The transit data are a fully calibrated set of observations, in principle allowing complicated models for the stars to be investigated. We have restricted ourselves to the assumption that the stars are constant in magnitude and that their motions have no acceleration. For each system we determine the position at epoch J1991.25 for each component, one or more parallaxes and one or more proper motions. Coordinates are given in the International Celestial Reference System (ICRS), nearly identical with the equator and equinox J2000.0.
The resulting set of equations is described in The Hipparcos and Tycho Catalogues, Sect. 2.9 of Vol. 1 ([ESA 1997]). We solved these non-linear equations iteratively through a linearization at the start of each iteration. The initial values were only updated by modest amounts in each iteration lest the proces should diverge rapidly.
Due to the periodicity of the focal plane grid, the equations have many solutions and the challenge is to find the correct one. The essential point is to have accurate initial values of position and proper motion of the stars. Several sources of information were used to achieve this goal. First of all, we always checked the star positions in the Tycho-2 Catalogue ([Høg et al. 2000]) and sometimes in Tycho-1 ([ESA 1997]) or in the Hipparcos Input Catalogue (HIC, [Turon et al. 1992]). HIC was often useful for getting initial values of proper motions, though it sometimes erroneously assumes the same proper motions for components of doubles, and also the Starnet catalogue ([Röser 1996]), the ACT ([Urban et al. 1998]) and the Tycho Reference Catalogue ([Høg et al. 1998a]) provided good proper motions. In a few cases the Guide Star Catalog ([Lasker et al. 1990]) and the Digitized Sky Survey provided initial positions.
An important source of positional information is the Tycho Data Base of Identified Counts which we always consulted and often used for looking for possible close companions. This data base contains photon counts for all transits of selected patches of the sky across the slit system of the Tycho instrument onboard the Hipparcos satellite. The patches are circles of diameter 5 arcsec, centred on the predicted positions of the 4 million brightest stars. Each transit gives a one dimensional scan of such a patch, represented in the data base with 31 photon count values. By combining many transits in many different scan directions a two dimensional image can be constructed. With only one transit, the map would consist of 31 parallel lines of varying intensity. When all such elementary maps are stacked, an image appears. Dominating scan directions or bright stars in the neighborhood will produce various kinds of stripes. Figures 1-4 show examples of such images, where either additional components or gross errors in the Hipparcos catalogue are revealed. Unfortunately, such maps are less useful for more complicated stellar systems, where sidelobes from e.g. two brighter stars may hide a fainter star. Also certain separations of double stars are difficult to deal with unless the magnitude difference is small.
For each of the investigated systems three different solutions were normally attempted. A full solution, with separate parallaxes and proper motions for each component; a solution with only one parallax but separate proper motions; and a solution with one common parallax and one common proper motion for all components. For triple systems also solutions with two proper motions, i.e. a common proper motion for two of the components, were sometimes used. Several factors were taken into account when deciding which of the solutions to accept, if any. First of all, the series of iterations must converge and the least squares minimum should be well defined. Secondly, a large number of unknowns should only be retained when they are meaningful.
We found, not surprisingly, that a shallow minimum was an indication of a false solution. The shallowness was tested by giving a small offset to the solution, of the order of 10-20 mas (or mas/yr) in each unknown, and starting the iteration again from that point. If the minimum is deep, we will be back at the solution in only a few steps. The method was not developed beyond a subjective decision on whether or not the solution felt right, and it was merely used for rejecting some systems from the present work. Some solutions were less clear than others. They are flagged with a "U''.
The present work deals with some of the difficult cases from Hipparcos where large pointing errors or large magnitude differences often reduce the possibility of fitting a complicated model to the data. We have only determined individual parallaxes if they were significantly different in the full solution; and when only one parallax was determined, we have only determined individual proper motions when they differed significantly.
In cases where only one proper motion or one parallax is given for a double star with one bright and one faint star, the values refer in reality to the brighter component and little can be concluded for the fainter star. Giving only one parallax does not necessarily mean that we have concluded that the system is physical rather than optical.
Copyright The European Southern Observatory (ESO)
to the west is clearly seen.
HIC mentions an orbiting companion, but gives no details. The system is
resolved in Tycho-2
