4.1 Large Arcs and Central Potential

In Section 2 we showed that the location and morphology of arcs provide information on the projected mass distribution within the corresponding critical line. The relevant parameters for the modeling of the lens are the position of the arcs with respect to a possible cluster center, the number of merging images together with observed parity changes, the break between each merging image which gives the position of the critical line, the length and width of arcs, the ratio of their curvature radius and their distance from the center, their orientation, and last but not least, their redshifts. For a given geometry of the universe it is necessary to scale the projected mass density by measuring the redshift of at least one large arc. Only then can the total mass in the cone of light delineated by the arc be determined. The mass can be well estimated just by using a point mass lens M(r < r_arc) c² r² D / 4G. For larger radii the total mass M(r) can be derived if the gradient of the potential is known as a function of r. It will be seen later that M(r) can be derived from observations of the arclets. In fact it would be useful to get the redshifts of several bright arc(let)s at various distances from the center, but the probability of having two bright arc(let)s in the same cluster is very small and no such cases have been observed yet.

Miralda-Escudé (1993a) has summarized how some generic features of arcs can guide the modeling. As an example, for single elliptical potentials, a steep mass density profile generates thin arcs oriented along the minor axis of the potential. Similarly, a shallower potential profile generates thick arcs along the main axis (Kovner 1989). Consequently, the ratio between the curvature radius and the radial distance is small for steep potentials, whereas it is large for shallower potentials. The curvature and orientation of tangential arcs only provide an upper limit for the core radius for an elliptical potential (second order dependence, Kneib 1993). This poor dependence on core radius is quite understandable, since no rays of light are deflected in the very center of the cluster for large tangential arcs. Only multiply-lensed images with a radial arc (Fig. 11) give strong constraints on the typical scale of the mass profile since a radial arc is formed in the steepest part of the potential near the core radius (Miralda-Escudé 1993a). Finally, the shape of arcs, such as straight arcs (A2390, MS0302), or the position of peculiar merging images (A370), can suggest several mass concentrations that should be modeled with multiple potentials. Kovner (1989) gave numerous theoretical examples with two elliptical potentials that is a precious guide for a quick qualitative understanding of observers data.

In practice, the mass distribution in clusters is deduced by fitting the image produced by an analytic potential (or an analytic projected mass density) to the observations. Smooth circular or elliptical models with or without point mass distributions have been used: galaxy point masses in the very first models (Hammer 1987) with a fast image reconstruction method proposed by Schramm & Kayser (1987) and Kayser & Schramm(1988), nearly elliptical mass distributions (Grossman & Narayan, 1989; Narasimha & Chitre, 1988; Mirada-Escudé 1993a); halos following a de Vaucouleurs law (Hammer & Rigaud 1989; Angonin, Hammer & Le Fèvre 1992). It is worth noting here that Mellier & Mathez (1987) gave an approximate 3D mass density profile corresponding to the de Vaucouleurs law over five orders of magnitude, that is relatively easy to use for lens modeling:

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The projected elliptical potential given in part 2 was extensively used by Kovner (1987a); Mellier et al. (1990); Kochanek (1990); Kassiola, Kovner & Blandford (1992); Mellier, Fort & Kneib (1993); Kneib et al. (1993) but is only valid for small ellipticities. Recently, Kassiola & Kovner (1993a), and Kormann, Schneider & Bartelmann (1994) computed analytic expressions with pseudo-isothermal elliptic mass distributions which are easily tractable and valid for large ellipticities. In practice, with the image resolution of ground based telescopes there are almost no significant discrepancies in the resulting matter distribution discussed below when different analytical expressions or fitting procedures are used. This will probably not be the case with future arc images expected with the refurbished Hubble Space Telescope. So far, our knowledge of the distribution of dark matter in distant clusters is rather indirect and poor. Therefore, most of the models are chosen to be as simple as possible with a minimum number of free parameters which do not exceed the number of observational parameters. The simplest approach is to verify if the single elliptical potential model described in part 2 fits the images of large arcs.

