At this stage we have a catalogue of
galaxy candidates. We now have
to make the "auto-crossidentification'' to merge a same object seen on different
plates. Because the information on the original plate will be lost in
such a merging process we have to apply now the corrections
which are plate-dependent, like the effects of the mean airmass extinction or the
distance to the center of the chart
(Rousseau et al. 1996;
Garnier et al. 1996).
We use a Principal Component Analysis method applied on pixels positions (i,j)of the matrix associated to an object. So, we derive for each object
a
covariance
matrix from which we calculate its eigenvalues (v1 and v2) and
corresponding eigenvectors.
The position angle
of the major axis is
determined from the direction of the first eigenvector
(eigenvector associated with the highest eigenvalue).
The major and minor axes are deduced from the square root of the
first and second eigenvalues. The apparent magnitude is deduced from
the sum of all pixel intensities. We thus obtain the following parameters:
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(5) |
![]() |
(6) |
![]() |
(7) |
![]() |
(8) |
In order to calibrate these equations we extracted from the LEDA database
the apparent blue diameter D25,
the major to minor axis ratio
R25= D25/d25
(axes are defined at the isophote 25 mag arcsec-2) and the total
magnitude
for the galaxies of the training sample.
These quantities are in the system of the Third Reference Catalogue
(RC3,
de Vaucouleurs et al. 1991).
We get the following results (
is the standard deviation, n is the
number of remaining objects after 3-
rejection:
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(9) |
![]() |
(10) |
![]() |
(11) |
For stars, a comparison with SAO magnitudes gives a preliminary calibration:
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(12) |
The catalogue of
galaxies is sorted according to the declination
(the search is easier and faster with such a sorting). Each galaxy is compared
with all the others. This is done four times because one galaxy may only appear
four times at the intersection of four charts. At each of these four iterations
only the locally two closest galaxies are merged if their separation calculated along
a great circle is smaller than a given limit. This procedure avoids the result
depending on the order the galaxies are considered (in other words, the merging
is done according to a physical measurement but not following an arbitrary order).
The limit of the separation is calculated from the actual uncertainty on
the position:
Thus, we will still work with the catalogue of
galaxy candidates
(a direct merging would have lead to a catalogue of
galaxies. No
inconsistency is found).
In view of this cross-identification we carried out a campaign of measurement of accurate coordinates. More than 34000 positions of LEDA galaxies were measured (Paturel et al. 1999; Paturel et al. 2000) and we studied the accuracy of the coordinates provided to us by large catalogues (Paturel & Petit 1999). We added some recent accurate measurements (Cotton et al. 1999). After this work we have a list of 194544 galaxies from LEDA with accurate coordinates and the main astrophysical parameters (diameter, axis ratio, position angle and magnitude).
The cross-identification is based essentially on coordinates using
a method similar to the one used for the auto-crossidentification. Nevertheless,
two modifications are introduced: 1) The limit of the separation is calculated
from the previous formula (Rel. 13) but the value of
is deduced
from the weighted mean of the coordinate accuracy
(Paturel & Petit 1999)
and quadratically increased by the uncertainty of the DSS coordinates
(
), because the coordinates we are comparing have independent errors
(this was not the case for auto-crossidentification).
2) When several galaxies match the position criterion we use astrophysical
parameters to choose the best one. For this purpose we calculate a generalized separation between
the objects according to
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(14) |
At this stage we build the mean catalogue where a galaxy appearing
several times is merged into one object. There is no practical difficulty
because each object has its internal number from the auto-crossidentification step.
Nevertheless, we must take into account that
some periodical parameters (like the right
ascension or the position angle) must be treated with special care.
For instance, two measurements of the position angle of a galaxy elongated
in the N-S direction may produce, e.g.,
and
.
The mean of both measurements is not
but
.
After
having merged all objects appearing on different plates we obtain
a catalogue of
objects.
Automatic program of galaxy recognition cannot differentiate a true galaxy from, e.g. a planetary nebula or a globular cluster. Further, filaments in a bright nebula, in a HII region, in the neighborhood of a very large galaxy or in the halo of a very bright star can well be recognized as a galaxy. In order to remove such artefacts we constituted a catalogue by collecting objects prone to create them. This catalogue of "forbidden zones" is built from the following objects:
For stars the forbidden
zone is the central circle (diameter
)
and the branches of
the diffraction cross. The total extension (with both arms) of one branch is estimated to
B=-3 mv +25 (arcmin).
For galaxies, the forbidden zone is the surface of the ellipse
defined by its axes D25 and d25 (at the isophote 25 mag arcsec-2)
and the position angle of the major axis
(from North towards East).
For all other objects the forbidden zone is the surface of the object assumed to be
circular of diameter D.
The forbidden zone catalogue gives for each object: the code (ST, GA, GC, OC, BN,
H2, PN), the right ascension and declination for equinox 2000, and the parameters
for the definition of the forbidden zone (
and B for stars, D25, d25and
for galaxies and D for others). This catalogue is sorted according to
declination and contains 21921 objects.
In Table 3 we give the number of rejected objects for the different
classes of forbidden zones.
Code | Object | Number of rejection |
ST | Stars (mv>7) | 4028 |
GA | Galaxies (>5') | 34 017 |
H2 | HII regions | 25 560 |
GC | Globular clusters | 1906 |
OC | Open clusters | 12 578 |
BN | Bright Nebulae | 112 318 |
PN | Planetary Nebulae | 196 |
Total | 190 603 |
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Figure 13:
Completeness curve for the main catalogue of
![]() |
The slope of the linear part is
.
This is significantly less than the theoretical value (0.6). This result
has been permanently found and has been interpreted in several ways (fractality,
incompleteness, flat distribution of galaxies).
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