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5. Investigation of magnitude dependent errors

Figure 3 (click here) and  4 (click here) show the proper motion differences H37cr-absolute for all 360 selected Hipparcos stars before the determination of the spin parameters. The data were binned in five intervals, with 27, 77, 145, 84 and 26 (from bright to faint) stars, respectively.

  figure325
Figure 3: Proper motion differences H37cr-absolute in tex2html_wrap_inline1138 over the B magnitude. All 360 stars with reliable measurements are plotted. Filled squares with error bars show the mean differences and their dispersions in five intervals: tex2html_wrap_inline1142 and tex2html_wrap_inline1144. Only the stars on the right side of the dashed line (tex2html_wrap_inline1056) were used in the final solution

  figure336
Figure 4: Same as Fig. 3, but for proper motion differences tex2html_wrap_inline1148. Filled rhombs and error bars show the mean proper motion differences and their dispersions over the B magnitude

  figure346
Figure 5: Change of the dispersion of the proper motion differences H37cr-absolute in tex2html_wrap_inline1138 (filled squares) and tex2html_wrap_inline1148 (filled rhombs) over the B magnitude. The 360 stars with reliable measurements were binned in five intervals: tex2html_wrap_inline1142 and tex2html_wrap_inline1144. Only the stars on the right side of the dashed line (tex2html_wrap_inline1056) were used in the final solution

  figure358
Figure 6: Values of tex2html_wrap_inline1164 computed from all stars with tex2html_wrap_inline1166. The lower diagrams show for the star solution the number of stars included and for the field solution the range of the number of stars per field and as crosses the mean number over all fields

Figure 5 (click here) shows the dispersion of the proper motion differences H37cr-absolute for the same magnitude intervals. A clear trend towards smaller dispersions with fainter stars can be seen. The larger dispersions of the proper motion differences H37cr-absolute of bright stars in Fig. 5 (click here) are caused by larger errors in our proper motion determination for these stars. There is a deviation of the mean differences H37cr-absolute from zero seen in Fig. 3 (click here), and especially in Fig. 4 (click here) for the bright stars. This possible systematic error in the obtained absolute proper motions of bright stars makes an exclusion of these stars from the final link plausible. A further argument for their exclusion is their nonuniform distribution over the link fields (compare the last two columns in Table 1 (click here)) which may lead to systematic effect in the determination of the spin parameters.

A simple test of the magnitude dependence of the link coefficients was made: The whole set was subdivided into 2 subsets of 180 bright (tex2html_wrap_inline1168) and 180 faint (B > 9.5) stars. The computation of the rotation parameters tex2html_wrap_inline1172, tex2html_wrap_inline1174, tex2html_wrap_inline1176 yielded significantly different results for these subsets. To study this magnitude dependent effect in more detail, the bright stars were successively omitted from the solution in steps of 0.1 mag (see Fig. 6 (click here)). Due to the relatively small number of bright stars (only 27 stars with B < 8) there are almost no changes in the left part of the diagrams in Fig. 6 (click here). But from tex2html_wrap_inline1180 to tex2html_wrap_inline1182 a trend can be seen, i.e. the bright stars increase the values of tex2html_wrap_inline1172 and tex2html_wrap_inline1174 systematically. The number of stars per field (see lower right diagram of Fig. 6 (click here)) decreases rapidly for tex2html_wrap_inline1188. From tex2html_wrap_inline1190 there are only one or no stars in some of the link fields, i.e. the solution becomes unstable. Therefore, the selected value tex2html_wrap_inline1192 represents a compromise between a sufficient number of stars (per link field) and a minimum influence of bright stars affected by larger (and possible systematic) proper motion errors.

Final solution with 256 stars in 24 fields
referred to the final Hipparcos system (see Sect. 6 (click here))
star solution field solution
residual spin parameters [mas/yr]
tex2html_wrap_inline1194 tex2html_wrap_inline1196
tex2html_wrap_inline1198 tex2html_wrap_inline1200
tex2html_wrap_inline1202 tex2html_wrap_inline1204
rms of solution
6.5 mas/yr 3.1 mas/yr
correlation coefficients
rxy = +0.20 rxy = +0.21
rxz = -0.04 rxz = -0.04
ryz = -0.04 ryz = -0.01.

The final solution for the spin parameters was obtained with 256 stars with tex2html_wrap_inline1218 distributed over 24 link fields. The minimum number of stars per field was 4.

The star solution and the field solution show the same behaviour in Fig. 6 (click here) although the errors of the spin parameters from the field solution are larger. In every case, the star solution and the field solution agree within their errors.


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