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

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2. Correlation technique

Let N1 stars of one population in M 33 have a surface density [FORMULA] while another population of N2 stars has a surface density [FORMULA] and [FORMULA] be the two-dimensional angular distance between the stars of the k-th stellar couple as defined in the Appendix. Supposing a random distribution of the stellar populations, the distance [FORMULA] between the two stars of the k-th couple has a probability (see the Appendix):

[EQUATION]

The quantity [FORMULA] gives the probability to find one star of population 1 and another star of population 2 within a radius equal to [FORMULA] in case both populations are randomly distributed. The associated stars of two different populations form couples with [FORMULA] and consequently [FORMULA]. The couples of foreground stars have large mutual distances [FORMULA] and [FORMULA]. Small values of [FORMULA] can be used as a good characteristic for associated couples. We chose upper and lower limits of the probability, [FORMULA] and [FORMULA]. [FORMULA], couples of associated stars are selected. Those stars for which [FORMULA] as "foreground couples" [FORMULA] are defined. Further in Sect. 4.1 will see that [FORMULA] and [FORMULA]. When the associated stars are selected by the criterion [FORMULA] the number of associated couples is indicated as N5. A stronger criterion for selecting the associated stars assumes [FORMULA] and the foreground couples [FORMULA]. Then the number of associated couples is denoted as N1. A simple way to evaluate the correlation between two populations is to obtain the percentage of associated objects

[EQUATION]

The ratios R1 and R5 are very suitable measures of the correlation between the stellar populations. If all the stars between two populations are associated, then [FORMULA] or [FORMULA]. In the opposite case there are no associated stars between the populations ([FORMULA] or [FORMULA]). The ratios of Eq. 2are analogous to the conventional coefficient of correlation in the statistics. Another way to evaluate the correlation between two stellar populations is to calculate the ratio of the number of associated objects to the expected number in a random distribution:

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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