A&A 442, 365-380 (2005)
DOI: 10.1051/0004-6361:20053003

Astrometric orbits of SB9 stars

S. Jancart - A. Jorissen[*] - C. Babusiaux - D. Pourbaix[*]

Institut d'Astronomie et d'Astrophysique, Université Libre de Bruxelles, CP 226, Boulevard du Triomphe, 1050 Bruxelles, Belgium

Received 7 March 2005 / Accepted 8 July 2005

Abstract
Hipparcos Intermediate Astrometric Data (IAD) have been used to derive astrometric orbital elements for spectroscopic binaries from the newly released Ninth Catalogue of Spectroscopic Binary Orbits ($S\!_{B^9}$). This endeavour is justified by the fact that (i) the astrometric orbital motion is often difficult to detect without the prior knowledge of the spectroscopic orbital elements, and (ii) such knowledge was not available at the time of the construction of the Hipparcos Catalogue for the spectroscopic binaries which were recently added to the $S\!_{B^9}$ catalogue.

Among the 1374 binaries from $S\!_{B^9}$ which have an HIP entry (excluding binaries with visual companions, or DMSA/C in the Double and Multiple Stars Annex), 282 have detectable orbital astrometric motion (at the 5% significance level). Among those, only 70 have astrometric orbital elements that are reliably determined (according to specific statistical tests), and for the first time for 20 systems. This represents a 8.5% increase of the number of astrometric systems with known orbital elements (The Double and Multiple Systems Annex contains 235 of those DMSA/O systems).

The detection of the astrometric orbital motion when the Hipparcos IAD are supplemented by the spectroscopic orbital elements is close to 100% for binaries with only one visible component, provided that the period is in the 50-1000 d range and the parallax is >5 mas. This result is an interesting testbed to guide the choice of algorithms and statistical tests to be used in the search for astrometric binaries during the forthcoming ESA Gaia mission.

Finally, orbital inclinations provided by the present analysis have been used to derive several astrophysical quantities. For instance, 29 among the 70 systems with reliable astrometric orbital elements involve main sequence stars for which the companion mass could be derived. Some interesting conclusions may be drawn from this new set of stellar masses, like the enigmatic nature of the companion to the Hyades F dwarf HIP 20935. This system has a mass ratio of 0.98 but the companion remains elusive.

Key words: astrometry - stars: binaries: spectroscopic - stars: fundamental parameters

1 Introduction

The Ninth Catalogue of Spectroscopic Binary Orbits ($S\!_{B^9}$; Pourbaix et al. 2004, available at http://sb9.astro.ulb.ac.be) continues the series of compilations of spectroscopic orbits carried out over the past 35 years by Batten and collaborators. As of 2004 May 1st, the new Catalogue holds orbits for 2386 systems. The Hipparcos Intermediate Astrometric Data (IAD; van Leeuwen & Evans 1998) offer good prospects to derive astrometric orbits for those binaries. Astrometric orbits are often difficult to extract from the IAD without prior knowledge of at least some among the orbital elements (e.g., Pourbaix 2004). As an illustration of the difficulty, only 45 out of 235 Double and Multiple Systems Annex Orbital solutions (DMSA/O, see ESA 1997 and Lindegren et al. 1997) were derived from scratch. For those $S\!_{B^9}$ binaries whose orbit has become available after the publication of the Hipparcos Catalogue, new astrometric orbital elements may be expected from the re-processing of their IAD. This is the major aim of the present paper, which belongs to a series devoted to the re-processing of the IAD for binaries (Pourbaix & Boffin 2003; Pourbaix & Jorissen 2000).

One of the major challenges facing astronomers studying binaries and extrasolar planets is to get the inclination of the companion orbit in order to derive the component masses. The orbital inclinations will be provided in this paper for 70 systems (Sect. 5). To get the component masses requires moreover the system to be spectroscopic binary with 2 observable spectra (SB2). Unfortunately, SB2 systems are not favourable targets to detect their astrometric orbital motion using the IAD. When the component's brightnesses do not differ much (less than about 1 mag), the orbital motion of the photocenter of the system around its barycenter might not be large enough to allow detection (see Eq. (8) below). This means that the astrometric orbit cannot in general be derived from the IAD for SB2 systems (neither can the component solutions - DMSA/C - when available, be reprocessed using the IAD, because the abscissa residuals of DMSA/C entries turn out to be abnormally large, even for non-binary stars), thus compromising our ability to derive stellar masses in a fully self-consistent way in the present paper. This difficulty will be circumvented by the use of the mass - luminosity relationship for main sequence stars, thus allowing us to derive at least the companion's mass (Sect. 6.1). This information will then be combined with the position of the system in the eccentricity - period diagram to diagnose post-mass-transfer systems (Sect. 6.2).

Another important motivation of the present paper is to test on the IAD, algorithms designed (i) to detect astrometric binaries and (ii) to determine their orbital parameters in the framework of the future ESA cornerstone mission Gaia. IAD are indeed very similar to what will be available at some stage of the Gaia data reduction process. The fit of an orbital model to the IAD is greatly helped with a partial knowledge of the orbital elements, coming from the spectroscopic orbit (Pourbaix 2004). In the present context, orbital elements like eccentricity e, orbital period P and one epoch of periastron passage T0 are provided by the spectroscopic orbits listed in $S\!_{B^9}$. With Gaia, these elements may come (in the most favourable circumstances) from the spectroscopic orbit derived from the on-board radial-velocity measurements.

