2 Numerical methods

In order to allow some re-processing 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 by-product 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:

• definition of an objective function to minimize;
• computation of the objective function from the astrometric parameters;
• global minimization of the objective function with the simulated annealing algorithm;
• local minimization of the objective function (with a quasi-Newton method);
• comparison of the values of the objective function for the Hipparcos 5-parameter model and for our 9-parameter model, and application of an F-test to evaluate whether the 9-parameter model is significantly better than the Hipparcos solution;
• check of the astrophysical consistency of the 9-parameter solution from some a priori knowledge of the properties of the system (i.e., component masses).

These steps are discussed in turn in the remaining of this section, first for IAD, and then for TD.

   
2.1 Intermediate Astrometric Data

   
2.1.1 Objective function

The first step in any data-fitting problem is to set up an objective function in order to compare different solutions (corresponding to different values of the model parameters). This function is usually constructed in such a way that its lowest value corresponds to the best solution. The problem then reduces to minimizing that function. The IAD already include some corrections (e.g. aberration and satellite-attitude corrections) and they are no longer stricto sensu raw data. They provide the abscissa residuals along a reference great circle for the Hipparcos 5-parameter solution (i.e. $\alpha_0, \delta_0, \varpi, \mu_{\alpha}, \mu_{\delta}$, respectively: the position in right ascension and declination in the equinox 2000.0 at the epoch 1991.25, the parallax, the proper motions in right ascension and declination; following the practice of the Hipparcos catalogue, $\alpha_0^* = \alpha_0 \cos\delta$ and $\mu_{\alpha}^*= \mu_{\alpha} \cos\delta$ will be used instead of $\alpha_0$ and $\mu_{\alpha}$). By slightly modifying these 5 parameters and by adding 7 orbital parameters (Binnendijk 1960), the aim is to further reduce the abscissa residuals $\Delta v$ below the values obtained from the Hipparcos 5-parameter model. In the case of the Hipparcos Intermediate Astrometric Data, the objective function is the $\chi ^2$ expressed by Eq. (17.12) of Vol. 3 in Hipparcos:

$$\chi^2=\left(\Delta v-\sum_{k=1}^M\frac{\partial v}{\partial ... ..._{k=1}^N\frac{\partial v}{\partial p_k}\Delta p_k\right)\!\!,$$ (1)

where the superscript t means transposed, $\Delta v$ are the abscissa residuals provided by the IAD file and corresponding to the Hipparcos 5-parameter solution, and $\frac{\partial v}{\partial p_k}$ is the partial derivative of the abscissa with respect to the k-th parameter expressing how the abscissa residual varies when a correction $\Delta p_k$ is applied to the value of the k-th parameter with respect to the Hipparcos solution. M is the number of parameters retained in the solution, and V^{- 1} is the inverse of the covariance matrix of the observations given by:

$$V=\left( \begin{array}{cccc} V_1 & 0 & \cdots & 0\\ 0 & V_2 ... ...ddots & \ddots & 0 \\ 0 & \cdots & 0 & V_n \end{array}\right)$$

with

$$V_j=\left( \begin{array}{cc} \sigma_{F_j}^2 & \rho\sigma_{F_... ...o\sigma_{F_j}\sigma_{N_j} & \sigma_{N_j}^2 \end{array}\right).$$ (2)

If an observation j was processed by the two consortia (FAST and NDAC), the residuals obtained by the two reduction consortia are correlated and V_j is the $2\times 2$ variance-covariance matrix for measurement j. On the other hand, when only one consortium processed observation j, V_j reduces to one number (the estimated uncertainty $\sigma_j$ of the measurement). $\Delta v$, together with $\frac{\partial v}{\partial p_k}$ (k = 1 to 5, with $p_1 \equiv \alpha_0^*, p_2 \equiv \delta, p_3 \equiv \varpi, p_4 \equiv \mu_{\alpha^*}, p_5 \equiv \mu_\delta$), the original astrometric parameters as well as $\rho, \sigma_{F_j}$ and $\sigma_{N_j}$ are given in the IAD file. In order to evaluate $\chi ^2$ (Eq. 1) for an orbital model, expressions for the partial derivatives of v with respect to the orbital parameters are required. They can be expressed as a function of the partial derivatives of v with respect to $\alpha_0^*$ and $\delta_0$ as follows (Hipparcos catalogue, Vol. 3, Eq. (17.15)):

$$\frac{\partial v}{\partial o}=\frac{\partial v}{\partial \al... ...partial v}{\partial \delta_0}\frac{\partial \eta}{\partial o}$$

where o is any orbital parameter and

$$\begin{array}{l} \xi=\alpha_0^*+\mu_{\alpha^*}(t-t_0)+R P_{\a... ...=\delta_0+\mu_{\delta}(t-t_0)+R P_{\delta}\varpi+x. \end{array}$$ (3)

