          Astron. Astrophys. 330, 963-974 (1998)

## 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 or 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 With these definitions and , one gets     © European Southern Observatory (ESO) 1998

Online publication: January 27, 1998 