Astron. Astrophys. 325, 383-390 (1997)

3. Data analysis

In astrometric studies involving extended sources, we need to choose reference points within the source structures, from which to measure positions or angular separations. In long-term monitoring studies, the comparison of the angular separations between two radio sources measured at different epochs will be most meaningful if those reference points are identical from epoch to epoch.

For simplicity, we describe the data analysis in two parts: the first treats source structure and the second the astrometric analysis for point-like sources.

3.1. Milli-arcsecond scale structures of 1038+528 A and B

We made maps of 1038+528 A and B at 3.6 cm and 13 cm with the Caltech VLBI package, which uses hybrid mapping techniques and deconvolution algorithms like CLEAN. Calibration factors for converting the measured interferometric amplitudes and phases into the source complex visibility function were derived from the information provided by the participating observatories (system temperatures and gain curves), the contemporaneous observations of 4C39.25, and an iterative procedure of self-calibration (Cornwell & Wilkinson 1981). The maximum correlated flux densities on intercontinental baselines at 3.6 cm and 13 cm were, respectively, about 450 and 300 mJy for 1038+528 A, and about 90 and 130 mJy for 1038+528 B, with estimated standard errors of about in each case. Fig. 1 shows the VLBI maps of 1038+528 A and B, at 3.6 cm and 13 cm, for June 1990 (Rioja 1993), and for previous epochs (Marcaide et al. 1985; Elósegui 1991). The pixel sizes of the maps at 3.6 cm and 13 cm are 170 and 550 µas, respectively. To parameterize the brightness distributions in Fig. 1, we used weighted least squares to fit models consisting of elliptical Gaussian components to the self-calibrated visibility data. Table 1 lists the resulting parameter values and their estimated standard errors at 3.6 cm for quasar B at the 3 epochs derived using the Caltech VLBI analysis programs MODELFIT and ERRFIT (Pearson 1991).

Fig. 1. (Top) Hybrid maps of the quasars 1038+528 A and B for epochs in March 1981, May 1983 and June 1990 at 3.6 cm. The contours are 1,2,5,10,20,40, and 80%, and 2,5,8,15,25,50, and 90% of the peak of brightness of each map for A and B, respectively. The restoring beam size (FWHM of a circular gaussian) is 0.6 mas. (Bottom) Corresponding hybrid maps at 13 cm. The contours are 2,5,10,20,40, and 80%, and 2,5,8,15,25,40, and 70% of the peak of brightness of each map for A and B, respectively. The restoring beam size (FWHM of a circular gaussian) is 1.5 mas. Nevertheless, note that the real interferometric beam size at 13 cm is nearly four times larger than at 3.6 cm.

Table 1. Parameterized description of the brightness distributions of quasar B, in 1981.2, 1983.4 and 1990.5 at 3.6 cm. We derived these parameters by fitting models consisting of elliptical gaussian components to the calibrated visibility data using the program modelfit, from the Caltech package (S: fluxes; R: distance between components; : PA. The subscripts correspond to the labelling of the components in Fig. 2). Errors are standard errors, derived using the program errfit, also from the Caltech package.

For long-term astrometric studies, the fiducial points in the brightness maps of the sources should be prominent features, whose positions can be precisely identified and measured from epoch to epoch. The reference points in the maps of 1038+528 A and B in June 1990 were chosen following the same criteria as for the analyses of the observations at the previous epochs. In particular, the choice of these locations was based on the most prominent delta function components used to construct the maps shown in Fig. 1. More specifically, each reference point corresponds to the centroid of the subset of delta functions which have flux densities exceeding 25% of the highest one and which lie within one beam area from it. Using these selection criteria, we expect the reference point in the map of 1038+528 A to coincide with the position of the core (, ), whereas for 1038+528 B the reference point is the eastmost component in the maps at 3.6 cm (). Fig. 2 shows the correspondence between the source components and the reference points used for the astrometric analysis, in both sources. The eastmost component for 1038+528 B at 13 cm is not detected at 3.6 cm; its spectral index must therefore satisfy ( Marcaide et al. 1985).

Fig. 2. Projection of quasars 1038+528 A and B on the sky. The components labelled as ref in both structures correspond to the reference points used in the astrometric analysis presented here. These are maps at 3.6 cm in June 1990.

The so-called structure contribution to the observed phase-delay can be estimated from the brightness distribution (Shapiro et al. 1979). We used the maps shown in Fig. 1 and the reference points (described above) to estimate the source structure contribution for each baseline for each observation. These contributions are as large as some tens of picoseconds (ps).

In Sect. 5 we consider other plausible criteria for choosing the reference points within the extended structures of both radio sources, along with the astrometric implications.

3.2. Astrometric analysis for point-like radio sources

VLBI observables are sensitive to the changing orientation of the telescope array with respect to the source as the Earth rotates. Essentially, the astrometric analysis consists of disentangling this geometric signature from other contributions to the observed interferometric quantities, and thereby estimating the position of a source or, for example, the angular separation between two sources.

At present, the a priori knowledge of baseline vectors, source position, clock behaviour, and propagation medium is not sufficiently accurate to allow the phase ambiguity in the measurements of phase delays to be eliminated. Hence, these quantities are ambiguous by an integral multiple of 120 and 450 ps at 3.6 cm and 13 cm, respectively. However, the differencing technique has been successfully used for pairs of radio sources up to several degrees apart, producing unambiguous differenced-phase-delay observables largely free from the main sources of error (Shapiro et al. 1979; Marcaide & Shapiro 1983; Bartel et al. 1986; Guirado et al. 1994, 1995; Lara et al. 1996).

The basic observable used in the astrometric analysis of the three epochs of observation of 1038+528 A and B is the differenced phase delay, formed by subtraction of the phase delay measured for one source from that measured for the second source observed simultaneously. The strategy followed in the data analysis was the same for all three epochs (see Marcaide & Shapiro 1983, for details). We used the differenced phase delays measured in June 1990 at 3.6 cm and 13 cm to estimate the angular separations between 1038+528 A and B at that epoch, in the barycentric celestial reference frame J2000.0, using a recent version of the astrometric program VLBI3 (Robertson 1975). The data analysis is carried out using theoretical models, plus weighted least squares, to estimate values for the relevant parameters. Values for the parameters which describe the Earth's orientation, the telescope locations, and the reference source coordinates, are taken from a global VLBI solution provided by Goddard Space Flight Center (GSFC) (Chopo Ma priv. comm.). VLBI3 uses time polynomials to model the atmospheric and instrumental contributions to the observables; alternative methods of astrometric analysis use stochastic models to account for these contributions (e.g., Kalman filters, used by OCCAM (Zarraoa 1993) and SOLVK (Herring 1990) softwares). The critical parameter determining the reliability of the ambiguity resolution is the relative separation of the two sources. In the case of 1038+528 A,B the effects of errors in the other parameters are scaled down by a factor of almost , that is, the relative separation (33") expressed in radians (Shapiro et al. 1979). We had no problems in achieving a correct resolution of these phase-delay ambiguities by using the relative separation measured in 1983.4, which implies in turn that the separation changed by less than 1 mas during the interval spanned between 1983.4 and 1990.5. After removal of the ambiguities, the coordinates of quasar A with respect to those of quasar B were estimated from weighted-least-squares analysis of the differenced phase delays.

Finally, we re-analysed the observations from previous epochs (1981.2 and 1983.4) using a consistent set of values (global VLBI solution GLB831, provided by GSFC) for all three epochs for the parameters that describe the Earth's polar motion and rotation, the baseline vectors, and the reference source position in the sky.

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

helpdesk.link@springer.de