2 The Hipparcos data

During 3 years and for about 118000 stars, the Hipparcos satellite (ESA 1997) measured tens of abscissae per star, i.e., 1-dimensional positions along precessing great circles. Corrections like chromaticity effects, satellite attitude, ... were then applied to these abscissae. It was decided that the residuals  $(\Delta v)$ of these corrected abscissae (with respect to a 5-parameter single-star astrometric model) would be released together with the Hipparcos Catalogue. They constitute the IAD (van Leeuwen & Evans 1998). In order to make the interpretation of these residuals unambiguous, the released values were all derived with the single-star model, no matter what model was used for that catalogue entry. It is then possible for anybody to fit any model to these IAD to seek further reduction of the residuals.

2.1 The orbital model

The fit of the IAD with an orbital model is achieved through a $\chi^2$ minimization:

 
                                 $\displaystyle \chi^2$ = $\displaystyle \left(\Delta v-\sum_k\frac{\partial v}{\partial p_k}\Delta p_k-\sum_i\frac{\partial v}{\partial o_i}o_i\right)^{{\rm t}}$  
    $\displaystyle \vec{V}^{-1}\left(\Delta v-\sum_k\frac{\partial v}{\partial p_k}\Delta p_k-\sum_i\frac{\partial v}{\partial o_i}o_i\right),$ (1)

where $\Delta p_k$ is the correction applied to the original (astrometric) parameter pk [where $(p_1, p_2, p_3, p_4, p_5) \equiv (\alpha, \delta, \varpi, \mu_{\alpha^*}, \mu_{\delta})$], oi are the orbital parameters and $\vec{V}$ is the covariance matrix of the data. $\Delta v_j, \partial v_j/\partial p_k,$ and $\vec{V}$ ( $j=1,\dots, n; k=1, \dots, 5$) and the Main Hipparcos solution are provided, n is the numberof IAD available for the considered star (see van Leeuwen & Evans 1998 for details). Equation (1) thus reduces to
 
                                 $\displaystyle \chi^2$ = $\displaystyle \left(\Delta v-\sum_k\frac{\partial v}{\partial p_k}\Delta p_k-y ... ...partial v}{\partial p_{1}}-x \frac{\partial v}{\partial p_{2}}\right)^{{\rm t}}$  
    $\displaystyle \vec{V}^{-1} \left(\Delta v-\sum_k\frac{\partial v}{\partial p_k}... ...-y \frac{\partial v}{\partial p_{1}}-x \frac{\partial v}{\partial p_{2}}\right)$ (2)

where (x, y) is the relative position of the photocenter with respect to the barycenter of the binary system given by

\begin{eqnarray*}x &=& AX + FY \\ y &=& BX + GY \end{eqnarray*}


with

\begin{eqnarray*}X &=& \cos E- e \\ Y &=& \sqrt{1-e^2} \sin E. \end{eqnarray*}


A, B, F, G are the Thiele-Innes constants (describing the photocenter orbit), e is the eccentricity and E the eccentric anomaly.

2.2 Outliers screening

Even in the original processing, not all the observations were used to derive the astrometric solution. Some of the observations were flagged as outliers and simply ignored if their residuals exceeded three times the nominal (a priori) error for those measurements. These outlying observations are identified by lower case "f'' or "n'' flags in the IAD file (instead of upper case "F'' or "N'' flags, corresponding to processing by the FAST or NDAC consortium, respectively). Since the model (and therefore the residuals) is going to be revised, so must be the outliers. Because the Thiele-Innes model is a linear one (see Eq. (2)), its solution is unique and it may therefore be used to screen out the outliers of the orbital model.

All observations are initially kept. The observation with the largest residual using the orbital model is removed and the model fitted again without it. If the original residual exceeds three times the standard deviation of the new residuals, the observation is definitively discarded (since the number of observations is always less than 300, random fluctuations should yield less than 1 observation with a residual larger than $3\sigma$). The process is then repeated with the new largest residual, and so on. Otherwise, the observation is restored and the whole process is terminated.

A total of 3486 observations (out of 84 766) are thus removed. 60% of these outliers turn out to come from the NDAC processing even though the two consortia essentially contribute for the same amount of data. The percentage of outliers is ten times larger than in the original Hipparcos processing.

3 The sample

Among the $\sim$118000 stars in the Hipparcos catalogue, some 17918 were flagged as double and multiple systems (DMSA) and 235 of them, the so-called DMSA/O, have an orbital solution. Our sample consists of the $S\!_{B^9}$ entries with an HIP number, excluding DMSA/C entries (i.e., resolved binaries not suited for IAD processing). The sample contains 1374 HIP+$S\!_{B^9}$ entries which cover an extensive period and eccentricity range (see Figs. 1 and 2).

  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig1.eps} \end{figure} Figure 1: Period-eccentricity diagram for the selected $S\!_{B^9}$ objects with an HIP entry.
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  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig2.eps} \end{figure} Figure 2: Distribution of the orbital periods for the selected $S\!_{B^9}$ objects with an HIP entry.
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Even though a grade characterizes the quality of the spectroscopic orbits listed in $S\!_{B^9}$, those grades were not considered a priori in the present processing, which uses the most recent orbit available. The quality of the spectroscopic orbit will be checked at the end of the process, in the discussion of Sect. 4 relative to the detection efficiency of the astrometric wobble.

   
4 Astrometric wobble detection

  
4.1 Detection assessment

We check whether an orbital motion lies hidden in the IAD using two mathematically equivalent methods of orbit determination, the Thiele-Innes and Campbell approaches. In both cases, the eccentricity, orbital period and the time of passage at periastron are taken from the spectroscopic orbit. For multiple systems, we always use the shortest period. This choice may not necessarily be the best one, but its validity is anyway assessed a posteriori by the "periodogram'' test (see below).

