**B.4.4. The Angular Structure of Lenses**

Assuming you have identified all the halos needed to model a particular
lens,
there are three sources of angular structure in the potential. The first
source is the shape of the luminous lens galaxy, the second source is
the dark matter in the halo of the lens, and the third source is
perturbations
from nearby objects or objects along the line of sight. Of these, the
only one which is easily normalized is the contribution from the stars
in the lens galaxy, since it must be tightly connected to the monopole
deflection of the stars. The observed axis ratios of early-type galaxies
show a deficit of round galaxies, a plateau for axis ratios from
*q* ~ 0.9 to *q* ~ 0.5 and then a sharp decline beyond
*q* ~ 0.5 (e.g. Khairul & Ryden
[2002]).
Not surprisingly, the
true elliptical galaxies are rounder than the lenticular (S0) galaxies
even if both are grouped together as early-type galaxies.
In three dimensions, the stellar distributions are probably close to oblate
with very modest triaxialities (e.g. Franx et al.
[1991]).
Theoretical models of galaxy formation predict ellipticities and
triaxialities larger than observed for luminous galaxies and show
that the shapes of the dark matter halos are significantly modified
by the cooling baryons (Dubinski
[1992],
[1994],
Warren et al.
[1992],
Kazantzidis et al.
[2004]).
Local estimates of the shape
of dark matter halos are very limited (e.g. Olling & Merrifield
[2001],
Buote et al.
[2002]).
Stellar isophotes also
show deviations from perfect ellipses (e.g. Bender et al.
[1989],
Rest et al.
[2001])
and the deviations of simulated halos
from ellipses have a similar amplitude (Heyl et al.
[1994],
Burkert & Naab
[2003]).

It is worth considering two examples to understand the relative
importance of the higher order multipoles of a lens. The first is the
singular isothermal ellipsoid (SIE) introduced in
Section B.3 (Eqns. B.38-B.40). Let
the major axis of the model lie on the
_{1} axis, in
which case only the
cos(*m*)
multipoles with
*m* = 2, 4,^{ ... } are non-zero. All non-zero poles also have
the same radial dependence, with
_{cm} =
*A*_{m} / and
_{cm} = -
2*A*_{m} /
(*m*^{2} - 1). The ratio of the internal
to the external multipole depends only on the index of the multipole,
_{cm,int} /
_{cm,ext} =
(*m* - 1) / (*m* + 1). Note, in particular,
that the quadrupole moment of an SIE is dominated by the matter
*outside* any given radius, with an internal quadrupole fraction of

(B.74) |

*For lenses dominated by dark matter halos that have roughly flat global
rotation curves, most
of the quadrupole moment is generated outside the Einstein ring
of the lens (i.e. by the halo!).*
This will hold provided any halo truncation radius is large
compared to the Einstein ring radius. The tangential deflection is larger
than the radial deflection, with
|_{cm,rad}
/ _{cm,tan}| = 1
/ *m*. The final question is
the relative amplitudes between the poles. The ratio of the angular
deflection from the *m* = 2 quadrupole to the radial deflection of the
monopole is

(B.75) |

while the ratio for the *m* = 4 quadrupole is

(B.76) |

where the axis ratio of the ellipsoid is
*q* = 1 - .
Each higher order multipole has an amplitude
_{m}
^{m/2} to
leading order.

The relative importance
of the higher order poles can be assessed by computing the deflections
for a typical lens with the monopole deflection
(essentially the Einstein radius) fixed to be one arc second. Using the
leading
order scaling of the power-series, but setting the numerical value to
be exact for an axis ratio *q* = 1/2, the angular deflection from the
quadrupole is
0."46 and that
from the *m* = 4 pole is 0."09
^{2}, while
the radial deflections will be
smaller by a factors of 2 and 4 respectively. Since typical astrometric
errors are of order 0."005, the quadrupole is quantitatively
important for essentially any ellipticity while the *m* = 4 pole
becomes quantitatively important only for
*q* 0.75
(and the *m* = 6 pole becomes quantitatively important for
*q* 0.50).

