5.1. Detection probabilities
The range of semi-major axes studied was to AU. Separate synthetic catalogues were generated for each decade in a and for the four basic mass-ratio distributions (, , , ). Table 2 gives the expected number of solutions of type C, G, O, X and if all the entries were actually binaries with in the indicated interval. In the synthetic catalogues the total number of entries satisfying the survey criterion [Eq. (1)] varies depending on the assumed distributions. For instance, the increasing mass-ratio distribution produces 25-30% more entries than the decreasing distribution , since there are many more binaries with small whose total magnitude may fall within the survey limit. A similar thing happens at the adopted FK5 limit (). To correct for this effect, the numbers in the table have been scaled to the observed total number of entries ( for C, G, O and X, and for ). In other words, the tabulated numbers divided by N or give the detection probabilities as function of a and the detection method.
Table 2. Expected number of binary solutions as function of the semi-major axis a and mass-ratio distribution . The first column gives the mid-point of the logarithmic interval, e.g. -1.5 means to 0.1. The next five columns give the expected number of detected C, G (quadratic and cubic), O and X systems among the 13 092 Hipparcos Catalogue entries considered, assuming that all the entries are actually binaries with specified and with uniformly distributed in one decade. The last three columns give the expected number of delta-mu systems among the 826 entries in the FK5 subset. Four different mass-ratio distributions are considered; in the last one the companion is assumed to be a white dwarf of mass 0.6.
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 distributions
The expected number of different solution types among the Hipparcos main-sequence survey stars can now be calculated by assuming distributions of q and a according to DM. The mass-ratio distribution found by DM is approximated by . The detection probabilities for this distribution are obtained by linear interpolation between the corresponding numbers in Table 2. For a total multiplicity of 0.57 and the same (log-normal) period distribution as found by DM, the multiplicities in the different intervals of would be 0.019, 0.056, 0.111, 0.145, 0.126, 0.072, and 0.027. (We transform periods to semi-major axes using the mean relation of our sample, .) Summing up the multiplicities times the detection probabilities gives the following predicted counts:
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.
Resolved systems (C): the ratio of observed to predicted counts is 1.54. Agreement could therefore be obtained by postulating a total multiplicity of in our model.
Long-period astrometric binaries (G and ): here the ratio of observed to predicted counts is much higher, 2.3 to 2.9, corresponding to a total multiplicity of about 1.4. The inconsistency with the C results is discussed below.
Orbital and stochastic solutions (O and X): compared with the model, there are too few O solutions in the Hipparcos Catalogue, and far too many X solutions. Both discrepancies are not unexpected considering how these solutions were derived in constructing the catalogue. A systematic search for orbital solutions among all the Hipparcos stars was not done. About half of the O solutions were previously known as spectroscopic binaries and it can be assumed that many of them received an astrometric orbital solution only because a spectroscopic period was available. Using the Hipparcos intermediate astrometry (van Leeuwen & Evans 1998) it will be possible to derive astrometric orbits for many more spectroscopic binaries discovered since the publication of the Hipparcos Catalogue. Thus the observed number of O systems from the Hipparcos Catalogue is probably much smaller than the actual number that can eventually be derived from the satellite data. - For the stochastic solutions the situation is the reverse: they are basically a waste-bin for all the cases where no other reasonable solution could be found. Falin & Mignard (1999) have shown that many of the X solutions are actually resolved systems, which could not be solved in the standard reductions due to bad initial positions of one or both of the components. Other X entries could be short-period astrometric binaries, which might be re-classified as O solutions if their periods become known.
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 counts
In 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 Q denote the fraction of binaries with a white-dwarf companion, i.e. the ratio of the number of MS+WD binaries to the total number of binaries with a main-sequence primary. The MS+WD binaries have usually developed from systems in which the WD started as the more massive component. If we consider a typical star of and assume that the original companion mass followed the IMF of field stars (Scalo 1998), then we find that roughly 5% of the companions had original masses . A large fraction of them would now be white dwarfs. We therefore expect Q to be of the order of a few per cent. For the subsequent calculations we consider the two extreme cases and .
Let us first assume that the distribution of a is log-normal with the same width as found by DM (, corresponding to ). The mean value and the total multiplicity are however left as free parameters. We determine these, for a given mass-ratio distribution, by a standard chi-square fitting of the predicted counts to the observed C, ,, , and . The resulting fit is unacceptable ( with 4 degrees of freedom) for the increasing and parabolic mass-ratio distributions (, ), as well as for a flat distribution (). This conclusion holds independent of the assumed Q. On the other hand, the decreasing mass-ratio distribution gives reasonable fits both for () and (). These distributions are shown as the solid curves in Fig. 2. The total multiplicity (including WD companions) is 1.2 in both cases. The median a is 12 AU () to 20 AU (). For a typical system mass of these semi-major axes correspond to median periods of 25 and 55 years, rather shorter than the 180 years found by DM for nearby solar-type stars.
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 C, , , , and are computed by means of Table 2 as linear combinations of the unknowns. Comparing with the observed counts gives a . We then minimise subject to the constraints . The results for the three central intervals are shown by the circles in Fig. 2 (filled for ; open circles for ). 95% confidence intervals were determined by investigating how much each could be varied (while leaving the other unknowns free to adjust) before the increased by 4 units from its minimum. These limits are shown as error bars in the figure. The results for and 3.5 are not shown, as their confidence intervals extend over the whole range of ordinates.
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 a the data are consistent with the DM distribution, although there is a suggestion that the multiplicity drops to zero quicker than the DM curve for AU. The predicted counts for the thick dashed curve in Fig. 2 are:
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 X and O in the fits described above. This was motivated by the difficulty to correctly model the criteria for these solutions in the actual data reductions. For the orbital solutions, an important factor in the actual reductions was whether the system was already known as a spectroscopic binary. In that case it may have received an orbital solution in the catalogue although e.g. a cubic fit would have been acceptable according to our model. On the other hand, if the system gave large residuals in the polynomial fits but had no spectroscopic period, it might be classified as stochastic in the catalogue although the model would classify it as orbital. Thus, depending on the availability of spectroscopic data, the distinction between X, O and G solutions in the real catalogue is much less precise than in our model. Because of the borderline cases between O and G, this could affect our analysis even though the X and O cases were not used. In other words, the observed used in the above analysis could include a fraction of systems with periods less than 3.5 yr, which in the model would by of type O. To investigate this possibility, we made an alternative fit in which the total count was modelled instead of . With this modification, the multiplicity in the decade around was reduced from 26% to 22%. Thus, only a small part of the high frequency of binaries derived for this interval could be due to inappropriate modelling of the distinction between cubic and orbital/stochastic solutions.
5.4. Predicted distributions of delta-mu and acceleration terms
In 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-1, and this is also the typical error for binaries with periods less than 1 yr or greater than yr. Periods from 3 to 30 yr often result in G solutions (shown as filled circles), while intermediate periods may result in undetected but significant errors (up to a few times 10 mas yr-1), i.e. delta-mu binaries.
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 (g) terms, while for the third-order () terms the model distribution is slightly too wide.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000