# The Design and Simulation of an Astronomical Clock
## 5 The Astronomical Data  
It is evident that all the intended features of the astronomical clock disclosed and explained in Section 3 can be obtained not just by the proper construction of the dial, but also by the correct design of the clock mechanism. The synthesis of the astronomical clock mechanism is based on certain well-known astronomical facts and numerical data. The ratio between these data will be interpreted and realized by the clock mechanism gear ratios. Therefore, all astronomical data which are necessary for the design of the clock mechanism must be carefully considered and properly selected.  
First, the Earth rotates around its axis which is currently tilted 23.444 degrees from its orbital axis [6]. From the viewpoint on the Earth, the Sun rotates from the east to the west with the period known as a solar day with a mean length of 24 h. The apparent path of the Sun during the year, as seen from the Earth, passes through the zodiac which is divided into 12 signs and form the ecliptic coordinate system. The Sun's position at the vernal equinox is the origin of ecliptic longitude. From an observer on the Earth, the zodiac rotates daily to the west within the period known as sidereal day whose mean length, determined for the year 2019, is 23 h 56 min 4.09053 s or 23.934469592 h [7].  
The period of the Moon's orbit with respect to the line which connects the Earth and the Sun, or the period of lunar phases, is called a synodic month. This period varies over time but its mean value calculated for a long period of time is extremely stable. Currently, the average duration of the synodic month is \(T_{Ph}=29.530589\) days [6; 8] and is decreasing in mean length by about 3.6 \(\times\) 10\({}^{-7}\) solar days per century [6] or by about 0.311 s per millennium.  
Since the astronomical clock displays the daily motion of the Moon, the average period of the Earth's rotation relative to the Moon, or tidal lunar day, must be calculated. From the fact that the angular distance between the Sun and the Moon determines the lunar phase, the mean value of the tidal lunar day can be determined from Equation (1):  
\[T_{L}=24\bigg{(}\frac{1}{T_{S}}-\frac{1}{T_{Ph}}\bigg{)}^{-1}=24\bigg{(}1- \frac{1}{29.530588}\bigg{)}^{-1}=24.8412024\;\text{h} \tag{1}\]  
The orbit of the Moon is inclined to the ecliptic and crosses the ecliptic in two lunar nodes: the ascending node and the descending node [9]. The lunar nodes revolve slowly around the ecliptic [9] in the direction opposite to the direction of the Earth's revolution. The period of the lunar node cycle determined for the year 2019 is \(T_{N}=18.612952\) years [8]. The mean interval of time between two successive conjunctions of the Sun with the same lunar node is called the eclipse year and lasts \(T_{E}=346.620067\) days [8]. It is a little bit shorter than the solar year because the directions of nodal precession and the Earth's revolution are opposite. The lunar nodes play an important role in the explanation of the phenomena called lunar and solar eclipses. From the viewpoint of an observer on the Earth, a solar eclipse can occur only when the Moon and the Sun are both near the same node, and a lunar eclipse can only happen when the Sun and the Moon are close to opposite lunar nodes. If the angle between the line of nodes and the Moon or the Sun is less than 9\({}^{\circ}\)30\({}^{\prime}\), a lunar eclipse must occur, while if it is greater than 12\({}^{\circ}\)15\({}^{\prime}\), a lunar eclipse is not possible (lunar eclipse limit). If the abovementioned angle is less than 15\({}^{\circ}\)23\({}^{\prime}\), a solar eclipse must happen, while if it is more than 18\({}^{\circ}\)35\({}^{\prime}\), a solar eclipse cannot occur (solar eclipse limit) [10].  
All the astronomical data important for the synthesis of the astronomical clock mechanism are given in Table 1.