MODERN COSMOLOGY

392 Gravitational lensing

to the critical curve (which must be a circle). We see readily that the first case
occurs for a tangential critical curve, and the second for a radial critical curve.
The image of a circle (in the source plane) which lies close to a tangential critical
curve will be deformed to an ellipse with major axis tangential to the critical
curve. However, if the image of a circle gets close to a radial critical curve it will
be deformed to an ellipse with major axis radial to the critical curve.
For a tangential critical curve(|θ|=θt)we get

m(θt)=

∫θt

0

2 θκ(θ)dθ=θt^2. (14.61)

With the definition ofκthis translates to
∫ξt

0

2 ξ&(ξ)dξ=ξt^2 &cr. (14.62)

The total massM(ξt)inside the critical curve is thus

M(ξt)=πξt^2 &cr. (14.63)

This shows that the average density〈&〉tinside the tangential critical curve is
equal to the critical density&cr. This can be used to estimate the mass of a
deflector if the lens is sufficiently strong and the geometry is such that almost
complete Einstein rings are formed.

14.3.1.2 Einstein radius

For a lens with axial symmetry we get, with (14.11), the following equation:

β(θ)=θ−

Dds

DsDd

4 GM(θ)

c^2 θ

, (14.64)

from which we see that the image of a source, which is perfectly aligned (that
meansβ = 0), is a ring if the lens is supercritical. By settingβ = 0in
equation (14.64) we get the radius of the ring

θE=

(

4 GM(θE)

c^2

Dds

DdDs

) 1 / 2

, (14.65)

which is calledEinstein radius. The Einstein radius depends not only on the
characteristics of the lens but also on the various distances.
The Einstein radius sets a natural scale for the angles entering the description
of the lens. Indeed, for multiple images the typical angular separation between
the different images turns out to be of order 2θE. Moreover, sources with angular
distances smaller thanθEfrom the optical axis of the system are magnified quite
substantially whereas sources which are at a distance much greater thanθEare
only weakly magnified.