Cluster membership can be estimated from kinematical data (i.e. proper
motions and/or radial velocities), from the number density of the stars
around an assumed cluster centre, from the CMD or
from a common analysis of these data which allows to obtain more reliable
results.
Astrometric membership probabilities from proper motion data can be
computed using different criteria. After shifting the origin of the
proper motion diagram to the center of the proper motion distribution
of the cluster stars, Ebbinghausen (1942) applied a
probability factor 
where
is the rms error of an individual stellar proper motion
and a is 
or 
for, as he called, high probable,
probable and low probable members, respectively. This approach can be
successfully used if the proper motion distribution of the cluster members
shows a concentration to a point different from the distribution of the
field stars. Vasilevskis (1962) fitted the proper motion
distribution with two bivariate Gaussians and calculated the membership
probability for each star as the relation between the distribution functions
of cluster members and of all stars. Sanders (1971) suggested
to use the maximum likelyhood method for the fitting and computed
distribution parameters of cluster and field stars and cluster membership
probabilities. Nevertheless, the determination of membership probabilities
by the method of Sanders (1971) can also lead to uncorrect
results if the distribution functions of cluster and field stars are
overlapping. Frequently, the proper motions of cluster and field stars
separated by this method are not normally distributed. This effect is
clearly visible with large numbers of stars included in the investigation
(cf. the results for the globular cluster M 3 by Scholz & Kharchenko
1994).
Especially if we use deep Schmidt plates with moderate scale and relatively
small epoch differences of a few decades a membership determination based
only on proper motions is difficult for distant clusters. That is due to the
low significance of the individual proper motions of the cluster stars and to
a large number of faint field stars which have proper motions of the same order
as the cluster stars. Kharchenko & Schilbach (1995) met these difficulties
when they were trying to obtain membership probabilities of four open clusters
located in the Saggitarius Carina spiral arm of the Galaxy.
As far as the fit of two
bivariate Gaussians to the proper motion distribution did not yield stable
results they suggested to use the information on both proper motion and
position distributions of the stars near the clusters. The parameters of
cluster and field star distributions were computed in a four-dimensional
space of proper motions 
and plate coordinates (X,Y).
We apply this method here to the globular cluster M 5 and compare the results
with those obtained from the membership determination by using only the
proper motions or only the positions of the stars.
The distribution function 
of the stars in the vicinity
of the cluster can be described as the sum of two distribution functions,
respectively of the field stars and of the cluster stars:
The cluster itself can in principle also be divided into two parts - the core
and the korona - with different velocity dispersions. If we speak about a core
region in our membership analysis we do not mean the real cluster core in the
astrophysical sense. The real core radius 
of the cluster is less than
1 arcmin (see Table 5 (click here)). This very central region of
the cluster is not resolved at all on the Schmidt plates. In all membership
solutions we used only the images outside a cluster radius of 5 arcmin (but
computed membership probabilities for all objects). As we will see later there
is no indication of two different cluster populations from our membership
determination. Note that in the star counts using the whole measuring
data of the Schmidt plates (see Sect. 8.1.) we applied a crowding correction
in order to reach the inner regions of the cluster.
The stellar population in the field is also
not homogeneous as far as we observe field stars at rather different distances
and, therefore, with different kinematical properties.
To a first approximation we can consider two groups of stars - main sequence
stars with spectral classes A, F, G, K and red giants of the disk,
respectively with absolute magnitudes 
and 
.
The mean distances of these two groups of stars 
in the given
magnitude interval differ by a factor of 5, whereas their mean space velocities
usually differ only by a factor of 2 (Kharchenko 1980). The
proper motions are proportional to v/r, therefore the distribution function
of the field stars is again the sum of at least two components. We can neglect
other field stars, i.e. high luminosity stars and stars of other luminosity
classes due to their small numbers. So we get
The parameters of the distribution function can be determined by the maximum
likelyhood method. These parameters are independent of each other in such a
system of coordinates (P,Q) and of proper motions 
, where
the corresponding correlation coefficients are equal to zero. That means,
our four-dimensional coordinate system must be rotated by angles 
and 
. These angles are determined by the condition that the
correlation coefficients between the parameters reach a minimum.
