3. Reduction of astrometric data at 8.4 and 2.3 GHz
The data-reduction procedure for the 8.4 and 2.3 GHz observations was similar to that followed for the 5 GHz data (G95); therefore, we will describe it only briefly, emphasizing those aspects of the reduction that differed significantly.
For each source and frequency band, we corrected the phase delays for their inherent (multiple) 2-ambiguities; to make this correction, we first constructed a model of the array geometry based on a consistent set of antenna coordinates, source coordinates, and Earth-orientation parameters from the International Earth Rotation Service (IERS) Annual Report for 1993 (IERS 1994). We calculated site coordinates at epoch 1988.83 from the 1993.00 tabulated coordinates and rates of the IERS Terrestrial Reference Frame; UT1-UTC, Earth's pole coordinates, and corrections to the model of the Earth's nutation in longitude and obliquity at the time of each observation were interpolated from the daily estimates provided by IERS. We used these parameters in an extensively improved version of the VLBI3 program (Robertson 1975) to remove most of the multiple 2-ambiguities in the phase delays via a weighted-least-squares analysis of the phase delays and phase-delay rates; afterwards, we used interactive PGPLOT-based software to correct the (few) remaining ambiguities and to check the "phase connection" by inspection of the phase closure of each antenna triangle.
We then corrected for the remaining "overall ambiguities" (integer multiples of the radio-frequency period by which the phase delays for one source might be offset from those of the other at each station and frequency band) by estimating them in our weighted-least-squares analysis. Since, at this stage of the data reduction, some important effects, such as plasma delay, were not yet included in the model, the estimates differed by up to one third of a period from integer numbers of 2's. Nevertheless, we made a first correction by changing these estimates to the nearest integral multiple. These values were revised after we made a plasma correction to the phase delay (see below).
3.1. Reference-point selection
To compare our results with those previously reported, we defined reference points in the images of the two radio sources. The structure and evolution of these sources have been extensively studied by Eckart et al. (1986, 1988), Witzel et al. (1988), Schalinski (1990), and Hummel et al. (1992). Our main interest here is to identify features in the sources that may serve as appropriate reference points for astrometry. For each source at each frequency, we present in Figs. 1 and 2 our hybrid maps computed with DIFMAP standard techniques (Shepherd et al. 1995). For each source and frequency band, we initially chose the reference point to correspond to the maximum of the brightness distribution obtained by convolving the delta components of the CLEAN model with the corresponding synthesized beam (but see below). To remove the effects of the ionosphere on our phase delays as accurately as possible, we tried to ensure that the reference points selected in the maps at 8.4 and 2.3 GHz corresponded to the same point in the sky, despite the frequency dependence of both opacity and resolution effects. For 1928+738 at 8.4 GHz, we have labelled the inner components (see Fig. 1) as X1, X2, and X3 (which may correspond to components A, A1, and C, respectively, in the coeval 22 GHz maps of Hummel et al. 1992). Component X1 is likely the core, and X2 and X3 travelling jet components; our reference point at 8.4 GHz is the apparent peak of X2, the brightest feature. At 2.3 GHz these three components are blended, and our reference point (S1) is likely weighted more by the jet components X2 and X3 than by component X1, since the last is associated with the core and likely to have a more inverted spectrum than either X2 or X3. Therefore, the peak of brightness at 2.3 GHz likely corresponds to some point between components X2 and X3 in the 8.4 GHz image. We adopted this registration and aligned the peak of brightness of the 2.3 GHz image with the midpoint between components X2 and X3, dropping thereby the initial assumption that the origins of the two maps correspond to the same point on the sky. We thus identified the origin of the 2.3 GHz map with a point in the 8.4 GHz map 0.5 mas away, along PA . We assigned an uncertainty of 0.5 mas to this registration to cover all positions between the 8.4 GHz components X2 and X3. For 2007+777 at 8.4 GHz, the peak of brightness seems to coincide with the easternmost component, probably the core (Fig. 2). At 2.3 GHz, with a coarser resolution and a smaller contrast between the core and jet components (due to the different spectral indices of the two components), the location of the peak of brightness is a weighted average of the positions of the core and the jet component. We thus registered the peak of brightness of the 2.3 GHz image with the midpoint between the core and jet components of the 8.4 GHz image, that is, 0.6 mas along PA . We assigned an uncertainty of 0.5 mas to the value above to cover all positions between the 8.4 GHz core and the jet component.
3.2. Propagation-medium corrections
We modelled the tropospheric zenith delay at each station as a piecewise-linear function characterized by values specified at epochs four hours apart. A priori values at these nodes were calculated from local surface temperature, pressure, and humidity, based on the model of Saastamoinen (1973). The use of more nodes produced insignificant changes in the final results. We used the dry and wet Chao mapping functions to determine the tropospheric delay at non-zenith elevations, i.e., along the line of sight for each observation at each site. Since the antenna elevations were always higher than , more elaborate mapping functions would have yielded similar astrometric results.
