In order to show clearly the advantage of the method used here for membership determination, we also calculated the result based on the method suggested by Zhao & He (1990).
Figure 5: The histogram of membership probabilities of M 11, which the
membership determination is based on Zhao & He (1990)
The histogram of the membership probabilities for this result is plotted in Fig. 5 (click here). Comparing Fig. 3 (click here) and Fig. 5 (click here), one sees that the separation for cluster stars and field stars from the improved method used in the present paper is much better than that from the old method. According to the efficiency for distinguishing cluster members from field members suggested by Shao & Zhao (1996), the efficiencies for these two kinds of membership determinations are 0.85 and 0.67, respectively. This also gives strong support to the improved method.
Furthermore, we can compare the surface density distributions for
cluster stars and field stars to check which method is better for
membership determination. The surface densities for cluster stars
and field stars can be defined by the following formulae:
and
respectively, where the sum is done for stars within the area
and Pi the
membership probability for ith star.
Figure 6: The surface density distributions (in unit of /arcmin2) for
cluster stars and field stars in the M 11 region: a) cluster stars;
b) field stars
Figure 7: The surface density distributions (in unit of /arcmin2) for
cluster stars and field stars in the M 11 region, for which the membership
determination is based on Zhao & He (1990).
a) cluster stars; b) field
stars
Figure 6 (click here) and Fig. 7 (click here) are the surface
density distributions for cluster stars and field stars obtained from
the improved method used in the present study and the old method
suggested by Zhao & He (1990), respectively. It can be seen clearly
that the density for field stars in Fig. 6 (click here) is more homogeneous than
that in Fig. 7 (click here). This gives a further support of our method. The reasons
for the distribution of field stars obtained from the old method, for
which the density increases significantly to the cluster center, are:
(1) as mentioned in Sect. 3, i.e., the dependencies of the space
distribution for cluster stars and magnitude are not taken into
account in the old method for membership determination; (2) the stellar
density is higher in the core and therefore more blended images are
present on the plates. Blends are hard to measure since the centroid
is very sensitive to exposure time, seeing, etc. They normally produce large, incorrect proper motions that are computationally
assigned as field stars.
We must also point out that the surface density model assumed in
Eq. (2)
in the present study is only a simple approach. The empirical density
distribution law of King (1962) can fit star clusters much better than
this one. But there are more parameters (central density 
, core
radius 
and tidal radius 
) in the King formulae, and it is too
complicated to solve with the maximal likelihood method. In the inner
part of a cluster, Eq. (2) is quite the same as the King formulae.
The major difference appears in the outer part. In Fig. 6 (click here), we see
that the density distribution for cluster stars plays a role of
exponential. So, Eq. (2) can be adopted reasonably here.
McNamara & Sanders (1977) analysed the internal motion of M 11 based on
the sample of
McNamara et al.(1977). They found that the
cluster radius increases with decreasing stellar mass after calculating
the densities of cluster members as functions of distance from the cluster
center and magnitude. They also found that there is a vast extensive halo
of relatively low-mass stars for M 11. Dynamically, they suggested that
equipartition of kinetic energy does not exist in M 11, but the relaxation
of its core has been intense, which means that mass segregation exists.
From Table 6 (click here) of the present study, the characteristic radius for cluster stars, which is one of the nine distribution parameters, increases with magnitude, i.e. the concentration to the cluster center for bright members is higher than that for faint members. There obviously exists a space mass segregation effect in M 11. The proper motion dispersion for members has a tendency to increase with magnitude also, but not as clearly as for the characteristic radius. Thus there exists velocity mass segregation to some extent. These two conclusions are consistente with McNamara & Sanders (1977) and also supported by Su (1994), Ying et al. (1996) and Su et al. (1997) by means of other statistic methods.
 | star number |  |  |  |
| <13.0 | 155 | 0.66 | 0.55 | 0.61 |
| 13.0-13.5 | 165 | 0.61 | 0.63 | 0.62 |
| 14.5-14.0 | 168 | 0.71 | 0.77 | 0.74 |
| 15.0-14.5 | 153 | 0.78 | 0.83 | 0.81 |
| >14.5 | 231 | 1.29 | 1.31 | 1.30 |
Here is a brief explanation for these two effects: There are two important dynamical time scales for an open cluster. One is the crossing time scale and the other a relaxation time scale. The crossing time scale is defined as the time spent for a typical member star to cross through the cluster, and the relaxation time scale is defined as the time for a cluster undergoing dynamical relaxation to approach equilibrium. Based on the distance of 1.6 kpc (Mathieu 1984; Su et al. 1994), the tidal radius of 24.5 pc (Su 1994; Su et al. 1997) for M 11 and the proper motion dispersion obtained above, we estimate the crossing time scale on the order of 1 107 yrs. For the relaxation time scale, we use the method suggested by Spitzer & Hart (1958) and Su (1994) to obtain 2 107 yrs. Given an age of 2 108 yrs (Mathieu 1984; Su 1994; Su et al. 1997), then M 11 is some 10 and 20 times older than its crossing time scale and relaxation time scale, respectively. Dynamical relaxation of M 11 must be well advanced. Mass segregation effects should have appeared.
| <12.4 | 12.4-13.0 | 13.0-13.6 | >13.6 |
| P>0.7 | 1.91 | 1.96 | 2.10 | 2.23 |
| P>0.8 | 1.75 | 1.80 | 1.83 | 1.86 |
| P>0.9 | 1.68 | 1.71 | 1.75 | 1.80 |
| P>0.95 | 1.66 | 1.70 | 1.73 | 1.79 |
The reason that its velocity-mass segregation is unclear may be due to low accuracies of proper motion measurements for faint stars. This can be seen from Table 7 (click here), where the proper motion errors versus magnitudes are listed. Fainter stars have lower accuracies.
Figure 8: Luminosity functions for M 11 in different region. a) inner
region 
; b) outer region(
)
To make these two effects for M 11 more certain, the luminosity functions for member stars in different regions and the normalized cumulative surface density distributions for different magnitude ranges of cluster stars are plotted in Fig. 8 (click here) and Fig. 9 (click here), respectively.
Figure 9: Normalized cumulative density distribution for M 11 clusters
stars. From top to bottom, the curves denote magnitude ranges of
, 12.5-13.5, 13.5-14.5 and >14.5 respectively
In Fig. 8 (click here), the luminosity function in the central region is flatter than that in the outer region for M 11, i.e. there are relatively fewer faint stars in the central region. From Fig. 9 (click here), it can be seen that the central concentration for bright stars is higher than that for faint stars. Thus the space-mass segregation for M 11 is very clear. In another aspect, the variation of the proper motion dispersions with different threshold membership probabilities for members in different magnitude ranges are listed in Table 8 (click here). One sees that a velocity-mass segregation effect still exists to some extent, although the values of dispersions decrease and the differences among the values decrease with threshold probability increasing. This probably reflects contamination by field stars.