# Computer Modelling of Heliostat Fields by Ray-Tracing Techniques: Simulating the Sun
## 5 Results and Discussion
### Angular Speed  
What is the angular speed of the Sun along its daily trajectory, seen from Earth? Knowing this is important to manage a heliostat field, because it indicates how often the orientation of the heliostat mirrors must be recalculated. The simple answer is \(360/24=15^{\circ}/\mathrm{h}\). To calculate a more accurate value, we can determine the Sun's position at two instants \(t_{0}\) and \(t_{1}\), separated by a small interval \(\Delta t=t_{1}-t_{0}\), determine the angle \(\alpha\) between the two \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Julich**} & \multicolumn{2}{c}{**Protaras**} \\ \cline{2-5}  & **Time 1** & **Date** & **Time 1** & **Date** \\ \hline Minimum time for sunrise & 3.4446 & 16 June & 4.5640 & 12 June \\ Maximum time for sunrise & 7.7344 & 30 December & 6.9505 & 7 January \\ Sunrise equal 2 at spring & 6.1018 & 9 March & 6.1068 & 9 March \\ Sunrise equal 2 at fall & 5.7330 & 3 October & 5.7393 & 3 October \\ Minimum time for sunset & 15.3589 & 12 December & 16.4549 & 5 December \\ Maximum time for sunset & 19.7620 & 25 June & 18.9529 & 29 June \\ Sunset equal 2 at spring & 17.9884 & 30 March & 17.9945 & 30 March \\ Sunset equal 2 at fall & 17.8938 & 10 September & 17.8895 & 10 September \\ Minimum daylight & 7.7016 & 21 December & 9.6459 & 21 December \\ Maximum daylight & 16.3034 & 20 June & 14.3595 & 20 June \\ \hline \end{tabular}  
* Time in decimal hours. 2 Equal means that sunrise or sunset occur simultaneously at Julich and Protaras.  
\end{table} 
Table 1: Relevant data for the four plotted curves in Figure 10.  
Figure 11: Altitude at Julich and Protaras calculated along a full day, for the summer (20 th June) and winter (21 st December) solstices in 2024.  
vectors and divide by \(\Delta t\) to obtain an instantaneous angular speed. This speed is essentially independent of the location (Northern or Southern hemisphere) as it mainly depends on the Earth's rotation around its axis. However, it does change with time, as shown in Figure 12 for the whole year of 2024, calculated at 12:00 LCT time. To understand this change, the angles \(\alpha\) can be added for a full day, from 00:00 to 24:00, using for example an interval \(\Delta t=1\)s. At the equinoxes, 20 th March and 22 nd September, the angle sum is 360.07\({}^{\circ}\) and 360.09\({}^{\circ}\), respectively, as expected for a full circumference. However, at the solstices, 20 th June and 21 st December, the angle sum is only 330.25\({}^{\circ}\) and 330.18\({}^{\circ}\), respectively. As the total angular path decreases but the 24 h interval remains the same, the angular speed decreases to just 13.76\({}^{\circ}\)/h. The average arc for the whole year is 345.28\({}^{\circ}\) and the average speed is 14.39\({}^{\circ}\). To understand this variation in the angle sum, Figure 13 reports the calculated daily Sun trajectory around Earth for the two equinoxes and the two solstices, for 2024 at Protaras (the results are similar for Julich, only the changes in daylight time during the year become more apparent). Seen from Earth, the plane of the Sun's daily circular trajectory only includes Earth at the equinoxes, moving progressively away until reaching a maximum offset at the summer and winter solstices. Therefore, the total daily scanned angular range, when following the Sun from Earth, decreases from a maximum of 360\({}^{\circ}\) at the equinoxes to a minimum of 330\({}^{\circ}\) at the solstices. Beyond the obvious scientific and pedagogic interest of this discussion, in astronomical terms, the overall conclusion regarding solar applications is: the heliostats must be realigned more often closer to the equinoxes, or alternatively, the maximum value of angular speed should be used throughout the year. Figure 12: Angular speed for Sun throughout 2024, calculated at 12:00 LCT time.