B.3.3. Non-Circular Lenses
The tangential pseudo-caustic at the origin producing Einstein ring images
is unstable to the introduction of any angular structure into the
gravitational potential of the lens. There are two generic sources of
angular perturbations. The first source of angular perturbations is
the ellipticity of the lens galaxy. What counts here is the
ellipticity of the gravitational potential rather than of the surface
density. For a lens with axis ratio q, ellipticity
= 1 - q, or
eccentricity e = (1 - q2)1/2, the
ellipticity of the potential is usually
~
/ 3 -
potentials are always rounder than densities.
The second source of angular perturbations is tidal perturbations from any
nearby objects. This is frequently called the "external shear" or the
"tidal shear" because it can be modeled as a linear shearing of the
deflections. In all known lenses, quadrupole perturbations (i.e.
cos(2
)
where
is the
azimuthal angle)
dominate - higher order multipoles are certainly present and they can
be quantitatively important, but they are smaller. For example, in an
ellipsoid the amplitude of the
cos 2m
multipole scales as
m
(see Section B.4.4 and
Section B.8).
Unfortunately, there is no example of a non-circular lens
that can be solved in full generality unless you view the nominally analytic
solutions to quartic equations as helpful. We can make the greatest
progress for the case of an SIS in an external (tidal) shear
field. Tidal shear is due
to perturbations from nearby objects and its amplitude can be determined by
Taylor expanding its potential near the lens (see Part 1 and
Section B.4). Consider a lens with Einstein
radius E
perturbed by an object with effective lens potential
a distance
p
away. For
E <<
p we can Taylor
expand the potential of the nearby object about the center of the primary
lens, dropping the leading two terms.
3 This leaves, as the first
term with observable consequences,
![]() |
(B.26) |
where p is
the surface density of the perturber at the center of the
lens galaxy and
p
> 0 is the tidal shear from the perturber. If the
perturber is an SIS with critical radius bp and
distance
p
from the primary lens, then
p =
p
= bp /
2
p. With this
normalization, the angle
p points toward
the perturber. For a circular lens, the shear
p
= <
> -
can be expressed in terms
of the surface density of the perturber, and it is larger (smaller) than the
convergence if the density profile is steeper (shallower) than isothermal.
The effects of
p are
observable only if we measure a time delay
or have an independent estimate of the mass of the lens galaxy, while the
effects of the shear are easily detected from the relative positions of the
lensed images (see Part 1). Consider, for example, one component of the
lens equation including an extra convergence,
![]() |
(B.27) |
and then simply divide by
1 - p to get
![]() |
(B.28) |
The rescaling of the source position
1
/ (1 -
p) has no
consequences since the source position is not an observable
quantity, while the rescaling of the deflection is simply a change
in the mass of the lens. This is known as the "mass sheet degeneracy"
because it corresponds to adding a constant surface density sheet to the
lens model (Falco, Gorenstein & Shapiro
[1985],
see Part 1), and it is an important systematic problem for both strong
lenses and cluster lenses (see Part 3).
Thus, while the extra convergence can be important for the quantitative
understanding of time delays or lens galaxy masses, it is only the shear
that introduces qualitatively new behavior to the lens equations.
The effective potential of an SIS lens in an external shear is
= b
+
(
/ 2)
2
cos 2
leading
to the lens equations
![]() |
(B.29) |
where for
> 0 the perturber is due North (or South) of the lens. The
inverse magnification is
![]() |
(B.30) |
where =
(
1,
2) =
(cos
,
sin
).
The first step in any general analysis of a new lens potential is to locate the critical lines and caustics. In this case we can easily solve µ-1 = 0 to find that the tangential critical line
![]() |
(B.31) |
is an ellipse whose axis ratio is determined by the amplitude of the shear
and whose
major axis points toward the perturber.
We call it the tangential critical line because the associated
magnifications are nearly tangential to the direction to the lens galaxy
and because it is a perturbation to the Einstein ring of a circular lens.
The tangential caustic, the image of the critical line on the source plane,
is a curve called an astroid (Fig. B.15, it is
not a "diamond" despite repeated use of the term in the literature). The
parametric expression for the astroid curve is
![]() |
(B.32) |
where the parameter
is the same as
the angle appearing in the critical curve (Eqn. B.31) and we have defined
± =
2b
/ (1 ±
)
for the locations of the cusp tips on the axes. The astroid consists of
4 cusp caustics on the symmetry axes
of the lens connected by fold caustics with a major axis pointing
toward the perturber. Like the SIS model without
any shear, the origin plays the role of the radial critical line and
there is a circular radial pseudo-caustic at
= b.
