and Cosmology

3.8 Galaxies as Gravitational Lenses
depends only on the distances to the lens and to the
source. Although Σ cr incorporates a combination of cosmological
distances, it is of a rather “human” order of
magnitude,
( )
Dd D −1
ds
Σ cr ≈ 0.35
gcm −2 .
D s 1 Gpc
A source is visible at several positions θ on the sphere,
or multiply imaged, if the lens equation (3.49) has several
solutions θ for a given source position β. Amore
detailed analysis of the properties of this lens equation
yields the following general result:
If Σ ≥ Σ cr in at least one point of the lens, then
source positions β exist such that a source at β has
multiple images. It immediately follows that κ is
a good measure for the strength of the lens. A mass
distribution with κ ≪ 1 at all points is a weak lens,
unable to produce multiple images, whereas one
with κ 1 for certain regions of θ is a strong lens.
For sources that are small compared to the characteristic
scales of the lens, the magnification μ of an image,
caused by the differential light deflection, is given by
(2.83), i.e.,
( )∣ μ =
∂β ∣∣∣
−1
∣ det ∂θ
. (3.53)
The importance of the gravitational lens effect for extragalactic
astronomy stems from the fact that gravitational
light deflection is independent of the nature and the state
of the deflecting matter. Therefore, it is equally sensitive
to both dark and baryonic matter and independent
of whether or not the matter distribution is in a state
of equilibrium. The lens effect is thus particularly suitable
for probing matter distributions, without requiring
any further assumptions about the state of equilibrium
or the relation between dark and luminous matter.
3.8.2 Simple Models
Axially Symmetric Mass Distributions. The simplest
models for gravitational lenses are those which are axially
symmetric, for which Σ(ξ) = Σ(ξ), where ξ =|ξ|
denotes the distance of a point from the center of the
lens. In this case, the deflection angle is directed radially
inwards, and we obtain
ˆα = 4GM(ξ)
c 2 , (3.54)
ξ
where M(ξ) is the mass within radius ξ. Accordingly,
for the scaled deflection angle we have
α(θ) = m(θ)
θ
:= 1 ∫θ
θ 2
0
dθ ′ θ ′ κ(θ ′ ), (3.55)
where, in the last step, m(θ) was defined as the dimensionless
mass within θ. Since α and θ are collinear, the
lens equation becomes one-dimensional because only
the radial coordinate needs to be considered,
β = θ − α(θ) = θ − m(θ) . (3.56)
θ
An illustration of this one-dimensional lens mapping is
shown in Fig. 3.35.
Example: Point-Mass Lens. For a point mass M, the
dimensionless mass becomes
m(θ) = 4GM D ds
c 2 ,
D d D s
reproducing the lens equation from Sect. 2.5.1 for
a point-mass lens.
Example: Isothermal Sphere. We saw in Sect. 2.4.2
that the rotation curve of our Milky Way is flat for large
radii, and we know from Sect. 3.3.3 that the rotation
curves of other spiral galaxies are flat as well. This indicates
that the mass of a galaxy increases proportional
to r, thus ρ(r) ∝ r −2 , or more precisely,
ρ(r) =
σ v
2
2πGr 2 . (3.57)
Here, σ v is the one-dimensional velocity dispersion of
stars in the potential of the mass distribution if the
distribution of stellar orbits is isotropic. In principle,
σ v is therefore measurable spectroscopically from the
line width. The mass distribution described by (3.57) is
called a singular isothermal sphere (SIS). Because this
mass model is of significant importance not only for the
analysis of the lens effect, we will discuss its properties
in a bit more detail.
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