## 4. Astrophysical applications## 4.1. Flux ratiosThe effects discussed above may be of some importance for the determination of by means of the time delay in QSO 0957+561A,B because of microlensing effects on the radio core. Our argument is based on the theoretical modelling by Grogin and Narayan (1996) who use radio observations of Garrett et al. (1994). One of the constraints of their model is the gradient along the jet (A image), , of the eigenvalue of the relative magnification matrix transforming the A image into B. The observed value of is measured upward along the A jet, corresponding to a difference in between core and jet (Component 5 in image A at an angular distance of from the core) of . We find that an error of 0.04 in in the core) amounts to an error of about 0.07 mag in the core, and vice versa since according to Garret et al. "the change in the relative magnification from the jet to the core is mainly caused by the spatial derivative of ". Therefore the uncertainty of should be changed: Hence, microlensing effects are important if is about 0.07 or larger. We must here take into account that the core images of both component A and B are affected, so that the relevant analytical approximation for the standard deviation is that of the difference between the two images where we have assumed , and is the Einstein radius projected on to the source plane. With a typical lens mass of and density parameter , we find . In view of our numerical results we roughly estimate the standard deviation in Eq. (6) to be about 20% smaller than in Eq.(7), i.e. We thus see from Eq.(6) that the estimated uncertainty in (one standard deviation) is significantly increased if is about one l.y. or less (), and that this could lead to a change in the best-fit model of Grogin and Narayan (1996) and thus a change in the estimated value of . In view of the observed variability of the radio core of more than in one year (5 months in the source system), a radius of less than one l.y. does not seem unrealistic. It is also interesting to note that the discrepancy between the observed and the model-predicted value of in the best-fit model og Grogyn and Narayan is rather large (nearly two standard deviations). We finally note that the relative importance of microlensing effects will be greater when more accurate radio observations are available. ## 4.2. Emission linesA constraint on the source size can be obtained by measuring the
equivalent widths of an emission line in the different images of a
quasar. An interesting example of this has been published by Lewis et
al. (1996) for QSO 2237+0305. Their data from 1991 on the For this particular quasar we arrived at a similar conclusion by analyzing the lightcurve (see Refsdal and Stabell (1993)). Note however, that this system is very special in that the lensing galaxy is very close (), so that the microlensing timescale is much shorter than for other known systems. For these systems a light curve covering about 50 years or more would probably be required, so that the method using the equivalent widths is the only one applicable at the moment. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |