Astron. Astrophys. 337, 39-42 (1998)

## Appendix A: derivation of Eq. (1)

Let objects of one population have a surface density while objects of another population have a surface density . Both populations occupy the same area of a galaxy. We assume the distribution of the stellar populations in the galaxy to be Poissonian. The coordinates of population 1 objects (i = 1,2,..., N1) and those of population 2 objects are (j = 1, 2,..N2). The two-dimensional angular distances are

The total number of stellar distances is . The quantities are used in order to identify the couples of closest neighbours between the two populations. The distance of the first couple constituted from the first population 1 star and its nearest neighbour of population 2 is:

The stars of this couple are excluded from the further analysis. Then the distances are obtained in the same way and the stars of these couples are consecutively excluded also. The distance between the stars of the k-th couple is:

In this way a series of increasing distances are obtained. The maximum possible number of couples is (if ) and (if ).

The probability to find at least one object of population 1 within a radius from its closest neighbour of population 2 can be defined by Eq. (see Appendix in Ivanov, 1996):

Similarly, the probability to find at least one object of population 2 within a radius from its nearest neighbour of population 1 is:

Then the probability that two neighbours - one from population 1 and another from population 2-fall within a radius from one of them is:

This is Eq. 1 of the paper).

© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998