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 and a -function one gets
The event rate contribution for timescales in the interval is given by .
Let us now compare different mass spectra which have only the mass µ', i.e.
If one assigns the same probability to any mass a-priori, one has , i.e.
More generally, one can use any explicit form of the mass spectrum, e.g. a power law for by using a weighting factor , i.e.
so that the case above corresponds to .
For the power-law mass spectra, one obtains
For a given µ', the probability for a timescale in the interval is given by as a function of . This fact has been used by GRI and RJM to compare the distribution of the timescales for different masses µ'. By exchanging the roles of µ' and one obtains the contribution of masses in the interval for events with to the event rate as (µ' is called µin the following)
so that gives the probability density for the mass µ.
The normalization factor is obtained by integration over µ, which gives
For the case that the velocity distribution does not depend on x, the function separates as
The probability density for µfollows as
Note that the width has cancelled out. This is due to the fact that the fit parameters are kept fixed and only the unknown quantities µ, x, and are varied. Implicitly, the same probability is assigned to each parameter for the different values of µ, x, and .
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998