In the above expression, $\xi$ and $\eta$ represent the Cartesian coordinates of the observed component on the plane tangent to the sky at the position ( $\alpha_0, \delta_0$). They combine the displacements due to the proper motion, the orbital motion and the parallax. R is the radius vector of the Earth's orbit in A.U. at time t. $P_{\alpha}$and $P_{\delta}$, the parallax factors, are given by Binnendijk (1960):

$$\begin{eqnarray*}P_{\alpha}&=&\cos\epsilon\cos\alpha\sin\hbox{$\odot$ }-\sin\alp... ...{$\odot$ }\\ &&{} -\cos \alpha \sin \delta \cos \hbox{$\odot$ } \end{eqnarray*}$$

where $\hbox{$\odot$ }$ and $\epsilon$ are respectively the longitude of the Sun and the obliquity of the ecliptic, both at time t. The variables x and y describe the apparent orbit (i.e., projected on the plane orthogonal to the line of sight) of the observable component around the center of mass of the system. They are usually expressed in terms of the Thiele-Innes constants A, B, F, G of the photocentric orbit as (Heintz 1978)

$$\begin{array}{l} x=AX + FY\\ y=BX + GY \end{array}$$ (4)

with

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

where E is the eccentric anomaly and (X,Y) are the coordinates in the true orbit. It should be noted that (x,y) and (X,Y) are referred to a Cartesian system with x pointing towards the North (Heintz 1978), contrary to $(\xi, \eta)$ where $\xi$ points towards increasing right ascensions. For all the systems considered in the present paper, the magnitude difference between the two components is larger than the Hipparcos detection threshold (since the companion to CPRS is a cool white dwarf; Sect. 5 and Jorissen et al. 1998). Thus, one can safely assume that there is no light coming from the secondary. Hence, the orbit of the photocenter of the primary component is the same as the absolute orbit of the primary around the center of mass of the system.

At this point, it is important to realize that the Hipparcos 5-parameter solution, that is used as a starting point for the new 12-parameter solution, is in fact equivalent to a 12-parameter solution where the semi-major axis of the orbit (a₀) is null. It is therefore enough to consider in Eq. (1) the correction term relative to the semi-major axis of the orbit (and in fact $\Delta a_0 = a_0$, since the initial value of a₀ is null). The other orbital parameters $i, \omega, \Omega, e, P$ and T enter Eq. (1) only through the partial derivatives $\frac{\partial \xi}{\partial a_0}$ and $\frac{\partial \eta}{\partial a_0}$ [equal, respectively, to $\frac{\partial y}{\partial a_0}$ and $\frac{\partial x}{\partial a_0}$ according to Eq. (3)] entering $\frac{\partial v}{\partial a_0}$. These other orbital parameters do not require explicit correction terms in Eq. (1) (and their starting values would be ill-defined anyway). Hence, the orbital solution is the one minimizing

$$\chi^2=\Xi^{t}V^{-1}\Xi$$ (5)

where

 

$$\Xi \Delta v-\sum_{k=1}^5\frac{\partial v}{\partial p_k}\Delta p_k - ... ...0}+\frac{\partial v}{\partial p_2}\frac{\partial \eta}{\partial a_0}\right) a_0$$ = (6)

is a vector of dimension N (N is the number of observations), and $\xi$ and $\eta$ are functions of $a_0, i, \omega, \Omega, e, P$ and T, and $\frac{\partial\xi}{\partial a_0} \equiv \frac{\partial y}{\partial a_0}, \frac{\partial\eta}{\partial a_0} \equiv \frac{\partial x}{\partial a_0}$. Thus, the 12 parameters enter the evaluation of $\chi ^2$ although there are only six correction terms subtracted from $\Delta v$.

Our experience has shown that, except for very special cases (i.e., parallaxes and semi-major 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 $\omega$ 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 $\omega$ free removes this difficulty, and offers moreover a way to check the consistency of the astrometric solution, since the astrometric $\omega$ 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 re-processing either. In a few cases, we noticed that because of the orbital contribution, some observations yield residuals larger than $3\sigma$ of the residuals. In these cases, these observations were removed and the fit re-iterated. We never had to iterate more than twice to remove all outliers.

2.1.2 Forcing parallaxes to be positive

One of the five astrometric parameters entering $\chi ^2$ in Eq. (1) is the parallax $\varpi$. 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 semi-major 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 $\varpi$) by an unconstrained one. An appropriate choice appears to be

$$\varpi'=\log \varpi$$ (7)

which is $C^{\infty}$ between $]0,+\infty[$ and $\Bbb {R}$. The variable $\varpi '$ is in fact equivalent to the distance modulus m-M since

$$m-M = -5 - 5 \varpi'.$$ (8)

The variable $\varpi '$ is thus used in the minimization process instead of $\varpi$ as one of the p_k parameters entering Eq. (1) [with $\frac{\partial v}{\partial \varpi'} = \frac{\partial v}{\partial \varpi} \varpi \ln 10$]. At the end of the minimization, Eq. (7) is reversed and the parallax $\varpi$ is derived from $\varpi '$. For any real number $\varpi '$, $\varpi$ necessarily lies in between 0 and $\infty$. Such a substitution is legitimate, since the parallax is not a directly measured quantity, but rather one among many parameters used in a model fit to the observations. Moreover, since maximum-likelihood estimators enjoy the invariance property (see e.g., Mood et al. 1974, p. 284), $\varpi$- and $\varpi '$-fitting must yield identical results for those cases where $\varpi$-fitting yielded a positive parallax (since the $\varpi'=\log \varpi$ transform may be used when $\varpi > 0$ to extract $\varpi '$).

