In order to allow some reprocessing of the Hipparcos data, the Hipparcos consortia provided the users with two kinds of data: the Transit Data (TD) and the Intermediate Astrometric Data (IAD). TD (Quist & Lindegren 1999) are a byproduct of the analysis performed by the NDAC consortium, and merge astrometric and photometric data. They basically provide the signal as modulated by the grid in front of the detector. They are therefore especially well suited to the analysis of visual double or multiple systems, since the separation, position angle and magnitudes of the various components may in principle be extracted from the TD. For that reason, TD are not provided for all Hipparcos entries, but only for those stars known or suspected of being double or multiple systems. IAD are lower level data which provide astrometric information only (the star's abscissa along a reference great circle, and the pole of that circle on the sky; van Leeuwen & Evans 1998), but for every Hipparcos entry. In the following, we describe how IAD and TD (when available) have been used to derive the astrometric parameters of the CPRS that have a known spectroscopic orbit. When TD are available, the astrometric parameters may be derived independently from the two data sets, and allow an interesting internal consistency check. In both cases, the method proceeds along the following steps:
At this point, it is important to realize that the Hipparcos 5parameter
solution, that is used as a starting point for the new 12parameter solution,
is in fact equivalent to a 12parameter solution where the semimajor axis
of the orbit (a_{0}) is null. It is therefore enough to consider in
Eq. (1) the correction term relative to the semimajor axis of
the orbit (and in fact
,
since the initial value of a_{0}
is null). The other orbital parameters
and T
enter Eq. (1) only through the partial derivatives
and
[equal,
respectively, to
and
according to Eq. (3)]
entering
.
These other orbital parameters do not require explicit
correction terms in Eq. (1) (and their starting values would be
illdefined anyway). Hence, the orbital solution is the one
minimizing
Our experience has shown that, except for very special cases (i.e., parallaxes and semimajor axes larger than about 20 mas, orbital periods significantly different from 1 year but smaller than 3 years; one example is HIP 50805 in Table 1), astrometric orbits could not be derived from the IAD without an a priori knowledge of some of the orbital elements, for instance the spectroscopic ones (e, Pand T). However, it appeared that fixing at the value derived from the spectroscopic orbit often led to orbital inclinations i unrealistically close to zero for spectroscopic binaries with radial velocity variations. Leaving the parameter free removes this difficulty, and offers moreover a way to check the consistency of the astrometric solution, since the astrometric should be consistent with its spectroscopic value.
As far as outliers are concerned, we almost always keep the same data set as that used by FAST and/or NDAC (Vol. 3, Sect. 17.6; Hipparcos). IAD that were not considered by FAST or NDAC (i.e., with the IA2 field set to "f'' or "n'') were thus not included in our reprocessing either. In a few cases, we noticed that because of the orbital contribution, some observations yield residuals larger than of the residuals. In these cases, these observations were removed and the fit reiterated. We never had to iterate more than twice to remove all outliers.
One of the five astrometric parameters entering in Eq. (1) is the parallax . With no other prescriptions as those described in Sect. 2.1.1, the minimization process may very well end up with a negative parallax. Indeed, negative parallaxes are not rare in the Hipparcos and Tycho catalogues.
Parallaxes cannot only be seen as inverse distances (which are defined positive) but also as the semimajor axis of the parallactic ellipse (see Eq. (3)). The direction of motion along the parallactic ellipse is of course imposed by the annual revolution of the Earth around the Sun, regardless of the actual dimension of the parallactic ellipse or of the observational uncertainties. In that sense, the parallactic ellipse is oriented, and negative parallaxes can be seen as corresponding to a parallactic ellipse covered in the wrong direction. That constraint being of physical nature, one should seek to fulfill it. In this section, we present a method which always delivers positive parallaxes. This method is especially useful for stars like those Mira variables or carbon stars that came out with large negative parallaxes in the Hipparcos Catalogue. In those cases, forcing the parallax to be positive has a strong impact on the derived proper motion (Pourbaix et al., in preparation), which may be supposed to be better determined with a physically sound model yielding positive parallaxes. However, the major drawback of the method is that the errors on the parallax do no longer follow a normal distribution. Therefore, the use of the parallaxes provided by this paper for e.g., luminosity calibrations should be done with care to avoid biases.
In order to force the parallax to be positive  and at the same time
avoiding the difficulties inherent to any constrained minimization techniques
 one may replace the constrained variable (the parallax )
by an
unconstrained one^{}. An appropriate choice appears to be
Most of the objects considered in the present paper have large parallaxes that would have come out positive by a direct fit of anyway. Thus, in the present case, the fitting of (instead of ) does not represent so much of an improvement. Nevertheless, the procedure of fitting has been introduced here for the sake of generality.
