From the digitized images the x-y-positions of the stars were
determined by means of a least-squares fit of a (modified)
two-dimensional Gaussian. The details of this approach have already
been described in Paper I.
In general, the stellar images
were well fitted by a spherically symmetric distribution. Images
with strong elongation appeared only near the edges and corners of
plates from the Greenwich normal astrograph. Extensive tests with
different plates and different image models showed that the effect
becomes detrimental to astrometry for distances larger than 50mm
from the plate center (angular distance 083). The reason is
that the images are not merely elongated, but have an asymmetric
profile along their major axis. A fit of an asymmetric model
profile did not lead to satisfactory results with regard to the position.
Therefore we decided to accept measurements from these plates only
within the above mentioned limit.
Depending on the brightness of the star and on the quality of the
exposure, the fitting procedure gave formal errors for the coordinates
of the image center in the range of 0.2
to 0.6
. This
"centering error" describes the goodness of the centering process but
does not account for the (large-scale) measuring
accuracy of the machine, the positional stability of the emulsion
etc. The total accuracy of the stellar positions as derived
by comparing results from different plates of the same epoch
is between 1.2
and 1.8
per plate.
Although the emphasis of this study is on proper
motions, we put some effort also on the photometric evaluation
of the plates, because photometry is an important complement to
proper motions and for most of the fainter stars in our list
dedicated photometric measurements are lacking.
Eight plates from the blue passband and two additional plates
taken in the visual band were selected for the determination of stellar
magnitudes. The instrumental magnitudes from the
the fitting procedure were transformed to standard B
and V by means of photoelectric and CCD measurements of up to 50 stars
from de Vaucouleurs et al. (1994), Doroshenko (1994),
Oja (1987),
Carney & Latham (1987) and HIC.
The agreement between the results from different plates was generally
better than 01. A comparison with independent measurements that
were not used in the calibration process also gave residuals of
not more than 01. Therefore we assume that the accuracy of our
results is at this level. From the distribution of measured magnitudes
it follows that our sample of stars is (in the central part
of the field) complete to about 125 in V an 135 in B, while the
magnitude limits of the sample are 145 and 155 respectively.
Three extragalactic sources in the field around M81 are bright
enough to be visible not only on Schmidt plates but also on early
astrograph exposures. They are located in the center of M81
and in the inner parts of NGC3077 and M82 (central 30'' to 120'').
The center of M81 is an exceptionally bright source
with a rather high degree of symmetry, i.e. the isophotes can
be represented by a system of concentric ellipses.
In order to see what can be achieved with a normal centering
method, we applied a fit of an elliptical model distribution
to the images of this source. Depending on the
quality of the individual images, internal accuracies between
0.3 and 0.7
for the position of the object center
were obtained. This indicates that the fitting
method is indeed applicable in this case.
The other two sources show complex and multiple structures.
In M82 we find two groups of knots (commonly interpreted as
starburst regions), which are separated by a strong dust filament.
Here, a fit of a suitable model function is practically impossible.
However, the positional information contained in the images of
these sources is accessible through direct comparison between
the images by means of maximum cross-correlation.
The practical implementation of this method works as follows:
1) All images are rebinned to a common plate scale.
2) The essential structure (the "signal") is extracted on all
plates in images of equal size.
3) One image of the series is chosen as a "reference image" and
an arbitrary point in it (e.g. the center of the frame) is selected
as "reference point".
4) Each image is shifted into the coordinate domain of the
reference image. The cross-correlation between the images
is calculated as a function of the shift vector. The optimum
shift, at which the crosscorrelation is at maximum, is searched
.
5) Finally, with the optimum shift vectors the location of the
reference point is transferred from the reference plate to all
other plates.
The cross-correlation (see Eq. (2) of Paper I) can be calculated
in different ways.
One can either consider the images as step functions and evaluate
C only for shifts, which are integer multiples of one pixel in
x and y. Then, the maximum of C must be found by interpolating
in the grid of values of C, for instance with a 2nd order polynomial.
Or one can consider the images as continous functions, for instance with
bilinear interpolation between adjacent pixel values. This
allows to evaluate C with continously varying shifts and to find the
maximum of C directly.
Both possibilities were tested on the images of M81, M82 and NGC3077.
The positions of the maximum of C from both methods
were found to be consistent on the level of 0.04 pixel or 0.4
in each coordinate (rms deviation).
We also tested how much the results of the cross-correlation depend
on the particular choice of the reference image and the frame size.
Again, the deviations between the different results were found to
be small and of minor importance in the total positional error budget.
The final results of both the cross-correlation method and the
fitting method (in the case of M81) are summarized in Table 2 (click here).
The figures given there were obtained at the end of the plate reduction
described in the next section.
An estimate of the positional accuracy can be read off from the
dispersions 
around the mean trend. For M81 and
M82e the positions from the different plates agree on the level
of 015, while for NGC3077 the dispersion in the positions
is somewhat larger and reaches 020 to 024. The latter
is probably not due to the source structure, but can be understood
as a consequence of the weakness of the source.