For each site, we combined our simultaneous, virtually source-structure-free, connected phase delays at 8.4 and 2.3 GHz to remove the plasma contribution (mainly due to Earth's ionosphere), taking advantage of its dependence. The absence of 2.3 GHz data from Bonn forced us to follow a different procedure to remove the ionospheric contribution from the observables of the baselines involving this station. An estimate of the ionospheric delay -with the proper sign- can in principle be made from the 8.4 GHz data alone by calculating half the difference between each group delay and its corresponding (connected) phase delay; however, even for the most sensitive baseline, MB, the rms scatter of these estimates was 0.2 ns, which is larger than a phase-delay ambiguity interval and, therefore, the use of such ionospheric estimates in our phase-delay analysis would have degraded the quality of our astrometric determinations. Instead, for each observation for the MB baseline, we estimated the ionospheric delay by shifting in time the dual-frequency ionospheric estimates for the baselines MT and ML to account for the differences in longitude between Onsala and Medicina on the one hand and Bonn on the other, and then averaging the shifted values for MT and ML. (Note that the latitude of Bonn, N, is about halfway between that of Medicina, N, and Onsala, N.) We are assuming that the ionosphere varies linearly across Europe. This assumption has proved to be successful for differential astrometry at other epochs (e.g., G95). In our case, further support for the smoothness of the ionosphere over the stations involved is provided by the similarity between the ionospheric contribution for the baselines ML and MT: the largest difference between the ionospheric delays for these two baselines is 0.1 ns, nearly an order of magnitude smaller than the maximum difference delay for each of these baselines. The ionospheric estimates for the other baselines involving Bonn were calculated by taking advantage of the closure of the ionospheric contribution to the phase delays, which must be zero for each triangle of stations. For example, the ionospheric delay for BF is just the difference between the contributions for the baselines MF, calculated by combining our dual frequency observations, and MB, calculated by shifting and averaging the dual-frequency estimates of MT and ML.
As stated above, after making the ionospheric correction, we again used weighted-least-squares to estimate, for each station, the overall number of ambiguities for our (nearly) structure-free, plasma-corrected phase delays; they turned out to be no more than about 0.1 (at 8.4 GHz) and so we fixed them to zero.
3.3. Relative position
We obtained our estimate of the relative position of the reference points in the two sources by weighted-least-squares analysis of the differenced phase delays, formed by subtracting the plasma-corrected phase delay of each observation of 2007+777 from the corresponding phase delay of the previous observation of 1928+738. These differenced delays are nearly free from errors due to inaccuracies in our models for the atmosphere, site coordinates, and Earth orientation. We also included the phase delays for 1928+738 in the analysis to estimate the relative behaviour of the station clocks (see G95). The postfit residuals of the differenced phase delays are shown in Fig. 3. The result for the coordinates of component X1 (see Fig. 1) of 1928+738 relative to the core of 2007+777 is shown in Table 2. The coordinates of the peak of brightness (associated with component C2; see below) of 1928+738 relative to the core of 2007+777 for the epoch 1985.77 at 5 GHz are also shown. To make a consistent comparison between these two positions, the 5 GHz data were re-analyzed in the same system as was used in the analysis of the 8.4/2.3 GHz data (i.e., a priori values of the antenna coordinates, source coordinates, and Earth-orientation parameters were taken from IERS 1994).
Table 2. Estimates of the coordinates (J2000) of 1928+738 minus those of 2007+7771.
3.4. Error analysis
From an error analysis similar to the one described in G95 we determined the contribution to the standard error of our relative position from the a priori uncertainty of quantities not estimated in the weighted-least-squares analysis, namely, core location, nutation, and ionospheric plasma delays. To determine the standard deviation associated with the core identification and with Earth nutation we followed an identical procedure to that described in G95 (see Table 3). The standard deviation associated with the ionosphere is mainly due to errors in the registration of the maps at 8.4 and 2.3 GHz. We estimated in Sect. 3.1 that the errors in the registration of the maps at the two frequency bands were 0.5 mas for each source; the contribution of this error to the final result is scaled down by the factor (R-1)-1 (0.079, where R is the square of the ratio between the reference frequencies of the two bands; see Guirado et al. 1995b for details). The corresponding contributions are shown in Table 3. We note that, if the ionospheric contribution in 1988.83 were comparable to that in 1985.77, we would have expected the ionospheric contribution to the standard error to be 0.03 (by frequency scaling the 0.07 value at 5 GHz to 8.4 GHz). In any case, the error contribution for the ionosphere turned out to be less than one third of the statistical standard error. The overall standard deviation of each component of the estimated relative position of 1928+738 and 2007+777 was taken to be the root-sum-square of the standard deviations shown in Table 3.
© European Southern Observatory (ESO) 1998
Online publication: July 7, 1998