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Astron. Astrophys. 337, 69-79 (1998)

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4. Multi-epoch intercomparison

4.1. Procedure

A serious problem with analysing data from VLBI observations is that there is significant freedom in making images. Data are degraded by instrumental errors, incomplete and different aperture coverages, and ambiguities in deconvolution and self-calibration, all of which can strongly affect the final results. For this reason intercomparison of models and maps made separately at each epoch is not a good method of detecting small changes in a source (Conway et al. 1992).

To minimise the above effects we used the 5th epoch model and CLEAN map (see Fig. 1a,b) as starting points in re-modelfitting and re-mapping all of the epochs. This method should limit the differences between the final images of all five epochs of 0710+439, so we can be sure that any differences seen are demanded by the data and are due to real changes in the structure of the source (see Appendix).

The detailed procedure in modelfitting at each epoch was, starting with the 5th epoch model, to first allow just the flux density of all components to vary; however in each case the fit remained poor. We next allowed changes of all parameters of the gaussians within the B component (e.g. flux, radius, [FORMULA], major axis, axis ratio, and [FORMULA]) which gave a somewhat better fit, but only after allowing all the components to move in position (which led to significant motion mainly in component A2) did we get a good fit. Finally the u-v data were amplitude self-calibrated against the model and one final iteration of modelfitting carried out in which component positions were again allowed to vary. We note that it was never necessary at any epoch to change the size or shape of the gaussians within the A or C components. The final models had good agreement factors to the data (for epochs 1 to 5 total agreement factors were QTOT=1.149, QTOT=1.047, QTOT=1.039, QTOT=1.145 and QTOT=1.224 respectively).

4.2. Component position and flux variations

From the modelfits we measured the separation of many pairs of components as a function of time and fitted linear regression lines to this data to estimate relative velocities (see Table 3, Fig. 3).


Table 3. Measured apparent motions of the gaussian components within 0710+439

[FIGURE] Fig. 3. Changes in separation with time of the components A2 and C2. Filled circles represent data obtained by MODELFIT, solid line shows linear regression fit to these data. Open squares represent data obtained by JMFIT, dashed line shows linear regression fit. See Sect. 4.2 for a discussion of the errors on the plotted points and the fits

Although in VLBI data analysis various attempts have been made to estimate a priori error bars on component positions these schemes are of doubtful reliability. Gaussian error bars estimated from the variability of reduced Chi-squared on moving the components critically depend on the number of degrees of freedom in the data which depends in turn on the unknown degree of correlation of phase and amplitude errors with time. In addition as noted in Sect. 4.1 even our best fitting model has a reduced Chi-squared which is much further from unity than would be expected given statistical arguments. Given this situation we chose instead to estimate errors on motions from the internal scatter of our separation versus time data using standard methods of linear regression analysis. These methods applied to our data suggest that for the brighter components the random errors on the relative separation in each epoch are of order 20µas. This is comparable to the estimates we obtained from imaging simulations (see Appendix A.2).

We find from our fitting that there is no evidence for relative motion between any of the components A1, A3, B1, B2, B4, C1 and C2 (we will refer to these components as the `stationary' group). In contrast the most significant separation we find is between the outer components A2 and C2. From linear regression analysis of MODELFIT data we find a separation rate between these two components of [FORMULA] µas yr-1 (see Fig. 3). These components are well separated on our CLEAN images which allows us to also use the AIPS task JMFIT to fit the position of A2 and C2 on the CLEAN images at each epoch, giving a similar separation rate of [FORMULA] µas yr-1.

Given the small number of degrees of freedom tests of significance are best made from examining the correlation coefficients obtained from our linear regression analysis. For the MODELFIT and JMFIT analysis of the A2-C2 separation we obtain correlation coefficients between epoch and separation of 0.981 and 0.998 respectively, which allows us to reject the null hypothesis of no motion at better than the 1% and 0.1% confidence levels. Analysis of other pairs of components (see Table 3) suggests that the significant change in A2-C2 separation is caused primarily by motion of A2 northward rather than motion of C2 southward. For instance we found the separation rate of A2-B2 to be [FORMULA] µas yr-1, very similar to the A2-C2 separation rate. In contrast the B2-C2 separation rate of [FORMULA] µas yr-1 is consistent with zero.

Amongst the other components the only other indication of motion is that B3 is moving northward relative to B2 (and B4 and other members of the stationary group) at a rate of [FORMULA] µas yr-1. We also searched for motions between component pairs in directions perpendicular to the vectors separating them but found no significant motions.

