          Astron. Astrophys. 342, 34-40 (1999)

## Appendix A: derivation of In order to find we have to calculate the integral in Eq. (9) (see Sect. 2). We change the integration variable in Eq. (9) from to using the following relations:  we have that Eq. (9) can be rewritten as: and if we express the product as a function of (CN43 - Eq. 18) we can write in the following form: where is given by: or equivalently: The first integral in Eq. (A6) can be easily calculated: with For Eq. (A7) gives: that coincides with the first term in the right hand side of Eq. (22) in CN43.

The second integral is more difficult to evaluate and is given by: We have obtained this last result using Eq. (10) and Eq. (12) (see Sect. 2). Since we are interested in the moments of for a given we need only the behaviour of the function for . To obtain this we expand the term in powers of and in Eq. (A10) and then we have: or where is given by being . Substituting these last equations in Eq. (A12) we have: Now, we evalutate this integral following CN43. We introduce a system of coordinates with the z-axis in the direction of and letting , and . The result of this integration is given by: where For Eq. (A15) becomes which coincides with Eq. (37) in CN43.

The second integral in Eq. (A12) is: If we define the variable and we introduce Eq. (A13) in Eq. (A18) we obtain: where and w is the azimuthal angle. The integral can be calculated regarding z and t as complex variables and using an appropriate chosen contour (see Chandrasekhar & von Newmann 1942). The result of the integration in terms of the original variable is: where then we have obtained: Now substituting Eq. (A28) in Eq. (A4) we have: where we have used bars to indicate that the corresponding quantities have been averaged with the weight function .

If the distribution of the velocities of the field stars, , is spherical and the test star moves with velocity we have: and also:   and if we use the system of coordinates introduced by CN43 that is , and where is the angle between and we can simplify Eq. (A29) and then the Eq. (8) (see Sect. 2) becomes: where . We define, now, the following constants and functions:       so we can re-write Eq. (A34) as:     © European Southern Observatory (ESO) 1999

Online publication: December 22, 1998 