Extrasolar Moons as Gravitational Microlenses Christine Liebig

CHAPTER 2. GRAVITATIONAL LENSING 9
distances covered by the light path are very large in comparison to the extent of
mass distribution of the lens.
DS
I2
DLS
DL
L
O
β
θ
η S I1
Figure 2.1: Sketch of a gravitational
lensing system. Notation
is explained in the text.
ξ
α
˜α
In figures 2.1 and 2.2 the relevant parameters
of point-mass lensing are sketched. I1 and I2
denote the apparent locations of the two images
of the source S. There are always images for a
point mass lens L and a single source S (compare
equation (2.2) and its derivation below), except
for the singular case of perfect alignment, when
an Einstein ring is visible. The observer O sits in
the observer plane. The hyperbolic light paths
are approximated by two straight rays with a
sharp bend at the lens plane in figure 2.1.
General relativity predicts that a light ray,
which passes by a point mass M at a minimum
distance ξ, is deflected by an angle
ˆα = 4GM
c2 . (2.1)
ξ
G denotes the Gravitational constant, c the speed
of light. The point mass is a safe approximation
for a Galactic lensing situation, where two stars
align to form a gravitational lens system. It is
justified to approximate the lensing body as a
point mass or point lens under the assumption
that all light rays pass at a distance from the centre of mass, which is larger than
the radius of the lensing star.
Considering simple geometry in figure 2.1 we find the lens equation
βDS = θDS − ˜αDLS
for θ, β, ˜α