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Astron. Astrophys. 336, 503-517 (1998)

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5. Luminosity functions and mass functions

5.1. Luminosity functions

Another approach to investigate the existence of mass segregation is the analysis of the radial variation of the LF. Mass segregation is expected to produce variation in their slopes. We present the luminosity functions (LF) of the main sequence of the two clusters (as e.g. in Mateo, 1988; Vallenari et al., 1993; Will et al., 1995).

A series of LFs were constructed for each cluster at different radial distances from the centre of the cluster. In SL 666 we used four areas with [FORMULA], [FORMULA], [FORMULA] and [FORMULA] arcmin. In NGC 2098 the grid was somewhat coarser due to the poorer quality of the data ([FORMULA], [FORMULA] and [FORMULA] arcmin).

The LFs were normalised to an area of a circle of radius of 1 arcmin. The appropriate completeness corrections were applied (see Sect. 2), and the corresponding field LF was subtracted from the cluster LF. The resulting LFs are shown in Figs. 10a & 10b.

[FIGURE] Fig. 9. Correlation of the slope of radial density profiles (vs) magnitude, for SL 666.

[FIGURE] Fig. 10 a Luminosity Functions in R for SL 666 for radial distances 0.6 to 2.4 arcmins.

[FIGURE] Fig. 10. b LFs in R for NGC 2098 for radial distances 1.0 to 4.0 arcmins.

The photometric errors and completeness corrections (maximum accepted completeness 0.75) give reliable LFs down to [FORMULA] mag for the innermost area of SL 666, and down to [FORMULA] mag for the less crowded areas. In the case of NGC 2098, the corresponding limits are by at least 1 mag brighter, whereas the innermost region a limit at completeness 0.65 is adopted at [FORMULA] mag.

We can represent the LF as a function of magnitude R following [FORMULA] (e.g. Vallenari et al. 1993), and then obtain the slope s. The resulting slopes are given in Table 4. There is an apparent radial increase of the slope of the LF, in SL 666 (Fig. 10) up to radius [FORMULA] arcmin. This slope shows a correlated trend with the corresponding radial distance giving a Spearman correlation coefficient equal to 0.8, with a correlation probability equal to 80%. NGC 2098 has poorer photometry therefore for the distance up to [FORMULA] arcmin we have only the two innermost regions to represent a similar increase. Therefore NGC 2098 does not give very strong results about this effect.


[TABLE]

Table 4. Radial gradient of the LF


5.2. Mass function

The mass function is generally defined as:

[EQUATION]

where x can be calculated from the slope of [FORMULA] vs [FORMULA]. The LFs of Sect. 5.1 were used along with the mass-luminosity relation derived from the adopted theoretical stellar models to calculate the main sequence mass functions for the program clusters. Table 5 gives the derived mass function for the stellar masses adopted. This transformation assumes necessarily that all stars in the cluster are of the same age, and whatever bright end differences of main sequence stars are basically due to mass differences.


[TABLE]

Table 5. Mass function for SL 666.


Subsequently the values of the slope x in the various regions were calculated, and are shown in Table 6. The Spearman correlation coefficient between the slopes and the radial distances was found to be equal to 0.8 with a correlation probability equal to that found from the LF slopes (i.e. 80%). The results for NGC 2098 have large errors, as expected from the much poorer quality of the data, and cannot be used reliably for a radial study of the variation of the mass function.


[TABLE]

Table 6. Values of mass function slope x in the various regions for SL 666


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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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