4 Analysis method

A global reduction method was applied for photoelectric astrolabe residuals. The approach is much simpler although more precise than the conventional method. The new form of the equations (Martin et al. 1996; Martin & Leister 1997) give us the possibility of using the entire set of the observing programme using the different data taken at two zenith distances.

In this method, the data and the unknowns are treated as symmetrically as possible with a determined statistical weighting. The used symmetric form aims to get the mean residuals with statistical independence and uniform variance.

Considering all the star observations (i) in the involved catalogues (j), we can write, for each transit (east and west - E and W), the following equations (Martin & Leister 1997):

$$\begin{eqnarray} \nonumber -R_{i{\rm E}}^{j}& = &- \Delta \alpha_{i} \mid \sin Z... ...{\rm W}}^{j} \mid + \eta^{j} \cos Z_{i{\rm W}}^{j} - \zeta^{j}\,,\end{eqnarray}$$ (2) (3)

where:

$R_{\rm E}$ and $R_{\rm W}$ are the east and west mean residuals in zenith distance after the 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 $\eta$ 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 $\zeta$ is related to the true zenith distance of observation (instrumental effects); Z is the star azimuth from North to East; $\phi$ the local latitude, and $\xi$ the constant related to the equinox correction.

Adding the mean residuals ($R_{\rm E}$ and $R_{\rm W}$) above, the stellar declination corrections can be computed from astrolabe observations by the formula (Débarbat & Guinot 1970):

$$\begin{eqnarray} \nonumber \Delta \delta_{i}& =& - [(R_{\rm E}+R_{\rm W})]_{i}^{... ...cos Z _{i}^{j}/\cos S_{i}^{j}+\\ & & + \zeta^{j}/\cos S_{i}^{j}\,,\end{eqnarray}$$ (4)

From Eq. (3), we can write:

$$[(R_{\rm E}+R_{\rm W})/2] _{i}^{j} = -\Delta \delta_{i}\cos S_{i}^{j} -\eta^{j} \cos Z_{i}^{j} + \zeta^{j}\,.$$ (5)

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 ($\rho_{i}^{j}$):

$$\begin{eqnarray} \nonumber \rho_{i}^{j} &=& -\Delta \delta_{i} \cos S_{i}^{j} + ... ...{2} + A_{1}I_{i}\\ & & + \Delta z)^{j} - \eta^{j}\cos Z_{i}^{j}\,,\end{eqnarray}$$ (6)

where:

$\rho$ = $(R_{\rm E} + R_{\rm W})/2$; B₁, B₂, A₁ are the magnitude and colour index coefficients; m_i is the visual apparent magnitude of the star; I_i is the visual colour index (B-V) of the star and $\Delta z$ is the zenith distance correction.

The possibility of computing absolute declinations rests on the determination of $\eta$.

This equation can be applied to all stars even to transit stars with only one zenith distance to get systematic differences $\Delta\delta$, which is not possible with the classical methods. For the stars which are at maximum digression condition the coefficients of cosS in the matrix vanishes, and these stars do not contribute to the $\Delta\delta$ determinations.

However, the system has a singularity, which is removed by the condition:

$$\eta_{\rm ASPHO1} = \eta_{\rm ASPHO2} = \eta.$$