394 Gravitational lensing
For the Schwarzschild lens, which is a limiting case of an axial symmetric one,
we can substituteβusing equation (14.71) and obtain this way the amplification
for the two images
μ±=[
1 −
(
θE
θ±) 4 ]− 1
=
u^2 + 2
2 u√
u^2 + 4±
1
2
. (14.73)
u=r/REis the ratio between the impact parameterr, that is the distance between
the lens and the line of sight connecting the observer and the source and the
Einstein radiusREdefined in equation (14.67).ucan also be expressed asβ/θE.
Sinceθ−<θEwe have thatμ−<0. The negative sign for the amplification
indicates that the parity of the image is inverted with respect to the source. The
total amplification is given by the sum of the absolute values of the amplifications
for each image
μ=|μ+|+|μ−|=u^2 + 2
u√
u^2 + 4. (14.74)
Ifr=REthen we getu=1andμ= 1 .34, which corresponds to an increase
of the apparent magnitude of the source ofm=− 2 .5logμ=− 0 .32. For
lenses with a mass of the order of a solar mass and which are located in the halo
of our galaxy the angular separation between the two images is far too small to be
observable. Instead, one observes a time-dependent change in the brightness of
the the source star. This situation is also referred to asmicrolensing.
Much research activity is devoted to studying microlensing in the context of
quasar lensing. Today, several cases of quasars which are lensed by foreground
galaxies, producing multiple observable images are known. The stars contained
in the lensing galaxy can act as microlenses on the quasar and, as a result, induce
time-dependent changes in the quasar brightness, but in a rather complicated way,
since here the magnification is a coherent effect of many stars at the same time.
This is an interesting field of research, which will lead to important results on
the problem of the dark matter in galaxies [15]. However, we will not discuss
extragalacticmicrolensing in detail (see, for instance, [16]), whereas we will
report in some depth ongalacticmicrolensing (see section 14.4).
The time delay between the two images of a Schwarzschild lens is given by
ct=4 GM
c^2(
1
2
u√
u^2 + 4 +ln√
u^2 + 4 +u
√
u^2 + 4 −u)
. (14.75)
The two images have a comparable luminosity only ifu ≤1 (otherwise the
difference is such that one image is no longer observable since it gets too
faint). Foru=1 one obtainst∼ 4 RS/c(typically for a galaxy with mass
M = 1012 M one findst ∼ 1 .3 years). Such measurements are important
since they allow to determine the valueH 0 of the Hubble constant (see section
14.5.1).