3 Results

Table 1 lists the various samples that have been considered, namely dwarf barium and subgiant CH stars (McClure 1997; North, priv. comm.), mild and strong barium stars (Jorissen et al. 1998), Tc-poor S stars (Jorissen et al. 1998) and CH stars (McClure & Woodsworth 1990). Their spectroscopic orbital parameters have been taken from the reference quoted.

Table 1 also provides various parameters that allow either to assess the quality of the derived orbital solution or to identify the reason why a reliable orbital solution could not be derived for some of the spectroscopic binaries considered. The following parameters potentially control our ability to derive an astrometric orbit from the IAD, with favorable circumstances being mentioned between the parentheses: the parallax (Col. 3; large parallax), the number of available IAD measurements (FAST+NDAC; Col. 4; large number of measurements), the ecliptic latitude (Col. 5; this parameter may play a role since it controls how different the orientations of the reference great circles are; favorable cases have absolute values larger than 47$^\circ$), the orbital period (Col. 6; in the range 1 - 3 yr to ensure a good sampling of the orbit), the orbital eccentricity (Col. 7; low eccentricity).

The following columns characterize the quality of the solution obtained from the minimization process. Column 11 provides $\chi ^2/(N_{\rm IAD}-9)$, which should be of the order of unity if the internal error $\sigma$(Eq. 2) on the abscissae has been correctly evaluated by the reduction consortia and if the model provides an adequate representation of the data. The first risk error associated with rejecting the null hypothesis that the orbital and Hipparcos 5-parameter solutions are identical (Eq. 11) is given in Col. 12. Low values of $\alpha$ are generally associated with Hipparcos solutions of the G, X or O types (as listed in Col. 9; see the caption to Table 1 for more details), since the orbital motion is then large enough to have been noticed already by NDAC or FAST. Columns 13 and 14 compare the $a_0/\varpi$ ratio derived from the orbital solution to its expected value from Eq. (13). In Cols. 11 to 13, the data refer to the orbital solution obtained by combining NDAC and FAST data. For dwarf barium stars, the masses used to estimate $a_0/\varpi$ according to Eq. (13) are $M_2 = 0.62\pm 0.04$ $M_{\odot}$, whereas M₁ is derived from the spectroscopic gravity, with an estimated error of 0.05 $M_{\odot}$ (North, priv. comm.). According to the statistical analysis of the spectroscopic mass functions performed by Jorissen et al. (1998), M₁ and M₂ pairs (expressed in $M_{\odot}$) of ( $1.7\pm 0.2, 0.62\pm 0.04$), ( $2.1\pm 0.2, 0.62\pm 0.04$) and ( $1.8\pm 0.2, 0.62\pm 0.04$) have been adopted for strong barium stars, mild barium stars and Tc-poor S stars, respectively. The same analysis performed by McClure & Woodsworth (1990) for CH stars yielded $M_1 = 1.0\pm 0.1$ $M_{\odot}$ and $M_2 = 0.62\pm 0.04$ $M_{\odot}$.

Astrometric orbits were accepted when $\alpha$ is smaller than 10 per cent and the expected $a_0/\varpi$ value falls within the $1\sigma$ confidence interval. A few cases not fulfilling these criteria were nevertheless accepted after visual inspection of the orbital arc.

Examination of Table 1 reveals that the following criteria need to be fulfilled in order to be able to extract a reliable astrometric orbit from the Hipparcos data:  mas,  yr, . The success rate is as follows for the various samples: dwarf barium stars (9/13), mild barium stars (3/24), strong barium stars (6/26), Tc-poor S stars (3/10) and CH stars (2/8). The high success rate for dwarf barium stars naturally results from the fact that these dwarf stars are on average closer from the sun than the giant stars.

  

Table 2: Astrometric orbital elements for various classes of CPRS. The first line provides the data from the HIPPARCOS catalogue or from the spectroscopic orbit (label H/S in Col. 2). The following lines provide the orbital solutions derived in the present work from the FAST, NDAC and combined FAST+NDAC IAD (labels F, N and A, respectively in Col. 2). When available, the orbital solution from the transit data (label T in Col. 2) or from the DMSA/O (label O) are given next. The columns $\varpi$, $a_0/\varpi$ and K₁contain the nominal value followed by the boundaries of the "$1\sigma$'' confidence interval (computed by propagating the 1$\sigma$ errors on a₀, $\varpi '$ and i as if they were uncorrelated). An asterisk after the HIP number refers to a note at the end of the table

 

Table 2: continued

 

Table 2: continued

 

Table 2: continued

 

Table 2: continued

 

Table 2: continued

 

Table 2: continued

Table 2 lists the astrometric and orbital parameters for the reliable orbits according to the criteria discussed above. The results from the different processing modes are collected in Table 2, according to the symbol given in Col. 2: H/S refers to the parameters from the Hipparcos catalogue and from the spectroscopic orbit [on that line $a_0/\varpi$ is the semi-major axis in A.U. estimated from Eq. (13)], F refers to the processing of the IAD from FAST only, N to the IAD from NDAC only, A from the processing of the combined FAST/NDAC data set, O to the orbital parameters from the DMSA/O, and T to the parameters resulting from the TD.