As mentioned earlier, there is no useful general solution for the image
positions and magnifications. We can, however, solve the equations for
a source on one of the symmetry axes of the lens. For example, consider
a solution on the minor axis of the lens
(2
= 0 for
> 0).
There are two ways of solving the lens equation to satisfy the
criterion. One is to put the images on the same axis
(
2 = 0)
and the other is to place them on the arc defined by
0 = 1 +
-
b /
.
The images with
2 = 0 are
simply the SIS solutions corrected
for the effects of the shear. Image 1 is defined by
![]() |
(B.33) |
and image 2 is defined by
![]() |
(B.34) |
Image 1 exists if
1
> - b, it is a saddle point for - b <
1
< -
+
and it is a minimum for
1
> -
+.
Image 2 has the reverse ordering. It exists for
1
< b, it is a saddle point for
+
<
1
< b and it is a minimum for
1
<
+.
The magnifications of both
images diverge when they are on the tangential critical line
(
1
= -
+
for image 1 and
1 =
+
+
for image 2) and approach zero as they move into the core of the lens
(
1
- b for image 1
and
1
+ b for image
2). These two images shift roles as the source moves through the origin.
The other two solutions are both saddle points, and they exist only
if the source lies inside the astroid
(|
1|
<
+
along the axis). The positions of images 3 (+) and 4 (-) are
![]() |
(B.35) |
and they have equal magnifications
![]() |
(B.36) |
The magnifications of the images diverge when the source reaches the cusp
tip
(|1|
=
+)
and the image lies on the tangential critical curve.
Thus, if we start with a source at the origin we can follow the changes in
the image structure (see Fig. B.15,
B.16).
With the source at the origin we see 4 images on the
symmetry axes with reasonably high magnifications,
| µi| =
(2 /
) / (1 -
2)
~ 10. It is a generic result that the least magnified four-image system is
found for an
on-axis source, and this configuration has a total magnification of
order the inverse of the ellipticity of the gravitational potential.
As we move the source toward the tip of the cusp
(
+,
Fig. B.15),
image 1 simply moves out along the symmetry axis with slowly dropping
magnification, while images 2, 3 and 4 move toward a merger on the
tangential critical curve at
=
(-
+,
0). Their
magnifications steadily rise and then diverge when the source reaches
the cusp. If we move the source further outward we find only images
1 and 2 with 1 moving outward and 2 moving inward toward the origin.
As it approaches the origin, image 2 becomes demagnified and vanishes
when
b.
Had we done the same calculation on the major axis
(Fig. B.16),
there is a qualitative difference. As we moved image 1 outward along the
2
axis, image 3 and 4 would merge with image 1 when the source reaches the
tip of the cusp at
2
=
_ rather
than with image 2.
Unfortunately once we move the source off a symmetry axis, there is no
simple
solution. It is possible to find the locations of the remaining images given
that two images have merged on the critical line, and this is useful for
determining the mean magnifications of the lensed images, a point we will
return to when we discuss lens statistics in
Section B.6. Here we simply illustrate
(Fig. B.17) the behavior of the images when we
move the source radially
outward from the origin away from the symmetry axes. Rather than three
images merging on the tangential critical line as the source approaches
the tip of a cusp, we see two images merging as the source approaches the
fold caustic of the astroid. This difference, two images merging versus
three images merging, is a generic difference between folds and cusps
as discussed in Part 1. All images in these four-image configurations are
restricted to an annulus of width
~ b
around the critical line, so the mean magnification of all four image
configurations is also of order
-1
(see Finch et al.
[2002]).
There is one more possibility for the caustic structure of the lens if the
external shear is large enough. For
1/3 < ||
< 1, the tip of the
astroid caustic extends outside the radial caustic, as shown if
Fig. B.18.
This allows a new image geometry, known as the cusp or disk geometry,
where we see three images straddling the major axis of a very flattened
potential. It is associated with the caustic region inside the astroid
caustic but outside the
radial caustic. This configuration appears to be rare for lenses produced
by galaxies, with APM08279+5255 as the only likely candidate, but
relatively
more common in clusters. The difference is that clusters tend to have
shallower density profiles than galaxies, which shrinks the radial
caustics relative to the tangential caustics to allow more cross section
for this image configuration and lower ellipticity thresholds before it
becomes possible (Oguri & Keeton
[2004]
most recently, but also see Kochanek & Blandford
[1987],
Kovner
[1987a],
Wallington & Narayan
[1993]).
![]() |
![]() |
Figure B.18. Example of a cusp or disk image geometry on the source (top) and image (bottom) planes. The shear is high enough to make the tangential caustic extend outside the radial caustic. For a source inside both caustics (triangles) we see a standard four-image geometry as in Fig. B.16. However, for a source outside the radial caustic but inside the tangential caustic (squares) we have three images all on one side of the lens. This is known as the cusp geometry because it is always associated with cusps, and the disk geometry because flattened disks are the only natural way to produce them. Once the source is outside the cusp tip (pentagon), a single image remains. |
In general, it is far more difficult to analyze ellipsoidal lenses, in part because few ellipsoidal lenses have analytic expressions for their deflections. The exception is the isothermal ellipsoid (Kassiola & Kovner [1993], Kormann, Schneider & Bartelmann [1994], Keeton & Kochanek [1998]), including a core radius s, which is both analytically tractable and generally viewed as the most likely average mass distribution for gravitational lenses. The surface density of the isothermal ellipsoid
![]() |
(B.37) |
depends on the axis ratio q and the core radius s. For
q = 1 - < 1
the major axis is the
1 axis and
s is the major axis core
radius. The deflections produced by this lens are remarkably simple,
![]() |
(B.38) |
The effective lens potential is cumbersome but analytic,
![]() |
(B.39) |
the magnification is simple
![]() |
(B.40) |
and becomes even simpler in the limit of a singular isothermal ellipsoid
(SIE) with s = 0 where µ-1
1 - b /
. In this case,
contours of surface density
are also contours of
the magnification, and the tangential critical line is the
= 1/2 isodensity contour
just as for the SIS model. The critical radius scale b can be related
to the circular velocity in the plane of the galaxy relatively
easily. For an isothermal sphere we have that bSIS =
4
(
v /
c)2 Dds / Ds where the
circular velocity is vc = 21/2
v. For the
projection of a three-dimensional (3D) oblate ellipsoid of axis ratio
q3 and inclination i, so that
q2 = q32
cos2i + sin2 i, the deflection scale is
b = bSIS(e3 /
sin-1 e3) where e3 = (1 -
q32)1/2 is the eccentricity
of 3D mass distribution. In the limit that
q3
0 the model becomes a Mestel
([1963])
disk, the infinitely thin disk producing a flat rotation
curve, and b = 2bSIS /
(see
Section B.4.9 and
Keeton, Kochanek & Seljak
[1997],
Keeton & Kochanek
[1998],
Chae
[2003]).
At least for the case of a face-on disk,
at fixed circular velocity you get a smaller Einstein radius as you make
the 3D distribution flatter because a thin disk requires less mass to
produce the same circular velocity.
We can generate several other useful models from the isothermal ellipsoids.
For example, steeper ellipsoidal density distributions can be derived
by differentiating with respect to s2. The most useful
of these is the first derivative with
-3/2 which is
related to the Kuzmin
([1956])
disk (see Kassiola & Kovner
[1993],
Keeton & Kochanek
[1998]).
It is also easy to generate models with
flat inner rotation curves and truncated halos by taking the difference
of two isothermal ellipsoids. In particular if
(s) is an
isothermal ellipsoid with core radius s, the model
![]() |
(B.41) |
with a > s has a central core region with a rising
rotation curve for
s, a flat
rotation curve for
s
a
and a dropping rotation curve for
a. In the
singular limit
(s
0),
it becomes the "pseudo-Jaffe model" corresponding to a 3D density
distribution
(r2
+ s2)-1(r2 +
a2)-1 whose name derives
from the fact that it is very similar the Jaffe model with
r-2(r + a)-2 (Kneib et al.
[1996],
Keeton & Kochanek
[1998]).
We will discuss other common lens
models in Section B.4.1.
The last simple analytic models we mention are the generalized singular
isothermal potentials of the form
=
F(
) with
surface density
(
,
) =
(1/2)(F(
)
+ F''(
))
/
. Both the SIS and SIE
are examples of this model. The generalized isothermal sphere has a number
of useful analytic properties. For example, the magnification contours
are isodensity contours
![]() |
(B.42) |
with the tangential critical line being the contour with
= 1/2, and
the time delays between images depend only on the distances from the images
to the lens center (see Witt, Mao & Keeton
[2000],
Kochanek, Keeton & McLeod
[2001],
Wucknitz
[2002],
Evans & Witt
[2003]).
3 The first term, a constant, gives an equal contribution to the time delays of all the images, so it is unobservable when all we can measure is relative delays. The second term is a constant deflection, which is unobservable when all we can measure is relative deflections. Back.