Most of the objects considered in the present paper have large parallaxes that would have come out positive by a direct fit of $\varpi$ anyway. Thus, in the present case, the fitting of $\varpi '$ (instead of $\varpi$) does not represent so much of an improvement. Nevertheless, the procedure of $\varpi '$-fitting has been introduced here for the sake of generality.

The price to pay is, however, that the errors on $\varpi$ 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:

$$\begin{array}{l} \hat{\varpi}'-\sigma\le\varpi'\le\hat{\varp... ...i}'-\sigma} \le \varpi\le 10^{\hat{\varpi}'+\sigma} \end{array}$$ (9)

where $\hat{\varpi}'$ is the value resulting from the $\chi ^2$- minimization, and $\sigma$ its estimated uncertainty. It should be noted that the errors on $\varpi '$ do not follow a normal distribution either, since the normality is only guaranteed for "linear models'' (Mood et al. 1974) and $\Xi$ (Eq. 5) does not depend linearly upon $\varpi '$. Therefore, the confidence interval corresponding to a given probability level is in general not symmetric around $\varpi '$. However, to ease the computations, the uncertainty $\sigma$ on $\varpi '$ will be computed from the inverse of the Fisher information matrix at the point minimizing $\chi ^2$ (see Sect. 2.1.3). This procedure implicitly assumes that the model is linear in the vicinity of the minimum, so that the adopted confidence interval is in fact one that is symmetric around $\varpi '$.

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 $\chi ^2$. That important property illustrates the superiority of this approach with respect to the constrained minimization.

   
2.1.3 $\chi ^2$ minimization

If the $\chi ^2$ 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, $\omega$ and $\Omega$ enter $\chi ^2$ in a highly non-linear way, so that the function expressing $\chi ^2$ in the 9-parameter 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 $\chi ^2$[Eq. (5)] in the working space $\Bbb {R}$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 $\chi ^2$ value) is finally adopted. Once (a neighborhood of) the global minimum is thus obtained, it is tuned with the BFGS quasi-Newton algorithm (Dennis Jr. & Schnabel 1995).

Unlike the Levenberg-Marquardt (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 (a₀, i, $\omega$ and $\Omega$) are extracted from a fit to the astrometric data, they can advantageously be replaced by the Thiele-Innes elements (A, B, F and G) so that $\chi ^2$ becomes a quadratic function of the model parameters. The minimum of $\chi ^2$ 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 $\omega$ from the spectroscopic orbit; see, however, the comment about fixing $\omega$ after Eq. (6) in Sect. 2.1.1). Such additional constraints would be much more difficult to impose through the Thiele-Innes elements.

   
2.1.4 F-test: 5-parameter vs. orbital model

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 5-parameter Hipparcos solution - requires the use of an F-test.

The method used here is inspired from the test devised by Lucy & Sweeney (1971). If $\chi^2_{\rm Hip}$ and $\chi^2_{\rm orb}$ denote the residuals for the Hipparcos 5-parameter model and for the orbital model (with 9 free parameters), respectively, the efficiency of the additional 4 parameters in reducing $\chi^2_{\rm orb}$ below $\chi^2_{\rm Hip}$ may be measured by the ratio:

$$F =\frac{N-9}{4} \frac{\chi^2_{\rm Hip} - \chi^2_{\rm orb}}{\chi^2_{\rm orb}},$$ (10)

where N is the number of available measurements.

If the hypothesis that there is no orbital motion (i.e., a₀ = 0 in Eq. (5)) is correct, then it may be shown (Bevington & Robinson 1992) that F follows a Snedecor $F_{\nu_1,\nu_2}$ distribution with $\nu_1 = 4$ and $\nu_2 = N-9$degrees of freedom. Thus, if Eq. (10) yields $F = \hat{F}$, then, on the assumption that there is no orbital motion, the probability that F could have exceeded $\hat{F}$ is

$$\alpha = \mbox{Prob}(F > \hat{F}) \equiv Q(\hat{F}\vert\nu_1,\nu_2).$$ (11)

In other words, $\alpha$ is the first risk error of rejecting the null hypothesis that the orbital and 5-parameter models are identical while it is actually true.

The residuals given in the IAD files always relate to a 5-parameter solution, even when a more sophisticated model (the so-called "acceleration'' 7- or 9-parameter models, or even orbital model) was published in the Hipparcos catalogue (see Table 1). For those cases, the $\alpha$ 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 5-parameter model).

   
2.1.5 Astrophysical consistency of the orbital solution

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 F-test, 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

$$\frac{a_0\sin i}{\varpi} = \frac{P\; K_1\; \sqrt{1-e^2}}{2\pi},$$ (12)

where K₁ is the radial-velocity semi-amplitude of the visible component. The left-hand side of the above identity entirely depends upon astrometric parameters, whereas its right-hand side contains only parameters derived spectroscopically.

