Lens equation 387
usingξ=Ddθ. In this case the lens equation (14.13) is linear, which means that
βis proportional toθ:
β=θ−β=θ−
4 πG
c^2
DdsDd
Ds
&θ=θ−
&
&cr
θ. (14.30)
From equation (14.30) we immediately see that for a lens with a critical surface
mass density we get for all values ofθ:β=0. Such a lens would perfectly focus,
with a well-defined focal length. Typical gravitational lenses behave, however,
quite differently. A lens which has&>&crsomewhere in it is defined as
supercritical, and has, in general, multiple images.
Definingk(θ):=&(θDd)/&crwe can write the lens equation as
β=θ−α ̃(θ), (14.31)
with
α ̃(θ)=
1
π
∫
R^2
d^2 θ′k(θ′)
θ−θ′
|θ−θ′|^2
. (14.32)
Moreover,
α ̃(θ)=∇θ+(θ) (14.33)
where
+(θ)=
1
π
∫
R^2
d^2 θ′k(θ′)ln|θ−θ′|. (14.34)
The Fermat potential is given by
(θ,β)=^12 (θ−β)^2 −+(θ) (14.35)
and we then obtain the lens equation from
∇θ(θ,β)= 0. (14.36)
Note that
+= 2 k≥ 0 (14.37)
(usingln|θ|= 2 πδ^2 (θ)), sincekas a surface mass density is always positive
(or vanishes).
The flux of a source, located inβ, in the solid angle d(β)is given by
S(β)=Iνd(β). (14.38)
Iνis the intensity of the source in the frequencyν.S(β)is the flux one would see
if there were no lensing. However, the observed flux from the image located inθ
is
S(θ)=Iνd(θ). (14.39)