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Astron. Astrophys. 355, 552-563 (2000)

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5. Comparison of astrometry at all 4 epochs

In this section we make a comparison of the astrometric measurements from the series of 4 epochs of observations. Any increase of the temporal baseline in the program of monitoring the separation between A and B should result in a more precise identification of any systematic trends, with an improved elimination of random contributions. In Sect. 5.1 we justify comparing the astrometric values measured at the various epochs, even though non-identical observing, post-processing and analysis procedures were involved. In Sect. 5.2 we present the astrometric results from the 4 epochs. In Sects. 5.3, 5.4, 5.5 and 5.6 we present various analyses of these results, and attempt to quantify, or put upper-limits to, proper motions within and between the A and B quasars.

5.1. Comparison between the techniques used at different epochs

Before attempting a comparison of the astrometric results from the 4 observing epochs, we need to show that any bias in the astrometric estimates introduced by the use of different procedures is small compared with other errors in the measurements for the individual epochs. The consistency between the results from previous epochs of observations has been exhaustively tested (Marcaide et al. 1994; Rioja et al. 1997a). We outline here the largest changes involved in the fourth epoch, 1995.9, with respect to previous ones:

  1. The observing array and frequency set-up used in the fourth epoch was different from previous epochs of observations (frequency range 8404.5 to 8436.5 MHz instead of 8402.99 to 8430.99 MHz at first 3 epochs). This results in a different coverage of the UV plane, leading to changes in the reconstruction of the source images. Investigations of such effects by Marcaide et al. (1994) show that the effect on the astrometric anaylsis is only a few µas. It is important to note that the observations at all 4 epochs have comparable resolutions and sample the same range of structural scales in the sources.

  2. The processing of the fourth epoch was done using the VLBA correlator, which uses a theoretical model derived from CALC 8.2; we used AIPS to analyse the data with visibility phases residual to that model. For previous epochs the correlation was done at the MPIfR (Bonn) MK3 correlator, and an analysis of the data using total phases was made with VLBI3 (Robertson 1975). The differences between CALC 8.2 and the one implemented in VLBI3 propagate into changes of only 1-2 µas in the astrometric analysis of the 1038+528 A-B separation (Rioja 1993, Rioja et al. 1997a). This is because any such differences are "diluted" by the source separation expressed in radians - [FORMULA] in the case of this very close source pair.

  3. The values used in the analysis of previous epochs for Earth Orientation Parameters (EOP), stations and reference source coordinates were consistently derived from a single global solution provided by Goddard Space Flight Center (GSFC). For the correlation of the fourth epoch, the values used for EOP were derived from IERS solutions, and the station coordinates from USNO catalogs. We have made a comparison of the values derived for all the parameters at the 4 epochs from a single global solution from IERS (namely IERS eopc04), with the actual values used in the individual epoch analysis. The difference between the corresponding EOP values is always less than 4 mas. Such discrepancies propagate into errors in the relative position estimates at each epoch of only a few µas.

  4. Our astrometric analysis in AIPS using a phase-referencing approach and HDM differs from the phase difference method used in VLBI3 analysis. Comparisons show that these procedures are equivalent (Porcas & Rioja 1996; Thompson et al. 1986). Both involve the definition of reference points in source maps; uncertainties in the reference point positions (as described in Sect. 4.3) arise in the same way.

  5. Finally, a minor VLBA correlator error (Romney priv. comm.) caused incorrect time labels to be attached to the visibility records, resulting in incorrect (u,v ) values. The effect on the relative visibility phases for our source pair is small ([FORMULA]o) and can be neglected.

The magnitudes of all of the effects reported in this section are much smaller than our estimate in Sect. 4.3 of the uncertainity in reproducing the reference point in the source, from epoch to epoch, and we are thus justified in comparing the astrometric results from all 4 epochs.

5.2. Astrometric separations at the 4 epochs

The astrometric measurements of the separations between the reference points in A and B at [FORMULA] 3.6 cm from 4 epochs are presented in Fig. 4. It includes our new 1995.9 measurement and those from three earlier epochs, in 1981.3, 1983.4 and 1990.5, reported in Marcaide & Shapiro (1984), Marcaide et al. (1994) and Rioja et al. (1997a), respectively. The origin of the plot represents the separation at epoch 1.

[FIGURE] Fig. 4. Measured separations between the A and B reference points at epochs 2 (1983.4), 3 (1990.5) and 4 (1995.9), with respect to epoch 1 (1981.2). Plotted error bars correspond to [FORMULA]as.

Changes with time in Fig. 4 represent the vector difference between any motions of the reference points in quasars A and B. The near-orthogonal nature of the source axes in 1038+52 A,B (along which one might expect any motion to occur) simplifies the interpretation of any trends seen. The new 1995.9 value follows the same steady trend towards the NW shown by the three previous epochs. Rioja et al. (1997a) interpreted this as an outward expansion of the reference component in quasar B at a rate of [FORMULA]as yr-1, and quoted an upper bound on any proper motion of quasar A of [FORMULA]as yr-1.

5.3. Vector decomposition

In this section we attempt to separate the individual contributions from the 2 quasars in the astrometric separation measurements presented in Fig. 4. We make no assumption about the stability of either component, but assume that any displacements of the A or B reference points from their positions at epoch 1 are along the corresponding source axis directions. This is a plausible assumption if the reference point coincides with a non-stationary component moving along a ballistic trajectory, or with the location of the peak of brightness within an active core or near the base of jet, where changes during episodes of activity are likely to occur along the jet direction. This approach is closely related to that used previously by Rioja et al. (1997a). For fixed assumed source axes for A and B, it results in a unique decomposition of the changes in the A-B separation into separate A and B displacements, from 1981 to 1995.