Most of the retained orbits are indeed characterized by $\chi^2/(N-9)$ values of the order of unity, as expected. The first-risk errors $\alpha$ are not always close to 0, but if the derived value for the semi-major axis is in good agreement with its expected value, that agreement has been considered as sufficient for retaining the orbit. The only cases where the reverse situation occurs (small $\alpha$ but discrepant $a_0/\varpi$) are the dwarf barium star HIP 60299, the mild barium star HIP 36042 and the CH star HIP 53763. Although the orbit of the latter is not well defined, it has been kept in our final list to illustrate the large uncertainty on $\varpi$ resulting from an orbital period close to 1 yr (Sect. 4).

The F, N and A solutions for the retained orbits are also generally in good agreement, the only exceptions being the dwarf Ba stars HIP 62409 and HIP 116233, and the mild Ba star HIP 117607. However, the model parameters of these systems are highly correlated, and the different measurement errors in the different data sets thus drive the solution in different directions. This statement may be expressed in a quantitative way using the concept of efficiency $\epsilon$ introduced by Eichhorn (1989) and Pourbaix & Eichhorn (1999). It is defined as

$$\epsilon = \sqrt[p]{\frac{\prod\nolimits^p_{k=1} \lambda_k}{\prod\nolimits^p_{k=1} q_{kk}}},$$ (17)

where $\lambda_k$ are the eigenvalues of the covariance matrix of the estimated parameters, q_kk are its diagonal elements, and p denotes the number of parameters in the model. If $\epsilon$is close to unity, there is obviously little correlation between the parameters. For the combined NDAC+FAST solution, it amounts to 0.32, 0.21 and 0.42 in the case of HIP 62409, HIP 116233 and HIP 117607, respectively, thus translating some degree of correlation between the model parameters.

In Table 2, the uncertainty on $a_0/\varpi$ has been computed by combining the upper and lower limits on a₀ and $\varpi$, thus neglecting any possible correlation between these two quantities (which is generally small - except for the three systems listed above - as derived from the efficiency being close to unity).

The orbital solutions derived in the present paper are too many to display the astrometric orbit for all cases. A few representative cases among the different subsets of Table 1 (orbital periods shorter or longer than the duration of the Hipparcos mission, small or large parallaxes...) have instead been selected and are presented in Fig. 1.

Figure 1: The orbital arc on the plane tangent to the line of sight for some representative orbits among those listed in Table 2. The segments connect the computed position on the orbit to the great circle (not represented, perpendicular to the segment) corresponding to the observed position (Hipparcos measurements are one-dimensional). HIP 50805 is the only system for which the DMSA/O provides an orbital solution from scratch; HIP 53763 has a small parallax (about 2 mas), not very well determined since the orbital period is close to 1 yr; only the NDAC solution is acceptable for HIP 62409; the orbit of HIP 103546 is at limit of what can be extracted from the IAD; HIP 105881 is an example of an incomplete, albeit well determined, orbital arc; HIP 116233 is one case where the NDAC and FAST solutions are rather different

3.1 Comparison with DMSA/O solutions

For HIP 17296 (Tc-poor S), 31205 (strong Ba) and 56731 (strong Ba), orbital solutions are provided in the DMSA/O and were derived using spectroscopic elements from the literature. HIP 50805 (dwarf Ba) is the only case in our sample where an orbit could be derived from scratch by the Hipparcos consortia. For all these systems, the astrometric orbits derived by the methods described in Sect. 2 are in excellent agreement with the DMSA/O elements, thus providing an independent check of the validity of our procedures. Further checks are presented in Sect. 3.2.

The large number (23) of systems for which orbital solutions could be extracted from the Hipparcos data (as compared to only 4 of those already present in the DMSA/O) illustrates the great potential that still resides in the Hipparcos IAD or TD.

   
3.2 Check of the astrometric orbit

Several checks are possible to evaluate the accuracy of the astrometric elements derived in the present paper. First, it is possible to compare the astrometric and spectroscopic values of $\omega$, the argument of periastron. In most cases, the two determinations agree within 2$\sigma$ (Fig. 2). However, even when the orbital period is shorter than the Hipparcos mission, the $\omega$ derived from the IAD is seldom as precise as the spectroscopic one.

