# The Design and Simulation of an Astronomical Clock
## 6 Approximations of Astronomical Data
Since the difference between angular velocities of the Sun and the lunar node pointers represents the approximation of the lunar node angular velocity relative to the Sun, the next Equation (8) determines the approximation of the eclipse year \(\overline{T}_{E}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Type** & **Approximation** & **Type** & **Approximation** \\ \hline C & \(0/1\) & SC & \(1003/1041=(17\times 59)/(3\times 347)\) \\ C & \(1/1\) & SC & \(1135/1178=(5\times 227)/(2\times 19\times 31)\) \\ SC & \(13/14=13/(2\times 7)\) & SC & \(1267/1315=(7\times 181)/(5\times 263)\) \\ SC & \(14/15=(2\times 7)/(3\times 5)\) & SC & \(1399/1452=1399/\left(2^{2}\times 3\times 11^{2}\right)\) \\ SC & \(1531/1589=(1531/7\times 227)\) \\ SC & \(24/25=32\times 3/5^{2}\) & C & \(1663/1729=1663/(7\times 13\times 19)\) \\ SC & \(25/26=5^{2}/(2\times 13)\) & C & \(1795/1863=(5\times 359)/\left(3^{4}\times 23\right)\) \\ C & \(26/27=(2\times 13)/3^{2}\) & C & \(3458/3589=(2\times 7\times 13\times 19)/(37\times 97)\) \\ C & \(53/55=53/(5\times 11)\) & SC & \(15,627/1629=(3\times 5209)/\left(7^{2}\times 331\right)\) \\ C & \(79/82=79/(2\times 41)\) & SC & \(19,085/19808,808=(5\times 11\times 347)/\left(2^{5}\times 619\right)\) \\ C & \(132/137=\left(2^{2}\times 3\times 11\right)/137\) & SC & \(22,543/23,395=22,543/(5\times 4679)\) \\ SC & \(871/904=(13\times 67)/\left(2^{3}\times 113\right)\) & C & \(26,001/26,986=(3^{5}\times 107)/(2\times 103\times 131)\) \\ \hline \end{tabular}
\end{table} 
Table 4: The best rational approximations of the ratio \(R_{MZ}\). \begin{table}
\begin{tabular}{c c c c} \hline
**Type** & **Approximation** & **Type** & **Approximation** \\ \hline C & \(1/1\) & C & \(30/29=(2\times 3\times 5)/\left(2^{2}\times 7\right)\) \\ SC & \(16/15=2^{4}/(3\times 5)\) & C & \(59/57=59/(3\times 19)\) \\ SC & \(17/16=17/2^{4}\) & SC & \(266/257=(2\times 7\times 19)/257\) \\ SC & \(18/17=\left(2\times 3^{2}\right)/17\) & SC & \(325/314=(5^{2}\times 13)/(2\times 157)\) \\ SC & \(\ldots\ldots\ldots\ldots\) & SC & \(384/371=\left(2^{7}\times 3\right)/(7\times 53)\) \\ SC & \(27/26=3^{3}/\left(2\times 13\right)\) & C & \(443/428=443/\left(2^{2}\times 107\right)\) \\ SC & \(28/27=\left(2^{2}\times 7\right)/3^{3}\) & C & \(502/485=(2\times 251)/(5\times 97)\) \\ C & \(29/28=29/\left(2^{2}\times 7\right)\) & SC & \(945/913=\left(3^{3}\times 5\times 7\right)/(11\times 83)\) \\ \hline \end{tabular}
\end{table} 
Table 3: The best rational approximations of the ratio \(R_{SM}\). \[\overline{T}_{E}=\left(\frac{2781}{1773}-T_{S}\right)^{-1}=\left(\frac{2781}{1773}- 1\right)^{-1}=346.6250162\text{ days} \tag{8}\]  
Thus, the mechanism of the astronomical clock approximates the eclipse year with an error whose absolute value is determined by the following expression (9):  
\[\left|\overline{T}_{E}-T_{E}\right|=\left|346.6250162-346.620067\right|=0.00494 92\text{ days}\approx 7.13\text{ min} \tag{9}\]  
The reciprocal value of the ratio \((T_{S}-T_{SID})/T_{SID}\) which appears in Equation (7) determines the number of days in the solar or tropical year. Consequently, the clock mechanism produces an approximation of the mean tropical year of \(T_{T}=365.242189\) days [6] with an error whose absolute value can be determined by Equation (10). \[\left|\begin{array}{c}T_{T}-T_{S}\frac{\overline{T}_{SID}}{T_{S}-\overline{ T}_{SID}}\right|=\left|365.242189-365.2422884\right|=9.94\times 10^{-5}\text{ days}\approx 8.6\text{ s} \tag{10}\]  
After all calculations and evaluations presented in this section, it can be concluded that the rational approximations of real ratios \(R_{MZ}\), \(R_{SPH}\), \(R_{MZ}\) and \(R_{SN}\) are not perfect, but are sufficiently good and practically applicable for the synthesis of the astronomical clock mechanism. The next section is dedicated to this task.