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

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4. The probability density for the mass

Eq. (15) gives the total event rate which includes events with all possible timescales from the mass spectrum and the distribution of the lens position and velocity. By adding an integration over [FORMULA] and a [FORMULA] -function one gets

[EQUATION]

The event rate contribution for timescales [FORMULA] in the interval [FORMULA] is given by [FORMULA].

Let us now compare different mass spectra which have only the mass µ', i.e.

[EQUATION]

If one assigns the same probability to any mass a-priori, one has [FORMULA], i.e.

[EQUATION]

More generally, one can use any explicit form of the mass spectrum, e.g. a power law for [FORMULA] by using a weighting factor [FORMULA], i.e.

[EQUATION]

so that the case above corresponds to [FORMULA].

For the power-law mass spectra, one obtains

[EQUATION]

For a given µ', the probability for a timescale in the interval [FORMULA] is given by [FORMULA] as a function of [FORMULA]. This fact has been used by GRI and RJM to compare the distribution of the timescales [FORMULA] for different masses µ'. By exchanging the roles of µ' and [FORMULA] one obtains the contribution of masses in the interval [FORMULA] for events with [FORMULA] to the event rate as (µ' is called µin the following)

[EQUATION]

so that [FORMULA] gives the probability density for the mass µ.

The normalization factor is obtained by integration over µ, which gives

[EQUATION]

where

[EQUATION]

For the case that the velocity distribution does not depend on x, the function [FORMULA] separates as

[EQUATION]

where

[EQUATION]

The probability density for µfollows as

[EQUATION]

Note that the width [FORMULA] has cancelled out. This is due to the fact that the fit parameters are kept fixed and only the unknown quantities µ, x, and [FORMULA] are varied. Implicitly, the same probability is assigned to each parameter for the different values of µ, x, and [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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