Astron. Astrophys. 329, 827-839 (1998)

5. New model of the Cloverleaf gravitational lens

5.1. The procedure

The modelling of the Cloverleaf is based on the minimization algorithm described previously in several papers (Kneib et al 1993, 1996) and used to model giant arcs in cluster-lenses. This algorithm adjusts the parameters of the model through a minimization of the differences in the position and the geometry of the lensed spots once they are sent back to the source plane. The fitting uses observational constraints like the position, intensity and shape of the lensed spots and eventually the light distribution associated with the deflector as additional information on the mass distribution. The model incorporates parameters of the lensing potential, through a simple analytical representation of the mass distribution. In the present case, we use a truncated elliptical mass distribution (EMD) defined as the difference of two pseudo-isothermal elliptical models (PIEMD) (Kassiola & Kovner 1993, Hjorth & Kneib 1997):

where a is the core radius and s is the truncature radius. Moreover we have , where () are the centre position, () the current position in the principal axis of the lens, and is the ellipticity of the mass distribution. The important characteristic of such a model is that the mass distribution has an elliptical symmetry whatever . Its dependency at large radii imposes a finite mass to the model, and is compatible with the theoretical prescription of violent relaxation models (Hjorth & Madsen 1995). Furthermore, the model treatment is fully analytical.

The constraints used for the gravitational lens modelling are as follows:

1. the relative positions of the quasar spots from the HST and the intensity ratio taken in the R and I band. These constraints will primarily enable us to determine the mass model.
2. the non-detection of a 5th spot, which puts a limit on the size of the lens core.
3. the position of the cluster center as measured from the overdensity of galaxies near the Cloverleaf.
4. both the cluster and the lensing galaxy are assumed (for convenience) to be at z = 1.7.

The relative intensity ratios and the measured shapes of the CO spots are used as a test of the model, and provide as well information on the size and geometry of the CO source.

5.2. Results

We have computed two types of model: model 1 which includes an individual galaxy with a dark halo at z = 1.7 and model 2 which considers an individual galaxy and a cluster both at z = 1.7. These models are not unique but give similar qualitative results. The parameters for the lensing galaxy and the cluster component are shown in Table 6. Model 2 is dominated by the shear of the cluster (as the center of the cluster is close to the Cloverleaf - PA of the cluster center is 35 deg), which explains why the PA of the lensing galaxy is negative and different than in model 1. The cluster component is poorly known since we were not able to derive significant constraints on the lensing galaxy from the photometry (see Sect. 3.3). Yet, once an upper limit has been found for the velocity dispersion of the lensing galaxy, any model including an additional lens-plane (such as that of the galaxy cluster) and implying a mass, hence a velocity dispersion for the individual lensing galaxy lower than this limit, is formally acceptable, provided the gravitational shear observed on the CO map is modelled equally well.

Table 6. Results of the lens modelling of the Cloverleaf. The first model does not include a cluster component, while the second one does. Note that the cluster model can be even more complex and it should be considered only as a one particular solution.

In order to test, to first order, the lens model with the CO(7-6) map, we have assumed the CO source to be elliptical with a gaussian profile. We have fitted its position, size and ellipticity so that it reproduces the observed CO image. The upper limit of its size is provided by the CO elongated spots A and B which are close to merging, but still clearly separable. Thus, the model must predict, in the image plane, disconnected isocontours of the A and B spots. However, we underline that this estimated size also depends on the ellipticity of the lens mass distribution (as the scale in the source is proportional to R_E where R_E is the Einstein radius). We found a typical size of 460 230 pc (FWHM) for model 1, and 155 110 pc (FWHM) for model 2. In the present case, in the source plane translates into kpc with the chosen cosmology. A summary of the CO modeling is displayed in Fig. 7. Fig. 7j shows the CO emission of Fig. 7b before convolving it by the interferometer natural beam. It clearly shows us that it will be difficult to reach a higher degree of precision in describing exactly the source morphology, unless higher-resolution CO images can be acquired. An additional effect of the convolution results in an apparent shift of the centroids of the CO spots with respect to the quasar point-like spots. This demonstrates that the displacements observed between the centroids of the CO and visible spots are artifacts of the distorted morphology of the CO emission in the image plane.

