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Annu. Rev. Astron. Astrophys. 1992. 30:
311-358
Copyright © 1992 by Annual Reviews. All
rights reserved |
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.