# The Design and Simulation of an Astronomical Clock
## 6 Approximations of Astronomical Data
The set of best rational approximations of the ratio \(R_{SPH}=\)29.530589 is given in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Astronomical Data** & **Duration** & **Astronomical Data** & **Duration** \\ \hline Solar day & \(T_{S}=24\) h & Lunar tidal day & \(T_{L}=24.8412024\) h \\ Sidereal day & \(T_{SID}=23.934469592\) h & Lunar nodal period & \(T_{N}=18.612952\) years \\ Synodic month & \(T_{Ph}=29.530588\) days & Eclipse year & \(T_{E}=346.620067\) days \\ \hline \end{tabular}
\end{table} 
Table 1: The astronomical data. The semi-convergent \(945/32\) is chosen to approximate the ratio \(R_{SPh}=29.530589\). The error of this approximation is \(0.000662\) days per lunation or less than 12 min per year. The ratio \(R_{SM}\) between angular velocities of the Sun and the Moon clock pointers is determined by Equation (2):  
\[R_{SM}=\frac{T_{L}}{T_{S}}=\frac{24.8412024}{24}=1.0350501 \tag{2}\]  
The set of best rational approximations of the ratio \(R_{SM}\) is given in Table 3.  
The semi-convergent \(945/913\) is selected as the approximation of the ratio \(R_{SM}\). The absolute value of the error of this approximation is calculated from the following expression (3):  
\[\left|24\times\frac{945}{913}-24.8412024\right|=1.949\times 10^{-5}\,\mathrm{h} \approx 0.07\,\mathrm{s} \tag{3}\]  
This error denotes that the mean position of the Moon will be determined by the clock mechanism with an error of approximately 0.07 s per day or less than 30 s per year. Regarding the operational scheme of the astronomical clock mechanism given in Figure 8, the ratio \(R_{\mathrm{MZ}}\) between the angular velocities of the Moon and the zodiac pointers is calculated by the following expression (4):  
\[R_{MZ}=\frac{T_{SID}}{T_{S}}\times R_{SM}{}^{-1}=\frac{23.934469592}{24}\times \frac{913}{945}=0.963499592 \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Approximation** & **Type** & **Approximation** \\ \hline C & \(29/1\) & SC & \(384/13=2^{7}\times 3/13\) \\ C & \(30/1=(2\times 3\times 5)/1\) & C & \(443/15=443/(3\times 5)\) \\ C & \(59/2\) & C & \(502/17=2\times 251/17\) \\ SC & \(266/9=2\times 7\times 19/3^{2}\) & SC & \(945/32=(3^{3}\times 5\times 7)/2^{5}\) \\ SC & \(325/11=(5^{2}\times 13)/11\) & C & \(1447/49=1447/7^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The best rational approximations of the ratio \(R_{SPh}\). Figure 8: The operational scheme of the astronomical clock mechanism. The set of convergents (C) and semi-convergents (SC) by which the ratio \(R_{MZ}\) is approximately determined is given in Table 4. The convergent \(26,001/26,986\) is selected as the approximation of the ratio \(R_{MZ}\). Since the ratio between this convergent and the semi-convergent \(913/945\) produces the rational approximation \(\overline{T}_{SID}\) of the sidereal day by the following expression (5),  
\[\overline{T}_{SID}=24\times\frac{26,001/26,986}{913/945}=23.93446961\;\text{h}, \tag{5}\]  
the absolute value of the error obtained by this approximation can be calculated from the expression (6)  
\[\left|\overline{T}_{SID}-23.934469592\right|=2\times 10^{-8}\;\text{h}=7.2 \times 10^{-5}\;\text{s} \tag{6}\]  
This value means that the clock mechanism determines the mean sidereal day with an error of approximately \(7.2\times 10^{-5}\;\text{s}\) per day or less than 0.03 s per year. In accordance with the operational scheme of the clock mechanism, astronomical data given in Table 1 and the fact that the lunar nodal cycle is k = 18.612952 times longer than a year, the ratio \(R_{SN}\) between angular velocities of the Sun and the lunar node pointers can be determined by the following expression (Equation (7)):  
\[R_{SN}=\left[\frac{T_{S}}{T_{SID}}+\left(\frac{T_{S}-T_{SID}}{T_{SID}}\right) \times\frac{1}{k}\right]^{-1}=0.997123293 \tag{7}\]  
The best rational approximations of the ratio \(R_{SN}=0.997123293\) are given in Table 5. The convergent 2773/2781 is chosen as the best applicable approximation of the ratio \(R_{SN}\).