Fig. 7. Results of the lens modelling of the Cloverleaf superimposed on the HST image. a  is the HST image overlaid with the CO observed (-225, +225 km/s); b  is the CO predicted for model 1, convolved by the interferometer beam. c  is similar to b but for model 2. j  is the the CO predicted for model 1, not convolved by the interferometer beam. d, e, f  is similar to the first row but for the blue emission. g, h, i  is similar to the first row but for the red emission. The dotted line is the corresponding critical line. k  gives the position of the best fitted CO sources for the blue (dashed), red (dotted) and total (solid) emission, for model 1. l  is similar to k but for model 2. The central diamond-shape curve (in b, e, h, k and l) is the internal caustic crossed by the lensed CO source at redshift z = 2.558.

The amplication ratios inferred from the modelling and the observations are displayed in Table 7. The agreement between the expected amplifications and the observed flux ratios is relatively good. The quasar intrinsic magnitude is about I 20.5. The comparison between the expected amplification and the observed ones in CO is not easy. Differential amplification due to the CO source extent and location with respect to the diamond caustic can explain the observed difference. The total amplification of the CO emission is 18 for model 1 and 30 for model 2. Following Barvainis et al (1997), these amplification factors translate for model 1 (resp. model 2) to molecular mass = 3. 10⁹ M (resp. 2.10⁹ M ) and I = 3. 10⁹ M (resp. 2.10⁹ M ). These mass estimates are in good agreement with the dynamical mass computed in the next section, providing uncertainties in the inclination and the conversion from to .

Table 7. Amplification ratios of the four spots as inferred from the lens modelling and compared with the visible and the CO (-225,+225 km/s) flux ratios. For the model we also give the m amplification within brackets. Accounting for the errors, the model reproduces the amplification ratios reasonably well. The last line gives the amplification ratio of the fifth spot. It is considerably demagnified. Since it is not detected on the visible data, we expect its magnitude to be fainter than , which implies that the quasar itself is fainter than .

5.3. Comparison with previous models

The comparison with previous models presented by K90 and recently by Yun et al (1997) and KKS is somewhat difficult. First, none of these included a lensing contribution from a distant galaxy cluster (our model 2). Regarding the properties of the individual lensing galaxy, the K90 model 1 (see Table 4 and Fig. 4a of K90) and KKS are "close" to our results of model 1. The relative position of the source and the lensing galaxy center are almost identical and the orientation of the gravitational potential is similar (PA of 20^o for K90, 21.6^o for KKS, and 21.5^o for our model 1). The upper limit on the velocity dispersion we find when no cluster component is introduced is 280 km/s, in excellent agreement with their upper limit ( 285 km/sec). But on the other hand, our ellipticity is significantly different. K90 used the standard definition of the eccentricity () for the projected mass density, the corresponding ellipticity is , which is 3.5 times larger than in our results. KKS found an ellipticity of 0.58 for an elliptical SIS model, for a two shear model (elliptical SIS plus external shear) while they found an ellipticity of 0.40 with a external shear of 0.19. This difference results from the analytical form of the mass profile chosen: K90 and KKS used a SIS while we are using a sum of two truncated EMDs with two parameters for the shape of the profile. Then, more freedom is given in the radial mass profile, leading to solutions with smaller ellipticities (cf. the model of H14176 and discussion in Hjorth & Kneib 1997).

On the contrary, our results do not agree with those by Yun et al (1997) who find a lensing elliptical potential perpendicular to that derived by K90 and to ours. This discrepancy is possibly due to the presence of a strong external shear. Indeed, the four CO spots observed (and predicted by our lens model) better constrain the orientation of the potential (although they do not insure the uniqueness of the solution).

© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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