When arc images contain too few significant pixels the reconstruction of the potential may be done using minimization techniques with a priori hypotheses which are often not based on observations. For instance, Kovner's models (1988) minimized the source size. The results were so convincing that they immediately appeared as additional proof of the gravitational nature of the newly discovered arcs. It can also be assumed that the distant source has a circular shape. But more sophisticated techniques can be used with images having high angular resolution. Good examples are given by Mellier, Fort & Kneib (1993), Kneib et al. (1993), and Miralda-Escudé (1994) for modeling MS2137-23 and A370. These clusters are among the most constrained because they show several multiple images. Each image is defined by its position (X_i, Y_i), its shape (a_i, _i, _i), and in some cases its redshift (z_i). The image parameters depend on the intrinsic parameters of the source but they also carry information about the lensing potential. Kochanek (1990), Miralda-Escudé (1991a), and Kneib (1993) defined some tractable estimators which relate simply the source and image parameters to the potential. Fig. 16 shows the relation between the shape parameters and the convergence and the shear. The total number of direct observational constraints is given by the number of multiple images. Additional indirect constraints are also used, such as the velocity dispersion of the cluster galaxies, the position and shape parameters of the giant central galaxies, and eventually X-ray maps. The center of the potential that is derived from the arc(let)s pattern is in fact a critical parameter for the modeling. When a huge cD galaxy is found at the cluster center it often coincides in position with the center of the potential, as inferred from lensing. But in most of the cases given in table 1 there is an uncertainty of at least several arcsec.

The fitting procedure for the modeling of a potential can be described as follows. For a given set of potential parameters, all the image segments of an arc system associated with a (presumed) unique source are projected back to the source plane (Fig. 10, bottom panel), and the position and shape of the projected images are considered together. Each image produces an individual object in the source plane that the procedure attempts to superimpose perfectly onto a single object of well-defined shape, by varying the potential parameters. When a good fit is obtained, the antecedent sources of each arc segment must be very close to each other. Projected back to the image plane, the source must generate at least as many images as are observed (Fig. 10, bottom panel). It can also give extra, eventually demagnified, images that can or cannot be observed on the CCD images, but which are crucial tests for any modeling. Even if the minimization is done in the source plane, we have to check the goodness of the fit in the image plane by looking at the differences between the images generated by the source in the model potential and the observed images. It is actually better to optimize the reconstruction directly in the image plane where we benefit from the magnification factor.

In general, the best models are not unique but with the resolution of ground based telescopes the family of solutions are similar whatever the optimization procedure used. Nevertheless, for some exceptional cases the observed number of gravitational images is large enough to (over)constrain the elliptical model described in part 2.4. For MS2137-23 and A370, the procedure has worked out so well that it was even possible to predict the location of faint additional images of arcs that have been detected a posteriori on deep CCD images. The prediction capability of a lens modeling is the most convincing way to prove the validity of a solution.

For future HST images of arcs, the spatial resolution will be far higher than the width of the arc (about one arcsec). One could use the Etherington theorem (1933) which prescribes that each corresponding element of a multiple image has the same surface brightness as its antecedent in the source plane. Kochanek et al. (1989) successfully used this approach on some high resolution images of radio rings (the so-called Ring-Cycle algorithm). For exceptional optical images obtained with ground based telescopes, Kochanek & Narayan (1992) have extended the Ring-Cycle approach with the lensclean algorithm which allows reconstruction of the source. This has not been done yet with real observations, but this is certainly the next step for modeling giant optical arcs from HST images.

Figure 10. A schematic example of the fitting procedure. The top panel shows the images projected back to the source plane for a given set of potential parameters. The best model shall first superimpose the three antecedent sources on a single object of well defined size and shape. Then the source is re-imaged in the image plane (bottom panel) and the differences between the observed and reconstructed magnified images are minimized.