The price to pay is, however, that the errors on
do not any
longer follow a normal distribution. Moreover, the confidence interval of
the parallax is no more
symmetric. It has been estimated by the following expression:
The second expression in Eq. (9) clearly shows that the parallax is positive everywhere in the confidence interval, which would not be guaranteed in a constrained minimization of . That important property illustrates the superiority of this approach with respect to the constrained minimization.
If the expressed by Eq. (5) were a quadratic expression of the unknown parameters p_{k}, its unique minimum could be found from the solution of a set of linear equations. However, the parameters i, and enter in a highly nonlinear way, so that the function expressing in the 9parameter space may have several local minima, and finding its global minimum is a much more arduous task.
Faced with such situations, one of us (DP) has already successfully worked out global optimization techniques such as simulated annealing (Pourbaix 1994; Pourbaix 1998b). Practical details about the implementation of the method to minimize the objective function [Eq. (5)] in the working space ^{9} may be found in Pourbaix (1998a). Simulated annealing being a heuristic method, one can only prove its convergence to the global minimum after an infinite time (which we cannot afford). We thus stop the procedure after a finite time. In order to nevertheless have a good chance to obtain (a neighborhood of) the global minimum, we repeat 40 times this highly non deterministic minimization process. The best solution ever met (i.e., the one leading to the smallest value) is finally adopted. Once (a neighborhood of) the global minimum is thus obtained, it is tuned with the BFGS quasiNewton algorithm (Dennis Jr. & Schnabel 1995).
Unlike the LevenbergMarquardt (Marquardt 1963) minimization algorithm, BFGS does not return the covariance matrix of the model parameters. The inverse of the Fisher information matrix at the minimum is therefore used as the best estimate of that covariance matrix (Pourbaix 1994).
The whole procedure has been applied separately on the data from the FAST consortium only, from NDAC only and from both combined, thus resulting in three different solutions, hopefully consistent with each other.
In a few instances, the solution obtained from the combined FAST+NDAC data set turns out to be very close to either the FAST or NDAC solution, but FAST and NDAC taken separately yield rather different solutions. That situation probably reflects the very different weights attributed to the two data sets for that particular object in the merging process applied to produce the output catalogue. For our analysis we always keep the covariance matrices of the observations as they are given in the electronic version of the catalogue.
As pointed out by an anonymous referee, in the case where all the Campbell elements (a_{0}, i, and ) are extracted from a fit to the astrometric data, they can advantageously be replaced by the ThieleInnes elements (A, B, F and G) so that becomes a quadratic function of the model parameters. The minimum of can then be found analytically and no minimization (neither global nor local) technique is needed. For the sake of generality, we nevertheless use the Campbell set (and thus the minimization scheme) because this more general scheme allows, if necessary, to easily incorporate external constraints (like for instance the knowledge of i for eclipsing binaries, or from the spectroscopic orbit; see, however, the comment about fixing after Eq. (6) in Sect. 2.1.1). Such additional constraints would be much more difficult to impose through the ThieleInnes elements.
The introduction of more free parameters in the orbital model as compared to the single star solution necessarily leads to a reduction of the objective function. To evaluate whether this reduction is statistically significant  or, equivalently, whether the orbital solution represents a significant improvement over the 5parameter Hipparcos solution  requires the use of an Ftest.
The method used here is inspired from the test devised by Lucy & Sweeney (1971).
If
and
denote the residuals
for the Hipparcos 5parameter model and for the orbital model (with
9 free parameters),
respectively, the efficiency of the additional 4 parameters in reducing
below
may be measured by
the ratio:
If the hypothesis that there is no orbital motion (i.e., a_{0} = 0 in
Eq. (5)) is correct, then it may be shown (Bevington & Robinson 1992)
that F follows a Snedecor
distribution with
and
degrees of freedom. Thus, if Eq. (10) yields
,
then, on the assumption that there is no orbital motion, the
probability that F could have exceeded
is
The residuals given in the IAD files always relate to a 5parameter solution, even when a more sophisticated model (the socalled "acceleration'' 7 or 9parameter models, or even orbital model) was published in the Hipparcos catalogue (see Table 1). For those cases, the value listed in Table 1 is always close to zero, although it does not really characterize the improvement of the orbital solution with respect to the solution retained in the Hipparcos catalogue (which goes already beyond a 5parameter model).