In Fig. B.22 we compare the SIE to ellipsoidal
de Vaucouleurs and NFW models. Unlike the SIE, these models are
not scale free, so the multipoles depend on the distance from the
lens center in units of the major axis scale length of the lens,
*R*_{major}. The behavior of the de Vaucouleurs model will be
typical of any ellipsoidal mass distribution that is more centrally
concentrated than an SIE.
Although the de Vaucouleurs model produces angular
deflections similar to those of an SIE on small scales (for the same axis
ratio), these are beginning to decay rapidly at the radii where
we see lensed images (1 - 2*R*_{major}) because most of the
mass is interior to the image positions and the amplitudes of the
higher order multipoles decay faster with radius than the monopole (see
Eqn. B.48).
Similarly, as more of the mass lies at smaller radii, the quadrupole
becomes dominated by the internal quadrupole. The NFW model
has a somewhat different behavior because on small scales it
is less centrally concentrated than an SIE (a
1 / *r*
central density cusp rather than
1 /
*r*^{2}). It
produces a somewhat bigger quadrupole for a given axis ratio,
and an even larger fraction of that quadrupole is generated
on large scales. In a "standard" dark matter halo model,
the region with <
*R*_{major} is also where we see the
lensed images. On larger scales, the NFW profile is more centrally
concentrated than the SIE, so the quadrupole begins to decay
and becomes dominated by the internal component.

It is unlikely that mass distributions are true ellipsoids producing
only even poles (*m* = 2, 4, ^{ ... }) with no twisting of
the axes
with radius. For model fits we need to consider the likely amplitude
of these deviations and the ability of standard terms to absorb and
mask their presence. It is clear from Fig B.22
that the
amplitude of any additional terms must be of order the *m* = 4
deflections expected for an ellipsoid for them to be important. Here we
illustrate the issues with the first few possible terms.

A dipole moment (*m* = 1) corresponds to making the galaxy lopsided
with more mass on one side of the lens center than the other.
Lopsidedness is not rare in disk galaxies (~ 30% at
large radii, Zaritsky & Rix
[1997]),
but is little
discussed (and hence presumably small) for early-type galaxies.
Certainly in the CASTLES photometry of lens galaxies we never
see significant dipole residuals.
It is difficult (impossible) to have an equilibrium system
supported by random stellar motions with a dipole moment
because the resulting forces will tend to eliminate the dipole. Similar
considerations make it difficult to have a dark matter halo offset from
the luminous galaxy. Only disks, which are supported by ordered rather than
random motion, permit relatively long-lived lopsided structures.
Where a small dipole exists, it will have little effect on the lens
models unless the position of the lens galaxy is imposed as a stringent
constraint. The reason is that a dipole adds terms to the
effective potential of the form
_{1}
*G*() whose
leading terms are degenerate with a change in the unknown source position.

Perturbations to the quadrupole (relative to an ellipsoid) arise from variations in the ellipticity or axis ratio with radius. Since realistic lens models require an independent external shear simply to model the local environment, it will generally be very difficult to detect these types of perturbations or for these types of perturbations to significantly modify any conclusions. In essence, the amplitude and orientation of the external shear can capture most of their effects. Their actual amplitude is easily derived from perturbation theory. For example, if there is an isophote twist of between the region inside the Einstein ring and outside the Einstein ring, the fractional perturbations to the quadrupole will be of order , or approximately / 3 of the monopole - independent of the ability of the external shear to mimic the twist, the actual amplitude of the perturbation is approaching the typical measurement precision unless the twist is very large. Only in Q0957+561 have models found reasonably clear evidence for an effect arising from isophotal twists and ellipticity gradients, but both distortions are unusually large in this system (Keeton et al. [2000]). In general, in the CASTLES photometry of lens galaxies, deviations from simple ellipsoidal models are rare.

Locally we observe that the isophotes of elliptical galaxies are not perfect ellipses (e.g. Bender et al. [1989], Rest et al. [2001]) and simulated halos show deviations of similar amplitude (Heyl et al. [1994], Burkert & Naab [2003]). For lensing calculations it is useful to characterize these perturbations by a contribution to the lens potential and surface density of

(B.77) |

respectively where the amplitude of the term is related to the usual
isophote parameter *a*_{m} =
_{m}| 1 -
*m*^{2}| / *m b* for a lens with Einstein
radius *b*. A typical early-type galaxy might have
| *a*_{4}| ~ 0.01,
so their fractional effect on the deflections,
|_{4}| /
*b* ~ | *a*_{4}| / 4 ~ 0.003,
will be comparable to the astrometric measurement accuracy.