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Astron. Astrophys. 343, L35-L39 (1999)

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3. Gravitational lens model

Compared with the original NTT image (Warren et al. 1996a) the new VLT image has much higher signal-to-noise ratio. The counter image predicted by our original model, but not convincingly detected in the NTT image, is now clearly seen. The same modelling procedure as described in Warren et al. (1996a), where the projected surface mass density was assumed to follow the (intrinsic) de Vaucouleurs profile, now measured from the K-band image (as described above), has been applied. The single free parameter is the global mass-to-light ratio (M/L).

Utilising the computational technique for arbitrary lenses with elliptical symmetry (Schramm 1994) the M/L ratio in the model was adjusted to produce the most compact configuration for the unlensed image in the source plane. Here we briefly review the key steps in the procedure; a position in the image plane, represented by the complex coordinate [FORMULA], is mapped onto the source plane position [FORMULA], according to


where [FORMULA]. The deflection potential, [FORMULA], is related to the projected surface mass density in the lens, [FORMULA], by


where [FORMULA] is the critical surface mass density to gravitational lensing and is given by


Here, [FORMULA] are the angular diameter distances between the source (s), lens (l) and observer (o) (Schneider et al. 1992).

A 61[FORMULA]61 pixel ([FORMULA]) region of the VLT narrow-band image, centred on the galaxy, was used for the computation. Assuming a fiducial value of M/L, the mapping given in Eq. 1 gives the coordinates in the source plane of any image-plane coordinate. The M/L was adjusted, focusing the emission over the source plane into a small region. This determines the centroid of the source. The source was then modelled as a Gaussian profile. The structure in the ring (i.e. the angular extent of the gaps, size of the counterimage) is dictated by the angular extent of the source. A source of FWHM of [FORMULA] when reimaged by the lensing potential was found to reproduce the structure in the ring well. To reimage the source each pixel was sub-pixelated into a 10[FORMULA]10 grid; these grid points were mapped to the source plane to measure the surface brightness at each grid point. Mapping of the surface brightness in this way is accurate provided the grid spacing mapped to the source plane is substantially smaller than the scale over which the surface brightness of the source varies.

Having fixed the source position and profile the lens M/L was then finely readjusted to provide the best fit, in terms of [FORMULA], of the model of the ring to the data. The results of this procedure are presented in Fig. 2. The upper left-hand panel presents the VLT image of the ring after subtraction of the model for the surface brightness distribution of the foreground galaxy. Below, the model source is shown on the same scale together with the caustic lines defined by the gravitational lens model; the caustics delineate the regions of multiple imaging over the source plane. The resultant image for this source configuration is presented in the lower right-hand panel. In the upper right-hand panel this image has been convolved with a Gaussian seeing profile to approximate the observing conditions. There is good correspondence between the structure in the model and the observed structure of the ring.

The measured angular radius of the ring in the VLT image is [FORMULA]. This more accurate value is smaller than the value measured from the NTT image ([FORMULA]) by [FORMULA] and this significantly lowers the mass estimate. Part of the discrepancy between the two measurements is due to the fact that the ring is elliptical in shape and that the counterimage (invisible in the old data) lies on the minor axis i.e. the old value for the radius was measured along the major axis of the ellipse. The two measurements are consistent therefore. The computed mass within the Einstein radius is [FORMULA] for [FORMULA]. The uncertainty in the mass estimate within the Einstein radius is dominated by the uncertainty in the radius rather than the form of the mass profile. Changing the radius by [FORMULA] (i.e. [FORMULA]) changes the computed mass by [FORMULA]. The M/L ratio for the model, corrected for luminosity evolution (Paper I), is [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999