as superdispersion. Phillips (1991a) subsequently demonstrated that the origin of this effect was spurious and originated from small errors in the dispersion measure (of the order of one part in 1000) possibly due to a combined effect of [FORMULA] -variability, fiducial point ambiguities and a certain choice of the actually used value of the dispersion constant, [FORMULA] (cf. Sect.  2).

Our approach to keep the [FORMULA] -value fixed, is justified by the following estimation. PW92 found long term variations of the dispersion measure as large as [FORMULA] cm-3 pc. During our analysis, we mainly tried to align profiles between 2.25 GHz and 32.00 GHz. Therefore, the same [FORMULA] possibly existing in the used timing model would produce a time offset of only [FORMULA] t(32.00 GHz - 2.25 GHz)=8.2 [FORMULA] s, which is, less than our typical measurement accuracy for long period pulsars of [FORMULA].

Supposing that pulsar ephemeris describes perfectly the observed timing behaviour and that the data are unaffected by retardation, aberration or magnetic field sweep-back. An adjustment of only the phase offset then means that the choice of the pulse longitude actually used for the TOA calculation will not affect the proper alignment at all. In fact, if the determined profile midpoint does not coincide with the proper fiducial point, the offset between these two longitudes will be just the resulting timing residual. Since each profile is shifted according to the associated timing residual, the pulse longitude corresponding to the adopted TOA will be obviously shifted into the proper position. This is, of course, only true in case of perfect timing models and absence of additional physical effects. On the other hand, if the effects which we are interested in, are measurable, the profile midpoint should indeed deviate from its expected position at phase zero. Thus, when we look for effects of retardation, aberration or magnetic field sweep-back, we look for deviations of the determined profile midpoint from the expected arrival time. In cases where the chosen fiducial point does obviously not coincide with the profile midpoint (e.g. for B1133+16 at 32 GHz where only the remaining component remains visible), the point marked in the plots (and its deviation from phase zero) does not have a physical meaning.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998