and Cosmology

2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
After multiplication with x, this becomes a quadratic
equation, whose solutions are
x = 1 (
|y|± √ ) y
4 +|y|
2
2 |y| . (2.81)
From this solution of the lens equation one can
immediately draw a number of conclusions:
• For each source position y, the lens equation for
a point-mass lens has two solutions – any source is
(formally, at least) imaged twice. The reason for this
is the divergence of the deflection angle for θ → 0.
This divergence does not occur in reality because of
the finite geometric extent of the lens (e.g., the radius
of the star), as the solutions are of course physically
relevant only if ξ = D d θ E |x| is larger than the radius
of the star. We need to point out again that we explicitly
exclude the case of strong gravitational fields
such as the light deflection near a black hole or a neutron
star, for which the equation for the deflection
angle has to be modified.
• The two images x i are collinear with the lens and the
source. In other words, the observer, lens, and source
define a plane, and light rays from the source that
reach the observer are located in this plane as well.
One of the two images is located on the same side of
the lens as the source (x · y > 0), the second image is
located on the other side – as is already indicated in
Fig. 2.21.
• If y = 0, so that the source is positioned exactly behind
the lens, the full circle |x|=1, or |θ|=θ E ,is
a solution of the lens equation (2.80) – the source
is seen as a circular image. In this case, the source,
lens, and observer no longer define a plane, and the
problem becomes axially symmetric. Such a circular
image is called an Einstein ring. Ring-shaped images
have indeed been observed, as we will discuss
in Sect. 3.8.3.
• The angular diameter of this ring is then 2θ E .
From the solution (2.81), one can easily see that
the distance between the two images is about
Δx =|x 1 − x 2 | 2 (as long as |y| 1), hence
Δθ 2θ E ;
the Einstein angle thus specifies the characteristic
image separation. Situations with |y|≫1, and hence
angular separations significantly larger than 2θ E ,are
astrophysically of only minor relevance, as will be
shown below.
Magnification: The Principle. Light beams are not
only deflected as a whole, but they are also subject
to differential deflection. For instance, those rays of
a light beam (also called light bundle) that are closer to
the lens are deflected more than rays at the other side of
the beam. The differential deflection is an effect of the
tidal component of the deflection angle; this is sketched
in Fig. 2.23. By differential deflection, the solid angle
which the image of the source subtends on the sky
changes. Let ω s be the solid angle the source would
subtend if no lens were present, and ω the observed
solid angle of the image of the source in the presence
of a deflector. Since gravitational light deflection is not
linked to emission or absorption of radiation, the sur-
Fig. 2.23. Light beams are deflected differentially, leading to
changes of the shape and the cross-sectional area of the beam.
As a consequence, the observed solid angle subtended by the
source, as seen by the observer, is modified by gravitational
light deflection. In the example shown, the observed solid angle
A I /Dd 2 is larger than the one subtended by the undeflected
source, A S /Ds 2 – the image of the source is thus magnified
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