The minimization process will yield a solution in all cases, but that solution may not be astrophysically relevant. A statistical check of the significance of the orbital solution, based on the Ftest, has been presented in Sect. 2.1.4. In this section, two criteria testing the validity of the orbital solution on astrophysical grounds are presented.
The first test is based upon the identity
This test has the advantage of being totally independent of any assumptions. However, it involves the orbital inclination which is not always very accurately determined (see Sect. 3.2), so that the above identity may not always be very constraining considering the often large uncertainty on i.
A somewhat more constraining identity to assess the astrophysical
plausibility of the computed astrometric orbit is the following:
Because of the assumptions involved, this test is only used as a guide to identify astrophysicallyunplausible solutions. It turns out that such cases are generally those with large error bars or with inconsistent N, F and A solutions, thus providing further arguments not to retain those solutions. In very few cases (HIP 36042, 53763 and 60299), valid data yielded solutions not consistent with Eq. (13). Those cases were nevertheless kept in our final list.
HIP  HD/DM  P  e  T  Sol  Transit?  Rem  
(mas)  ()  (d)  (JD 2400000)  (%)  astrom.  expected  
CH stars  
168  224959  1.95 1.34  62  2.8  1273.0  0.179  46064  5  y  1.23  36  1.6  1.08 0.53  
4252  5223  1.12 1.17  50  16.8  755.2  0  45535.6  5  0.69  0  1e+09  0.76 0.38  
22403  30443  1.63 1.7  28  12.5  2954.0  0  46306  5  y  0.90  9  24  1.89 0.94  
53763  5.29 1.47  70  32.2  328.3  0  46542.1  5  0.84  3  1.8  0.44 0.22  Accepted,*  
62827  A  5.3 2.15  40  11.8  571.1  0  46467.8  5  1.10  89  0.42  0.63 0.31  *  
102706  198269  3.16 1.11  68  34.2  1295.0  0.094  46358  5  1.18  34  2.5  1.09 0.54  
104486  201626  4.93 0.84  100  40.6  1465.0  0.103  45970  7  1.35  0  1.5  1.18 0.58  Accepted  
108953  209621  1.47 1.3  46  30.6  407.4  0  45858.3  5  1.02  94  0.8  0.50 0.25 


Dwarf Ba stars  
8647  11377  6.38 1.14  42  26.3  4140.0  0.16  45240  5  y  0.95  34  9.5  2.20 0.97  
32894  50264  14.11 1.96  78  52.2  912.4  0.098  46791  X  1.05  0  0.84  0.89 0.46  Accepted  
49166  87080  7.9 1.39  90  42.2  273.4  0.177  48373  5  1.03  34  0.47  0.39 0.19  Accepted  
50805  89948  23.42 0.93  60  36.6  667.8  0.117  46918  O  y  0.82  0  0.67  0.67 0.30  Accepted 
60299  107574  5.02 1.06  50  14.7  1350.0  0.081  46342  9  0.93  0  1.8  0.83 0.26  Accepted  
62409  +17  8.2 1.28  58  20.1  1796.0  0.14  46291  5  1.29  25  0.87  1.34 0.65  NDAC accepted  
69176  123585  8.75 1.39  54  29.3  457.8  0.062  48207  5  1.24  0  0.47  0.44 0.16  Accepted  
71058  127392  10.63 1.7  46  15.4  1498.7  0.071  47070  5  0.85  73  0.56  1.21 0.61  
104785  202020  9.53 1.5  52  6.1  2064.0  0.08  47122  5  1.16  14  1.4  1.51 0.77  
105969  204613  16.61 1.78  80  64.8  878.0  0.13  47479  X  1.02  0  0.75  0.77 0.33  Accepted  
107818  207585  7.53 1.51  52  10.5  670.6  0.03  47319  5  0.82  7  0.39  0.58 0.22  Accepted  
116233  221531  8.83 1.21  56  8.3  1416.0  0.165  47157  7  0.84  0  1.2  0.97 0.37  Accepted  
118266  224621  6.95 1.45  80  32.6  307.8  0.048  49345  5  1.00  45  0.27  0.40 0.19  

HIP  HD/DM  P  e  T  Sol  Transit?  Rem  
(mas)  ()  (d)  (JD 2400000)  (%)  astrom.  expected  
Mild Ba stars  