It is clear that the dominant contribution to the separation changes seen in Fig. 4 comes from quasar B, in which the source axis is well defined by the 127o PA of the separation between core and reference components. For quasar A the source axis bends, from the inner "core" region (PA = 15o) to the outer jet components, and it is not so clear which direction should be chosen.

In our analysis we tried a range of values for fixing the A source axis (0 to 45o in steps of 5o). For each, we calculated A and B reference-point displacements at epochs 2, 3 and 4 with respect to epoch 1. Then we performed a least-squares fit to the B displacements with time to estimate a linear expansion rate for the B reference feature along PA 127o. In Fig. 5 we plot the deconvolved B reference point displacements from the analysis with the A source axis fixed at PA 25o (the value adopted by Rioja et al. 1997a). The fitted expansion rate is [FORMULA]as yr-1; the error and associated rms values take account of the small number of points and 2 degrees of freedom. This rate agrees well with the value of [FORMULA]as yr-1 deduced by Rioja et al. (1997a). The rms residual from the fit (7 µas) is low, and vindicates our use of measurements derived from differing techniques for investigating the relative proper motion between A and B.

[FIGURE] Fig. 5. Changes in position of reference component in B along PA 127o, deduced from deconvolution of the A-B separation measurements. Assumed source axis for A is 25o. Plotted error bars correspond to [FORMULA]as.

5.4. Structural evolution within 1038+528 B

Our deconvolution analysis of the changes in separation measured between all 4 epochs supports the finding, previously proposed, that the B reference component moves along the source axis, away from the B core. In this section we make an independent determination of the separation rate between the core and reference component in B from measurements within the maps at the 4 epochs.

Fig. 6 shows the separation between the core and reference component in B at the four epochs plotted against time. For epochs 1-3 we used the values given in Rioja et al. (1997a). For 1995.9 we used AIPS task UVFIT to estimate a separation from the B visibility data directly, in order to follow the methodology used for the other epochs as closely as possible; the value obtained was 1.895 mas. The slope from a least-squares fit corresponds to an expansion rate of [FORMULA]as yr-1. In the standard picture of extragalactic radio sources, the "core" is stationary, so this corresponds to an outward expansion of the reference component along PA 127o.

[FIGURE] Fig. 6. Changes in the position of the reference component in B along PA 127o, from measurements of its separation from the core in hybrid maps of B. Plotted error bars correspond to [FORMULA]as.

The rms of the fit (8 µas) is again surprisingly low, implying typical errors in the separation measurements at each epoch (both within the B structure and between the reference points) of only about 10-12[FORMULA]as along the direction of the B source axis. This is considerably less than the estimate of position separation errors given in Sect. 4.3.

5.5. Relative proper motion

The analysis presented in the previous sections demonstrate clearly that the chosen reference component within quasar B is unsuitable for use as a marker for tracing any relative proper motion between quasars A and B. The value of its expansion velocity derived in Sect. 5.4 appears to differ significantly from that deduced by vector-decomposition in Sect. 5.3. Although the difference between these estimates, if real, could be interpreted as motion of the core of B at a rate of [FORMULA]as yr-1, this is not a conclusive result since differences of this order arise from choosing different values of PA for the motion in A in the vector decomposition method.

A more suitable tracer of relative proper motion between the quasars is the variation of the separation between the cores of A and B. We have used the separations between the core and reference component measured in the B map at each epoch, and the astrometric separations between A and B, to calculate the separations between the A and B cores at each epoch; these are plotted in Fig. 7. The area occupied by the points defines an upper limit of [FORMULA]as yr-1 for any relative proper motion between the A and B cores, and hence between the quasars themselves, during the period of nearly 15 years for which the separation has been monitored with VLBI. The limit seems to be set by the relatively large deviation of the 1995.9 epoch point in the direction of the A source axis, presumably arising from the difficulty in defining the reference point at the A "core" from epoch to epoch.

[FIGURE] Fig. 7. Separation between the cores of A and B (with respect to epoch 1), derived by correcting the A-B reference point separation measurements with the core-reference separations measured in the B hybrid maps. Plotted error bars correspond to [FORMULA]as.

5.6. Possible "core" motions?

Finally, we investigate any possible residual motions of the "cores" in A and B. The most likely causes of any such apparent motions are changes in the relative brightness or positions of features in the source structures at a resolution below that of the maps. One might expect that these, too, would produce effects predominantly along the source axis directions. We therefore used the vector deconvolution method on the plot of core-core separation with time to study displacements of the cores along their source axis directions. Fig. 8a and b show plots of the separated contributions from B and A, for an assumed A source axis PA 25o. The displacements for the B core seem to increase systematically. The fitted rate is [FORMULA]as yr-1, indicating a possible slow outward motion. The displacements for the A core do not seem to vary systematically - the fitted slope is [FORMULA]as yr-1. Here the scatter is considerably larger, reflecting both the difficulties of defining the reference point along the A core-jet axis, and also, perhaps, real "jitter" of the position of the peak due to variations in the "core" substructure. These plots indicate the level of stability of the individual core positions; the fits represent realistic upper limits to any possible systematic core motion in the A and B quasars along their source axis directions.

[FIGURE] Fig. 8a and b. Residual motions of the cores using the deconvolution approach. a Displacements of B core along PA 127o; b Displacements of A core along PA 25o. Plotted error bars correspond to [FORMULA]as.

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Online publication: March 9, 2000
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