In the Thiele-Innes approach, the remaining four orbital parameters are derived through the Thiele-Innes constants  A, B, F, G obtained from the $\chi^2$ minimization of the linear model expressed by Eq. (2). The semi-major axis of the photocentric orbit (a0), the inclination (i), the latitude of the ascending node ($\Omega$) and the argument of the periastron ($\omega$) (also known as Campbell's elements) are then extracted from the Thiele-Innes constants, using standard formulae (Binnendijk 1960). In the Campbell approach, on the other hand, two more parameters, $\omega$ and the semi-amplitude of the primary's radial-velocity curve K1are adopted from the spectroscopic orbit. Here, only two parameters of the photocentric orbit (i and $\Omega$) are thus derived from the astrometry. This model is non-linear. The Campbell approach implicitly assumes that there is no light coming from the companion, since the spectroscopic elements constrain a1 according to

 \begin{displaymath} a_1 \sin i = \varpi \; \frac{K_1\; P\; \sqrt{1-e^2}} {2 \pi}\cdot \end{displaymath} (3)

The IAD, on the other hand, give access to the photocentric orbit characterized by a0, and we assume that a1 = a0. If this assumption does not hold, the solutions derived from the Thiele-Innes and Campbell approaches will be inconsistent, and will be rejected a posteriori by the consistency check described in Sect. 5.

We quantify the likelihood that there is an orbital wobble in the data with a F-test evaluating the significance of the decrease of the $\chi^2$ resulting from the addition of four supplementary parameters (the four Thiele-Innes constants) in the orbital model (Pourbaix & Arenou 2001):

 \begin{displaymath} {\it Pr}_2 = {\it Pr}(F(4,n-9) > \hat{F}), \end{displaymath} (4)

where $ \hat{F} = \frac{n-9}{4} \frac{\chi^2_{\rm S} - \chi^2_{\rm T}}{\chi^2_{\rm T}}$ follows a F-distribution with (4, n-9) degrees of freedom, n is the number of available IAD for the considered star, $\chi^2_{\rm T}$ and $\chi^2_{\rm S}$ are the $\chi^2$ values associated with the orbital and single-star models, respectively. Pr2 is the probability that the random variable F(4, n-9) exceeds the given value $\hat{F}$, it is thus the first-kind risk associated with the rejection of the null hypothesis "there is no orbital wobble present in the data''. The Pr2 test is a $\chi^2$-ratio test; it is therefore insensitive to scaling errors on the assumed uncertainties.

An alternative - albeit non-equivalent - way to test the presence of an orbital wobble in the data is to test whether the four Thiele-Innes constants are significantly different from 0. The first kind risk associated with the rejection of the null hypothesis "the orbital semi-major axis is equal to zero'' may be expressed as

\begin{displaymath}Pr_3 = Pr(\chi^2_{ABFG} < \chi^2_4), \end{displaymath} (5)

where $ \chi^2_{ABFG} = \vec{X}^{{\rm t}} \vec{C}^{-1} \vec{X}$, $\vec{X}$ is the vector of components A,B,F,G and $\vec{C}$ is its covariance matrix. Pr3 is thus the probability that $\chi^2_4$, the $\chi^2$ random variable with 4 degrees of freedom, exceeds the given value  $\chi^2_{ABFG}$. The Pr3 test, being based on the $\chi^2_{ABFG}$ statistics, is an absolute test, and it is therefore sensitive to possible scaling errors on the assumed uncertainties.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig3.eps} \end{figure} Figure 3: Comparison of the Pr2 and Pr3 statistics for the whole sample of 1374 stars, showing that Pr2 and Pr3 are not equivalent. Crosses correspond to systems with $F2_{\rm TI} > 2.37$, where  $F2_{\rm TI}$ is the goodness-of-fit for the Thiele-Innes model (Eq. (6)); open squares correspond to systems with $F2_{\rm TI} < -1.95$.
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Because a0 vanishes when there is no wobble present in the data (and conversely), it may seem that the Pr2 and Pr3tests are equivalent (notwithstanding the fact that the former test is relative, whereas the latter is absolute). As revealed by Fig. 3, this is not necessarily so, though, for the reasons we now explain. Since the model is linear, the equality $\chi^2_{\rm T}=\chi^2_{\rm S}-\chi^2_{ABFG}$ holds. Therefore, $\hat{F} = \frac{n-9}{4} \frac{\chi^2_{\rm S} - \chi^2_{\rm T}}{\chi^2_{\rm T}} = \frac{n-9}{4} \frac{\chi^2_{ABFG}}{\chi^2_{\rm T}}$, so that Pr2 and Pr3 are basically equivalent as long as  ${\chi^2_{\rm T}} \sim n-9$, i.e., when the Thiele-Innes model fits the data adequately. This latter fit may be quantified by the goodness-of-fit statistics  $F2_{\rm TI}$ (Kovalevsky & Seidelmann 2004; Stuart & Ord 1994), defined as:

 \begin{displaymath} F2_{\rm TI} = \left(\frac{9 \nu}{2}\right)^{1/2} \left[\left... ...chi^2_{\rm T}}{\nu}\right)^{1/3} + \frac{2}{9\nu} - 1 \right], \end{displaymath} (6)

