## Appendix A: derivation ofIn 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: 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: 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 which coincides with Eq. (37) in CN43. The second integral in Eq. (A12) is: and we introduce Eq. (A13) in Eq. (A18) we obtain: where and 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 |