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

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6. The method of mass moments

For the moments of the [FORMULA] -distribution for several events, one obtains following RJM

[EQUATION]

where

[EQUATION]

[FORMULA] yields for the case [FORMULA], [FORMULA], and for [FORMULA] the average mass (in units of [FORMULA]) as

[EQUATION]

One sees that the moments of the mass spectrum can be extracted from the moments of the time-scale distribution if the spatial distribution of the mass density and the velocity distribution is given (RJM).

For the moments of the mass spectrum one gets

[EQUATION]

This gives for [FORMULA]

[EQUATION]

which yields

[EQUATION]

so that

[EQUATION]

For [FORMULA], one obtains

[EQUATION]

since [FORMULA], which is the same value as obtained for the expectation value of the lens mass for [FORMULA] as derived above, if one considers a single observed event.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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