4. Analysis of the maps
4.1. Source structures
The 1995.9 hybrid maps of quasars A and B at 3.6 cm (Fig. 1) show the core-jet structures typical of quasars at mas scales. They may be compared with maps from previous epochs given in Rioja et al. (1997a). The structure of A in the new map shows no major changes with respect to previous epochs. There is a prominent peak at the SW end of the structure (the "core") and a jet extending in PA 15-25o containing at least two "knot" components (k1 and k2).
The new map of B at 3.6 cm is qualitatively similar to those from previous epochs. It shows 2 point-like components separated by just under 2 mas in PA 127o. Spectral arguments support the identification of the NW component as a "core" (Marcaide & Shapiro 1984); the SE component, corresponding to a knot in the jet, has been used as a reference component in previous astrometric studies. The separation between these 2 components in 1995.9 has increased, continuing the expansion along the axis of the source, as discovered from previous epochs of observations at this wavelength (Rioja et al. 1997a). There is no trace of the third, extreme SE component, seen in maps of this source at 13 cm. This feature is evidently of lower surface brightness at 3.6 cm and is resolved out at the resolution of these observations.
We used AIPS task IMSTAT to estimate total flux densities for quasars A and B (within windows surrounding the sources in the hybrid maps). The values are given in Table 1. Table 1 also lists the fluxes and relative positions of the most prominent features in the maps of A and B, obtained using task JMFIT to find parameters of elliptical Gaussian functions which best fit the various source sub-components. The formal errors from the fits, however, do not give realisitic values for the parameter uncertainties. The distribution of flux between the core and k1 in quasar A, and their relative separation, are quite uncertain, for example.
Table 1. Parameters derived from the Hybrid Maps of 1038+528 A and B.
4.2. Estimating positions of reference features
The astrometric measurement of a separation between two non-point sources must always refer to the measured positions of reference points within maps (or other representations) of the source structures. The selection of suitable reference points is crucial in monitoring programs, where the results from the analysis of a multi-epoch series of observations are compared. Ideally, a reference point should correspond to the peak of a strong, unresolved component, which is well separated from other radio emission within the source structure.
For the 1995.9 epoch observations of 1038+528 A,B we selected the same reference features as those used for the analyses of previous observing epochs. These are the "core" component for quasar A, and the prominent SE component for quasar B. These features are labelled with a cross in Fig. 2a and b. The core of A is indeed strong and compact, but has the disadvantage that it merges with knot k1. Although the SE component of B is no longer the strongest feature at 3.6 cm, it has always been strong at both 3.6 and 13 cm wavelengths, is reasonably compact and is easily distinguishable in maps made at longer wavelengths, thus facilitating spectral studies. Our astrometric analysis refers to the measured positions of the peaks of these components in A and B. We used the AIPS task MAXFIT to measure the position of these peaks in the PRA and HDM maps. MAXFIT defines the location of a peak in a given map region by fitting a quadratic function to the peak pixel value and those of the adjacent pixels. A comparison of this method of defining the peak position with that used for earlier epochs is described in the next section.
4.3. Position error analysis
An analysis of errors presented in Rioja et al. (1997a) shows that the dominant uncertainty in the astrometric measurements of the separation between this close pair of quasars comes from the limited reproducibility of the reference point positions in the VLBI maps, from epoch to epoch. The magnitude of this effect is hard to quantify, however, since it depends on the nature of the source brightness distribution surrounding the reference point, and the method used to define the position of the peak, in addition to the resolution of the array and the signal-to-noise ratio of the peak in the map.
A rough estimate of the error due solely to finite signal-to-noise in the maps is given by dividing the beam size by the ratio of the component peak to the rms noise level in the maps (see e.g. Thompson et al. 1986). This yields values of 3.3, 3.5 and 1.4 µas for the A and B PRA maps and the HDM map. These may be taken to represent lower limits to the reference point position errors; realistic errors will be larger, and will depend on the nature of the reference features and the manner in which the position is estimated.
It is important to choose a definition of the reference point position such that it can be reproduced reliably from epoch to epoch, and is as independent as possible from the parameters used in making the map (e.g. cell size and beam width). The AIPS task JMFIT can be used to fit an elliptical Gaussian to a component in a CLEAN map, for example. However, the position of the peak of the Gaussian depends on how asymmetric the component brightness distribution is, and the area of the map to which the fit is restricted. MAXFIT fits just to the local maximum around the peak map value, and is thus less sensitive to the rest of the distribution.
We have attempted to quantify some limits to reproducibility arising from the use of MAXFIT for defining the peak position in CLEAN maps. We investigated the effect of changing the true position of a point-like source with respect to the pixel sampling (here 3.3 pixels per beam) by offsetting the source position in 10 increments of 1/10 of a pixel in the visibility domain, mapping and CLEANing the new data sets, and estimating the new positions in the CLEAN maps using MAXFIT. The maximum discrepancy found between the values of the artificial offset and the shift derived by MAXFIT was 1/20 of a pixel. This corresponds to 8 µas in our 3.6 cm maps.
For the analysis of previous observing epochs, the reference points were defined to be the centroid of the most prominent delta functions from which the CLEAN source map was derived (Rioja et al. 1997a). We examined possible systematic differences resulting from these different definitions of reference points. One might expect the largest discrepancies to arise when the underlying source structure near the reference point is asymmetric, as in quasar A. We investigated such differences by determining "centroid" positions for both A and B reference components, using various criteria for excluding clean components from the calculation; this included the "25 percent of the value at the peak" threshold used for earlier epochs. For A the difference between this centroid position and the MAXFIT value was 0.12 pixel (18 µas). For B the difference was less than 0.1 pixel. These are probably the largest potential sources of error arising from using different methodologies at different epochs.
Our use of two different mapping procedures - phase-reference mapping and HDM - also gives some insight into the size of position errors resulting from standard CLEAN + phase self-cal mapping algorithms. The differences between the separation estimates from the PRA maps and the HDM map are 27 and 28 µas in RA and Dec respectively. This would suggest that differences in the positions of peaks in maps reconstructed in different ways may vary at the 14 µas level.
After considering the various possible effects which can limit the accuracy of position estimates, we adopt a "conservative" value for the error in estimating the peak position in our 3.6 cm maps, embracing all the effects detailed above, of as (this corresponds to a thirtieth of the CLEAN beam). The associated estimated error for a separation measurement between the two sources is as.
4.4. Astrometry results
Table 2 lists the results of our astrometric measurements of reference point positions in the maps. They are presented as changes in measured separation between the reference points in A and B in 1995.9, with respect to their separation in 1981.3. The values given from the phase-reference technique correspond to the difference between the A and B reference feature position offsets in their respective PRA map. The values derived from HDM have been corrected for the artificial offset introduced before adding the A and B source visibilities.
Table 2. Change in the separation between quasars A and B in 1995.9 with respect to 1981.2, estimated using standard phase referencing (PRA) and hybrid double mapping (HDM) techniques.
All these values have been corrected for a small error in the AIPS
calculation of the u,v coordinates in the frequency-averaged data set.
(Distances measured within the maps must be adjusted by a small
correction factor of .)
Table 3 lists the coordinates of the reference source (A) adopted in the analysis and the measured coordinate separation between quasars A and B in 1995.9.
Table 3. Fixed source coordinates used for quasar A in the astrometric analysis (these coordinates correspond to GSFC global solution GLB831 (Chopo Ma, priv. comm.)), and separation between quasars A and B measured in 1995.9.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000