3. Event rate and mass spectrum
Consider a coordinate system where the lens is at rest and let the source move on a straight line projected onto the lens plane with velocity . Following Mao & Paczynski (1991), the characteristic width w is then defined as the range of impact parameters for which a microlensing event occurs. Clearly, the width w is proportional to the Einstein radius , so that . The event rate is given by the product of the area number density of the lenses, the perpendicular velocity and the characteristic width of the considered type of event:
For variable lens position, the area number density of the lenses has to be replaced by an integral of the volume number density n over the line-of-sight direction x. For a general lens population, the number density depends also on the mass µ of the considered objects, so that one gets
as area number density of the lenses. If the mass spectrum does not depend on x, one can separate the x and µ-dependence by
where the function follows the volume mass density as , so that at the reference distance where .
The total volume number density of lenses at the reference distance is
so that the probability for a mass µin the interval is
which gives the mass spectrum.
With being the probability of finding the perpendicular velocity in the interval , one obtains for the event rate
with the average mass .
Let be a characteristic velocity and . The probability density for is then given by
Note that may depend on x. For any x, is normalized as
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998