Assuming normal distributions of the proper motions of all stellar groups
and of the coordinates of the cluster stars and a uniform distribution of
the coordinates of the field stars within a radius 
we can write the
following equations in the new coordinate system for the cluster stars
(core or corona):
and for the field stars (nearby or distant stars):
with
and
If we consider the proper motions and coordinates of N stars as an
N-dimensional random vector with independent components its probability
density will be determined by the likelihood function
where
and 
is the vector of the parameters to be determined. The function (2)
reachs the maximum at 
.
With these conditions we get a system of
j equations for the determination of the components of the vector 
:
If 
corresponds to (1), we have j=1,2,...,28.
After rotating the coordinate system back to the system (X,Y) and
the membership probabilities of cluster stars were
computed as
with
and
Membership probabilities 
and 
, respectively computed only
from proper motions or only from the positions are
Table 2: Parameters of distribution functions
Stars with cluster membership probabilities P higher than 61, 14 and 1
per cent are located at distances less than 
and 
from the maximum of the four-dimensional distribution function. Respectively,
they were considered as high probable, probable and low probable cluster
members.
The number of unknown quantities in the system of Eqs. (3) is too large
for getting a stable solution even with large numbers of stars. Additional
assumptions concerning for instance the symmetric distribution of the
cluster members in the space of coordinates and proper motions
(cf. Kharchenko & Schilbach 1995) are needed.
In our case we did not made assumptions concerning the symmetry of the proper
motion dispersion or of the dispersion in the positional distribution of the
cluster stars, but we could reduce the number of the components
of the distribution function (1). As expected we did not find any indication
of two different distributions of cluster stars in the region investigated
(r > 5 arcmin) in preliminary membership determinations using only the
proper motions. With a heliocentric distance of 7.6 kpc (Peterson 1993) for M 5
and the probably small internal velocities of the cluster stars (
2 km/s)
corresponding to about 0.06 mas/yr we must observe similar proper motion
distribution functions of a cluster core and korona dominated by the rms
errors of the stellar proper motions. On the other side we did also not find
different parameters for two cluster populations in the membership
solution using only the coordinates. Therefore, we excluded the core
distribution function 
from Eq. (1) and left with 19 unknowns
in the case of a membership solution by use of proper motions and
coordinates (i.e. for the cluster stars: the mean proper motion components and
their dispersions, the centre coordinates and their dispersions and the number
of cluster stars; for near field stars: their mean proper motion components
and dispersions and their number; for distant field stars: their mean
proper motion components and dispersions and their number).
Figure 3: Colour-magnitude diagrams for cluster
members obtained in the four-dimensional space of proper motions and
coordinates (upper left), in the two-dimensional membership determinations
using only the proper motions (upper right) or using only the coordinates
(lower left) and for the field stars (lower right).
The high probable cluster members, probable members and
low probable members (respectively, within 
and 
from the centre of the distribution) are marked by crosses, small crosses and
dots, respectively. In the lower right diagram the dots represent all stars
in an annulus around the cluster
For the solution of Eq. (3) we excluded obvious non-members, i.e. stars located outside a cluster radius of 50 arcmin and high proper motions stars with total proper motions larger than 40 mas/yr. We also excluded stars with large colour indices (B-V) > 1.8. In order to get more reliable results we did not use the faintest stars with B > 19 and the stars inside a cluster radius of 5 arcmin, strongly affected by image crowding on the Schmidt plates. By the last two conditions we excluded most of the stars with large proper motion errors. Among the remaining 5073 stars we determined 1573 cluster stars, 1276 distant field stars and 2224 near field stars with the parameters given in Table 2 (click here).