In this section, we discuss three ways to determine the absolute declination corrections: Krejnin's Method, the Method of Differences, and the Global Reduction from astrolabe data.

Considering all the star observations (*i*) in the involved
catalogues (*j*),
we can write the following equations for each transit (east and
west - E and W):

where:

and are the east and west mean residuals
in zenith distance after addition of group corrections for east
and west
transits of the star (homogenisation of the stellar groups), and
*S* is the parallactic angle. The constant can be
understood as the difference between the
real and observed values of the latitude after the homogenisation
of the stellar groups (related with the equator correction);
the constant is related to the true zenith distance of
observation
(instrumental effects);
*Z* is the star azimuth; the local latitude, and
the constant related to the equinox correction.

Adding the mean residuals ( and ) above, the stellar declination corrections can be computed from astrolabe observations by the formula (Débarbat & Guinot 1970):

Using the stars at maximum digression condition (cos), the Eq. (3) can be rewritten as a function of only the constant :

where *C* is a constant independent of the star coordinates
(*C*=*f*(*I*_{i},
*m*_{i}), *I*_{i} = visual colour index (*B*-*V*) and *m*_{i} =
visual apparent magnitude), and *z* is the zenith
distance of observation. The possibility
of computing absolute declinations rests in the determination of
.

Let and be the
declination correction
values of a given star (*i*) observed at the zenith distances
*z*_{A} () and *z*_{B} () at the same latitude
.

From Eq. (4), we can write:

where:

Therefore:

or

The left-hand side is independent of the selected star and so
must be the
right-hand side. Therefore, we are authorised to take the average
of the data
from all available stars, so that we arrive at a single equation
connecting
and .

On the other hand, the instantaneous value of the latitude can be written from the computed values and , after application of group corrections, as:

In a similar way as we have done for Eq. (5), the right-hand side of Eq. (6) can be averaged along the observing time span. We thus get an independent relationship which allows the determination of and and, consequently through Eq. (4), of the star declination corrections. This is the principle of Krejnin's method.

We reduced Eq. (3) by least squares method only for the
stars common to the programmes taken at (*A*) and
(*B*) zenith distances, i.e., the stars common to the
VL1 and VL3 catalogues and the stars common to the VL2 and VL3
catalogues.

However, as the programmes were taken in the same declination zone and were referred to the same fundamental catalogue (FK5), we can hope that and are small and similar, otherwise the system has a singularity.

In this case, = ; = ; = and = , which were confirmed by the results from Krejnin's method.

Thus, we can write for each star common to both catalogues (*A* and
*B*):

where: and is the declination correction.

It is important to note that the zenith distance correction () contains colour and magnitude effects.

From the difference between the Eqs. (7), we can write:

The advantages of an analysis of the system of Eqs. (7) are to obtain by the least squares method the terms and , the absolute declination corrections (), and a better estimate of the correlation between the unknowns.

We used global reduction to determine the absolute declinations with a modern method of astrometric reduction that includes all of the information in a set of equations, with all of the unknowns.

In this method, the data and the unknowns are treated as symmetrically as possible with a determined statistical weighting. The symmetric form to use the data is directly to get the mean residuals for which it is reasonable to claim statistical independence and uniform variance.

An example of this application is the overlap reduction (Eichhorn 1960) and global reduction techniques (Benevides-Soares & Clauzet 1986; Basso 1991; Benevides-Soares & Teixeira 1992; Chollet & Najid 1992; Chollet 1993; Najid 1993).

From Eq. (3), we can write:

Considering all of the star observations with double passage
(E/W), and
the colour index and magnitude function *C*(*I*_{i},*m*_{i}),
we get the average of the mean residuals ():

where: *B*_{1}, *B*_{2}, *A*_{1} = magnitude and colour index
coefficients;
*m*_{i} = visual apparent magnitude of the star; *I*_{i} = visual
colour index (*B*-*V*) of the star and = zenith distance
correction.

Equation (9) can be applied to all stars. For those which are
at
maximum digression condition (0.3,
), the coefficient of cos*S* in the matrix
vanishes,
and these stars do not contribute to the
determination.

This method uses all of the stars, even those which were observed
in
only
one zenith distance (*z*). These stars do not contribute to the
solution (equator correction)
or the colour-magnitude effects, but they contribute to the
determination and to the computed standard deviation.

The importance of this method is that it allows the utilization even of transit stars at only one zenith distance to get systematic differences , which is not possible with the other two methods presented in this paper.

The method has the advantage to obtain simultaneously the
differences in positions, the equator correction and the
instrumental effects. The systematic
effects are dependent of the star position and the physical
characteristics
that directly influence the quality of the observation.
However, the system
has a singularity, which is removed with the condition:

The results for latitude imply that is the same for all
catalogues involved, and so the singularity of the system is
removed.

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