19816  26886  2.74 1.05  48  21.9  1263.2  0.395  48952.12  5  0.55  1  5.2  0.77 0.23  
20102  27271  6.01 1.13  28  18.6  1693.8  0.217  47104.38  5  0.66  46  1.8  0.93 0.28  
26695  288174  2.89 1.32  42  21.3  1824.3  0.194  47157.62  5  0.80  66  5.1  0.98 0.29  
32831  49841  1.05 1.44  44  17.2  897.1  0.161  48339.71  5  y  1.15  44  1.8e+09  0.61 0.18  
34143  53199  3.65 1.3  44  9.3  7500.0  0.212  41116.2  5  y  0.99  72  26  2.51 0.75  
35935  58121  2.82 0.95  64  15.7  1214.3  0.14  46811.21  5  1.12  53  0.98  0.75 0.22  
36042  58368  2.36 0.97  56  14.3  672.7  0.221  45617  5  1.00  1  2  0.50 0.15  Accepted  
36613  59852  1.86 1.24  60  25.8  3463.9  0.152  46841.03  5  1.25  100  6.2e+51  1.50 0.45  
43527  0.41 1.4  56  30.9  3470.5  0.217  48828.06  5  y  0.84  59  49  1.50 0.45  
44464  77247  2.86 0.97  46  34.6  80.5  0.0871  48953  5  y  1.05  93  0.39  0.12 0.04  
51533  91208  4 0.95  48  24.4  1754.0  0.171  45628.36  5  1.45  72  3.5  0.95 0.28  
53717  95193  2.3 1.03  44  19.0  1653.7  0.135  46083.62  5  1.15  89  5.3  0.92 0.27  
73007  131670  2.33 1.22  34  9.0  2929.7  0.162  46405.11  5  1.17  45  21  1.34 0.40  
76425  139195  13.89 0.7  38  28.5  5324.0  0.345  44090  5  0.90  61  25  2.00 0.59  
78681  143899  3.6 1.29  36  1.2  1461.6  0.194  46243.43  5  1.65  93  1.2  0.84 0.25  
94785  180622  3.37 1.04  54  22.5  4049.2  0.061  50534.41  5  1.03  52  26  1.66 0.50  
103263  199394  6.33 0.63  84  59.6  4382.6  0  50719.34  5  0.81  76  4  1.75 0.53  
103722  200063  0.73 1.02  48  17.2  1735.5  0.073  47744.64  5  1.00  3  14  0.95 0.28  
104732  202109  21.62 0.63  98  43.7  6489.0  0.22  40712  7  0.71  0  6  2.28 0.68  
105881  204075  8.19 0.9  54  7.0  2378.2  0.2821  45996  5  y  0.92  6  1  1.17 0.35  Accepted 
106306  205011  6.31 0.68  68  36.2  2836.8  0.2418  46753.59  5  0.71  3  1.7  1.31 0.39  
109747  210946  3.42 1.14  36  11.7  1529.5  0.126  46578.18  5  0.89  14  2.2  0.87 0.26  
112821  216219  10.74 0.93  56  23.3  4098.0  0.101  44824.92  5  0.85  25  3  1.96 0.74  *  
117607  223617  4.61 0.95  62  2.9  1293.7  0.061  47276.68  7  0.75  0  0.87  0.78 0.23  Accepted 
HIP  HD/DM  P  e  T  Sol  Transit?  Rem  
(mas)  ()  (d)  (JD 2400000)  (%)  astrom.  expected  
Strong Ba stars  
4347  5424  0.22 1.42  52  30.9  1881.5  0.226  46202.8  5  y  1.29  51  3.7e+09  1.12 0.40  
13055  16458  6.54 0.57  80  60.2  2018.0  0.099  46344  7  1.20  0  1.6  1.17 0.42  Accepted  
15264  20394  2.16 1.14  48  15.3  2226.0  0.2  47929  5  1.39  77  8  1.25 0.44  
17402  24035  3.72 0.8  70  76.9  377.8  0.02  48842.65  5  1.12  49  3.3e+09  0.38 0.14  *  
23168  31487  4.54 1.21  40  29.1  1066.4  0.045  45173  5  0.45  86  0.47  0.77 0.27  
25452  36598  3.32 0.68  72  85.3  2652.8  0.084  45838.95  5  0.99  5  4.3  1.41 0.50  
29099  42537  1.13 0.93  64  75.9  3216.2  0.156  46147.32  5  y  1.30  82  1.3e+15  1.60 0.57  
29740  43389  1.25 1  52  25.8  1689.0  0.082  47222.46  5  0.53  0  33  1.04 0.37  *  
30338  44896  1.56 0.7  90  56.9  628.9  0.025  48464.3  5  0.88  0  1.8  0.54 0.19  
31205  46407  8.25 0.92  80  34.3  457.4  0.013  47677.45  O  y  0.77  0  0.71  0.44 0.15  Accepted 
32713  49641  0.73 0.88  48  19.2  1768.0  0  46306  5  0.63  0  7.5  1.08 0.38  
32960  50082  4.71 0.99  40  16.2  2896.0  0.188  45953.12  5  1.39  44  3.9  1.49 0.53  
36643  60197  1.67 0.84  92  50.5  3243.8  0.34  46015.97  5  0.64  59  21  1.61 0.57  
