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

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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 [FORMULA]. 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 [FORMULA], so that [FORMULA]. The event rate [FORMULA] 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:

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where the function [FORMULA] follows the volume mass density [FORMULA] as [FORMULA], so that [FORMULA] at the reference distance where [FORMULA].

The total volume number density of lenses at the reference distance is

[EQUATION]

so that the probability for a mass µin the interval [FORMULA] is

[EQUATION]

which gives the mass spectrum.

With [FORMULA] being the probability of finding the perpendicular velocity in the interval [FORMULA], one obtains for the event rate

[EQUATION]

or

[EQUATION]

with the average mass [FORMULA].

Let [FORMULA] be a characteristic velocity and [FORMULA]. The probability density for [FORMULA] is then given by

[EQUATION]

Note that [FORMULA] may depend on x. For any x, [FORMULA] is normalized as

[EQUATION]

With these definitions and [FORMULA], one gets

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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