where $\nu = n - 9$ is the number of degrees of freedom. If the Thiele-Innes model holds, we expect $F2_{\rm TI}$ to be approximately normally distributed with zero mean and unity standard deviation[*]. Bad fits correspond to large F2 values, abnormally good fits to large negative values. Solutions with F2 > 2.37 should be discarded at the 5% threshold.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig4.eps} \end{figure} Figure 4: F2 (goodness of fit) versus Pr3 for systems complying with Pr2 < 5%. The envelope of these points is well reproduced with the theoretical curve (solid line) assuming Pr2 = 5% and 59 observations (which corresponds to the average number of observations for the considered systems). The dashed line corresponds to Pr2 = 1%.
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Figure 4 compares F2 with Pr3 and reveals that the two tests are not simple substitutes of one another: there are systems which fail at the Pr3 test but comply with the F2 test and conversely. The situation becomes clearer when one realizes that the upper envelope corresponds to the condition Pr2 < 0.05, which may be translated into a lower bound on  $~\chi^2_{ABFG}/\chi^2_{\rm T}$: solutions retained by the Pr2 test have large  $\chi^2_{ABFG}/\chi^2_{\rm T}$ ratios. There are two ways to fulfill such a condition: If $~\chi^2_{\rm T}$ is small (i.e., F2 is small, or abnormally good fits), then even small $~\chi^2_{ABFG}$ values (i.e., large Pr3) comply with the Pr2 test. This explains why the Pr2 test does not eliminate systems with large Pr3 when their Thiele-Innes fit is abnormally good. Conversely, if  $\chi^2_{ABFG}$ is large (i.e., Pr3 is small), then even large  $~\chi^2_{\rm T}$ values (i.e., large F2 or bad Thiele-Innes fits) comply with the Pr2 test. This explains why at small Pr3 values, even bad Thiele-Innes fits (large F2 values) are retained. This would typically be the case of a DMSA/X system where the Thiele-Innes model brings a substantial improvement with respect to the single-star model (i.e., $\chi^2_{ABFG}=\chi^2_{\rm S}-\chi^2_{\rm T}$ is large, or Pr3 is small), but the overall quality of the Thiele-Innes fit remains poor (large F2).

Table 1: The 282 stars flagged as astrometric binaries ( Pr1, Pr2, Pr3 < 0.05 and $F2_{{\rm TI}}< 2.37$; see text). Italicized entries identify the 122 stars passing the Pr1, Pr2 and Pr3 tests at the more stringent 0.006% level, and $F2_{{\rm TI}}< 2.37$.

In the Campbell approach, the situation is somewhat more complicated since the model expressed by Eq. (1) does not depend linearly upon the model parameters i and $\Omega$. Therefore, the quantity  $\chi^2_{\rm C}$ extracted from the minimization of Eq. (1) does not follow a $\chi^2$ distribution with n-2 degrees of freedom (Lupton 1993). Since the non-linear model may be linearized at the expenses of adding more parameters (e.g., the coefficients of a Fourier or Taylor expansion), n-2 overestimates the number of degrees of freedom (Pourbaix 2005). Overestimating the number of degrees of freedom affects all the statistical tests using the $\chi^2_{\rm C}$ value. In particular, the first kind risk Pr1extracted from an equation similar to Eq. (4) (substituting  $\chi^2_{\rm T}$ by  $\chi^2_{\rm C}$) is underestimated (Pourbaix 2005). Since this threshold is used to reject solutions which have Pr1 larger than the adopted threshold, it may nevertheless be used, keeping in mind that not enough solutions are in fact discarded by the Pr1 test. It is very likely, though, that these unacceptable solutions will be screened out by the other tests.

The combination of these four statistical indicators allows us to flag 282 stars as astrometric binaries at the 5% level (i.e., Pr1, Pr2, Pr3 < 0.05 and $F2_{{\rm TI}}< 2.37$) among the 1374 HIP+$S\!_{B^9}$ sample stars defined in Sect. 3.

  
4.2 Detection rate

The 282 astrometric binary stars passing the four tests described in Sect. 4.1 at the 5% level are listed in Table 1. Italicized entries correspond to the 122 stars passing the Pr1, Pr2 and Pr3 tests at the more stringent 0.006% level and F2 < 2.37. These stars thus represent prime targets for future astrometric observations or, if both components are visible, interferometric observations (see also Table 1A of Taylor et al. 2003), as they are astrometric binaries, but with orbital elements not always reliably determined (see Sect. 5).

Table 2: Detection rate (expressed in %) as a function of orbital period and parallax. The percentage is given along with its binomial error; the total number of stars in the bin is listed between parentheses. For $\varpi > 5$ mas and $100 \le P(\rm d) \le 3000$, the detection rate comes close to 100% when removing SB2 systems, systems with composite spectra or with a poor-quality spectroscopic orbit.

We present the detection rate as a function of the parallax $\varpi$ and the orbital period P in Table 2 and Figs. 5 and 6. A striking property of the astrometric-binary detection rate displayed in Fig. 5 is its increase around P = 50 d, due to the Hipparcos scanning law which does not favour the detection of shorter-period binaries. Similarly, the detection rate drops markedly for periods larger than 2000 d, corresponding to twice the duration of the Hipparcos mission. Worth noting are therefore the 5 astrometric orbits detected with periods larger than 5000 d: HIP 116727 ( $P = 24\thinspace135$ d), HIP 5336 (P = 8393 d), HIP 7719 (P = 7581 d), HIP 11380 (P = 6194 d) and HIP 33420 (P = 6007 d). The reason why the astrometric motion of HIP 116727 could be detected despite such long an orbital period, is that Hipparcos caught it close to periastron (e = 0.39), when the orbital motion is the fastest. Table 2 further reveals that the detection rate exhibits little sensitivity to the parallax (provided it is larger than 5 mas; otherwise, the IAD are not precise enough to extract the orbital motion), but rather that it is the orbital period which plays the most significant role. The detection rate in the most favourable cases lies in the range 50 to 80%. It must be stressed, however, that all the undetected astrometric binaries in those bins are either SB2 systems, systems with a composite spectrum or with a spectroscopic orbit of poor quality (the SB2 and composite-spectrum systems have components of similar brightness, so that in most cases, the photocenter of the system does not differ much from its barycenter, making the orbital motion difficult to detect; see Eq. (8) below). If we remove these entries from the sample, the detection rate is close to 100%. The orbital parameters of the detected binaries are further analyzed in Sect. 5. Such an analysis is made necessary when one realizes that the orbital inclinations derived by the Thiele-Innes and Campbell approaches do not always yield consistent values (Fig. 7), contrary to expectations. Section 5 therefore presents further criteria used to evaluate the reliability of the derived astrometric orbital elements (and, in particular, the consistency between the two sets of orbital parameters, Thiele-Innes versus Campbell).


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig5.eps} \end{figure} Figure 5: Percentage of astrometric binaries detected among $S\!_{B^9}$ stars as a function of orbital period. The error bars give the binomial error on each bin.
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4.3 The DMSA/O entries

Among the 1374 binaries from $S\!_{B^9}$, 122 are flagged as DMSA/O in the Hipparcos catalogue. We detect 89 of these (or 75%) (irrespective of $\varpi$). The detection rate climbs to 81.7% (85/104) for orbital periods longer than 100 d. It is worth examining why not all DMSA/O solutions were retrieved by our processing. A close look at the rejected systems reveals that there is nothing wrong with our analysis, since all but one among the 33 DMSA/O systems not recovered by our reprocessing belong to one of the following categories:

HIP 85749 is the only DMSA/O solution not belonging to any of the above categories. HIP 85749 has not been flagged as an astrometric binary by our reprocessing, because Pr1 = 0.26, although Pr2, Pr3 and F2 do qualify the star as an astrometric binary.


  \begin{figure} \par\mbox{\resizebox{5.5cm}{!}{\includegraphics{3003fi6a.eps}}\hf... ...eps}}\hfill \resizebox{5.5cm}{!}{\includegraphics{3003fi6c.eps}} }\end{figure} Figure 6: Left panel: period-parallax diagram for the selected $S\!_{B^9}$ objects with an HIP entry. Middle panel: period-parallax diagram for non-detected objects. Right panel: stars flagged as astrometric binaries by the Pr1, Pr2 and Pr3 tests at the 5% level and with $F2_{{\rm TI}}< 2.37$.
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5 Orbit assessment

 

The upper panel of Fig. 7 reveals that, even though orbital solutions pass the Pr1, Pr2and Pr3 tests, meaning that an astrometric orbital motion has been detected, these solutions do not necessarily yield Thiele-Innes and Campbell orbital elements that are consistent with each other. The inverse-S shape observed in Fig. 7 results from the following properties: (i) in the absence of an orbital signal in the IAD and when the spectroscopic radial-velocity semi-amplitude K1 is small, the Campbell solution tends to have $i_{\rm C} \sim 0^\circ$ or 180$^\circ$, while the Thiele-Innes solution tends to $i_{\rm TI} \sim 90^\circ$(Pourbaix 2004); (ii) the physical solutions fall on the diagonal, although this diagonal is polluted with unphysical solutions having $i_{\rm TI} \sim 90^\circ$. The lower panel of Fig. 7 displays the 122 stars complying with the Pr1, Pr2 and Pr3 tests at the 0.006% level. It clearly shows that the consistency between the Thiele-Innes and Campbell solutions may be improved considerably by decreasing the probability threshold to 0.006%.


  \begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics{3003fi7a.eps}}\par\vspace*{2mm} \resizebox{8.8cm}{!}{\includegraphics{3003fi7b.eps}} \end{figure} Figure 7: Comparison of the orbital inclinations derived by the Thiele-Innes and Campbell approaches. The 282 stars displayed in the upper panel all comply with the 4 criteria for astrometric wobbledetection (namely Pr1, Pr2 and Pr3 < 0.05 and $F2_{{\rm TI}}< 2.37$; see text), but their astrometric orbital elements are not always reliably determined as not all points fall along the diagonal. The lower panel displays the 122 stars complying with the Pr1, Pr2 and Pr3 tests at the 0.006% level.
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To remove the remaining inconsistent solutions, it is necessary to assess the reliability of the derived orbital elements. This may be done in at least two ways: The empirical approach has been preferred here, with the two tests involved now described in turn.

First, the consistency between the astrometric period and the adopted spectroscopic period is checked through a periodogram-like test. For 600 periods uniformly distributed in $\log P$ between 0.1 and 1200 d, the best 9-parameter (Thiele-Innes) fit is computed (the eccentricity and periastron time are kept unchanged). The resulting $\chi^2$ is plotted against the period, thus generating a Scargle-like periodogram (Scargle 1982). Its standard deviation $\sigma$ is computed. An orbital motion with the expected (spectroscopic) period is then supposed to be present in the IAD if the $\chi^2$ at that period is smaller than the periodogram mean value by more than  $\xi \sigma$, with $\xi $ chosen of the order of 3.

Second, the correlation existing between the Thiele-Innes orbital elements may be estimated through the efficiency parameter $\epsilon $ (Eichhorn 1989), expressed by

\begin{displaymath}\epsilon = \sqrt[p]{\frac{\Pi_{k=1}^p \lambda_k}{\Pi_{k=1}^p \vec{V}_{kk}}}, \end{displaymath} (7)

where $\lambda_k$ and $\vec{V}_{kk}$ are respectively the eigenvalues and the diagonal terms of the covariance matrix $\vec{V}$ and p denotes the number of parameters in the model. For an orbital solution to be reliable, its covariance matrix should be dominated by the diagonal terms, and the efficiency $\epsilon $ should then be close to 1 (Eichhorn 1989).

The 70 orbital solutions retained when adopting $\xi = 3$ and $\epsilon > 0.4$ are listed in Table 3, 20 of them being new orbital solutions not already listed in the DMSA/O annex. Figure 8 presents the distribution of their orbital periods. In Fig. 9 comparing the inclinations derived from the Thiele-Innes and Campbell solutions, the retained orbits now fall close to the diagonal, as expected.

Neither the parallax nor the proper motions differ significantly from the Hipparcos value for the stars of Table 3. They have therefore not been listed.

To increase the science content of this paper, Table 4 lists the astrometric orbital elements for a second category of systems: 31 newly derived orbits (i.e., not already present in the DMSA/O annex), not already listed in Table 3, from the list of 122 stars passing the Pr1, Pr2 and Pr3 tests at the 0.006% level (they are among the italicized stars in Table 1). These orbits are (possibly) of a slightly lower accuracy than the ones listed in Table 3 because they do not comply with the two empirical tests described in this section. Nevertheless, these newly derived orbits are worth publishing.

As already discussed in Sect. 4.3, there are 122 systems in our sample of 1374 which have a DMSA/O entry. Of these 122, 89 pass the Pr1, Pr2, Pr3 and F2 tests at the 5% level (Sect. 4.3) and 71 pass the Pr1, Pr2, Pr3 and F2 tests at the 0.006% level but only 50 have reliable orbital elements according to the 2 empirical tests described in this section. The 39 rejected DMSA/O systems are listed in Table 5, along with the failed test(s). Figure 10 compares the Thiele-Innes and Campbell inclinations for those systems with orbital elements not validated by the consistency tests.

The orbits derived in the present analysis and the DMSA/O ones generally agree well. For HIP 677 (=$\alpha$ And), a visual and SB2 system, there are astrometric orbits based on ground-based interferometric measurements already available (Pourbaix 2000; Pan et al. 1992). The inclination of $103^\circ\pm10^\circ$ found here is consistent with the value $105.7^\circ\pm0.2^\circ$ obtained by Pan et al. (1992). The only new constraint of interest provided by the IAD-derived photocentric orbit lies in a consistency check between that photocentric semi-major axis $a_0 = 7.3\pm0.4$ mas (Table 3) and the relative semi-major axis $a = 24.1\pm0.1$ mas (Pourbaix 2000; Pan et al. 1992), with the following relation to be satisfied (Binnendijk 1960):

 \begin{displaymath} a_0 = a (\kappa - \beta), \end{displaymath} (8)

where $\kappa = M_2 / (M_1 + M_2) = 0.331$and $\beta = (1 + 10^{0.4 \Delta m})^{-1}$, and $\Delta m$is the magnitude difference between the two components. Equation (8) then implies $\beta = 0.027$ or $\Delta m = 3.9$ mag, which is much larger than the value of 2.0 mag measured by Pan et al. (1992) or 2.19 mag derived by Ryabchikova et al. (1999). With $\Delta m = 2$ mag, $\beta = 0.137$, so that a0/a = 0.19 or a0 = 4.7 mas, which is inconsistent with the value of $7.3\pm0.4$ mas listed in Table 3 or $a_0 = 6.47\pm1.16$ mas from the DMSA/O. The origin of this discrepancy is unknown.

Table 3: The 70 orbital solutions (Campbell solutions) passing all consistency tests. The column labelled "Ref.'' provides the reference for the spectroscopic orbit used. In the case where a system is listed in the DMSA/O annex, the column labelled "DMSA'' compares the orbital semi-major axes and the inclinations from the DMSA/O annex and from this work.

In Table 5, cases where the efficiency test is the only one to fail generally correspond to rather wide orbits which cannot be accurately determined with Hipparcos data only (e.g., HIP 5336, 68682, 75695, 110130). When only the periodogram test fails, it means that either the spectroscopic period does not correspond to the astrometric motion, or that the IAD do not constrain its period well enough.

6 Some astrophysical implications

   
6.1 Masses

Masses of the components of spectroscopic binaries with one visible spectrum (SB1) are encapsulated in the mass function

\begin{displaymath}f(M_1,M_2) \equiv \frac{M_2^3 \sin^3 i}{(M_1 + M_2)^2} \equiv Q \sin^3 i, \end{displaymath} (9)

where M1 and M2 are the masses of the primary and secondary components, respectively. The knowledge of the inclination as given in Table 3 gives directly access to the generalized mass ratio Q listed in Table 6. To go one step further and have access to the masses themselves, supplementary information must be injected in the process. For main-sequence stars, this may come from the mass - luminosity relationship. The mass of the main-sequence primary component is estimated directly from its Hipparcos B-V color index, converted into an absolute magnitude MV using Table 15.7 of Cox (2000), and then into masses using Table 19.18 of Cox (2000). The corresponding masses are listed in Table 6. The major uncertainty on M2 comes from the uncertainty on  M1rather than from i. To fix the ideas, an uncertainty of 0.1 mag on B-V translates into an uncertainty of 0.2 (or 0.1, 0.05) $M_{\odot}$ on M1, and of 0.045 (0.032, 0.027) $M_{\odot}$ on M2 for  $0 \le M_V < 4$ (or  $4 \le M_V < 6$, $6 \le M_V$, respectively). The position of stars from Table 3 on the main sequence has been checked from the Hertzsprung-Russell diagram drawn from the Hipparcos data. In particular, it has been checked that the B-V color is not the composite of the two components (in which case, the above procedure to derive M1 may not be applied). Only HIP 47461 (=HD 83270) belongs to that category (as confirmed by Ginestet et al. 1991), so that neither masses are given in Table 6.

Individual systems of interest are discussed in Sect. 6.1.1.

The distributions of M1, M2 and q = M2/M1 for the 29 systems with main sequence primaries are displayed in Fig. 11. The q distribution appears to be strongly peaked around q = 0.6, but this feature very likely results from the combination of two opposite selection biases. Our sample is biased against systems with $q \sim 1$ (since these systems would generally be SB2 systems with components of almost equal brightness, whose astrometric motion is difficult to detect; see the discussion of Sect. 4.2) and against systems with low-mass companions (which induce radial-velocity variations of small amplitude, difficult to detect, and thus not present in $S\!_{B^9}$).

The M1 and M2 distributions also clearly reflect the bias against q = 1 since the distributions exhibit adjacent peaks. Although one would be tempted to attribute the M2 = 0.6 $M_{\odot}$ peak to a population of white dwarf (WD) companions, it is more likely to result from the two selection biases described above.

In the absence of a mass - luminosity relationship for giants, the mass of the companion cannot be derived reliably.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig8.eps} \end{figure} Figure 8: Distribution of the orbital periods for the 70 solutions retained.
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6.1.1 Masses for some specific systems

HIP 677 = $\alpha$ And



As already discussed in Sect. 5, HIP 677 is known to be a SB2 and visual binary (Pourbaix 2000; Ryabchikova et al. 1999). Masses are thus already available in the literature, namely  $M_1 = 3.6\pm0.2$ $M_{\odot}$, $M_2 = 1.78\pm0.08$ $M_{\odot}$ (Ryabchikova et al. 1999) or $M_1 = 3.85\pm0.22$ $M_{\odot}$, $M_2 = 1.63\pm0.074$ $M_{\odot}$ (Pourbaix 2000).



HIP 20935 = HD 28394



This F7V star is a member of the Hyades cluster. It has a mass ratio q = M2/M1 of 0.98. However, it falls exactly along the main sequence as defined by the other stars of our sample. There is thus no indication that this star has composite colors, as it should if the companion is a main sequence F star as well. A white dwarf (WD) companion of mass 1.1 $M_{\odot}$ is not without problems either. Böhm-Vitense (1995) has searched the IUE International Ultraviolet Explorer archives for spectra of F stars from the Hyades, in order to look for possible WD companions. No excess UV flux is present at 142.5 nm for HIP 20935, which implies that the WD must be cooler than about 10 000 K. For a 1.1 $M_{\odot}$ WD, this implies a cooling time of more than 1 Gyr (Chabrier et al. 2000), incompatible with the Hyades age of $800\times 10^6$ y. The remaining possibility is that the companion is itself a binary with two low-luminosity red dwarfs.



HIP 105969 = HD 204613



This star is known as a subgiant CH star (Luck & Bond 1982) and the system should therefore host a WD companion (McClure 1997). Interestingly enough, the mass inferred for this WD companion is 0.49 $M_{\odot}$, just large enough for a 2 $M_{\odot}$ AGB progenitor to have gone through the thermally-pulsing asymptotic granch branch phase (see Fig. 3.10 in Groenewegen 2003) to synthesize heavy elements by the s-process of nucleosynthesis. Those heavy elements were subsequently dumped onto the companion (the current CH subgiant) through mass transfer.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fig9.eps} \end{figure} Figure 9: Comparison of the inclinations derived from the Thiele-Innes constants and from the Campbell elements for the 70 systems retained. Compare with Fig. 7.
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6.2 $e - \log~P$ diagram

With the availability of extensive sets of orbital elements for binaries of various kinds (e.g., Duquennoy & Mayor 1991 for G dwarfs, Matthieu 1992 for pre-main sequence binaries, Mermilliod 1996 for open-cluster giants, Carney et al. 2001 for blue-straggler, low-metallicity stars, Latham et al. 2002 for halo stars), it has become evident that long-period (P > 100 d), low-eccentricity (e < 0.1) systems are never found among unevolved (i.e., pre-mass-transfer) systems. This indicates that binary systems always form in eccentric orbits, and the shortest-period systems are subsequently circularized by tidal effects. On the contrary, binary systems which can be ascribed post-mass-transfer status because they exhibit signatures of chemical pollution due to mass transfer (like barium stars, some subgiant CH stars, S stars without technetium lines...) are often found in the avoidance region (P > 100 d, e < 0.1) of the ( $e, \log P)$ diagram. Mass transfer indeed severely modifies their orbital elements, which often end up in this region (Jorissen 2003; Jorissen & Van Eck 2005).

The companion masses derived in Sect. 6.1 offer the opportunity to check whether systems falling in the avoidance region of the ( $e, \log P)$ diagram could be post-mass transfer systems (most probably then with a WD companion). In total, 8 systems fall in this region, as displayed in Fig. 12: HIP 6867 (=HD 9053 = $\gamma$ Phe; M0 III), HIP 8922 (=HD 11613 = HR 551; K2), HIP 10514 (=HD 13738; K3.5 III), HIP 24419 (=HD 34101; G8 V), HIP 32768 (=HD 50310 = HR 2553; K1 III), HIP 99965 (=HD 193216; G5 V), HIP 101093 (=HD 195725; A7 III) and HIP 101847 (=HD 196574; G8 III).

