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

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Appendix A: derivation of Eq. (1)

Let [FORMULA] objects of one population have a surface density [FORMULA] while [FORMULA] objects of another population have a surface density [FORMULA]. 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 [FORMULA] (i = 1,2,..., N1) and those of population 2 objects are [FORMULA] (j = 1, 2,..N2). The two-dimensional angular distances are

[EQUATION]

The total number of stellar distances is [FORMULA]. The quantities [FORMULA] 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:

[EQUATION]

The stars of this couple are excluded from the further analysis. Then the distances [FORMULA] 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:

[EQUATION]

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

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

This is Eq. 1 of the paper).

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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