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:

$$\frac{a_0}{\varpi} = P^{2/3} \frac{M_2}{(M_1 + M_2)^{2/3}},$$ (13)

where M₁ and M₂ are the masses of the visible and invisible components, respectively. However, it relies upon assumptions regarding the stellar masses. In some cases (e.g., dwarf barium stars), the mass of the observable component may be estimated directly from the spectroscopically-derived gravity and from an estimate of the stellar radius from the spectral type (North et al. 1999). For the other samples, an average mass derived from a statistical analysis of the spectroscopic mass function (Jorissen et al. 1998) is adopted. For all the samples considered here, the unseen component is almost certainly a white dwarf (the only possible exception being the Tc-poor S star HIP 99312 = HD 191589), whose mass may be taken as $0.62\pm0.04$ $M_{\odot}$ (Jorissen et al. 1998).

Because of the assumptions involved, this test is only used as a guide to identify astrophysically-unplausible 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.

   

Table 1: The various samples of CPRS that have been considered in the present work, along with various parameters characterizing the quality of the orbital solution derived from the IAD (see text for details). $\varpi _{\rm HIP}$ is the parallax from the Hipparcos catalogue, $N_{\rm IAD}$ is the number of IAD available for the star considered, $\beta$ is the ecliptic latitude, T is the time of periastron passage (or of maximum velocity for circular orbits). Sol $_{\rm HIP}$ is the number of parameters used in the Hipparcos solution. It may be one of the following types: "5'' = 5-parameter solution, "7'' = 7-parameter solution, i.e., including a linear acceleration term in the proper motion (see Sect. 2.3, Vol. 1 of ESA 1997), "9'' = 9-parameter solution, i.e., including a quadratic acceleration term in the proper motion ("7''- and "9''-entries are listed in the "G'' section of the Double and Multiple Star Annex - DMSA - of the Hipparcos and Tycho Catalogues), "X'' = stochastic solution (described in DMSA/X), and "O'' = orbital solution given in the DMSA/O. The column labeled "Transit?'' indicates whether TD are also available for the corresponding star. The column labeled $\chi ^2/(N_{\rm IAD}-9)$ lists the unit-weight variance for the orbital-parameter model. $\alpha$ is the first risk error of rejecting the null hypothesis that the orbital and single star models are identical while it is actually true (Eq. 11). The expected value of the orbital separation $a_0/\varpi$is computed from Eq. (13). The $1\sigma$ confidence interval is given next to the computed astrometric $a_0/\varpi$ value. Column "Rem'' indicates whether the orbital solution derived from the IAD has been accepted. An asterisk in column "Rem'' refers to a note at the bottom of the table

HIP	HD/DM	$\varpi _{\rm HIP}$	$N_{\rm IAD}$	$\beta$	P	e	T	Sol $_{\rm HIP}$	Transit?	$\frac{\chi^2}{(N_{\rm IAD}-9)}$	$\alpha$	$a_0/\varpi$		Rem
		(mas)		($^\circ$)	(d)	(JD- 2400000)					(%)	astrom.	expected
CH stars
168	224959	1.95 $\pm$ 1.34	62	-2.8	1273.0	0.179	46064	5	y	1.23	36	1.6 $\begin{array}{l} 5.1\\ 0.41\end{array}$	1.08 $\pm$ 0.53
4252	5223	1.12 $\pm$ 1.17	50	16.8	755.2	0	45535.6	5		0.69	0	1e+09 $\begin{array}{l}+\infty\\ 0\end{array}$	0.76 $\pm$ 0.38
22403	30443	1.63 $\pm$ 1.7	28	12.5	2954.0	0	46306	5	y	0.90	9	24 $\begin{array}{l}67\\ 3.2\end{array}$	1.89 $\pm$ 0.94
53763	$+42^\circ2173$	5.29 $\pm$ 1.47	70	32.2	328.3	0	46542.1	5		0.84	3	1.8 $\begin{array}{l}8.3\\ 0.3\end{array}$	0.44 $\pm$ 0.22	Accepted,*
62827	$+8^\circ2654$A	5.3 $\pm$ 2.15	40	11.8	571.1	0	46467.8	5		1.10	89	0.42 $\begin{array}{l}1.5\\ 0\end{array}$	0.63 $\pm$ 0.31	*
102706	198269	3.16 $\pm$ 1.11	68	34.2	1295.0	0.094	46358	5		1.18	34	2.5 $\begin{array}{ l}5.6\\ 0.78\end{array}$	1.09 $\pm$ 0.54
104486	201626	4.93 $\pm$ 0.84	100	40.6	1465.0	0.103	45970	7		1.35	0	1.5 $\begin{array}{l}2.3\\ 0.84\end{array}$	1.18 $\pm$ 0.58	Accepted
108953	209621	1.47 $\pm$ 1.3	46	30.6	407.4	0	45858.3	5		1.02	94	0.8 $\begin{array}{l}9.3\\ 0\end{array}$	0.50 $\pm$ 0.25
Dwarf Ba stars
8647	11377	6.38 $\pm$ 1.14	42	-26.3	4140.0	0.16	45240	5	y	0.95	34	9.5 $\begin{array}{l} 27\\ 0\end{array}$	2.20 $\pm$ 0.97
32894	50264	14.11 $\pm$ 1.96	78	-52.2	912.4	0.098	46791	X		1.05	0	0.84 $\begin{array}{l}1\\ 0.7\end{array}$	0.89 $\pm$ 0.46	Accepted
49166	87080	7.9 $\pm$ 1.39	90	-42.2	273.4	0.177	48373	5		1.03	34	0.47 $\begin{array}{l} 0.85\\ 0.2\end{array}$	0.39 $\pm$ 0.19	Accepted
50805	89948	23.42 $\pm$ 0.93	60	-36.6	667.8	0.117	46918	O	y	0.82	0	0.67 $\begin{array}{l}0.73\\ 0.6\end{array}$	0.67 $\pm$ 0.30	Accepted
60299	107574	5.02 $\pm$ 1.06	50	-14.7	1350.0	0.081	46342	9		0.93	0	1.8 $\begin{array}{l}2.5\\ 1.2\end{array}$	0.83 $\pm$ 0.26	Accepted
62409	+17 $^\circ2537$	8.2 $\pm$ 1.28	58	20.1	1796.0	0.14	46291	5		1.29	25	0.87 $\begin{array}{l}1.7\\ 0.28\end{array}$	1.34 $\pm$ 0.65	NDAC accepted
69176	123585	8.75 $\pm$ 1.39	54	-29.3	457.8	0.062	48207	5		1.24	0	0.47 $\begin{array}{l}0.69\\ 0.31\end{array}$	0.44 $\pm$ 0.16	Accepted
71058	127392	10.63 $\pm$ 1.7	46	-15.4	1498.7	0.071	47070	5		0.85	73	0.56 $\begin{array} {l}1.4\\ 0\end{array}$	1.21 $\pm$ 0.61
104785	202020	9.53 $\pm$ 1.5	52	6.1	2064.0	0.08	47122	5		1.16	14	1.4 $\begin{array}{l}3 .2\\ 0.14\end{array}$	1.51 $\pm$ 0.77
105969	204613	16.61 $\pm$ 1.78	80	64.8	878.0	0.13	47479	X		1.02	0	0.75 $\begin{array}{l}0.84\\ 0.66\end{array}$	0.77 $\pm$ 0.33	Accepted
107818	207585	7.53 $\pm$ 1.51	52	-10.5	670.6	0.03	47319	5		0.82	7	0.39 $\begin{array}{l}0.65\\ 0.2\end{array}$	0.58 $\pm$ 0.22	Accepted
116233	221531	8.83 $\pm$ 1.21	56	-8.3	1416.0	0.165	47157	7		0.84	0	1.2 $\begin{array}{l}1.5\\ 0.89\end{array}$	0.97 $\pm$ 0.37	Accepted
118266	224621	6.95 $\pm$ 1.45	80	-32.6	307.8	0.048	49345	5		1.00	45	0.27 $\begin{array} {l}0.55\\ 0.081\end{array}$	0.40 $\pm$ 0.19

