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Astron. Astrophys. 333, 531-539 (1998)

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3. Determination of absolute proper motions

With respect to the available plate material, i.e. with a relatively wide spread of observation epochs, we followed the same principles of the proper motion determination as in our extended work in the field of the globular cluster M 3 (Scholz, Meusinger & Irwin 1997). The idea is to build up a time series of coordinates for each single object measured on n plates. Within the plate matching process objects measured on the comparison plates are identified with objects measured on a deep reference plate. The plate matching is done iteratively starting with bright objects and large search radii (up to several mm), finally iterating down to faint objects within a target search radius of about [FORMULA] m, corresponding to 2 arcsec in the scale of the UKST reference plate.

All matched objects are used in a plate-to-plate solution for transforming the measured coordinates on the comparison plate into the coordinate system of the reference plate. After removing residual mean positional differences between the transformed coordinates and the coordinates on the reference plate as a function of the field position (for more details see Sect.  3.1), a zero point correction is applied by subtracting the mean coordinate difference of all available galaxies measured on the comparison plate and on the reference plate (see Sect.  3.2). The absolute proper motion of each object is then determined from the linear regression of the coordinates [FORMULA] over the epochs [FORMULA], with [FORMULA], where the number of plates [FORMULA] for a given object is mainly dependent on its magnitude. The advantage of this method are the large numbers of objects used in the differential plate-to-plate solutions. It is independent of an external reference catalogue (and the errors of such a catalogue).

3.1. Systematic error removal

The error correction is a substantial part of the reduction in the differential astrometry with full-scanned Schmidt plates. For a detailed description of the error reduction technique developed for the correction of APM scans of Schmidt plates see Evans (1988) and Evans & Irwin (1995). We followed the same principles in our systematic error removal, i.e. we applied a two-dimensional and one-dimensional correction of errors as a function of field position. The basic assumption of this technique is a constant motion over the whole field of all objects used in the error reduction. The central globular cluster region ([FORMULA]  arcmin), all faint objects ([FORMULA]) as well as the bright objects ([FORMULA]) were excluded from the sample of objects used for the determination of the corrections.

In this process we remove periodic errors of the measuring machine as well as residual distortions after a global plate-to-plate solution. These distortions are expected to be larger in the case of Schmidt plates from different telescopes taken with different plate centres. The measuring coordinate frame of each comparison plate was therefore transformed to that of the reference plate by using a quadratic [FORMULA] instead of the linear [FORMULA] plate constant polynomial relationship, usually applied in plate-to-plate solutions of Schmidt plates of the same telescope taken with similar conditions.

Although the residual systematic errors after the quadratic polynomial transformation are considerably reduced in comparison to the linear polynomial transformation we first used a two-dimensional error correction. The two-dimensional correction is done by binning the whole field into subareas and simply subtracting the mean shift [FORMULA] and [FORMULA] separately for each subarea and for each pair of comparison/reference plates. In our case we binned the 1 square degree field into 8  [FORMULA]  8 subareas.

After the two-dimensional error correction removing large scale systematic effects we applied a one-dimensional error correction removing the periodic errors of the APM in dependence on the x -coordinate. A detailed description of these errors can be found in Evans (1988) and Evans & Irwin (1995). There are no periodic errors of the APM as a function of the y -coordinate.

As an example, Fig. 2 shows the uncorrected coordinate differences as a function of the x -coordinate for the plate or13078 with respect to the reference plate j5193. The result of the two- and one-dimensional error correction is seen in Fig. 3.

[FIGURE] Fig. 1. The 1 square degree field around the globular cluster Pal 5 used in the proper motion determination. About 8000 objects measured on at least 3 out of 11 Schmidt plates are shown. The dashed circle shows the cluster region with a radius of 6 arcmin. The open circles denote the positions of the galaxies used to define an inertial reference frame.

[FIGURE] Fig. 2. Residual errors after the plate-to-plate solution with quadratic [FORMULA] plate constant polynomial and before any correction. As an example the differences dx and dy between the coordinates on the reference plate j5193 and the comparison plate or13078 as a function of the x -coordinate are shown. There are no y -dependent periodic errors due to the APM.

[FIGURE] Fig. 3. Residual errors after the two- and one-dimensional error removal (cf. previous figure).

Note that all corrections are carried out with respect to the reference plate, i.e. differentially. This technique works well in the differential determination of proper motions from plate-to-plate comparisons but does not improve the determination of equatorial coordinates by the use of an external reference catalogue with an object density far below the measured object density on the Schmidt plates.

Throughout the paper we use x - and y -proper motion components, which are equal to [FORMULA] and [FORMULA] in this zero declination field.

3.2. Selection of reference galaxies

The reference galaxies defining the absolute reference frame in our 1 square degree field were carefully selected using the 4 deepest R plates with first priority. The APM image classification becomes uncertain near the plate limits. Therefore, we selected all objects classified as galaxies on at least three out of four deep R plates. All objects classified as merged on at least one of the 10 APM measured plates were excluded from the sample of galaxies. The FITS frame of the POSS 2 plate was used for a visual check of the automated image classification using the MRSP software (Horstmann et al. 1989). 384 reference galaxies were selected. Their distribution over the field is shown in Fig. 1.

The central region ([FORMULA] arcmin) of the globular cluster was excluded in the selection of reference galaxies due to crowding effects. For comparison with our selection of reference galaxies, we have also looked for galaxies in the NASA/IPAC extragalactic database (NED). North from the globular cluster, at [FORMULA] there is a known cluster of galaxies, Abell 2050, also seen in our selection (Fig. 1).

The absolute zero point shift was determined for each comparison plate with respect to the reference plate using all available galaxies (from 384) measured on the comparison plate. The formal zero point error varied in dependence on the number of galaxies (only 50 galaxies on the Tautenburg plates but from 230 to 375 galaxies on the other comparison plates) and on their measuring accuracy on the comparison plate and on the reference plate. The zero point error was about 70 mas for the Tautenburg plates and [FORMULA]  mas for all other comparison plates.

3.3. Internal accuracy of proper motions

The x - and y -components of the absolute proper motion of each individual object were obtained from the linear regression of all available x - and y -coordinates of this object over the time. The coordinates were used independently of the image classification of the object. No weights were given to the different plates the object was measured on.

Measurements on at least three plates were required for the determination of the proper motion of an object. The final catalogue of proper motions contains nearly 8000 objects in a 1 square degree field centered on the globular cluster Pal 5. Their internal errors as obtained from the linear regression of min. 3 to max. 11 data points are shown in Fig. 4 (mean errors as a function of magnitude) and Fig. 5.

On the average the errors of objects with stellar classification on the reference plate are about 1 mas/yr smaller than the overall proper motion errors.


[FIGURE] Fig. 4. Mean proper motion errors as a function of magnitude. The individual proper motion errors of the bright objects ([FORMULA]) were binned in two intervals of 2 mag width, whereas the binning for fainter objects was done using a width of 1 mag. The [FORMULA] and [FORMULA], respectively show the internal proper motion errors in [FORMULA] and [FORMULA] for all objects measured on at least three plates. Smaller symbols represent mean proper motion errors of objects with stellar classification on the reference plate.

[FIGURE] Fig. 5. Histogram of internal proper motion errors in [FORMULA] and [FORMULA] for all objects measured on at least three plates (solid line), objects with stellar classification on the reference plate and with measurements on at least three plates (dotted line) and star-like objects measured on more than 6 plates (dashed line).

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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