Table 4: The 31 new orbital solutions (Campbell solutions) passing the Pr1, Pr2 and Pr3 tests at the 0.006% level, but failing at least one of the consistency tests. The column labelled "Ref.'' provides the reference for the spectroscopic orbit used. The columns labelled $\xi $ and $\epsilon $ provide the values of the corresponding empirical tests. The column labelled "D'' refers to DMSA.

Table 5: The 39 systems with a DMSA/O entry which do not fulfill the 2 tests assessing the reliability of the astrometric orbital elements, namely $\xi < 3$ and $\epsilon > 0.4$ and the probability tests at the 0.006% level (see text). Columns with "n'' correspond to failed tests.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fi10.eps} \end{figure} Figure 10: Comparison of the inclinations derived from the Thiele-Innes constants and from the Campbell elements for the 1304 systems not retained. The 39 rejected systems with a solution in the DMSA/O annex (Table 5) are represented by a filled square.
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  \begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics{3003f11a.eps}}\par\vspace*{2mm} \resizebox{8.8cm}{!}{\includegraphics{3003f11b.eps}} \end{figure} Figure 11: Upper panel: distribution of the mass ratio (M2/M1) for systems from Table 3 with a main-sequence primary star. Lower panel: distributions of M1 (dashed line) and M2 (solid line).
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None of these "avoidance-region'' systems offer conclusive evidence for hosting a WD companion, but at least do not contradict it either.

HIP 6867 has a circular orbit and a rather short orbital period (193.8 d) given its late spectral type. The orbit is therefore likely to have been circularized by tidal effects rather than by mass transfer (Jorissen et al. 2004). In this specific case, there is therefore no need for the companion to be a WD.

HIP 24419 has too small a companion mass (0.21 $M_{\odot}$) to host even a He WD. This system could nevertheless have gone through a so-called "case B'' mass transfer (occurring when the primary was on the first giant branch).

For HIP 8922, HIP 10514, HIP 32768, HIP 99965, HIP 101093 and HIP 101847, we could not find in the literature any information that could help us in assessing the nature of their companion. In the case of HIP 99965 though, the companion's mass of 0.56 $M_{\odot}$ would certainly not dismiss it of being a WD.

One should mention as well that HIP 10514 and HIP 101847 are listed in the Perkins catalog of revised MK types for the cooler stars (Keenan & McNeil 1989) without any mention whatsoever of spectral peculiarities. They are therefore definitely not barium stars, despite falling in the "avoidance region'' of the eccentricity - period diagram generally populated by barium stars. If we are to maintain that the "avoidance region'' can only be populated by post-mass-transfer objects - thus implying that the companion to HIP 10514 and all the stars discussed in the present section must be WDs - then we must accept at the same time that systems following the same binary evolution channel as that of barium stars do not necessarily end up as barium stars! Or in other words, binarity would not be a sufficient condition for the barium syndrome to develop (these systems would thus add to the non-barium binary systems listed in Jorissen & Boffin 1992).

7 Conclusions

Table 6: Masses and mass ratios for the 29 systems with main-sequence primaries passing all consistency tests.


  \begin{figure} \par\includegraphics[width=8.8cm,clip]{3003fi12.eps} \end{figure} Figure 12: The ( $e, \log P)$ diagram for the 70 systems with reliable astrometric orbital elements. Systems with giant primaries are represented by black squares, and main-sequence primaries with crosses. The point labels refer to the companion mass.
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The major result of this paper is that the detectability of an astrometric binary using the IAD is mainly a function of the orbital period (at least when the parallax exceeds 5 mas, i.e., about 5 times the standard error on the parallax): detection rates are close to 100% in the period range 50-1000 d (corresponding to the mission duration) for systems not involving components with almost equal brightnesses (i.e., SB2 systems or systems with composite spectra). These are more difficult to detect, because the photocenter motion is then much smaller than the actual component's motion.

A consistency test between Thiele-Innes and Campbell solutions has been designed that allowed us to (i) identify wrong spectroscopic solutions, and (ii) retain 70 systems with accurate orbital inclinations (among those, 29 involve main sequence primaries and 41 giant primaries). Among those 70 retained solutions, 20 are new astrometric binaries, not listed in the DMSA/O.

This number of 70 systems passing all quality checks seems small with respect to the 122 DMSA/O systems with an $S\!_{B^9}$ entry. A detailed check reveals, however, that many systems present in the DMSA/O either have inaccurate astrometric orbits that would not fulfill our statistical tests, or have inaccurate spectroscopic orbital elements that make the astrometric solution unreliable anyway, or have only a0 derived from the IAD, all other elements being taken from spectroscopic and interferometric/visual orbital elements.

Masses M2 for the companions in the 29 systems hosting a main-sequence primary star have been derived, using the mass-luminosity relation to estimate M1. This was not possible for systems hosting giant primaries.

The possibility that the region e < 0.1, P > 100 d of the ( $e, \log P)$ diagram is exclusively populated by post-mass transfer systems has been examined, but could not be firmly demonstrated.

Acknowledgements
A.J. and D.P. are Research Associates, FNRS (Belgium). This research was supported in part by the ESA/PRODEX Research Grants 90078 and 15152/01/NL/SFe. We thank M. Hallin and A. Albert for discussions. We would like to thank the referee of this paper, Prof. L. Lindegren, for his very valuable comments and suggestions.

References

 
Copyright ESO 2005