and Cosmology

2. The Milky Way as a Galaxy
70
the effect is measurable in principle. In the general case
that source, lens, and observer are all moving, v has
to be considered as an effective velocity. Alternatively,
the motion of the source in the source plane can be
considered.
Light Curves. In most cases, the relative motion can be
considered linear, so that the position of the source in
the source plane can be written as
β = β 0 + ˙β(t − t 0 ).
Using the scaled position y = β/θ E ,fory =|y| we
obtain
√
( ) t − 2 tmax
y(t) = p 2 +
, (2.89)
t E
where p = y min specifies the minimum distance from
the optical axis, and t max is the time at which y = p
attains this minimum value, thus when the magnification
μ = μ(p) = μ max is maximized. From this, and using
(2.85), one obtains the light curve
S(t) = S 0 μ(y(t)) = S 0
y 2 (t) + 2
y(t) √ y 2 (t) + 4 . (2.90)
Examples for such light curves are shown in Fig. 2.26.
They depend on only four parameters: the flux of the
unlensed source S 0 , the time of maximum magnification
t max , the smallest distance of the source from the
optical axis p, and the characteristic time-scale t E . All
these values are directly observable in a light curve. One
obtains t max from the time of the maximum of the light
curve, S 0 is the flux that is measured for very large
and small times, S 0 = S(t →±∞), orS 0 ≈ S(t) for
|t − t max |≫t E . Furthermore, p follows from the maximum
magnification μ max = S max /S 0 by inversion of
(2.85), and t E from the width of the light curve.
Only t E contains information of astrophysical relevance,
because the time of the maximum, the unlensed
flux of the source, and the minimum separation p
provide no information about the lens. Since t E ∝
√ MDd /v, this time-scale contains the combined information
on the lens mass, the distances to the lens and
the source, and the transverse velocity: Only the combination
t E ∝ √ MD d /v can be derived from the light
curve, but not mass, distance, or velocity individually.
Paczyński’s idea can be expressed as follows: if the
halo of our Milky Way consists (partially) of compact
objects, a distant compact source should, from time to
time, be lensed by one of these MACHOs and thus show
characteristic changes in flux, corresponding to a light
curve similar to those in Fig. 2.26. The number density
of MACHOs is proportional to the probability or abundance
of lens events, and the characteristic mass of the
MACHOs is proportional to the square of the typical
variation time-scale t E . All one has to do is measure
the light curves of a sufficiently large number of background
sources and extract all lens events from those
light curves to obtain information on the population of
potential MACHOs in the halo. A given halo model
predicts the spatial density distribution and the distribution
of velocities of the MACHOs and can therefore be
compared to the observations in a statistical way. However,
one faces the problem that the abundance of such
lensing events is very small.
Probability of a Lens Event. In practice, a system of
a foreground object and a background source is considered
a lens system if p < 1 and hence μ max > 3/ √ 5
≈ 1.34, i.e., if the relative trajectory of the source passes
within the Einstein circle of the lens.
If the dark halo of the Milky Way consisted solely of
MACHOs, the probability that a very distant source is
lensed (in the sense of |β|≤θ E ) would be ∼ 10 −7 , where
the exact value depends on the direction to the source.
At any one time, one of ∼ 10 7 distant sources would be
located within the Einstein radius of a MACHO in our
halo. The immediate consequence of this is that the light
curves of millions of sources have to be monitored to
detect this effect. Furthermore, these sources have to be
located within a relatively small region on the sphere to
keep the total solid angle that has to be photometrically
monitored relatively small. This condition is needed to
limit the required observing time, so that many such
sources should be present within the field-of-view of
the camera used. The stars of the Magellanic Clouds
are well suited for such an experiment: they are close
together on the sphere, but can still be resolved into
individual stars.
Problems, and their Solution. From this observational
strategy, a large number of problems arise immediately;
they were discussed in Paczyński’s original paper. First,