Figure 2: Comparison of the astrometric (ordinate) and spectroscopic (abscissa) determinations of the argument of periastron $\omega$

In a few cases, the spectroscopic orbit is assumed to be circular. In that case, the time T of passage at periastron becomes meaningless, and is replaced by the time of the nodal passage (or, equivalently, the time of maximum radial velocity). This is equivalent to setting $\omega$ equal to 0. Non-zero values for $\omega$ would correspond to other conventions for the origin epoch. Among our systems with acceptable orbits, two have circular orbits: HIP 53763 (CH star) and HIP 52271 (strong barium star). For these systems, our fit leads to values of $\omega$ significantly different ($2\sigma$) from 0, indicating that at the reference epoch, the star is in fact far from the node where it was expected to be.

It is also possible to compare the astrometric value of K₁ (using in Eq. (12) the value of a, i and $\varpi$ from the astrometry and e and P from the spectroscopy) with the spectroscopic value. Figure 3 shows that, even if $a_0/\varpi$ is well defined, the inclinations are generally not very accurately determined, thus leading to uncertain values of K₁. This unfortunate property of i is well illustrated in Figs. 4 and 5. One example where the accuracy of the astrometric value of i must be questioned is the dwarf barium star HIP 105969: despite the fact that the astrometric $a_0/\varpi$ ratio perfectly agrees with its estimate based on the masses, the astrometric prediction of the semi-amplitude of the radial-velocity variations differs by almost of factor of 2 as compared to the actual spectroscopic value (Table 2). The only way to resolve that discrepancy is to assume that the orbital inclination is largely in error.

The semi-major axis $R = a_0/\varpi$ as derived from its astrophysical estimate (Eq. 13) is compared to its astrometric value in Fig. 6, and the two values are often consistent with each other. Although the value of a₀ is likely to be affected by a positive bias (i.e., a positive a₀ value is derived even when the data consist of pure noise, as clearly apparent from the astrometric $a_0/\varpi$ values listed in Table 1), this bias does not markedly affects the retained solutions displayed in Fig. 6, except for solutions with $\varpi< 3$ mas, which all have $R/\hat{R} > 1$. Solutions for larger $\varpi$ values are almost equally distributed around unity.

Figure 3: Comparison of the semi-amplitude of the radial velocity curve derived from the astrometry (using Eq. 12) and the spectroscopic value. The three panels show the results derived from FAST, NDAC and FAST+NDAC (from top to bottom respectively)

Figure 4: Uncertainty on the inclination i as a function of i. All points lying inside the region delineated by the oblique dashed lines have orbital inclinations consistent with i = 90$^\circ$. Filled triangles correspond to orbital solutions derived from NDAC, filled squares to FAST and crosses to NDAC+FAST

Figure 5: Correlation between the uncertainties on the inclination i and the semi-major axis a0. The symbols have the same meaning as in Fig. 4

Figure 6: Comparison of the semi-major axis $R = a_0/\varpi$ derived from the astrometry and from Eq. (13). The upper panel displays the results from the FAST data, the middle panel the results from the NDAC data, and the lower panel the results from FAST and NDAC combined. The error bars assume that M1, M2 and P are error-free

3.3 Comparison TD vs. IAD

Among the 81 systems studied in this paper, 22 have TD and only 5 belong to the list of accepted orbits (HIP 17296, 25092, 31205, 5080, and 105881). They hence can be used to check the consistency of the two orbit-determination methods. The astrometric orbits derived from the TD are given in Table 2 as lines labeled with "T'' in Col. 2. The agreement between the results obtained with the two data types is quite good (within 1$\sigma$ error bars). The number of TD measurements for those five systems is up to twice as large the number of IAD measurements. As a consequence, the confidence intervals of the parameters is systematically narrower. One should however keep in mind that more precise does not imply more accurate. For instance, in the case of the mild Ba star HIP 105881, the 86 TD measurements yield $a_0/\varpi = 1.59$ (with 1$\sigma$ limits of 1.45 and 1.74) as compared to $1.2\pm0.35$ estimated from Kepler's third law. The 54 IAD measurements from FAST and NDAC combined yield a more accurate result of 0.99 (between 0.44 and 1.69).

For the sake of completeness, we checked the astrophysical consistency of the orbit derived for the 22 systems for which TD are available. All systems but the five already accepted were rejected. In essence, this confirms that the astrometric content of the TD is basically equivalent to the IAD and that nothing new can come out from TD if IAD do not yield a reliable solution.

Figure 7: Comparison of the position angle (expressed in radians) of the Hipparcos proper motion with that derived when account is made of the orbital motion (present work), as a function of the orbital period. The upper panel displays the results from the FAST data, the middle panel the results from the NDAC data, and the lower panel the results from FAST and NDAC combined. Error bars do not include the uncertainty on the Hipparcos values

Figure 8: Same as Fig. 7 for the proper-motion modulus

Figure 9: Comparison of the Hipparcos parallax with that derived when account is made of the orbital motion (present work), as a function of the orbital period. As in Fig. 7, results from FAST, NDAC and FAST+NDAC are presented in the upper, middle and lower panels, respectively. Error bars include only the uncertainty on the parallaxes derived from the present work, and are computed according to Eq. (9)