Simple lens models 393
In several lens models the Einstein radius delimits the region within which
multiple images occur, whereas outside this region there is a single image. By
comparing equation (14.26) with equation (14.65) we see that the surface mass
density inside the Einstein radius precisely corresponds to the critical density. For
a point-like lens with massMthe Einstein radius is given by
θE=
(
4 GM
c^2
Dds
DdDs
) 1 / 2
, (14.66)
or instead of an angle one often also uses
RE=θEDd=
(
4 GM
c^2
DdsDd
Ds
) 1 / 2
. (14.67)
To get some typical values we can consider the following two cases: a lens of
massMlocated in the galactic halo at a distance ofDd∼10 kpc and a source in
the Magellanic Cloud, in which case
θE=( 0. 9 ′′× 10 −^3 )
(
M
M
) 1 / 2 (
D
10 kpc
)− 1 / 2
(14.68)
and a lens with the mass of galaxy (including its halo)M∼ 1012 M located at a
distance ofDd∼1Gpc
θE= 0. 9 ′′
(
M
1012 M
) 1 / 2 (
D
Gpc
)− 1 / 2
, (14.69)
whereD=DdDs/Dds.
14.3.2 Schwarzschild lens
A particular case of a lens with axial symmetry is the Schwarzschild lens, for
which&(ξ)=Mδ^2 (ξ)and thusm(θ)=θE^2. The source is also considered as
point-like, this way we get, for lens equation (14.13), the following expression
β=θ−
θE^2
θ
, (14.70)
whereθEis given by equation (14.66). This equation has two solutions:
θ±=^12
(
β±
√
β^2 + 4 θE^2
)
. (14.71)
Therefore, there will be two images of the source located one inside the Einstein
radius and the other outside. For a lens with axial symmetry the amplification is
given by
μ=
θ
β
dθ
dβ