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3. Membership determination

The firststep in astrophysical research of an open cluster is to make a reasonable membership determination, as mentioned in Sect. 1. The popular methods can be summed up in two aspects: photometric and kinematics. But, as pointed out by Mathieu (1984), the uncertainty for photometric determination is quite large especially for binaries. Although Cabrera-Canio & Alfaro (1990) suggested another method, the computation is very complicated.

The most popular way to distinguish cluster stars from field stars now is based on their kinematical data, especially on proper motions. The fundamental work is suggested by Vasilevskis et al. (1958) and Sanders (1971) using the maximum likelihood principle. Zhao & He (1990) improved the method to be used for the condition of different accuracies of proper motions for individual stars.

It must be pointed out that there still are two shortcomings in these studies. On the one hand, the space distribution of cluster stars is not considered. The results obtained from this method must have biases to stars in the outer part of the cluster, i.e., the outer stars will have larger membership probabilities than they should have. In general, fainter stars are in the outer region, so that their probabilities will be overestimated. On the other hand, the distribution parameters are dependent upon magnitudes of stars, but only the average magnitude is concerned in their method of membership determination. There are more faint stars than bright stars in an open cluster. This will also lead to biases to fainter stars. These two aspects will enlarge the uncertainty in membership determination, especially for a cluster whose age is sufficient for dynamical relaxation.

Jones & Walker (1988) developed some improvements in this field. While the two factors mentioned above are considered in the distribution function of cluster stars, the influence for field stars has not been taken into account reasonably (See Eq. (8) and Eq. (9) in their paper). Su et al. (1995) made some corrections for them, used successfully for the open cluster M 67. In the present study, we will use the method of Su et al. (1995) to do membership determination. A brief introduction is given below.

According to van den Bergh & Sher (1960) and Francic (1989), the surface number density distribution for cluster stars can be assumed as

and the surface density distribution for field stars

Where tex2html_wrap_inline1770 is the central surface density of the cluster, r0 the characteristic radius of the cluster, r the distance of individual stars from the cluster center, f only depending upon magnitudes. Now, the normalized factor for cluster stars and field ones are



So the frequency functions of proper motions for cluster members and field stars, considering the space distribution and magnitudes of stars can be written as


respectively. In Eq. (6) and (7), we have nine distribution parameters to be solved by means of the maximum likelihood method. They are tex2html_wrap_inline1778, where tex2html_wrap_inline1780 is the ratio of central surface density for cluster stars to that for field stars; r0 the characteristic radius; tex2html_wrap_inline1784 and tex2html_wrap_inline1786 the proper motion centers for cluster members and field stars respectively; tex2html_wrap_inline1788 the intrinsic proper motion dispersion for members; tex2html_wrap_inline1790 the intrinsic proper motion dispersions for field stars in X, Y directions and tex2html_wrap_inline1796 the error of the proper motion of ith individual star. It must be kept in mind that all these nine parameters are functions of magnitude.

In the present study, stars with radial distances within 25 arcminutes centred to M 11 are chosen for membership determination. The number of these stars is 785. Because the distribution parameters now are functions of magnitude, we must divide our sample into several subsamples with different magnitude ranges. The principle of grouping the stars is that there should be roughly the same number of stars in each subsample and that the number of stars in each subsample should be large enough for statistical analysis.


group No. 1 2 3 4 5 6 7 8
tex2html_wrap_inline1696 <12.6 12.6-13.0 13.0-13.4 13.4-13.8 13.8-14.2 14.2-14.6 14.6-15.0 >15.0
star No. 77 62 116 119 122 114 85 90
Table 5: Subsamples for the stars in different B magnitude ranges in the M 11 region



group No. 1 2 3 4
tex2html_wrap_inline1780tex2html_wrap_inline1822 tex2html_wrap_inline1824 tex2html_wrap_inline1826 tex2html_wrap_inline1828 
r0tex2html_wrap_inline1832tex2html_wrap_inline1834tex2html_wrap_inline1836 tex2html_wrap_inline1838
tex2html_wrap_inline1840tex2html_wrap_inline1842 tex2html_wrap_inline1844 tex2html_wrap_inline1846 tex2html_wrap_inline1848 
group No. 5 6 7 8
tex2html_wrap_inline1780tex2html_wrap_inline1912 tex2html_wrap_inline1914 tex2html_wrap_inline1916tex2html_wrap_inline1918
tex2html_wrap_inline1840tex2html_wrap_inline1932 tex2html_wrap_inline1934 tex2html_wrap_inline1936tex2html_wrap_inline1938 
tex2html_wrap_inline1850tex2html_wrap_inline1942 tex2html_wrap_inline1944 tex2html_wrap_inline1946tex2html_wrap_inline1948
tex2html_wrap_inline1870tex2html_wrap_inline1962 tex2html_wrap_inline1964 tex2html_wrap_inline1966tex2html_wrap_inline1968 
Table 6: Distribution parameters for the stars in different magnitude ranges of M 11


The 785 stars in the M 11 region are divided into eight subsamples based on their magnitudes in the B band, which are shown in Table 5 (click here). By means of the maximum likelihood method, nine unknown distribution parameters for each subsample are determined. The results and the corresponding uncertainties are listed in Table 6 (click here), in units of arcmin and arcsec/century. The membership probabilities for individual stars can be calculated as follows

where Pi is the membership probability for the ith star. In Table 4, the membership probabilities for individual stars in the M 11 region are listed in Col. 12. The histogram of the membership probabilities for the 785 stars is shown in Fig. 3 (click here). It can be found that the separation for cluster stars and field stars is very good. The total integrated membership probabilities of these 785 stars is 547 and the number of the stars with membership probabilities higher than 0.7 is 541. From the point of statistics, there will be about 1tex2html_wrap_inline1684 field star contamination if these 541 stars are treated as the members of M 11. That is to say, it is a good sample of M 11 for detailed astrophysical researches. In Fig. 4 (click here), we also plot the vector-point-diagram of these 541 stars. From this figure, we can see that the average motion of these 541 member stars is close to zero. Thus the reference frame we chose has no unwanted distortions.

Figure 3: The histogram of membership probabilities of M 11

Figure 4: The vector-point diagram (VPD) of 541 member stars of M 11

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