2. Reduction method

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.

2.1. Krejnin's method

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.

2.2. Method of differences

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.

2.3. Global reduction

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: B₁, B₂, A₁ = 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 cosS 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.