Remarks:
HIP 53763 & 62827: large uncertainty on $\varpi$ in the 9-parameter solution (see Table 2), since the orbital period is close to 1 yr.

 

Table 1: continued

HIP	HD/DM	$\varpi _{\rm HIP}$	$N_{\rm IAD}$	$\beta$	P	e	T	Sol $_{\rm HIP}$	Transit?	$\frac{\chi^2}{(N_{\rm IAD}-9)}$	$\alpha$	$a_0/\varpi$		Rem
		(mas)		($^\circ$)	(d)	(JD- 2400000)					(%)	astrom.	expected
Mild Ba stars
19816	26886	2.74 $\pm$ 1.05	48	-21.9	1263.2	0.395	48952.12	5		0.55	1	5.2 $\begin{array}{l}15\\ 0.61\end{array}$	0.77 $\pm$ 0.23
20102	27271	6.01 $\pm$ 1.13	28	-18.6	1693.8	0.217	47104.38	5		0.66	46	1.8 $\begin{array}{l}5.6\\ 0\end{array}$	0.93 $\pm$ 0.28
26695	288174	2.89 $\pm$ 1.32	42	-21.3	1824.3	0.194	47157.62	5		0.80	66	5.1 $\begin{array}{ l}15\\ 0.74\end{array}$	0.98 $\pm$ 0.29
32831	49841	-1.05 $\pm$ 1.44	44	-17.2	897.1	0.161	48339.71	5	y	1.15	44	1.8e+09 $\begin{array}{l}+\infty\\ 0\end{array}$	0.61 $\pm$ 0.18
34143	53199	3.65 $\pm$ 1.3	44	-9.3	7500.0	0.212	41116.2	5	y	0.99	72	26 $\begin{array}{l}1 10\\ 0\end{array}$	2.51 $\pm$ 0.75
35935	58121	2.82 $\pm$ 0.95	64	-15.7	1214.3	0.14	46811.21	5		1.12	53	0.98 $\begin{array}{l}2.4\\ 0.22\end{array}$	0.75 $\pm$ 0.22
36042	58368	2.36 $\pm$ 0.97	56	-14.3	672.7	0.221	45617	5		1.00	1	2 $\begin{array}{l}3.8\\ 0.97\end{array}$	0.50 $\pm$ 0.15	Accepted
36613	59852	-1.86 $\pm$ 1.24	60	-25.8	3463.9	0.152	46841.03	5		1.25	100	6.2e+51 $\begin{array}{l}+\infty\\ 0\end{array}$	1.50 $\pm$ 0.45
43527	$-14^\circ2678$	0.41 $\pm$ 1.4	56	-30.9	3470.5	0.217	48828.06	5	y	0.84	59	49 $\begin{array}{l}3000\\ 0.14\end{array}$	1.50 $\pm$ 0.45
44464	77247	2.86 $\pm$ 0.97	46	34.6	80.5	0.0871	48953	5	y	1.05	93	0.39 $\begin{array}{l }1.3\\ 0\end{array}$	0.12 $\pm$ 0.04
51533	91208	4 $\pm$ 0.95	48	-24.4	1754.0	0.171	45628.36	5		1.45	72	3.5 $\begin{array}{l}8. 7\\ 0.41\end{array}$	0.95 $\pm$ 0.28
53717	95193	2.3 $\pm$ 1.03	44	-19.0	1653.7	0.135	46083.62	5		1.15	89	5.3 $\begin{array}{l} 22\\ 0\end{array}$	0.92 $\pm$ 0.27
73007	131670	2.33 $\pm$ 1.22	34	9.0	2929.7	0.162	46405.11	5		1.17	45	21 $\begin{array}{l}1 30\\ 0\end{array}$	1.34 $\pm$ 0.40
76425	139195	13.89 $\pm$ 0.7	38	28.5	5324.0	0.345	44090	5		0.90	61	25 $\begin{array}{l} 47\\ 5.3\end{array}$	2.00 $\pm$ 0.59
78681	143899	3.6 $\pm$ 1.29	36	1.2	1461.6	0.194	46243.43	5		1.65	93	1.2 $\begin{array}{l}4 .6\\ 0\end{array}$	0.84 $\pm$ 0.25
94785	180622	3.37 $\pm$ 1.04	54	22.5	4049.2	0.061	50534.41	5		1.03	52	26 $\begin{array}{l} 67\\ 3.3\end{array}$	1.66 $\pm$ 0.50
103263	199394	6.33 $\pm$ 0.63	84	59.6	4382.6	0	50719.34	5		0.81	76	4 $\begin{array}{l}11\\ 0\end{array}$	1.75 $\pm$ 0.53
103722	200063	0.73 $\pm$ 1.02	48	17.2	1735.5	0.073	47744.64	5		1.00	3	14 $\begin{array}{l}96\\ 0.25\end{array}$	0.95 $\pm$ 0.28
104732	202109	21.62 $\pm$ 0.63	98	43.7	6489.0	0.22	40712	7		0.71	0	6 $\begin{array}{l}10\\ 1.8\end{array}$	2.28 $\pm$ 0.68
105881	204075	8.19 $\pm$ 0.9	54	-7.0	2378.2	0.2821	45996	5	y	0.92	6	1 $\begin{array}{l}1.7\\ 0.43\end{array}$	1.17 $\pm$ 0.35	Accepted
106306	205011	6.31 $\pm$ 0.68	68	36.2	2836.8	0.2418	46753.59	5		0.71	3	1.7 $\begin{array}{l}+\infty\\ 0\end{array}$	1.31 $\pm$ 0.39
109747	210946	3.42 $\pm$ 1.14	36	11.7	1529.5	0.126	46578.18	5		0.89	14	2.2 $\begin{array}{ l}4.9\\ 0.68\end{array}$	0.87 $\pm$ 0.26
112821	216219	10.74 $\pm$ 0.93	56	23.3	4098.0	0.101	44824.92	5		0.85	25	3 $\begin{array}{l }6.8\\ 0\end{array}$	1.96 $\pm$ 0.74	*
117607	223617	4.61 $\pm$ 0.95	62	2.9	1293.7	0.061	47276.68	7		0.75	0	0.87 $\begin{array}{l}1.5\\ 0.47\end{array}$	0.78 $\pm$ 0.23	Accepted

Rem: HIP 112821 is also sometimes classified as a dwarf barium star.

 

