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

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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

[EQUATION]

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

[EQUATION]

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

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

[EQUATION]

with [FORMULA] the mass density of microlensing matter at the distance [FORMULA] 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)

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

For the average lensing duration one gets the following relation

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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