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Astron. Astrophys. 361, 770-780 (2000)

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4. Model assumptions

4.1. The Galaxy model

The Galaxy model is a slightly updated version of the one described by Holmberg et al. (1997). It is a non-dynamical model resulting in a synthetic catalogue of (primary) stars down to a given limiting magnitude. The data available for each star include: heliocentric coordinates (r, [FORMULA], b), intrinsic colour and absolute magnitude ([FORMULA], [FORMULA]), apparent magnitudes and colour index (V, Hp, [FORMULA]). The interstellar extinction was modelled according to Hakkila et al. (1997). The colour index [FORMULA] was used because it has a more well-defined relation to the Hipparcos magnitude Hp than [FORMULA]. When required, the approximate photometric transformations from ESA (1997, Vol. 1, Sect. 1.3) were used.

Since only main-sequence stars are considered, a unique mass-luminosity relation was assumed. This was based on Andersen (1991) for masses between 22.9 and 0.59 solar masses, and on the theoretical models by Chabrier et al. (1996) for masses down to 0.06[FORMULA].

4.2. Binary distributions

In generating a synthetic catalogue, all the stars are assumed to be binaries with [FORMULA] uniformly distributed in a given range (e.g. [FORMULA] 1-2, corresponding to a [FORMULA] distribution in the range [FORMULA] 10-100 AU). This facilitates later scaling to any desired multiplicity. Other orbital parameters are assumed to follow the distributions described below.

4.2.1. Mass ratio

The observed distribution of mass ratios [FORMULA] is an important diagnostic for binary formation processes. Earlier studies variously derived increasing, decreasing, peaked or flat distributions, perhaps mainly reflecting the different selections of objects considered (Trimble 1990). DM find that [FORMULA] decreases monotonically with increasing q. However, the detailed shape is rather uncertain, and it is plausible that it depends both on the period, as found e.g. by Mazeh et al. (1992), and on the primary mass (Fischer & Marcy 1992; Reid & Gizis 1997). We adopt a simple parametrisation of [FORMULA], viz. as a non-negative superposition of three basic forms: a linearly increasing density function [FORMULA], a linearly decreasing function [FORMULA], and a parabolic function [FORMULA]. For instance, the distribution found by DM (their Fig. 10) is reasonably approximated by [FORMULA], at least for [FORMULA]. These distributions are for the case when both components are main-sequence stars. Since some fraction of the binaries would have a degenerate (white-dwarf) companion, we consider also a distribution [FORMULA] where the companion is a white dwarf of mass 0.6[FORMULA] and negligible luminosity.

4.2.2. Eccentricity

It is well known (Duquennoy & Mayor 1992) that the eccentricity distribution depends strongly on the orbital period and age. We again use the results from DM, which should be representative for the disk population dominating the Hipparcos survey. For semi-major axis [FORMULA] AU ([FORMULA] d) we assume circularised orbits, i.e. [FORMULA]. Otherwise, we assume distributions in the interval [FORMULA] with


From limited experiments with other (e.g. uniform) distributions we have concluded that the assumed [FORMULA] is not critical for the results of present method.

4.2.3. Other assumptions

Using a random-number generator, a set of orbital parameters is generated for each system. The mass of the primary is known from the Galaxy model. From the mass-ratio distribution, the mass of the secondary follows and hence the total mass. Together with the semi-major axis this gives the period P from Kepler's third law. The time of periastron is then chosen randomly between 0 and P. In addition, the orbital plane of the binary is randomly oriented in space. This allows then to compute the positions of the components relative to the centre of mass for arbitrary times.

The distance to the binary is known from the Galaxy model, which determines the apparent orbit. For simplicity, the motions of the centre of mass due to parallax and proper motion were however not modelled. This means that any non-zero parallax or proper motion found by analysis of the simulated observations must be attributed to a combination of orbital motion and observational errors.

4.3. Hipparcos observations

The astrometric observations by Hipparcos consisted of one-dimensional positional measurements of the resolved components or, for unresolved systems, of the photocentres in the wide wavelength band Hp. Each scan of either field of view across an object produced such a measurement, which for the present purpose is characterised by the epoch and position angle of the scan and the standard error of the positional measurement.

Our modelling of the Hipparcos observations takes into account the variation of observational accuracy with Hp magnitude, the non-uniform sky coverage from the Hipparcos scanning law, and the irregular distribution of the epochs of observation due to the scanning law, including time gaps caused by the eccentric satellite orbit and intervals of malfunction.

The Hipparcos scanning geometry was modelled by the `nominal scanning law' described in ESA (1997, Vol. 2, Ch. 8). Time gaps were introduced by considering the distribution of the actual times of observation as recorded in the Hipparcos Transit Data (ESA 1997, Vol. 17, Disk 6). Using this scanning law, the mean standard errors of the five astrometric parameters could be simulated as function of ecliptic latitude [FORMULA], up to a multiplicative factor depending on the Hp magnitude. That factor, which equals the one-dimensional standard error per observation ([FORMULA]), was determined by matching the model accuracies to Tables 3.2.2 through 3.2.6 in the Hipparcos Catalogue (Vol. 1). An empirical fit gave


where [FORMULA]. The three terms represent the asymptotic variance for the bright stars due to attitude and calibration errors, the main variation caused by photon noise, and additional noise e.g. from the sky background. That the magnitude is multiplied by 0.32 (rather than 0.4) in the expression for u reflects the increasing observing time per star, as function of magnitude, which was built into the Hipparcos observation strategy. The mean number of observations per star was 112. On the whole, the model correctly describes the median standard errors of all the astrometric parameters as functions of Hp and [FORMULA] to within [FORMULA]%.

