Further plate characteristics influencing the final astrometric accuracy
have been found. The presence of a grid in the plates prevented us from
considering stars lying on or very close to the lines (Fig. 8 (click here)).
Because of this we loose approximately 
of the total number of stars in
the plate. These lines introduce changes in the density of the images and in
the positions of the nearby ones due to the Kostinsky effect
(Kostinsky 1907; Ross 1921). This effect shows up
also when images of stars lie very close to each other.
The developer is exhausted in the zones where two images are in contact
or very close and the amount of the products from the development process
is larger there than in other zones.
These products inhibit the chemical reaction which takes place during the
development and the final result is an apparent repulsion of the images.
Due to the small separation between
the three exposures (
m), they begin to merge in the case
of stars of magnitude 
, and consequently, the Kostinsky effect
introduces changes in the "true'' positions of the images. These can
be quite large: changes of 
m have been
detected (see Fig. 9 (click here)). This effect is very difficult to model: it
changes from plate to plate as it depends on the development process
of each plate individually. The determination of its variation as a function
of the stellar magnitude is one of the most important targets in order to be
able to correct for it.
Figure 8: Example of a triple star image of which one is on a grid line
Figure 9: Distance between exposures 1 and 2 of each star as a function of a
measure of the star brightness. The apparent repulsion between exposures 1
and 2 due to the Kostinsky effect is more evident for the brightest stars
To obtain the internal accuracy of the plate, we have transformed the
positions of two of the three exposures to the reference system of the
third one by means of a linear transformation, and the deviations from
the new computed positions were obtained. It is equivalent to consider
three different single exposure plates. The linear transformation was
computed by least squares fitting of the positions of 552 stars
in the field whose magnitudes ranged from B=12 to B=14.5 to avoid
large errors introduced by very bright or very faint stars.
Assuming that all exposures
equally contribute to the deviations, the rms of these deviations can be
calculated for each individual exposure using 1108 stars with
(thereby excluding the ten ones with larger errors). Since
these plates were taken originally to be complete up to the
magnitude, images fainter than
have poor quality. Almost all the stars with larger errors
were fainter than B=13. The results are presented in the first row in
Table 2 (click here).
| field a | model b |  c
|  d | no. of stars e |
| whole plate | 22 |  |
| 1108 |
 | 22 |  |  | 394 |
| plate centre | 22 |  | | 648 |
| whole plate | 12 |  |
| 1108 |
| plate centre | 12 |  | | 648 |
|
|
This analysis was also performed over the smaller region of
in the plate
covering the zone in which the cluster M67 is located, since a good external
catalogue is available for this area (Girard et al. 1989).
The linear model was in this case computed using a total of 215 stars
with magnitudes ranging from B=12 to B=14.5. The final rms were
obtained from a set of 394 stars brighter than 15.5 (second row in
Table 2 (click here)).
One can see that are larger than those for
. This may indicate that there are some kinds of local
distortions which show up when studying only a small plate region, while they
are absorbed by the model when considering the whole plate.
They can be due to slight emulsion displacements during the plate drying
process or because storage of the plate in vertical position for almost
one hundred years, to small errors in the telescope tracking, or to
optical distortions as the small field considered around M67 is at the plate
edge, where these distortions are expected to be stronger. Also the
background, and consequently the noise, was found to be larger at the
plate borders (see Fig. 12 (click here)).
To determine if these "plate border effects'' could account for the fact that
an internal reduction using stars
closer than 5 cm to the plate centre was carried out. Accuracies of
were found. This result
clearly reflects the poorer quality of plate borders. Therefore, it seems
that the triple Gaussian model can deal with this trend of when considering the whole plate, although the final
accuracies are poorer than when only considering the plate central region.
For the region mentioned before containing the
cluster a separate reduction was performed. The external comparison was
carried out with an astrometric catalogue specifically built for this
field (Girard et al. 1989). It has an external error of
in the star positions for 1950.8, the catalogue weighted mean
epoch. The cluster is located 10' north of the field centre. The plate
model used consisted of polynomials in x and y up to the second order
and the reduction was performed using the mean position over the three
positions corresponding to the three exposures we have per star. In this
way it is possible to avoid to some degree the consequences of the
Kostinsky effect over the brightest stars. The rms of the deviations are
shown in the second row in Table 3 (click here). In these calculations three
stars with exceptionally large deviations were not taken into account and
the star showing the largest deviation after the reduction was also
rejected.
| catalog a | model b |  c
|  d | no. of stars e |
| PPM | 22 |  |  | 10 |
| Girard | 22 |  | | 196 |
| PPM | 12 | |  | 10 |
|
|
The reduction of the full plate was performed using the PPM as an external
catalogue. Only 10 stars (with magnitudes 
)
could be used in the reduction as in the PPM stars are normally brighter than
V=11. The triple images of these stars are quite blended in our plates and
the errors for the parameters obtained in their Gaussian fitting are a bit worse
than the ones corresponding to fainter stars. The plate model used consisted of
polynomials in x and y up to the second order and the rms of the deviations
are shown in the first row in Table 3 (click here). The larger values for
than for can be due to the fact that
the reference stars happen to be in regions of the plate where the optical
distortions are stronger along x-axis than y-axis and the fitting algorithm
is slightly poorer in the case of the brightest stars because when the
exposures merge the determination of the exposure centres is less accurate.
Errors during the telescope tracking can also be an explanation.
In Fig. 10 (click here) the residuals in the mean position are plotted as a function of the plate position. They are uncorrelated so it seems that there are no global systematic errors present in the plate although ten reference stars are clearly too few to categorically affirm that there are no systematic errors left.
Figure 10: Residuals for the mean position of the three exposures after
the plate reduction with the PPM catalog as a function of the plate position.
Arrow lengths have been enlarged 1500 times