4 Individual solutions

  Most individual solutions of the 5-parameter type are unlikely to be improved upon. Exceptions are stars for which it can safely be assumed that their parallax is much too small to be measured by Hipparcos, in which case the parallax can be set to zero. This could be relevant when a relatively large correlation exists between the parallax and proper motion determination. In order to do so, all abscissa residuals are to be corrected for the difference between the determined parallax and the assumed parallax:  

$$\Delta v^\prime = \Delta v - {\partial v\over \partial \pi}\Delta \pi,$$ (10)

where $\Delta\pi = \pi_{\rm new} - \pi_{\rm old}$. The new set of residuals are then solved using the mechanism described in Sect. 3 for only four parameters a_i: position and proper motion.

The situation is much more complicated when a very precise proper motion is available from an external source. The problem is that before such a constraint can be incorporated, it needs to be represented in exactly the same reference frame as the Hipparcos data. The same applies when positions are obtained on photographic plates at a different epoch than the Hipparcos data. The transformation to the Hipparcos reference frame can only be obtained if a sufficiently large number of objects is available for the determination of the transformation parameters, e.g. when the measurement has been obtained in a sufficiently well determined reference frame, and has been corrected for any systematic differences between that reference frame and the Hipparcos reference frame, both in positions and in proper motions. In general, the uncertainty of the systematic errors will inhibit incorporation of proper motion data. Here, however, the Tycho catalogue can be of assistance in some cases. With a much higher density of stars than the Hipparcos catalogue, the Tycho positions can provide a sufficient number of reference points to determine accurate plate transformations if plates were obtained at an epoch close to the mean Tycho epoch. For clusters like the Pleiades and Praesepe one can use the fact that the internal proper motions of the cluster members are very small, and that by using only cluster members, larger epoch differences can be allowed.

Another possible application is for stars with orbital motions, where prolonged observations can provide additional measurements. It is likely that some of the 7- and 9-parameter solutions refer to orbital motions over a time-span much longer than the 3.5 years of the Hipparcos mission. The same applies to some stochastic solutions, where the orbital motion time-scale could be shorter than 3.5 years. Further observations may constrain the possibilities of non-linear proper motion fits. It will be difficult, however, to obtain positional measurements with accuracies comparable with the Hipparcos data. In such cases, the addition of radial velocity data may sometimes help resolve the solution of the Hipparcos data.

Important in the construction of orbital parameter solutions is the definition of the parameters a_i and their derivatives. This problem is very similar to that of the solar system objects, described in the next section, except that the angle $\theta$ (Fig. 3), defining the local direction to the equatorial pole, still needs to be calculated. Although it can be derived from the positions of the star and the pole of the reference great circle, it is far easier to obtain the angle $\theta$ directly from fields IA3 and IA4 (Vol. 1, Table 2.8.3) in the intermediate astrometric data file:

$$\tan\theta = {\rm IA3} / {\rm IA4},$$ (11)

while the epoch of observation is taken from the Reference Great Circle file (fields IR2 and IR5 in Vol. 1, Table 2.8.1) or from:

$${\rm epoch} - 1991.25 = {\rm IA6} / {\rm IA3} = {\rm IA7}/{\rm IA4}.$$ (12)

Using the same mechanism as employed for solar system objects, described in the next section, further models can be developed and implemented.