## 5. Results## 5.1. Detection probabilitiesThe range of semi-major axes studied was
to
AU. Separate synthetic
catalogues were generated for each decade in
In producing Table 2, no observation noise was simulated; this was necessary in order to eliminate false detections which would otherwise bias the detection probabilities. It is noted that the detection probabilities can be quite high (35 to 77%, depending on the mass-ratio distribution) for the resolved binaries with -1000 AU. This is not unexpected. More interesting is perhaps that some 20 to 30% of the binaries in the 1 to 10 AU range are detected as astrometric binaries with quadratic, cubic or orbital solutions (). The highest detection probability for astrometric binaries is % for MS+WD systems in the 1-10 AU range. Also the delta-mu method is relatively powerful in detecting binaries in the range from 1 to 100 AU. ## 5.2. Predicted numbers from DM distributionsThe expected number of different solution types among the Hipparcos
main-sequence survey stars can now be calculated by assuming
distributions of Most of these numbers are much smaller that the observed counts in Eqs. (4) and (5). The most straightforward interpretation of this discrepancy is that the actual multiplicity in our sample is significantly higher than the assumed 0.57. Alternative explanations are discussed in Sect. 6. Turning to a more detailed comparison of the observed and predicted counts, we shall consider separately the resolved systems, the long-period astrometric binaries, the orbital and the stochastic solutions.
In view of this it is reasonable to disregard altogether the observed counts for the O and X solutions. There remains however the inconsistency between the C results and the long-period astrometric binaries. The observed ratio between these two categories could in principle be reproduced by postulating an additional, rather large fraction of binaries with a white-dwarf companion (MS+WD systems), which would contribute to the G and counts, but not to the C counts. For instance, an assumed multiplicity of 0.88 for the MS+MS systems and 0.35 for the MS+WD systems would yield approximately the right numbers (except for the O and X solutions). However, such a high frequency of white dwarfs far exceeds what can reasonably be expected from evolutionary considerations. As will be shown below, the inconsistency between the number of resolved and astrometric binaries is more likely a consequence of our constraining the period distribution to that of DM. ## 5.3. Limits on the distributions from the observed countsIn the previous section we assumed (roughly) the same period and mass-ratio distributions as found by DM for solar-type stars. We now successively relax these assumptions by investigating alternative distributions. In the end, we attempt to invert the problem by deriving the binary frequencies in the different intervals from the observed counts. For this purpose we disregard the O and X systems according to the previous discussion (but see the discussion at the end of this section). We also need to make reasonable assumptions concerning the relative
number of systems with a white-dwarf companion. Let Let us first assume that the distribution of
If the standard width of the log-normal curve is left as a free parameter, we generally obtain a much more peaked distribution. The increasing, parabolic and flat mass-ratio distributions are again rejected at 95% confidence level, while good fits are obtained for the decreasing mass-ratio distribution both for ( with 3 degrees of freedom) and for (). The resulting distributions (dashed curves in Fig. 2) peak at -25 AU and correspond to a total multiplicity of 0.9 and 1.0, respectively. The standard widths are and 1.0. Finally, we have also attempted to estimate the distribution in
by direct inversion of the observed
counts for an assumed mass-ratio distribution
(). We only consider the interval in
from -1 to
, since it can be seen from
Table 2 that the detection probabilities are practically zero
outside this interval. Let to
be the unknown multiplicities in
the intervals centred at . The
expected counts We conclude from these experiments that the determination of the
multiplicity in the interval
-10 AU is fairly insensitive to
various assumptions, including the relative frequency of white-dwarf
systems. The estimated multiplicity in this interval is
% (95% confidence), to be compared
with 11% from the DM distribution. For the interval
-100 AU our estimate is
% (DM: 14.5%). For larger The predicted number of orbital solutions is a factor three higher than observed, which suggests that many more orbits can be derived from the Hipparcos intermediate data. According to the discussion in Sect. 5.2, we did not use the
observed counts ## 5.4. Predicted distributions of delta-mu and acceleration termsIn our analysis of the delta-mu binaries we used only the
intersection with the FK5 catalogue in order to compare with the best
ground-based proper motions. Naturally, proper-motion errors caused by
orbital motion are expected for many more stars in the Hipparcos
Catalogue. This is illustrated in Fig. 3, which shows the total
proper-motion errors from orbital motion and observation noise in a
synthetic catalogue using the DM distribution of orbital periods. The
typical (rms) proper-motion error for single stars is
1 mas yr
As described in Sect. 4.3, the observation model was made to
reproduce the average standard errors of the astrometric parameters
for single stars as given in the Hipparcos Catalogue. An important
additional verification of the observation model and detection
criteria is through the distributions of acceleration terms resulting
from the G solutions, which are sensitive to the tuning of the
observation errors. These distributions are shown in Figs. 4
( and
) and 5
( and
). The shaded histograms are the
observed distributions in our sample from the Hipparcos Catalogue
(Sect. 3), the curves are the distributions from five synthetic
catalogues (in order to reduce statistical scatter) scaled to the same
total number of G solutions. There is excellent agreement between the
observed and calculated distributions for the second-order (
© European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |