and Cosmology

3. The World of Galaxies
122
Fig. 3.34. As a reminder, another sketch of the lens geometry
ical description of such a lens system for an arbitrary
mass distribution of the deflector is obtained from the
following considerations.
If the gravitational field is weak (which is the case
in all situations considered here), the gravitational effects
can be linearized. 7 Hence, the deflection angle of
a lens that consists of several mass components can
be described by a linear superposition of the deflection
angles of the individual components,
ˆα = ∑ i
ˆα i . (3.45)
We assume that the deflecting mass has a small extent
along the line-of-sight, as compared to the distances between
observer and lens (D d ) and between lens and
source (D ds ), L ≪ D d and L ≪ D ds . All mass elements
can then be assumed to be located at the same
distance D d . This physical situation is called a geometrically
thin lens. If a galaxy acts as the lens, this condition
is certainly fulfilled – the extent of galaxies is typically
∼ 100 h −1 kpc while the distances of lens and source
are typically ∼ Gpc. We can therefore write (3.45) as
a superposition of Einstein angles of the form (2.71),
ˆα(ξ) = ∑ i
4Gm i
c 2
ξ − ξ i
|ξ − ξ i | 2 , (3.46)
7 To characterize the strength of a gravitational field, we refer to
the gravitational potential Φ. The ratio Φ/c 2 is dimensionless and
therefore well suited to distinguishing between strong and weak gravitational
fields. For weak fields, Φ/c 2 ≪ 1. Another possible way to
quantify the field strength is to apply the virial theorem: if a mass
distribution is in virial equilibrium, then v 2 ∼ Φ, and weak fields are
therefore characterized by v 2 /c 2 ≪ 1. Because the typical velocities
in galaxies are ∼ 200 km/s, for galaxies Φ/c 2 . 10 −6 . The typical
velocities of galaxies in a cluster of galaxies are ∼ 1000 km/s, so that
in clusters Φ/c 2 . 10 −5 . Thus the gravitational fields occurring are
weak in both cases.
where ξ i is the projected position vector of the mass
element m i ,andξ describes the position of the light ray
in the lens plane, also called the impact vector.
For a continuous mass distribution we can imagine
subdividing the lens into mass elements of mass
dm = Σ(ξ)d 2 ξ, where Σ(ξ) describes the surface mass
density of the lens at the position ξ, obtained by projecting
the spatial (three-dimensional) mass density ρ
along the line-of-sight to the lens. With this definition
the deflection angle (3.46) can be transformed into an
integral,
ˆα(ξ) = 4G
c 2
∫
d 2 ξ ′ Σ(ξ ′ ) ξ − ξ′
|ξ − ξ ′ | 2 . (3.47)
This deflection angle is then inserted into the lens
equation (2.75),
β = θ − D ds
D s
ˆα(D d θ) , (3.48)
where ξ = D d θ describes the relation between the position
ξ of the light ray in the lens plane and its apparent
direction θ. We define the scaled deflection angle as in
(2.76),
α(θ) = D ds
ˆα(D d θ),
D s
so that the lens equation (3.48) can be written in the
simple form (see Fig. 3.34)
β = θ − α(θ) . (3.49)
A more convenient way to write the scaled deflection is
as follows,
α(θ) = 1 ∫
d 2 θ ′ κ(θ ′ θ − θ ′
)
π
|θ − θ ′ | 2 , (3.50)
where
κ(θ) = Σ(D dθ)
(3.51)
Σ cr
is the dimensionless surface mass density, and the socalled
critical surface mass density
Σ cr =
c 2 D s
4πG D d D ds
(3.52)