and Cosmology

3. The World of Galaxies
124
The density (3.57) diverges for r → 0asρ ∝ r −2 ,
so that the mass model cannot be applied up to the
very center of a galaxy. However, the steep central in-
crease of the rotation curve shows that the core region
of the mass distribution, in which the density function
will deviate considerably from the r −2 -law, must be
small for galaxies. Furthermore, the mass diverges for
large r such that M(r) ∝ r. The mass profile thus has
to be cut off at some radius in order to get a finite
total mass. This cut-off radius is probably very large
( 100 kpc for L ∗ -galaxies) because the rotation curves
are flat to at least the outermost point at which they are
observable.
The SIS is an appropriate simple model for gravitational
lenses over a wide range in radius since it seems
to reproduce the basic properties of lens systems (such
as image separation) quite well. The surface mass density
is obtained from the projection of (3.57) along the
line-of-sight,
Σ(ξ) = σ 2 v
2Gξ , (3.58)
which yields the projected mass M(ξ) within radius ξ
∫ ξ
M(ξ) = 2π
0
dξ ′ ξ ′ Σ(ξ ′ ) = πσ 2 v ξ
G . (3.59)
With (3.54) the deflection angle can be obtained,
Fig. 3.35. Sketch of an axially symmetric lens. In the top panel,
θ − α(θ) is plotted as a function of the angular separation θ
from the center of the lens, together with the straight line
β = θ. The three intersection points of the horizontal line at
fixed β with the curve θ − α(θ) are the three solutions of
the lens equation. The bottom image indicates the positions
and sizes of the images on the observer’s sky. Here, Q is the
unlensed source (which is not visible itself in the case of light
deflection, of course!), and A, B1, B2 are the observed images
of the source. The sizes of the images, and thus their fluxes,
differ considerably; the inner image B2 is particularly weak in
the case depicted here. The flux of B2 relative to that of image
A depends strongly on the core radius of the lens; it can be so
low as to render the third image unobservable. In the special
case of a singular isothermal sphere, the innermost image is
in fact absent
ˆα(ξ) = 4π
α(θ) = 4π
( σv
) 2
,
c
) ( )
2 Dds
≡ θ E . (3.60)
c D s
( σv
Thus the deflection angle for an SIS is constant and
equals θ E , and it depends quadratically on σ v . θ E is
called the Einstein angle of the SIS. The characteristic
scale of the Einstein angle is
(
θ E = 1 . ′′ σ
) ( )
v
2 Dds
15
200 km/s D s
, (3.61)
from which we conclude that the angular scale of the
lens effect in galaxies is about an arcsecond for massive
galaxies. The lens equation (3.56) for an SIS is