MODERN COSMOLOGY

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