The mean absolute proper motion of the cluster given in Scholz et al. (1996) had been determined without the knowledge of membership probabilities for individual stars. The contamination with field stars in the cluster region was considered statistically on the basis of the number density and mean proper motion of the field stars outside the cluster. The ratio of the number of cluster stars to the number of field stars in the cluster region defined in Scholz et al. (1996) as 4 arcmin < r < 14 arcmin was 845/215. Note that in this case only the faint objects with stellar classification were used in the averaging of the proper motions. Stars brighter than V = 17.5 were not used in order to correspond to the magnitude interval of the reference galaxies and to prevent possible magnitude dependent systematic errors.
Figure 4: Colour-magnitude diagram for cluster members with 
in three radial zones: r < 9.5 arcmin, 9.5 < r < 19.3 and
19.3 < r < 27.6 (cf. with CMD of field stars = lower right CMD in Fig. 3)
Here we look according to the conditions used in the membership calculations
only for objects with r > 5 arcmin and B < 19. As a compromise between the
number of stars used in the averaging and the expected contamination with
field stars the mean cluster proper motion given in the second row of
Table 3 (click here) seems to represent the most reliable result.
Comparing the vector-point
proper motion diagrams of the stars listed in Table 3 (click here) we find
the most symmetric distribution in the case of 
. In all three cases
the median values of 
tend to be slightly larger than the mean values
given in Table 3 (click here). For 
the median of 
is at
6.15 mas/yr. Nevertheless, there remains a somewhat smaller value of
in comparison to the result
of Scholz et al. (1996). This difference may be the result of a small
magnitude dependent systematic error in 
, previously not detected.
In the other proper motion component, 
a magnitude equation had already
been mentioned and corrected for in Scholz et al. (1996). A smaller value of
(e.g. +5.8 instead of +6.7 mas/yr) does not change the
character of the Galactic orbit of M 5, unless it
leads to a decrease in the total value of the space motion of the
cluster (from 421 to 391 km/s), and therefore, to a smaller apogalactic
distance (44 instead of 61 kpc) of the cluster (Odenkirchen 1996).
Nevertheless, the former result of Scholz et al. (1996) remains more
reliable, as far as it dit not include the bright cluster stars
probably affected by a systematic magnitude dependent error with
respect to the mean magnitude of the reference galaxies (
).
As result from the membership analysis in the four-dimensional space of
coordinates and proper motions we find cluster members at a maximum radius
of about 28 arcmin (cf. Tables 3 (click here) and 4 (click here)).
If we plot the CMDs for all cluster members within 
from the
centre of the four-dimensional distribution function (
) in three
different zones around the cluster (Fig. 4 (click here)), so we obtain in all three
zones a CMD which is different from the
CMD of the field stars (cf. Fig. 4 (click here)). With increasing distance from
the cluster centre the contamination with field stars increases. However, even
in the outer zone (19.3 arcmin < r < 27.6 arcmin) the diagram looks
typical of a globular cluster CMD.
Table 4 (click here) shows the mean proper motions of cluster members
determined in the four-dimensional membership analysis (with 
)
and of cluster members obtained from a membership analysis using the
proper motions only (with 
) in comparison to the non-members
with total proper motions less than 18 mas/yr.
In the inner zone between 5 and 10 arcmin there is a higher number density of
non-members. This effect is due to the larger individual proper motion
errors in the inner cluster region resulting from stronger crowding
effects. With increasing cluster radius
we can see a systematic change in the mean proper motion of the cluster
members in direction of the mean proper motion of the field stars (cf.
Table 2 (click here)). But comparing the data of Tables 4 (click here)
and 2 (click here) even in the outer zone between 25 and 27.6
arcmin we see a significant difference between the mean cluster and mean
field star proper motion for both membership probabilities 
and
.
In Table 4 (click here) we compared the results for 
and
in order to investigate the most reliable membership
probabilities, particularly in the outer cluster region. With
there are no cluster members outside 20 arcmin from the
cluster centre. On the other side with 
the mean proper
motion components change more rapidly from +5.73 to 
for
and from -8.07 to 
for 
,
respectively from the 
zone to the 
zone.
The number of field stars 
contaminating the mean cluster motion can
be estimated from
where 
is the mean proper motion of the cluster
contaminated with field stars. The number of field stars included in
the numbers of cluster members given in Cols. 2 and 5 of Table 4 (click here)
is about 20 per cent and does not exceed 30 per cent even in the
outer zone (19.3 arcmin < r < 27.6 arcmin). Therefore, we
conclude from the membership determination alone that the cluster radius is
not less than 27.6 arcmin.