5 Geometrical transformations

 

5.1 Polynomial degree of the transformations

  The polynomial degree of the plate models was selected individually for each source plate. The tested plate models were based on Eqs. (4) and (5), without magnitude-dependent terms and without taking into account the proper motions. Since the expected density of cluster members in our field is small, ignoring proper motions and magnitude effects could introduce a larger dispersion (mainly in old epoch plates), but not global systematic effects.

We tried out different polynomial degrees, and we analyzed the residuals of the transformations by computing their difference among consecutive degrees. The n-th degree was accepted when the following two conditions were fulfilled:

1.

The changes of the residuals from the n-th to the (n+1)-th degree are around or below 1 $\mu$m, and

2.

The residuals from the n-th degree fit do not show any spatial systematic behavior.

As an example of the assignation of transformation degrees, Fig. 4 shows the difference of residuals of the transformation among several consecutive degrees for one plate (A 550), and Fig. 5 displays the spatial pattern of the residuals for the same plate and the selected degree (4$^{\rm th}$), averaged in 1 cm squares. Usually, third or fourth degree was enough for accomplishing both required conditions.

  

Figure 4: Change in the residuals of the geometric transformation of the source plate A 550 to the master plate OCA 3305 for successive pairs of polynomial transformations with increasing degrees. The selected degree was 4$^{\rm th}$

  

Figure 5: Spatial behaviour of the residuals of the fourth degree transformation polynomial from the source plate A 550 to the master plate OCA 3305. Each arrow represents the average residual in a 1 cm $\times$ 1 cm square. An arrow with length equivalent to 1 cm represents a mean residual of 0.25 $\mu$m

5.2 Anomalous plates

  While testing the degree for the transformation equations, the low astrometric quality of two plates became evident: A 575 and OCA 3314. The CAHA plate displayed a quite asymmetric systematic distribution of the residuals. These features were difficult to explain from usual optical or mechanical effects in Schmidt plates, and led to residuals of the order of 1 $\mu$m and systematic trends even when applying a 5-th degree polynomial. The reduction of the residuals required a very high degree (8-th or even more). We preferred to eliminate this plate for astrometric applications, although it was used as source of our V photographic photometry.

OCA 3314 plate is film-based. With the stopping of glass-plate production, the use of film substrates for big format photographic emulsions is being considered a possible alternative, at least until CCD technology will be able to cover wide fields with enough astrometric quality. The inclusion of one film among our material provided a chance for testing the astrometric and photometric performance of this kind of detectors compared to classical glass plates.

  

Figure 6: Spatial behaviour of the residuals of the transformation from the film-based source plate OCA 3314 to the master plate OCA 3305. The plate models were complete 2-D polynomials of different degrees, without magnitude terms. Each arrow represents the average residual in a 1 cm $\times$ 1 cm square. An arrow with length equivalent to 1 cm represents a mean residual of 1 $\mu$m (this scale is four times bigger than in Fig. 5)

The photometric accuracy of plate OCA 3314 is excellent. Its filter/emulsion combination (Table 2) matches R band, and the emulsion sensitivity was good enough to yield really small residuals in the photographic calibration (Table 3). However, the astrometric transformation to the master plate system made evident the presence of a horizontal deformation pattern mixed up with some local systematics. Likely, this deformation pattern is present not only in our $13\times13$ cm scanned area, but all over the plate. High degree polynomials reduced, but did not eliminate this pattern (Fig. 6) and some systematic trends still remained even with an 8th degree transformation.

We conclude that the film OCA 3314 has a general spatial deformation, in a 1-2 cm band pattern parallel to x axis, with some other local strong systematics (sink and source points). The systematic trends induced by this deformation are not easy to eliminate, and the best solution is to discard the film plate for astrometric purposes. Whether these conclusions could be extended to all film-based plates or not, is a question worth of attention and should be investigated performing extensive tests with different kinds of film-based photographic plates.