Simple lens models 397
radius of a galaxy.p=0 corresponds to the Plummer distribution, whereas for
p= 1 /2 we get the isothermal sphere for large values ofξ.
Definingx=ξ/ξ 0 andk 0 =& 0 /&crwe can write equation (14.88) as
k(x)=k 0
1 +px^2
( 1 +x^2 )^2 −p
. (14.89)
The deflection potential is given by
+(x)=
k 0
2 p
[( 1 +x^2 )p− 1 ], (14.90)
which is valid forp=0, whereas forp=0weget
+(x)=
k 0
2
ln( 1 +x^2 ). (14.91)
Thus the lens equation is
y=x−α(x)=x−
k 0 x
( 1 +x^2 )^1 −p
. (14.92)
If√k 0 > 1 there is one tangential critical curve forx = xt,wherext =
k^10 /^1 −p−1, and a radial critical curve forx =xr, which is defined by the
equation
1 −k 0 ( 1 +xr^2 )p−^2 [ 1 +( 2 p− 1 )xr^2 ]= 0. (14.93)
The corresponding caustics are given byyt≡y(xt)=0, whereas
yr≡|y(xr)|=
2 ( 1 −p)x^3 r
1 −( 1 − 2 p)xr^2
. (14.94)
Sources with|y|<yrlead to the formation of three images, whereas for|y|>yr
there is only one image. The three images are at: x >xt(image of type I),
−xt<x<rr(image of type II) and−xr<x<0 (image of type III).
14.3.5 Extended source
The magnification for an extended source with surface brightness profileI(y)is
given by
μe=
∫
I(y)μp(y)d^2 y
∫
I(y)d^2 y
, (14.95)
whereμp(y)is the magnification of a point source at positiony.Asanexample
let us consider a disk-like source with radiusRcentred inywith a brightness