Finally in our analysis of the multi-epoch data we searched for variations in component flux densities. In order to eliminate the effects of errors on the overall flux density scale at each epoch we measured the ratio of each component's flux density to that of component C2. None of the `stationary' components showed significant flux variations relative to each other or to C2, strongly arguing that all these components stayed constant in flux density over the observing period. We did however detect strong variability of the flux ratio for A2/C2 (see Fig. 4), implying changes in the A2 flux density and also a possible steady increase in the flux density of B3 of about [FORMULA] between the first and the last epochs.

[FIGURE] Fig. 4. Measurements of the flux density ratio of A2 and C2 components as a function of time

4.3. Bulk motion or internal structure changes?

The change in separation of the outer components of 0710+439 over 13 years is approximately [FORMULA] of the beam FWHM in the North-South direction. However we note that this shift is 1/3 of the FWHM of the A2 component in the same direction. This large shift combined with the fact that it appears to be consistent from epoch to epoch (see Fig. 3) and is the same when measured relative to several gaussian components, strongly argues that the motion of component A2 is real.

One possibility that must be eliminated is that the apparent change in the centroid position of A2 is not due to motion of the whole component northward but instead is due to changes in its internal structure. There could for instance be changes in the relative flux densities of stationary subcomponents within A2. However the non-monotonic change of the total flux density of A2 (see Fig. 4) seems to be inconsistent with a linear change of the centroid position of A2 (see Fig. 3). It also seems unlikely that if there were two subcomponents within A2 separated by a large enough distance to explain the detected centroid shift that we could still get a good fit at every epoch with a single gaussian component. Furthermore we would expect to see changes in the apparent size of the component with epoch which we do not see. Another possibility is that A2 consists internally of a true stationary hotspot and a jet knot which moves toward it. Again in this case we would have difficulty fitting A2 with a single gaussian and would expect to see the width of the whole component becoming smaller with time, which we do not see. While it always possible to construct `Christmas Tree' models in which the brightening and dimming of stationary components mimics bulk motion (Scheuer 1984), such models would need to be contrived to fit the observed changes in 0710+439. We argue that the changes seen are instead due to bulk motion of A2.

4.4. Motions relative to the core

Our analysis of the component motions showed (see Sect. 4.2) a group of components (A1, A3, B2, B4, C1, C2) which are stationary relative to each other. Because we were not able to detect the core position in our maps or models (except tentatively in the 5th epoch, see Sect. 3.3), we do not know for certain the motion, if any, of these components relative to the core. Despite this it seems most likely that this group is stationary relative to the core and defines a rest frame. If this is not the case then all these components, on different sides of the source, must move in unison, in a coordinated way relative to the core; a situation which appears very unlikely. Consider for instance if component C2 and hence the rest of the stationary group moves southward from the core at 0.126 [FORMULA] (exactly half of the A2-C2 separation rate). This scenario has the advantage that both A2 and C2 components are then advancing away from the core at the same speed. But in this case features B2 and B4 would be moving southward towards the core at 0.126 [FORMULA]. Since we believe that B2 and B4 are jet features the probability that both would be moving inward toward the core at the same speed, with an amplitude exactly matching the advance speed of A2 and C2 outward, would seem to be very low. The stationary group of components might conceivably move northward, but then component C2 would be moving inwards to the core, which seems to be unphysical if C2 is, as we expect, the southern hotspot.

We conclude that the most likely scenario is that all the components in the `stationary group' are also stationary with respect to the core as well as with respect to each other. In this case B2 and B4 are naturally interpreted as stationary shocks within the jet, and B3 as a possible travelling shock moving outward along the jet at 0.1[FORMULA] (see Sect. 4.2). However a consequence of this model is that the advance speeds of the two outer components, A2 and C2 through the surrounding medium are likely to be different. Consider the separation rates of A2-B2 and C2-B2, which give the lowest estimated errors (see Table 3) of A2 and C2 motion relative to a stationary component. From these measurements we estimate an advance speed for A2 of [FORMULA] and for C2 of [FORMULA]. If jet component B2 has after all a small undetected velocity outward relative to the core then the implied asymmetry in the advance speeds of A2 and C2 relative to the core would be increased. We note however that despite the apparent velocity difference statistical tests can only exclude the null hypothesis of no difference between the A2-B2 and C2-B2 advance speeds at the 10% confidence level; this result must therefore be confirmed by future observations.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998