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 is the central surface density of the cluster, *r*_{0}
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

and

respectively.

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

and

respectively. In Eq. (6) and (7), we have nine distribution parameters to
be solved by means of the maximum likelihood method. They are
,
where is
the ratio of central surface density for cluster stars to that for field
stars; *r*_{0} the characteristic radius;
and the proper motion centers for
cluster members and field stars respectively;
the intrinsic proper motion
dispersion for members;
the intrinsic proper motion dispersions for field
stars in *X*, *Y* directions and the error of the
proper motion of *i*th
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 |

<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 |

group No. | 1 | 2 | 3 | 4 |

r_{0} | ||||

group No. | 5 | 6 | 7 | 8 |

r_{0} | ||||

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 *P*_{i} is the membership probability for the *i*th 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 1 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