Astron. Astrophys. 336, 411-424 (1998)

2. Basics of microlensing

For completeness we give here a short summary of the most important formulae of gravitational microlensing; for more details see for instance Jetzer (1997).

The time dependent magnification of a light source due to gravitational microlensing is given by

with , where is the minimal distance of the MACHO from the line of sight between the source and the observer. is the Einstein radius, defined as

with , D and s are the distances to the source and the MACHO, respectively. is the relative transverse velocity of the involved objects. is the characteristic time for the lens to travel the distance .

The probability , that a source is found within a radius of some MACHO, is defined as

with the mass density of microlensing matter at the distance from us along the line of sight.

The microlensing event rate is given by (De Rújula, Jetzer, Massó 1991, Griest, Alcock, Axelrod et al. 1991)

where and is the transverse velocity distribution. The maximal impact parameter is related to the threshold magnification by Eq. (1). In the following we take which corresponds to . is the MACHO number density, which for a spherical halo is given by

here denotes the angle between the line of sight and the direction towards the galactic center. is assumed not to depend on x and is normalized such that

equals the local dark matter mass density. is the distance from the Sun to the galactic center and is the core radius of the halo.

For the average lensing duration one gets the following relation

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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