## 4. The probability density for the massEq. (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
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 so that gives the probability density for
the mass The normalization factor is obtained by integration over
For the case that the velocity distribution does not depend on
The probability density for 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 © European Southern Observatory (ESO) 1998 Online publication: January 27, 1998 |