and Cosmology

2. The Milky Way as a Galaxy
66
The deflection angle α(θ) depends on the mass distribution
of the deflector. We will discuss the deflection
angle for an arbitrary density distribution of a lens in
Sect. 3.8. Here we will first concentrate on point masses,
which is – in most cases – a good approximation for the
lensing effect on stars.
For a point mass, we get – see (2.71)
|α(θ)|= D ds
D s
4 GM
c 2 D d |θ| ,
or, if we account for the direction of the deflection (the
deflection angle always points towards the point mass),
α(θ) = 4 GM
c 2
D ds
D s D d
θ
|θ| 2 . (2.78)
Fig. 2.22. Geometry of a gravitational lens system. Consider
a source to be located at a distance D s from us and a mass
concentration at distance D d . An optical axis is defined that
connects the observer and the center of the mass concentration;
its extension will intersect the so-called source plane, a plane
perpendicular to the optical axis at the distance of the source.
Accordingly, the lens plane is the plane perpendicular to the
line-of-sight to the mass concentration at distance D d from
us. The intersections of the optical axis and the planes are
chosen as the origins of the respective coordinate systems.
Let the source be at the point η in the source plane; a light
beam that encloses an angle θ to the optical axis intersects the
lens plane at the point ξ and is deflected by an angle ˆα(ξ).All
these quantities are two-dimensional vectors. The condition
that the source is observed in the direction θ is given by
the lens equation (2.74) which follows from the theorem of
intersecting lines
or, after dividing by D s and using (2.72) and (2.73):
β = θ − D ds
ˆα(D d θ). (2.75)
D s
Due to the factor multiplying the deflection angle in
(2.75), it is convenient to define the reduced deflection
angle
α(θ) := D ds
D s
ˆα(D d θ) , (2.76)
so that the lens equation (2.75) attains the simple form
β = θ − α(θ) . (2.77)
Multiple Images of a source occur if the lens equation
(2.77) has multiple solutions θ i for a (true) source
position β – in this case, the source is observed at the
positions θ i on the sphere.
Explicit Solution of the Lens Equation for a Point
Mass. The lens equation for a point mass is simple
enough to be solved analytically which means that for
each source position β the respective image positions θ i
can be determined. If we define the so-called Einstein
angle of the lens,
θ E :=
√
4 GM
c 2
D ds
D s D d
, (2.79)
then the lens equation (2.77) for the point-mass lens
with a deflection angle (2.78) can be written as
β = θ − θ 2 E
θ
|θ| 2 .
Obviously, θ E is a characteristic angle in this equation,
so that for practical reasons we will use the scaling
y := β θ E
; x := θ θ E
.
Hence the lens equation simplifies to
y = x −
x
|x| . (2.80)
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