Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
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3.6 Observables and Model Fitting

3.6.1 COMPACT SOURCES It is straightforward to trace rays through gravitational lens models to determine the image locations, magnification tensors and time delays corresponding to a given source position. It is less easy to infer the lensing potential from the observations. The lens Equation 2 provides one vector equation (two components) for each of the n images. Since the true position of the source is not known, this leads to 2(n-1) constraints that may be applied to the lensing potential. The relative magnifications of the images provide 4(n-1) constraints when the images are resolved and the transformation matrices can be determined, or (n-1) in the more usual unresolved case when only flux ratios are available. Finally, time delays may provide additional constraints.

When there are galaxies in the field it is common to assume that their inner parts can be modeled with King, De Vaucouleurs, or similar profiles with a few adjustable parameters, or with standard mass-to-light ratios where photometry of the lens is available [as in Q2237+031 (Schneider et al. 1988)]. The disposition of dark matter in the outer halos, in clusters and presumably also in groups of galaxies, is more problematical (Young et al. 1981b). This is usually modeled as either an isothermal sphere (with a finite core) or a quadratic lens. The parameters in the lens model are determined by using the constraints from the data, e.g. the image locations, magnifications, and time delays.

An additional consideration is usually included subjectively: the lensing geometry should not be too improbable. However, this has to be evaluated with considerable care as there are selection effects whereby seemingly unlikely alignments may be highly favorable for discovery.

The most important selection effect is magnification bias (sometimes also referred to as amplification bias) (Turner 1980, Turner et al. 1984), which occurs for images that are magnified as a consequence of lensing. Although high magnification configurations may have a small a priori cross section, they may actually dominate flux-limited samples if faint sources are sufficiently more numerous than bright sources (cf Sections 5.2.2, 6.5). Similar biases are possible with marginal lenses (Kovner 1987d).

In general, the largest lensing cross sections are associated with arrangements in which there are two images located on either side of the galaxy as in Q0957+561 and those where there are four images lying roughly on the Einstein ring as in Q2237+031 and Q1115+080 (Blandford & Kochanek 1987b, Pojmanski & Szymanski 1988, Nemiroff 1989). If the galaxy is part of a cluster, the image separation will be enlarged and the magnification will be enhanced, and there will be a magnification bias associated with these cases (Turner et al. 1984). One complication that is probably relevant to Q2016+112 is that strong lenses may be localized at more than one redshift. This opens up a much larger parameter space (Nottale & Chauvineau 1986, Kochanek & Apostolakis 1988, Jaroszynski 1989).

Simple lens models based on elliptical potentials suffice to account for most of the multiply-imaged quasars (e.g. Narayan & Grossman 1989, Blandford et al. 1989, Kayser 1990, Kochanek 1991a). More complex potentials, derived from more realistic galactic mass distributions, are used for detailed modeling (e.g. Dyer & Roeder 1981, Narasimha et al. 1982, Kayser & Schramm 1988, Schramm 1990, Schneider & Weiss 1991), but the models are not well constrained at present (Kochanek 1991a).

3.6.2 EXTENDED SOURCES When the source is extended there are effectively as many images to consider as there are resolution elements covered by the source. Consequently, arcs and rings can potentially provide more information about the lensing potential than do multiply-imaged quasars. With a resolved source, different images of the same source element should have identical surface brightness and this can be used as constraints in the modeling. The radio rings are particularly well-suited for such modeling. Typically, some parts of these sources are singly imaged while other parts are multiply imaged (with either three or five images, of which one in the center may be highly demagnified). Plausible lens and source models can be derived iteratively by matching the observed intensities over the multiple-image regions. This has been done for MG1131+0456 (Kochanek et al. 1989), MG1654+1346 (Kochanek 1990a), and 1830-211 (Kochanek & Narayan 1992). The rings could turn out to be the best sources for accurate determination of the lens potential.

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