Table 1: continued

HIP	HD/DM	$\varpi _{\rm HIP}$	$N_{\rm IAD}$	$\beta$	P	e	T	Sol $_{\rm HIP}$	Transit?	$\frac{\chi^2}{(N_{\rm IAD}-9)}$	$\alpha$	$a_0/\varpi$		Rem
		(mas)		($^\circ$)	(d)	(JD- 2400000)					(%)	astrom.	expected
Strong Ba stars
4347	5424	0.22 $\pm$ 1.42	52	-30.9	1881.5	0.226	46202.8	5	y	1.29	51	3.7e+09 $\begin{array}{l}+\infty\\ 0\end{array}$	1.12 $\pm$ 0.40
13055	16458	6.54 $\pm$ 0.57	80	60.2	2018.0	0.099	46344	7		1.20	0	1.6 $\begin{array}{l}2\\ 1.2\end{array}$	1.17 $\pm$ 0.42	Accepted
15264	20394	2.16 $\pm$ 1.14	48	-15.3	2226.0	0.2	47929	5		1.39	77	8 $\begin{array}{l} 46\\ 0\end{array}$	1.25 $\pm$ 0.44
17402	24035	3.72 $\pm$ 0.8	70	-76.9	377.8	0.02	48842.65	5		1.12	49	3.3e+09 $\begin{array}{l}+\infty\\ 0\end{array}$	0.38 $\pm$ 0.14	*
23168	31487	4.54 $\pm$ 1.21	40	29.1	1066.4	0.045	45173	5		0.45	86	0.47 $\begin{array}{l }1.7\\ 0\end{array}$	0.77 $\pm$ 0.27
25452	36598	3.32 $\pm$ 0.68	72	-85.3	2652.8	0.084	45838.95	5		0.99	5	4.3 $\begin{array}{l}8.7\\ 1.3\end{array}$	1.41 $\pm$ 0.50
29099	42537	-1.13 $\pm$ 0.93	64	-75.9	3216.2	0.156	46147.32	5	y	1.30	82	1.3e+15 $\begin{array}{l}+\infty\\ 0\end{array}$	1.60 $\pm$ 0.57
29740	43389	-1.25 $\pm$ 1	52	-25.8	1689.0	0.082	47222.46	5		0.53	0	33 $\begin{array}{l}2400\\ 0.4\end{array}$	1.04 $\pm$ 0.37	*
30338	44896	1.56 $\pm$ 0.7	90	-56.9	628.9	0.025	48464.3	5		0.88	0	1.8 $\begin{array}{l}3.5\\ 0.92\end{array}$	0.54 $\pm$ 0.19
31205	46407	8.25 $\pm$ 0.92	80	-34.3	457.4	0.013	47677.45	O	y	0.77	0	0.71 $\begin{array}{l}1\\ 0.46\end{array}$	0.44 $\pm$ 0.15	Accepted
32713	49641	0.73 $\pm$ 0.88	48	-19.2	1768.0	0	46306	5		0.63	0	7.5 $\begin{array}{l}22\\ 2.3\end{array}$	1.08 $\pm$ 0.38
32960	50082	4.71 $\pm$ 0.99	40	-16.2	2896.0	0.188	45953.12	5		1.39	44	3.9 $\begin{array}{l }8.8\\ 0.62\end{array}$	1.49 $\pm$ 0.53
36643	60197	1.67 $\pm$ 0.84	92	-50.5	3243.8	0.34	46015.97	5		0.64	59	21 $\begin{array}{l}6 9\\ 0.66\end{array}$	1.61 $\pm$ 0.57
50006	88562	3.13 $\pm$ 1.17	50	-25.1	1445.1	0.204	45781.71	5		0.95	4	3.2 $\begin{array}{l}6.5\\ 1.3\end{array}$	0.94 $\pm$ 0.33
52271	92626	3.4 $\pm$ 0.71	104	-50.5	918.2	0	49147.83	5		1.05	9	0.59 $\begin{array}{l}0.99\\ 0.31\end{array}$	0.69 $\pm$ 0.25	Accepted
56404	100503	0.67 $\pm$ 1.2	64	-30.8	554.4	0.061	46144.83	5		0.63	26	2.3 $\begin{array}{l} 21\\ 0.15\end{array}$	0.50 $\pm$ 0.18
56731	101013	7.07 $\pm$ 0.68	74	43.2	1710.9	0.195	43934	O		1.19	0	1.1 $\begin{array}{l}1.4\\ 0.88\end{array}$	1.05 $\pm$ 0.37	Accepted