50006  88562  3.13 1.17  50  25.1  1445.1  0.204  45781.71  5  0.95  4  3.2  0.94 0.33  
52271  92626  3.4 0.71  104  50.5  918.2  0  49147.83  5  1.05  9  0.59  0.69 0.25  Accepted  
56404  100503  0.67 1.2  64  30.8  554.4  0.061  46144.83  5  0.63  26  2.3  0.50 0.18  
56731  101013  7.07 0.68  74  43.2  1710.9  0.195  43934  O  1.19  0  1.1  1.05 0.37  Accepted  
60292  107541  5.78 1.36  68  29.5  3569.9  0.104  44388.16  5  0.96  2  27  1.72 0.61  
68023  121447  2.21 1.02  52  6.0  185.7  0  46922.35  5  y  1.14  19  3.5  0.24 0.08  
69290  123949  .97 1.32  34  5.6  9200.0  0.972  49144.96  5  1.50  92  470  3.23 1.14  
94103  178717  2.9 0.95  52  32.5  2866.0  0.434  44258  5  y  0.89  71  16  1.48 0.53  
101887  196445  1.49 1.54  50  21.2  3221.3  0.237  46037.95  5  y  1.06  63  9.3  1.60 0.57  
103546  199939  3.16 0.75  72  57.6  584.9  0.284  45255.1  5  0.79  26  0.44  0.51 0.18  Accepted  
104542  201657  4.49 1.07  64  31.6  1710.4  0.171  46154.95  5  1.06  73  0.81  1.05 0.37  
104684  201824  0.56 1.56  50  7.4  2837.0  0.342  47413  5  1.79  87  2.1e+03  1.47 0.52  
110108  211594  4.59 1.18  26  4.4  1018.9  0.058  48538.19  5  1.11  17  0.99  0.74 0.26  Accepted


Tcpoor S stars  
5772  7351  3.21 0.82  28  19.1  4592.7  0.17  44696  5  y  0.92  88  6.9  1.97 0.67  
8876  1.92 1.5  38  9.5  4137.2  0.209  43578.31  5  y  1.13  87  2.4e+13  1.84 0.66  
17296  22649  6.27 0.63  78  42.2  596.2  0.088  42794.5  O  y  1.22  1  0.37  0.51 0.17  Accepted 
25092  35155  1.32 0.99  52  31.7  640.6  0.071  48092.41  5  y  0.79  51  0.94  0.53 0.18  Accepted 
32627  49368  1.65 1.11  60  17.4  2995.9  0.357  45145.37  5  y  1.03  57  16  1.48 0.50  
38217  63733  0 0.99  84  39.2  1160.7  0.231  45990.92  5  0.95  58  1.7e+14  0.79 0.27  
90723  170970  1.83 0.67  86  59.4  4392.0  0.084  48213.21  5  y  0.98  39  27  1.91 0.66  
99124  191226  0.39 0.71  84  54.9  1210.4  0.19  49691.78  5  0.98  23  11  0.81 0.27  
99312  191589  2.25 0.77  84  52.0  377.3  0.253  48844.02  5  0.80  82  0.58  0.37 0.13  Accepted  
115965  1.72 1.26  64  29.6  1252.9  0.091  48161.32  5  0.78  72  1.4  0.83 0.28 
Unlike the IAD, TD are only available for a small subset of the Hipparcos catalogue, e.g., for those stars that were known to be (or suspected of being) double or multiple systems at the time of the data reduction by the Hipparcos consortia. TD are a byproduct of (or, more precisely, an input for) the multiplestar processing by the NDAC consortium. Another difference with respect to the IAD concerns photometry. Whereas IAD contain astrometric information only, the brightness of the (different) observable component(s) of the system can be retrieved from the TD.
In the most general case (Quist & Lindegren 1999), each entry in the TD file
corresponds to
five numbers b_{1}, ..., b_{5} which represent the coefficients of the first
terms in the Fourier series modeling the observed signal as modulated by the
detector grid:
For the SB1 systems we are interested in, the situation simplifies a lot since
I_{2} may be taken equal to 0. The above system of equations is rankdeficient.
From the second and third equations, one can rewrite:
The remaining of the method follows the same steps as described in relation with the IAD, i.e. global and local optimization, positiveness of the parallax, ...
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