A usual way to check the completeness of any catalogue is to verify if
the number of objects N(d) closer than a given distance d does
indeed increase as 
. The N(d) for MSC is given in Fig. 2 (click here). The
departure from 
law begins already at the distance of 6 pc. The
straight line is an attempt to fit the first points by the cubic law.
It predicts the number of multiple systems within 10 pc to be 19. The
total number of stars within 10 pc being 236 (Gliese & Jahreiss
1991), the fraction of multiple systems can be estimated as 6%.
Thirteen systems within 10 pc with mean
primary mass of 
are actually listed in MSC.
The limit of the
Catalogue of nearby stars is shown by the arrow in Fig. 2 (click here).
The comparison of actually
listed (62) and predicted (217) numbers of multiple systems within
22.5 pc shows that only about 1/3 of them are discovered.
If the primary's mass dependence on distance is examined, it
becomes evident that multiple systems with primary mass less
than 
are all concentrated within 30 pc, and the
lower mass limit increases gradually with the distance.
The little that was said on the MSC incompleteness suffice to affirm that the "population'' of MSC must not be identified with the real population of multiple stars, and any statistical inferences from the MSC must be taken with caution. For example, the relative proportion of different multiplicities in MSC (3:4:5:6:7 = 470:108:27:5:2) is not necessarily representative of the true proportion, because the systems are not complete (cf. Sect. 2.4). The following sub-sections describe the statistical properties of the catalogue rather than the statistics of multiple stars, because no allowance for observational selection was made.
In Fig. 3 (click here) the period of wide sub-system 
is plotted in a
logarithmic scale against the period of the corresponding close
sub-system 
at the adjacent hierarchical levels. A system of
multiplicity higher than 3 provides more than one point in this figure.
The line marks equal periods. The absence of systems near
this line clearly indicates that stability criteria are satisfied: for
the majority of multiple systems 
. The smaller 
ratios exist, but they correspond mostly to visual
multiples and are
likely to be the result of uncertain period estimates from projected
separations.
The upper limit on 
was discussed above (Sect. 2.3).
On the other hand, there are no 
shorter
than 0.3 d, a typical period of main sequence contact
binary.
It can be noted that multiple systems in the 
plane occupy almost all space in the triangle
defined by these limits. Apart from this,
no evident relation between 
and 
can be seen. In
particular, the period ratio 
(its logarithm is the
distance of a point above the limiting line 
) does
not appear to have any preferential value. This conclusion
seems to be immune to selection effects and it must
be kept in mind in attempts to understand the formation
mechanisms of multiple systems.
The distribution of points in the limiting triangle is not uniform.
In Fig. 4 (click here) the distribution of 
is plotted. It shows a marked
depression for 
. For comparison, the period
distribution in the SB catalogue is also plotted. It is clear that
severe selection effects are influencing the shape of 
distribution for 
d. The same explanation may be valid for
the apparent maximum at 
y. At a typical
distance of 30 pc the corresponding semi-major axis (12 A.U. for 
mass sum) is equal to 
, suspiciously close to the
resolution limit of visual discovery techniques. A
speckle-interferometric survey of nearby stars could be particularly
helpful, providing the order-of-magnitude increase in angular
resolution needed to decouple the observational limit from
possible real features in the period distribution.
The diagram 
for triple systems was
discussed by Duquennoy & Mayor (1986). They noted a bimodal
distribution of 
with a gap at 
d and
stated that this gap can hardly be explained by the discovery
biases. Examining Figs. 3 (click here), 4 (click here), we do find the deficiency of points at
. However, as mentioned above, it may still be
related to discovery limitations.
There seems to be no dependence of 
and 
on the mass
of primary component.
When only systems with unevolved primaries from 0.8 to 1.5
are selected for the plot, the general aspect of Fig. 3 (click here)
does not change.
The discussion of 
relation may be
influenced by the fact that some components still remain
undiscovered:
the periods presently attributed to the
adjacent levels of hierarchy may be actually separated by
undiscovered intermediate levels.
