4 Method of data reduction

The Fifth Fundamental Catalogue (FK5) and the IAU (1976) System of Astronomical Constants have been used in the data reduction to obtain the residuals.

As for previous reductions (see for instance Guinot 1958), the chain method was used to get group corrections in this work. Since winter groups have been observed more times, and summer groups less, we use precise formulae to calculate group corrections (Thomas 1966):

$$C[i] = -\frac{1}{n}\sum_{r=0}^{n-1}(r-F[i]) \triangle x[i+r] .$$ (1)

Here C[i] is correction of group i (i=1,...,n; n = 12), $\triangle x[i+r] = x[i+r+1] - x[i+r]$, and

$$F[i] = \sum_{r=0}^{n-1}r E^{2}[i+r]/E_{k}^{2}.$$ (2)

Here E[i+r] is the mean error of group difference $\triangle x[i+r]$ and E_k the error of closing error k. The error of group correction C[i] is given by:

$$Ec[i] = \pm \frac{1}{n} \{ \sum_{r=0}^{n-1}(r-F[i])^2 E^2 [i+r]\}^{\frac{1}{2}}.$$ (3)

The mean residual of star j in group i with respect to the mean group is:

$$M[j] \!=\! V[j] \!+\! Cu[i] 15 \cos \phi \sin A[j] \!+\! Cy[i] \cos A[j] \!+\! Cz[i] .$$ (4)

Here V[j] is the average of residuals of star j in group i.

Through observing double transits of a star (one in the East of almucantar, the other in the West), we could have two residuals $M_{\rm e}$ and $M_{\rm w}$ which could give corrections to both right ascension and declination of the star (Guinot 1958):

$$\triangle \alpha = \frac{M_{\rm e} - M_{\rm w}}{30\vert\sin A\vert \cos \phi} ,$$ (5)

$$\triangle \delta = - \frac{M_{\rm e} + M_{\rm w}}{2\cos Q} .$$ (6)

Here $\phi$ is the latitude of the instrument position.