Usually, the distribution of proper motions is considered as a good astrometric criterion for selection of cluster members. However, for distant clusters, the individual proper motions of their members are rather small and differ insignificantly from the proper motions of field stars. In other words, due to numerous field stars with in the cluster area, the number of cluster members estimated from the proper motion distribution is never zero at any distance from the cluster center. Therefore, additional and independent criteria are needed for a confident selection of cluster members.
In Paper I we considered the distribution of star positions as a further selection criterion and determined the formal membership probability for each CC star. This method, however, requires predefined parameters of the cluster structure (e.g., cluster radius). Insufficiently accurate input parameters may introduce biases both in the computed membership and in the final cluster size. A further possibility is to use photometric criteria i.e., to analyse the loci of cluster candidates in the CMD. This approach allows, with a certain confidence, to reject stars falling below the cluster MS. Finally, as a selection criterion, one can consider the distribution of stars as a function of apparent magnitude which, as was shown in the previous section, should be different for field and cluster stars. With different weight, all these criteria were applied in this study to support the proper motion membership.
Among other data, the CC contains probabilities for stars to belong to the core or corona of NGC 6611. The probabilities were deduced from the distributions of positions and/or proper motions. In the following, we use the probabilities , and for the final cluster membership determination.
We divided all CC stars into five groups with respect to their proper motions and positions.
As we already discussed above, the stars located outside the cluster core represent a mixture of corona members and field stars. A clear separation of these stars in two groups (i.e., in the groups 4 and 5 as described in the previous section) from their positions and proper motions is rather difficult. Therefore, we applied an additional selection criterion to these stars taking into account specific features of the corona population.
According to Sect. 2, the apparent structure of the corona is highly variable due to irregularities of the absorption over the cluster surface. Therefore, we considered a relation between the absorption and a "threshold" membership probability derived from the proper motion distribution within a given sub-area. Such a relation can be expected for the following reasons. Due to variations of the absorption across the cluster, the real luminosity function of the cluster stars is transformed to an observed brightness function depending also on coordinates. Since the accuracy of proper motions depends on the apparent brightness of stars, the resulting distribution of proper motion errors will also depend on coordinates. Consequently, the threshold membership probability which is related to the error distribution is also a function of positions. On the other hand, with increasing absorption both numerous faint background stars and faint companions of apparently close pairs (unresolved on photographic plates) are lacking from a survey. The visible photocentre of such a pair moves towards the bright component and the astrometric accuracy is improving. As the number of stars becomes rapidly small with decreasing magnitude, we may expect, on average, a decrease of the proper motion errors as well of the corresponding threshold membership probability for stars of a given apparent magnitude in areas with higher .
Based on these considerations and on the difference in the distributions of field and cluster stars with apparent magnitude, we proposed a statistical procedure of the corona members selection. We estimated local membership probabilities by comparing the local BF with the brightness function of stars from the external CC region () which is dominated by field stars (cf. Sect. 2). The CC area was divided into 32 small cells with a step of . For stars satisfying the condition , we determined the local BF function in each cell and compared it with :
Function reflects the degree of agreement between the field and local BFs for a given . One can predict the behaviour of with increasing from 0 to 100 per cent. At small the functions and differ randomly due to a small number of stars selected in a cell. This difference tends to a minimum when increases up to a certain due to the inclusion of new field stars and a decrease of statistical fluctuations. In the vicinity of , the value which could be considered as a threshold proper motion probability for the cluster membership, starts again to increase due to the admixture of cluster stars.
As an example for the efficiency of the method, we show in Fig. 4 the BFs computed in the cell with = (9.0 arcmin, 112.5 deg). A threshold probability of = 61% was estimated for this cell. The stars were separated in two groups according to for cluster members and for field stars. From Fig. 4 we can see that the BF derived for field stars in this cell is in excellent agreement with the general field BF, whereas the cluster BF in the cell differs significantly from the field BFs and is in agreement (in its slope) with the core BF shown in Fig. 3. In Table 1 we give the threshold probabilities estimated for each cell. As expected, their distribution is strongly correlated with the absorption distribution shown in Fig. 1.
Table 1. Threshold membership probabilities (in%) for the cluster corona
Generally, we consider two color-magnitude diagrams (, ; , ) and a color-color diagram (, ) for a photometric selection ( values were not used as a direct criterion, but they are included implicitly in the photometric selection as a secondary criterion when the reddening-free CMDs were constructed). If no U- magnitude was available in the CC, only the , diagram was used. Taking into account the magnitude errors given in the CC and the error of the distance modulus ( in V), we considered the loci of suspected members in each diagram with respect to the ZAMS by Schmidt-Kaler (1982). A star was rejected as a cluster member if it fell below and to the left of the ZAMS in at least one of the diagrams. In order to derive the reddening-free diagrams, we applied the same procedure and adopted the same parameters as in Paper I: the stars with the available values of and were dereddened individually, whereas the average values (, ) were applied otherwise. From a comparison of different approaches of reddening determinations discussed in Sect. 2 we found that these values of the extinction law provide the least fuzzy and most plausible CMDs. The distance modulus determined in Paper I from a sample of stars with the most accurate magnitudes and colors was used for the construction of cluster's CMDs.
In Figs. 5,6,7 we show the results of the photometric selection in the membership groups 1, 3 and 4. The final results of the selection are presented in Table 2. The distribution of cluster members in the -plane is shown in Fig. 8. A non-symmetric distribution of cluster members over the cluster surface is certainly related to the irregular distribution of the absorbing matter.
Table 2. Statistics of the membership determination
The complete catalogue including the astrometric and photometric data as well as membership determination is available in machine-readable form at cdsarc.u-strasbourg.fr (CDS) via anonymous ftp.
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000