MODERN COSMOLOGY

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

. (14.72)