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4.4 The Weak Distortion Regime

After the discovery of small distorted objects in A370 (Fort et al. 1988), Tyson, Valdes & Wenk (1990) detected ~ 50 arclets in A1689 and Fort (1992), and Kneib et al. (1994) found ~ 60 similar objects in A370. More recently, Luppino et al. (1993) found a superb case of lensing with many arclets in MS0440+0204 (Fig. 15). One of the most extensive studies of an arclet population has been done for A2218 (Pelló et al. 1992; and Fig. 14). Arclets are distant galaxies belonging to Tyson's population (1988) which are weakly lensed by the foreground cluster (Fig. 4 and 5). They could be contaminated by cluster members or by foreground galaxies, like edge-on spirals aligned along the shear axis. Since these objects are often fainter than arcs, we cannot determine their redshifts spectroscopically. Furthermore, in the weak lensing regime, the gravitational potential may only change the intrinsic ellipticity of the sources by a small amount. The effect is so small that it can only be detected from a statistical analysis of the overall background population.

Figure 14
Figure 14. Deep CCD image of the rich cluster A2218 obtained at the Calar-Alto 3.50m telescope (Spain). Many arc(let)s surround the giant cD galaxy (top left). A second arc system is visible around another bright galaxy (bottom right) which is associated with a second concentration of mass (courtesy R. Pelló and J.-F. Leborgne).

The theoretical analyses by Kochanek (1990) and Miralda-Escudé (1991a) show that the distribution of arclets gives the center and the velocity dispersion of the cluster. They study the local correlation between the shapes of faint objects. The basic assumptions of their analysis are: conservation of the surface brightness for the mapping of the sources onto the image plane, the valid assumptions that the amplification matrix is constant between nearby images and also that the center of the image of a source is the image of the center of the source. Since images are weakly distorted, they are identified as small ellipses of which the ellipticity and orientation are given by their first and second moments weighted by the surface brightness.

Figure 15
Figure 15. Beautiful arcs and arclets distributed around the dense cluster MS0440+02. This is a very nice case of a circular distribution of gravitational images. (Courtesy G. Luppino)

The analysis is summarized by the following equations:

Equation 14 (14)

Equation 15 (15)

Equation 16
Equation 16 (16)
Equation 1c

Equation 17 (17)

The relationship between the shape parameters of the source fs and the shape parameters of the image fi is given by:

Equation 18 (18)

where (M-1)T is the transposed matrix.

Tyson, Valdes & Wenk first analyzed the statistical distortion of faint galaxies induced by a cluster (A1689). Their original idea was simply to calculate from an arbitrary center of reference a mean distortion parameter <T (x, y)> for all of the objects located within a given circular window at a distance r from this reference center. The distortion map was defined as:

Equation 19 (19)

where a and b are the lengths of the major and minor axes of the image. They are related to the second order moments by the simple relation

Equation 20 (20)

where I, S and Rtheta are respectively, the identity matrix, the diagonal matrix, and the rotation matrix. The averaged orientation and redshifts of the image parameters are then related to the source parameters through the shear gamma and the convergence kappa (Kochanek 1990):

Equation 21
Equation 21 (21)

The best center of the cluster is found for the (x0, y0) set of values that gives the largest distortion. For a singular isothermal sphere, the mean image distortion can be written as:

Equation 22 (22)

where thetac is the critical radius xic / Dd (see section 2). However, the distortion map is almost but not quite the dark matter map (Kaiser & Squires 1993).

Note that the shear increases from the center to the critical radius, and then decreases outside. The Tyson et al. study concerned the region within ~ 500 h50-1 kpc from the cluster center. They found that this cluster should have a typical velocity dispersion of ~ 1000 km s-1, and they gave an upper limit to the core radius of ~ 50 h50-1 kpc, in good agreement with arcs modeling. Kochanek (1990), Miralda-Escudé (1991a), and Kneib, Mellier & Longaretti (1992) emphasized that the technique has some limitations because the redshifts and the intrinsic ellipticities of the sources are unknown. Later, a rigorous analysis of the problem was done by Kaiser & Squires (1993). In principle, the projected mass density can be recovered by analyzing the weak shear on faint galaxies, even at distances as large as 5 h50-1 Mpc from the center. In that case, the asymptotic behavior of the mass profile (i.e. the slope of the potential) could be found, and possibly the cutoff radius of the cluster. However, measuring the weak distortion in the outermost regions of rich clusters is a technical challenge because the ellipticity induced by the lens on each individual galaxy is only a few percent, and must be measured on the faintest objects of the field. Systematic effects associated with the telescope (bad guiding), the atmosphere (seeing, atmospheric diffraction), the instrument (optical distortion, bad CCD charge transfer) induce instrumental elongations on the galaxies. In spite of these difficulties, Bonnet, Mellier & Fort (1994) attempted to measure the weak shear in the field of Cl0024+1654. They found this cluster to be the best candidate because it has the largest angular scale among the known cluster lenses and is therefore a good target to detect weak gravitational shear, even at large distances. The cluster was observed at CFHT within ± 2 hours from zenith, in subarcsecond seeing conditions during all the observing runs (< 0.7"). A comparison stellar field was observed in similar conditions in order to check the instrumental distortion all over the CCD field and the validity of the method.