Figure 4:
The distribution of the logarithm of short periods plotted in
Fig. 3 (click here). The scaled distribution of the periods of spectroscopic
binaries from the catalogue of Batten et al. (1989) is plotted as
dashed line for comparison. It is possible that the depression in the
period range from 5 to 
days is due to observational selection
Figure 5: Cumulative
distribution of Q, the ratio of the distant component mass to the
primary mass in a close sub-system.
The number of systems N with mass ratio less than Q is plotted against
in full line. The simulated data for random mass
selection from the Salpeter mass function with cutoff at Q < 0.1 are
plotted in dashed line
Even a cursory examination reveals that binary trees representing multiple systems are generally asymmetric. It is the primary that most often contains close sub-systems. Table 5 (click here) gives the number of different levels in MSC. Restricting ourselves to apparently triple systems, we find 381 cases of level 11 and 83 cases of level 12, i.e. 82% of the "triple'' stars have the close sub-system associated with the primary. It is not clear to which extent this preference is due to selection effects (bright primary stars are usually better studied).
Table 5: Number of sub-systems at different levels
This level asymmetry is evidently related to the mass ratio distribution. Let us consider the parameter Q, the ratio of distant component mass to the mass of the primary in a close sub-system. If the distant component is itself non-single, its mass sum is used in calculating Q, and such a quadruple system provides 2 values of Q, one for each of close sub-systems. The distribution of 702 values of Q is given in Fig. 5 (click here) (full line). The small "step'' at Q = 1 is an artifact of mass estimation procedure used in MSC.
If the masses of multiple star components were independently selected
from some mass distribution and then randomly combined, one would
expect to find 1/3 of systems with Q > 1, because
the most massive component is more likely to belong to the pair.
Surprisingly, the curve in Fig. 5 (click here) is not far from this prediction,
with only slight deficiency of Q > 1 systems. This encourages
further modelling of mass distribution. The dashed curve in Fig. 5 (click here)
corresponds to the simulated sample of triple stars with component
masses independently chosen from the Salpeter mass function
proportional to
in the range from 0.1 to 20 solar masses. The systems
with Q < 0.1 were excluded from the histogram.
The simulation predicts 11% of such systems;
presumably they are absent from the observed sample due to discovery
bias.
The agreement of simulated and real distributions is reasonably good
and gives some support to the idea that mass ratio distribution in
multiple stars may correspond to random mass combination. However, a
more refined study taking into account the observational selection
effects is needed.
The level asymmetry is perhaps related to maximum multiplicity of stable
systems. A fully populated binary tree with 3 levels contains 
components, the systems with 4 levels also exist which can
contain up to 16 components. The maximum multiplicity
of a hierarchical system
actually found
is 7 (16062-1912 = 
Sco = HR 6027/26). Undoubtedly, the
number of high-multiplicity systems and the highest known
multiplicity can be increased by using modern observing techniques,
but the conclusion that only a fraction of available hierarchical
levels is actually filled seems to be already firmly established.
The problems of orbit coplanarity and relative sense of rotation in
multiple stars has attracted much attention in the past (e.g. Fekel
1981; Worley 1967). Both problems can be treated jointly if one considers the
distribution of the angle 
between angular momentum vectors of
long- and short-period sub-systems. This angle can of course change
due to the interaction between the orbits, but at least to the first
approximation in a triple system the angular momentum of the
short-period pair precesses around the total angular momentum vector
which is almost parallel to the long-period momentum, so the relative
angle 
must not change very much.
An attempt to use the preliminary version of MSC to probe the
distribution of 
was done earlier (Tokovinin 1993) and here
the results of this study are summarized. Three kinds of multiple
systems were considered:
case, but strict alignment of momentum
vectors is also
excluded.
are possible. The
statistical arguments lead to the same conclusion, i.e. that
both complete alignment and complete independence
of angular momentum vectors are excluded.
which is found to be 
for 54 systems,
i.e. significantly different from either 
or 
.
Thus all three different samples behave in a similar way with respect
to relative angular momentum orientation. The 
statistics
corresponds to loosely aligned vectors, or, alternatively, to an equal
mixture of well aligned and randomly aligned systems. This result has
evident bearing to the formation history and dynamics of multiple
systems.