5 Precision of corrections

Precision of corrections in right ascension depends largely on the azimuth of the star observed, and precision of corrections in declination depends on $\cos Q$. For most astrolabes and middle latitudes, we often have $\cos Q = 0$or else $\cos Q$stays between -0.35 and +0.35 for many stars that have either no corrections or no precise corrections in declination. Those stars occupy quite wide zones in declination, which is often called the blind zone in declination measurements. But the PHA I in Irkutsk always has $\cos Q \geq 0.5$, so it has no such blind zone (Li Dongming et al. 1983). To see this, we could use another expression of $\cos Q$ (Xu Jiayan et al. 1994):

$$\cos Q = (\sin \phi /\cos Z - \sin \delta )/\tan Z/\cos \delta .$$ (7)

When

$$\phi + Z \gt 90^\circ,$$ (8)

we always have $\cos Q \gt 0$. Since for the PHA I, $Z = 45^\circ$,$\phi = 52\hbox{$.\!\!^\circ$}2$, $\phi+Z = 97\hbox{$.\!\!^\circ$}2$,we always have $\cos Q \geq 0.5$. This is how we eliminate the blind zone in declination determination.

We calculated precisions of corrections $\triangle \alpha \cos \delta$ and $\triangle \delta$ theoretically. In this calculation, we use mean RMS of 0.23'' and the assumption of observing each star 70 times for $\phi = 52\hbox{$.\!\!^\circ$}2$ at different declinations. The results are shown in Fig. 1, where curve 1 stands for $E_{\triangle \alpha} \cos \delta$varying with $\delta$, and curve 2 stands for $E_{\triangle \delta}$ varying with $\delta$. The trend of $E_{\triangle \alpha} \cos \delta$and $E_{\triangle \delta}$ varying with declination in the appendix are in good accordance with the two curves in Fig. 1.

 

Figure 1: Precision varies with declination for $\phi = 52\hbox{$.\!\!^\circ$}2, Z = 45^ \circ$