Figure 16
Figure 16. Relations between the shapes of the images and the potential. For each object we define the shape parameters tauI = (a2 - b2) / 2ab and deltaI = (a2 + b2) / 2ab, taupot (zs) = - 2 kappa gamma / (kappa2 - gamma2) and deltapot (zs) = (kappa2 + gamma2) / (kappa2 - gamma2). where a and b are the major and the minor axes of the individual images, and taupot (zs) and deltapot (zs) the potential parameters. For the case of circular sources (top panel), the image is an ellipse oriented on the main axis of magnification, and the link with the potential is: tauI = taupot, and sigmaI = A sigmaS, where sigmaI is the area of the image, sigmaS of the object, and A the magnification. For an elliptical source, the images are not oriented on the main axes of magnification, because of the intrinsic source orientation. The final image depends on the potential in a more complicated way, but the principle is basically the same as for a circular source (see Kneib et al. 1994).

A special algorithm for faint image analysis was developed and tested on simulations (Bonnet & Mellier in preparation). The analysis was performed centered first on the cluster center and then on a larger field off-centered ~ 10' from the cluster center. The most distant region investigated is at 5 h50-1 Mpc from the center!

Figure 17
Figure 17. Orthoradial distribution of arclets around the cluster center. In this example, sources are almost circular so that their main axis is the magnification axis. For circular sources, all the perpendiculars would intersect together at the cluster center giving an efficient procedure to determine the cluster center with high accuracy. For elliptical sources, the intrinsic shape of galaxies is equivalent to an increase in the dispersion of the measurement of the cluster center because the main axis of the ellipse does not correspond exactly to the magnification axis. The statistical analysis of the arclets consists in getting the shapes of each individual image, computing the cluster center, and in measuring the averaged shape parameters of images within bins around the cluster center. The radial evolution of the shear depends on the total mass profile of the deflector. The main uncertainties come from the intrinsic shape and redshift distributions of the sources, and also from the seeing degradation and pixelisation of CCD frames.

The preliminary results are impressive: the distortion is detected all around the cluster center, and a weak shear is also measured in the off-centered field at 3 h50-1 Mpc (Fig. 18). The coherent pattern shows an orthoradial distribution of ellipticity of background sources with respect to the cluster center, as expected for a gravitational shear. From this analysis, it is clear that Cl0024+1654 extends at least to 3h50-1 Mpc. The shear matches what is expected from an isothermal sphere, even in the outermost region, which gives a large mass of 4 x 1015 Msmsun. The shear is also compatible with a de Vaucouleurs profile (Bonnet, Mellier & Fort 1994). Unfortunately, we only observed one off-axis field. Other distant fields around the center are needed to confirm this result, and also to have a more precise idea on how the shear decreases with radius. The study of the weak shear far from cluster centers should be a real breakthrough in our knowledge of these systems. In particular, the strong lensing regime (arcs) combined with the radial evolution of the weak shear now gives compelling evidence that clusters are relatively well described by isothermal spheres with finite cores. Flores & Primack (1994) discuss the physical implications of these results for structure formation and show that they can be severe rejection tests for theoretical models.

Similar studies are now in progress in various places. By using their own faint image analysis (Kaiser & Squires in preparation), Fahlman et al (1994) observed the shear in the cluster MS1224 and attempted to infer the mass profile from the Kaiser & Squires (1993) technique. They found that the mass is about 3 times larger than the virial mass expected from a constant M/L and suggest that dark clumps should be present in the cluster. Smail et al. (1994b) analyzed with a similar technique the cluster Cl1455+22 and Cl0016+16 but they rather suggest that mass is well traced by light.

Figure 18
Figure 18. Detection of the shear field around Cl0024+1654. The figure is composed of two deep CCD images obtained at CFHT which have been rotated and co-added. The inner region of the cluster is on the right and concerns a small field. The off-centered field is on the left and covers a much larger field. The thick full lines indicate the local average ellipticity computed by using about 20 of the nearest faint distant galaxies (selected according to their colors, and with B > 26). The orientation of the shear is given by the orientation of the line. The pattern is typical of a coherent gravitationally induced shear centered on the cluster center. Note also the perturbation of the shear field in the upper left. This effect is due to a secondary deflector locally modify the shear field.

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