4.4. Data reduction and binary detection model

The Hipparcos Catalogue (ESA 1997), Vol. 1, Sect. 2.3.1 describes briefly how the different kinds of solutions (C, G, O, V, X or single-star) were arrived at depending on various criteria. The actual reduction process was however much more complicated, as described in Vol. 3 of the Catalogue. To model this process in detail is not feasible. The scheme described here is very simplified, but we think it is a reasonable representation of the actual process, with some notable exceptions (Sect. 5.2). The scheme is shown in Fig. 1 and only briefly described below.

[FIGURE] Fig. 1. Flow chart for simulating the data reductions and binary detection. The process results in a classification of each object as solution type C, G2, G3, O, X (Sects. 2.1-2.5), or as apparently single (S) as observed by Hipparcos. Each solution also provides a proper-motion error, which allows to identify delta-mu binaries (Sect. 2.6) in parallel with the above scheme.

For each star in the model catalogue, the times of observation and the scanning angles were computed from the scanning law according to the ecliptic coordinates of the object. The orbital positions of the binary components were then calculated at each time of observation. Together with the component masses (from the binary distribution model) and the known distance (from the Galaxy model), this gave the magnitude difference [FORMULA] (in the Hp magnitude system) and angular separation [FORMULA] on the sky. If the mean angular separation was less than 10 arcsec, a total magnitude (converted to V) was computed. The object was retained as a main-sequence survey star if the total magnitude satisfied the same criteria as applied on the Hipparcos Catalogue data, viz. Eqs. (1) and (2).

The first test was whether or not the binary was resolved. The criterion for a component (C) solution was that [FORMULA] and [FORMULA] arcsec, where [FORMULA] is the mean separation of the binary during the mission lifetime. Here,


The detection limits in [FORMULA] and [FORMULA] were estimated by means of a diagram similar to Fig. 3.2.106 in Vol. 1 of the Hipparcos Catalogue, but restricted to the sample described in Sect. 3. For separations greater than 10 arcsec, the individual components were added to the category of apparently single stars (S) depending on whether their magnitudes satisfied the survey criterion.

If the object was not of type C, its binary nature could still be detected through the combined non-linear motions of the components. The effective centre observed by Hipparcos coincides with the photocentre for separations less than about 0.3 arcsec, but is much closer to the primary component for separations greater than 0.7 arcsec (ESA 1997, Vol. 1, Sect. 13.7). The photocentre is related to the mass ratio (q) and magnitude difference through


where [FORMULA] is the angular distance from the mass centre to the photocentre. We used this expression for binaries with [FORMULA] arcsec. For more widely separated binaries we used instead the position of the primary, obtained by dropping the second term in Eq. (9). The photocentre or primary position, projected in the scanning direction, was calculated for every observation, and a Gaussian error with standard deviation from Eq. (7) was added.

A sequence of tests was then applied to see whether linear, quadratic or cubic motions as function of time could fit the data, and if they fitted, whether the highest-order polynomial term was statistically significant. A parallax term was always included in these solutions. The order of these tests (Fig. 1) may at first seem illogical. However, it was designed to simulate the corresponding tests during the Hipparcos double-star processing, in which cubic or quadratic solutions were favoured whenever they were found to be significant.

Bad fits were defined as having a chi-square value greater than the 99.87th percentile (corresponding to [FORMULA]). The significance of the cubic and quadratic terms was tested according to Eqs. [2.3.3] and [2.3.4] in Vol. 1 of the Hipparcos Catalogue.

In principle, cubic solutions should not be able to reproduce orbits with period less than about 3.5 yr, unless the photocentre motion is essentially linear e.g. because [FORMULA]. Such cases should instead result in orbital solutions. However, it was also found by inspecting the Hipparcos Catalogue that very few orbital solutions have periods less than 0.1 yr. Thus it was assumed that all binaries with periods between 0.1 and 3.5 yr, and which gave a bad linear fit, could be classified as solutions of type O. In this scheme, stochastic solutions (type X) correspond either to short-period binaries with a bad linear fit, or to long-period binaries with a bad cubic fit.

For non-resolved binaries, the proper motion is basically the average linear motion of the photocentre. Even if the binary nature of the system was detected through the acceleration, stochastic or orbital solutions, the residual proper motion represents an error in the true proper motion (Sect. 5.4).

In order to model the number of delta-mu binaries found by comparison with FK5, it was assumed that the latter catalogue is essentially complete for [FORMULA] mag. This limit was adjusted to give the correct total number of main-sequence stars in FK5. Here, the number of systems was counted in which the mean proper motion of the photocentre relative to the mass centre exceeds a given limit, irrespective of whether the system was also classified as another type of binary solution.

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Online publication: October 2, 2000