## 4. Model assumptions## 4.1. The Galaxy modelThe 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 ( 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. ## 4.2. Binary distributionsIn generating a synthetic catalogue, all the stars are assumed to be binaries with uniformly distributed in a given range (e.g. 1-2, corresponding to a distribution in the range 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 ratioThe observed distribution of mass ratios
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 decreases monotonically with
increasing ## 4.2.2. EccentricityIt 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 AU ( d) we assume circularised orbits, i.e. . Otherwise, we assume distributions in the interval with From limited experiments with other (e.g. uniform) distributions we have concluded that the assumed is not critical for the results of present method. ## 4.2.3. Other assumptionsUsing 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 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 observationsThe 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 Our modelling of the Hipparcos observations takes into account the
variation of observational accuracy with 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 , up
to a multiplicative factor depending on the where . 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 ## 4.4. Data reduction and binary detection modelThe 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.
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 (in the
The first test was whether or not the binary was resolved. The criterion for a component (C) solution was that and arcsec, where is the mean separation of the binary during the mission lifetime. Here, The detection limits in and 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 ( where is the angular distance from the mass centre to the photocentre. We used this expression for binaries with 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 ). 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 . 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 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. © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |