402 Gravitational lensing
Figure 14.5.Light curves for the different cases of figure 4.2. The maximal magnification
ism= 0 .32 mag, if the star just touches the Einstein radius (p= 1 .0). For smaller
values ofpthe maximum magnification gets larger.tis the time in units oft 0 (from [22]).
We can easily computeτassuming that the mass distribution in the galactic
halo is of the following form
ρ(r)=ρ 0 (a^2 +RGC^2 )
a^2 +r^2, (14.101)
which is consistent with a flat rotation curve.|r|is the distance from Earth,ais
the core radius,ρ 0 the local density nearby the solar system of dark matter and
RGCthe distance to the galactic centre. Standard values for these parameters are:
ρ 0 = 0 .3 GeV cm−^3 = 7. 9 × 10 −^3 M pc−^3 ,a= 5 .6 kpc andRGC= 8 .5 kpc.
Assuming a spherical halo made entirely of MACHOs, one finds an optical
depth towards the LMC ofτ= 5 × 10 −^7. This means that at any one moment
out of 2 million stars, one is being lensed. From this number it can be seen that
in order to obtain a reasonable number of microlensing events, an experiment has
to monitor several million stars in the LMC or in other targets such as the galactic
centre region (also referred to as the galactic bulge).
The magnification of the brightness of a star by a MACHO is a time-
dependent effect, since the MACHO, which acts as a lens, changes its location
relative to the line of sight to the source as it moves along its orbit around the
galaxy. Typically, the velocity transverse to the line of sight for a MACHO