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

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5. Moments of the probability distributions for physical quantities

The expectation value <µ> follows as

[EQUATION]

and for a general quantity [FORMULA] one obtains

[EQUATION]

This means that one averages over x with the density function [FORMULA] and over [FORMULA] with the density function [FORMULA].

For the quantity G being of the form

[EQUATION]

one obtains

[EQUATION]

For the mass, one has

[EQUATION]

so that [FORMULA] and [FORMULA], and

[EQUATION]

For [FORMULA], this gives

[EQUATION]

which is the same value as for the average mass in the mass spectrum (if one uses only the information from a single event), which can obtained by the method of mass moments described in Sect.  6. Note however that the probability density for µ for a specific event and the mass spectrum are different quantities and that the higher moments are different.

If the velocity distribution does not depend on x, one gets for the expectation value of the mass

[EQUATION]

[FORMULA] is related to the surface mass density [FORMULA] by

[EQUATION]

and [FORMULA] is related to the optical depth [FORMULA] by

[EQUATION]

Using these results, and noting that

[EQUATION]

the expectation value of the mass can be written in the convenient form

[EQUATION]

using the optical depth [FORMULA], the area number density of the lenses [FORMULA], the average square of the velocity [FORMULA], and the timescale [FORMULA]. Note that <µ> depends only on [FORMULA], not on the form of the velocity distribution.

To investigate the distribution of G around [FORMULA], I define

[EQUATION]

Using

[EQUATION]

one obtains for [FORMULA] the probability density

[EQUATION]

Note that this distribution depends only on the form of the distributions of x and [FORMULA] and not on any physical parameters like [FORMULA] or [FORMULA].

The distribution of [FORMULA] is given by

[EQUATION]

All moments of the probability distribution of G can be reduced to the function [FORMULA], which separates as the product of [FORMULA] and [FORMULA] if the velocity distribution does not depend on x. For the n -th moment one has

[EQUATION]

The relative deviation of G is given by

[EQUATION]

With

[EQUATION]

one obtains

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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