5 Errors

5.1 Total internal errors

The total internal errors $\epsilon_{\rm T}$ of the positions of the sources, given in Cols. 3 and 5 of Table 1, were estimated by means of the relation: $\epsilon_{\rm T}^{2} = \epsilon_{\rm r,ccd}^{2} + (\epsilon_{\rm t,ccd}^{2} + \overline{\epsilon}_{\rm t}^{2})$ $\Sigma D_{\rm r,ccd}^{2}$, in which $\epsilon_{\rm r,ccd}$ and $\epsilon_{\rm t,ccd}$ are the errors in the determination of the source and tertiary stars image centroids positions on the CCD frame, respectively, and $\overline{\epsilon}_{\rm t}$ is the mean value of the total internal errors of the positions determined for the tertiary stars on the long exposure plate. $\Sigma D_{\rm r,ccd}^{2}$ is the sum of the squares of the dependences (Schlesinger 1911) for the source on the CCD frame. $\overline{\epsilon}_{\rm t}$ was estimated by means of the analogous relation: $\overline{\epsilon}_{\rm t}^{2} = \epsilon_{\rm t,l}^{2} + (\epsilon_{\rm s,l}^{2} + \overline{\epsilon}_{\rm s}^{2})\overline{\Sigma D_{\rm t,l}^{2}}$, in which $\epsilon_{\rm t,l}$ and $\epsilon_{\rm s,l}$ are the errors in the measurement of the tertiary and secondary stars positions on the long exposure plate, respectively, and $\overline{\epsilon}_{\rm s}$is the mean value of the total internal errors of the positions determined for the secondary reference stars on the short exposure plate. $\overline{\Sigma D_{\rm t,l}^{2}}$ is the mean value of the sum of the squares of the dependences for the tertiary stars on the long exposure plate. Finally, $\overline{\epsilon}_{\rm s}$ was in turn estimated from: $\overline{\epsilon}_{\rm s}^{2} = \epsilon_{\rm s,s}^{2} + (\epsilon_{\rm p,s}^{2} + \epsilon_{\rm C}^{2})\overline{\Sigma D_{\rm s,s}^{2}}$, in which $\epsilon_{\rm s,s}$ and $\epsilon_{\rm p,s}$ are the errors in the measurement of the secondary and primary (catalogue) star positions on the short exposure plate, respectively, and $\epsilon_{\rm C}$ is the total error of the Hipparcos Catalogue in the corresponding zone of the sky. $\overline{\Sigma D_{\rm s,s}^{2}}$ is the mean value of the sum of the square of the dependences for the secondary stars on the short exposure plate. Although it was predictable that the error contribution of the Hipparcos Catalogue $(\epsilon_{\rm C}$, which comprises the error in the positions and the errors accumulated due to uncertainties in the proper motions) would be negligible compared to the measurement errors, we nevertheless did not use mean values, and calculated $\epsilon_{\rm C}$ on a zonal basis. It was determined that $\epsilon_{\rm C}(\alpha)$ and $\epsilon_{\rm C}(\delta)$ varied between 4.7 and 10 mas, and 1.1 and 8.9 mas, respectively.

The total internal errors of the positions derived from direct photography were estimated by means of the relations given in Sect. 4.1 of Costa & Loyola (1992).

5.2 Measurement errors

The errors in the measurement of the catalogue, secondary and tertiary reference stars positions on the photographic plates used in the above relations were average values previously determined, this because in practice only one setting was made on all reference stars. The values adopted were $\epsilon_{\rm p,s}(X)=\pm0.19\hbox{$^{\prime\prime}$}$, $\epsilon_{\rm p,s}(Y)=\pm0.18\hbox{$^{\prime\prime}$}$,$\epsilon_{\rm s,s}(X)=\pm0.10\hbox{$^{\prime\prime}$}$,$\epsilon_{\rm s,s}(Y)=\pm0.11\hbox{$^{\prime\prime}$}$,$\epsilon_{\rm s,l}(X)=\pm0.13\hbox{$^{\prime\prime}$}$,$\epsilon_{\rm s,l}(Y)=\pm0.14\hbox{$^{\prime\prime}$}$,$\epsilon_{\rm t,l}(X)=\pm0.12\hbox{$^{\prime\prime}$}$ and $\epsilon_{\rm t,l}(Y)=\pm0.13\hbox{$^{\prime\prime}$}$.

As error in the determination of the tertiary stars image centroid positions on each combined CCD frame, $\epsilon_{\rm t,ccd}$, we adopted the standard deviation of the mean of the X, Y positions of these stars in the four individual frames that were used to produce the combined CCD image for each target, from which their final X, Y positions were determined. $\epsilon_{\rm t,ccd}$ varied between 3 and 25 mas in X, and between 3 and 13 mas in Y.

The same method was used to estimate the error in the determination of the source image centroid position on each combined CCD frame, $\epsilon_{\rm r,ccd}$. It should be noted however, that because of the extreme faintness of some of the optical counterparts, their PSF were clearly less well defined in the individual frames compared to the combined image (this was not the case for the much brighter tertiary stars). For this reason, we consider that this procedure yields in general an upper limit of $\epsilon_{\rm r,ccd}$. This error varied between 4 and 80 mas in X, and between 3 and 90 mas in Y.

For those cases in which the CERS was visible on the long exposure plate, the standard deviation of the mean of multiple settings made on the optical counterpart was adopted as the measurement error. This error varied between 70 and 220 mas in X, and between 110 and 160 mas in Y.

In any case, it must be kept in mind that all of the above Sigmas are based on only four (and in a few cases on only three) independent measures.

  

Figure 2

  

Figure 3

  

Figure 4

  

Figure 5

  

Figure 6

  

Figure 7