Four optical aberrations would mainly be expected to show up in an optical system of the kind used in the CdC astrographs. These are coma, spherical aberration, field curvature and chromatic aberration. In particular, coma makes the images to become asymmetric as it gives them a comet-like appearance. Spherical aberration and field curvature generate extended and elongated images, respectively, from pointlike ones. How much elongated and asymmetric the images are is of importance in order to see how the optical aberrations are influencing the final results in the plate reduction.
Images at a radial distance from the plate centre greater than 
\
cm show a significant ellipticity, which is defined as:
where ak,bk are the image semimajor and semiminor axes of exposure k, respectively, given by Eqs. (2 (click here)) and (3 (click here)).
In Fig. 2 (click here) the ellipticity as a function of the plate position is
shown. Segment lengths are proportional to the mean ellipticity of the three
exposures, while their orientation with respect to the X axis is
the mean of the three angles 
. We expect to encounter this
effect in all CdC plates because they all were taken with
a similar kind of optics. Two examples of these images are shown in
Fig. 3 (click here).
Figure 2: Field curvature in the plate. The length of the
lines is proportional to the mean ellipticity of the three exposures of
each stellar image. The orientation with respect to the x-axis is
proportional to the mean of the three angles
Figure 3: Effects of optical aberrations are shown for two stars of different
magnitudes
No significant dependence of the position error on the ellipticity can be found after performing the plate reduction with the external catalogue (Fig. 4 (click here)). This suggests that the model is therefore able to deal correctly with elongated images.
Figure 4: Image ellipticities as a function of the
residuals in x and y for the three exposures after performing
the plate reduction with an external catalogue. No dependence on the
ellipticity value is evident here.
Ellipticity values of exposure 2 have been used as an example.
Analog diagrams are obtained with the ellipticity values of exposures 1 and 3
A study on the asymmetry of the images in the plate has been carried out by means of analysing the skewness of the marginal distribution along the "possible'' asymmetry axis:
where f(xi) is the marginal distribution along the image semimajor axis
and 
stands for the marginal distribution standard deviation.
Figure 5: Skewness values as a function of the distance to the plate centre.
of the values lie within 
Most of the skewness values () lie on the interval
(Fig. 5 (click here)), which
seems to indicate that the images can be considered symmetric and,
in consequence, that coma is not the dominant cause for the distortion
but rather the field curvature and spherical aberration. The last
one makes the pointlike images to appear as disks while the first one
is responsible for the elongation of these disks.
Figure 6: Histograms of skewness values for stars fainter (dashed
line) and brighter (solid line) than 
. The histogram
for the fainter stars is broader than the one for brighter stars
There are two other facts which reinforce the conclusion that asymmetry has no influence on the data:
1) The largest skewness values correspond to the fainter images, which
are much sensitive to plate defects and noise which might alter their
density profile. Figure 6 (click here) shows the histograms for the skewnesses of
stars with magnitude 
and 
.
The standard deviation of both histograms has been calculated and the results
show that the distribution for the brightest stars around 0 is narrower
than the corresponding one for the faintest ones, as one can see from
Table 1 (click here). At the bright end the non-linear response of the plate
also
contributes to this effect as the stars are increasingly broader thus masking
features visible in fainter star images.
| B magnitude a | data b |  c |  d |
 e |
 | 337 | 0.16 | 0.16 | 0.15 |
 | 910 | 0.25 | 0.30 | 0.27 |
This suggests that the images are symmetric and asymmetries only arise because of noise in faint stars. This is a normal effect in photographic plates and it is not a specific problem of the CdC plates.
2) The residuals for positions on the plate after the plate reduction with the catalogue (PPM) are not a function of the skewness values (Fig. 7 (click here)).
Figure 7: Residuals for star positions after the plate reduction with an
external catalogue vs. skewness values. Residuals are independent of the
skewness. As an example we show here the skewness computed for exposure 2.
Analog diagrams are obtained for skewness values of exposures 1 and 3
As a conclusion one can say that at least the asymmetry in the images is not important enough to have a detectable influence on the data. Anyway, it has been found that optical aberrations in CdC plates can be very complicated when the centres of the telescope lenses were misaligned, being dependent on magnitude and colour (Zacharias, private communication). And, as usual when working with photographic plates, plate borders are of lower quality than the plate central region as they introduce some trends in the final position residuals when performing the plate reduction.