60292	107541	5.78 $\pm$ 1.36	68	-29.5	3569.9	0.104	44388.16	5		0.96	2	27 $\begin{array}{l}51\\ 12\end{array}$	1.72 $\pm$ 0.61
68023	121447	2.21 $\pm$ 1.02	52	-6.0	185.7	0	46922.35	5	y	1.14	19	3.5 $\begin{array}{l}7.6 \\ 1.3\end{array}$	0.24 $\pm$ 0.08
69290	123949	.97 $\pm$ 1.32	34	-5.6	9200.0	0.972	49144.96	5		1.50	92	470 $\begin{array}{l} 9000\\ 0\end{array}$	3.23 $\pm$ 1.14
94103	178717	2.9 $\pm$ 0.95	52	32.5	2866.0	0.434	44258	5	y	0.89	71	16 $\begin{array}{l} 63\\ 0\end{array}$	1.48 $\pm$ 0.53
101887	196445	1.49 $\pm$ 1.54	50	-21.2	3221.3	0.237	46037.95	5	y	1.06	63	9.3 $\begin{array}{l}100\\ 0\end{array}$	1.60 $\pm$ 0.57
103546	199939	3.16 $\pm$ 0.75	72	57.6	584.9	0.284	45255.1	5		0.79	26	0.44 $\begin{array}{ l}0.88\\ 0.15\end{array}$	0.51 $\pm$ 0.18	Accepted
104542	201657	4.49 $\pm$ 1.07	64	31.6	1710.4	0.171	46154.95	5		1.06	73	0.81 $\begin{array} {l}2.3\\ 0\end{array}$	1.05 $\pm$ 0.37
104684	201824	0.56 $\pm$ 1.56	50	7.4	2837.0	0.342	47413	5		1.79	87	2.1e+03 $\begin{array}{l}+\infty\\ 0\end{array}$	1.47 $\pm$ 0.52
110108	211594	4.59 $\pm$ 1.18	26	4.4	1018.9	0.058	48538.19	5		1.11	17	0.99 $\begin{array}{ l}1.8\\ 0.43\end{array}$	0.74 $\pm$ 0.26	Accepted
Tc-poor S stars
5772	7351	3.21 $\pm$ 0.82	28	19.1	4592.7	0.17	44696	5	y	0.92	88	6.9 $\begin{array}{l}57 \\ 0\end{array}$	1.97 $\pm$ 0.67
8876	$+21^\circ255$	-1.92 $\pm$ 1.5	38	9.5	4137.2	0.209	43578.31	5	y	1.13	87	2.4e+13 $\begin{array}{l}+\infty\\ 0\end{array}$	1.84 $\pm$ 0.66
17296	22649	6.27 $\pm$ 0.63	78	42.2	596.2	0.088	42794.5	O	y	1.22	1	0.37 $\begin{array}{l}0.52\\ 0.24\end{array}$	0.51 $\pm$ 0.17	Accepted
25092	35155	1.32 $\pm$ 0.99	52	-31.7	640.6	0.071	48092.41	5	y	0.79	51	0.94 $\begin{array}{ l}2.7\\ 0.18\end{array}$	0.53 $\pm$ 0.18	Accepted
32627	49368	1.65 $\pm$ 1.11	60	-17.4	2995.9	0.357	45145.37	5	y	1.03	57	16 $\begin{array}{l }110\\ 0.91\end{array}$	1.48 $\pm$ 0.50
38217	63733	0 $\pm$ 0.99	84	-39.2	1160.7	0.231	45990.92	5		0.95	58	1.7e+14 $\begin{array}{l}+\infty\\ 0\end{array}$	0.79 $\pm$ 0.27
90723	170970	1.83 $\pm$ 0.67	86	59.4	4392.0	0.084	48213.21	5	y	0.98	39	27 $\begin{array}{l }59\\ 8.8\end{array}$	1.91 $\pm$ 0.66
99124	191226	0.39 $\pm$ 0.71	84	54.9	1210.4	0.19	49691.78	5		0.98	23	11 $\begin{array}{l}6 20\\ 0.14\end{array}$	0.81 $\pm$ 0.27
99312	191589	2.25 $\pm$ 0.77	84	52.0	377.3	0.253	48844.02	5		0.80	82	0.58 $\begin{array}{l }2.4\\ 0\end{array}$	0.37 $\pm$ 0.13	Accepted
115965	$+28^\circ4592$	1.72 $\pm$ 1.26	64	29.6	1252.9	0.091	48161.32	5		0.78	72	1.4 $\begin{array}{l}8.9\\ 0\end{array}$	0.83 $\pm$ 0.28

