3. Determination of the rotation of the Hipparcos catalogue system
The systematic part of the proper motion differences which can be ascribed to a rotation difference between the Hipparcos catalogue system (H) and the extragalactic frame (E) can be expressed by means of the observation equations (using the notations according to Lindegren & Kovalevsky (1995):
Here , are the equatorial coordinates, , are the proper motion differences in the sense Hipparcos-Catalog (GPM1 or MEGA) in right ascension and declination, respectively, and , , , are the components of the rotation vector. Equations can be formed and solved in two cases using 1) proper motion differences of each star and 2) mean proper motion differences (normal points) for each field. In other words we will have a so-called "star solution" and a "field solution" respectively. Both solutions must coincide if the stars were homogeneously distributed within the fields and the fields were homogeneously distributed over the sky. Formally, the mean square errors of unknowns in the "star solution" are less than those in the "field solution" because of the larger number of observation equations.
The results of the solution of the system of equations for combined and separate use of the GPM1 and MEGA data are given in Table 3 for the "field solutions" (upper lines) and "star solutions" (lower lines).
Table 3. Preliminary solutions of the equations
Table 3 shows that the KSZ and MEGA solutions differ from each other significantly. Applying the same equations for the proper motion differences in the sense "KSZ minus MEGA" the solution given in
the last line of Table 3 has been found. As one can see, actually proper motions of stars in the GPM1 and MEGA catalogues belong to different reference frames. Therefore, one should not use these data jointly. It should be also noted that the distribution of the MEGA fields on the sky is not an optimum to determine the rotation parameters of the Hipparcos system.
For this reason, later on, only the GPM1 catalogue data were used for the Hipparcos link purpose. The initial GPM1 data were analysed more carefully and new combined solution of the system of equations as well as separate solutions for proper motion differences in right ascension ( cos ) and declination () were obtained (see the first three upper lines in Table 4). The "field solutions" and "star solutions" were found to be practically the same. However, the solutions for ( cos ) and () differ significantly. It has been supposed that such differences were caused by the errors of determination of absolute proper motions in separate fields on the one hand and by small weights of unknowns and determed using only the ( cos ) differences.
Table 4. GPM1 "star solutions"
Random errors of absolute proper motions of stars in the GPM1 catalogue consist of two main components: random errors of relative proper motions of stars and random errors of their absolutisation. The last ones may be considered as systematic errors within the same field determining the zero point offsets for each field. Two additional unknowns (the zero point offsets of proper motions for each field in ( cos ) and () were added to the link equations. The errors of absolutisation of the other fields were supposed to be equal zero on average. After determining the offset corrections the catalogue of absolute proper motions has been improved and afterwards the corrected catalogue was applied for the determination of the Hipparcos link angles. This solution was called the "corrected catalogue solution". The corresponding results are given in Table 4 (lower lines). As one can see from Table 4 (comparing upper and lower lines) the combined solutions using the original (uncorrected) and corrected catalogues differ insignificantly (except their errors) but hte differences between ( cos ) and () solutions practically disappeared if the corrected catalogue is used. Therefore, later on, only the corrected catalogue was used.
In addition, some solutions of the system were carried out using different methods of weighting the observation equations. Since the accuracies of the Hipparcos and KSZ proper motions are known, the link equations may be weighted as well as different weights may be fixed, depending on the stellar magnitudess. The last line in Table 4 gives the weighted combined solution of equations for the corrected GPM1 catalogue. One can see that both the weighted and unweighted joint solutions are practically the same and afterwards all solutions were carried out without the weighting.
As a next step towards a discussion of the results the residuals of the link equations were examined and tested for non-randomness. From consideration of the covariance matrix follows that correlations between unknowns are practically absent (the correlation coefficients in modulus do not exceed 0.2) and the residuals in ( cos ) and () solutions are normally distributed. In order to examine any systematic dependence on star positions, their magnitudes and colours, the residuals of the link equations were developed in series of orthogonal functions using products of Hermite and Legendre polynomials as well as Fourier terms. No significant above-named dependencies have been obtained.
According to the recommendation of the "Working group on the link of the Hipparcos catalogue to the extragalactic reference frame", an additional analysis of the solutions was carried out in spite of the solutions were not changing practically with inclusion to the link equations of the terms depending from brightness of stars. In general, the Hipparcos rotation angles may be distorted by a magnitude equation. Therefore, the dependencies of the components of angle on brightness of stars were investigated in different way. Some solutions were carried out excluding gradually bright stars. As we found the solutions were different, the angle was strongly depedent on the magnitude range of the stars used in the solution (see Table 5).
Table 5. Change of the angle with exclusion of bright stars
From Table 5 follows that the angle changes its sign beginning from the magnitude which divides the whole file of stars into two approximately equal parts - bright stars and faint stars.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998