HIP 29740: $\alpha=0$ but the solutions are totally unrealistic.
HIP 17402: $\varpi$ not accurately determined, since orbital period is close to 1 yr.

2.2 Transit Data

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 by-product of (or, more precisely, an input for) the multiple-star 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 b₁, ..., b₅ which represent the coefficients of the first terms in the Fourier series modeling the observed signal as modulated by the detector grid:

$$\begin{eqnarray*}b_1&=&\sum_iI_i\\ b_2&=&\tilde M\sum_iI_i\cos\phi_i\\ b_3&=&-... ...um_iI_i\cos(2\phi_i)\\ b_5&=&-\tilde N\sum_iI_i\sin(2\phi_i)\\ \end{eqnarray*}$$

where I_i is the intensity of the i-th component of the system, and the phase $\phi_i$ corresponds to its abscissa along the reference great circle. $\tilde M=0.7100$ and $\tilde N=0.2485$ are the adopted reference values for the modulation coefficients of the first and second harmonics (Vol. 1, Sect. 2.9; ESA 1997). In terms of the Cartesian coordinates $(\xi_i, \eta_i)$ (see Eq. (3)) of component i in the plane tangent to the sky at the reference point specified in the TD file, the phase $\phi_i$ writes

$$\phi_i=f_x\xi_i+f_y\eta_i+f_{\rm p}(\varpi - \varpi_{\rm ref}).$$ (14)

The reference point is assigned an arbitrary parallax $\varpi_{\rm ref}$ and proper motion, as given in the TD file. The phase derivatives f_x, f_y and $f_{\rm p}$ with respect to $\xi, \eta$ and $\varpi$ are also provided by the TD file.

For the SB1 systems we are interested in, the situation simplifies a lot since I₂ may be taken equal to 0. The above system of equations is rank-deficient. From the second and third equations, one can rewrite:

$$\stackrel{\rm o}{\phi}=\arg(b_2,-b_3)$$ (15)

and also express the uncertainty on $\phi$ as

$$\sigma_{\phi}^2=\left(\frac{\partial \stackrel{\rm o}{\phi}}{... ...ackrel {\rm o}{\phi}}{{\partial} b_3}\right)^2\sigma_{b_3}^2.$$

With the above definitions, the objective function whose minimum is sought is given by:

$$D=\frac{1}{N}\sum_{k=1}^N\sigma_{\phi_k}^{-2}(\stackrel{\rm o}{\phi_k}-\phi_k)^2$$ (16)

where N is the number of TD. In the above expression, $\stackrel{\rm o}{\phi_k}$is the observed phase at time t_k as derived from Eq. (15), whereas $\phi_k$ is the phase computed from Eqs. (3) and (14) (noting that $f_x R P_\alpha + f_y R P_\delta \equiv f_{\rm p}$) for a given set of astrometric and orbital parameters.

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, ...