# Computer Modelling of Heliostat Fields by Ray-Tracing Techniques: Simulating the Sun
## 5 Results and Discussion
### Altitude, Azimuth  
Figures 8 and 9 show how altitude and azimuth change throughout the year for Julich and Protaras, at 08:00, 12:00 and 16:00 LCT time. At Julich, the Sun is below the horizon at 16:00 in winter (from 3 November to 21 January 2024). The overall altitude range is (\(-\)4.98\({}^{\circ}\), 62.14\({}^{\circ}\)) for Julich and (4.84\({}^{\circ}\), 78.02\({}^{\circ}\)) for Protaras. For Julich (longitude = +6.3878\({}^{\circ}\)) and Protaras (longitude = +34.0583\({}^{\circ}\)), the altitude at 08:00 is always higher than at 16:00 LCT time, because both locations are on the east side of their respective time zones. For places along the time zone meridians (longitude = 0\({}^{\circ}\), +15\({}^{\circ}\), +30\({}^{\circ}\), +45\({}^{\circ}\), etc. ), the altitude at 08:00 and 16:00 should be similar. However, places in the east border of each time zone see things happening 30 min earlier (the true midday occurs at 11:30), so if, for instance, the Sun rose at 08:00 and set at 16:00, an observer at the east border of the time zone would see the Sun rising at 07:30 and setting at 15:30 (an observer at the west side would see the Sun rising at 08:30 and setting at 16:30). Therefore, to have a properly equilibrated, optimized time schedule for the heliostat field, its longitude should be taken into account. For Julich, the time shift is (6.3878/7.5) \(\times\) 30 min = 26 min, while for Protaras is (4.0583/7.5) \(\times\) 30 min = 16 min. For example, a  
Figure 9: Calculated azimuth (horizon coordinates) for Julich, Germany (latitude = +50.9133\({}^{\circ}\)) and Protaras, Cyprus (latitude = +35.0125\({}^{\circ}\)) during the 366 days of 2024, at 08:00, 12:00 and 16:00 LCT time. 08:00-16:00 schedule should actually be translated as 07:34-15:34 at Julich and 07:44-15:44 at Protaras. The striking asymmetry of the altitude curves (more visible at Protaras, but the same trends can be seen at Julich) is the result of the apparent weak correlation between morning and afternoon Sun events. In February, altitude rises faster at 16:00 than at 08:00, so the two curves become closer. In May (3 months later), altitude rises faster at 08:00, so the two curves move away. In August (3 months later) the two curves become closer again, as altitude at 08:00 decreases faster than at 16:00, while the opposite occurs in November (3 months later). Azimuth also shows a complex pattern, particularly at 12:00, showing a maximum in June and another, smaller, in October. Therefore, the Sun's pointing direction is not simply changing monotonically during the year, as can be seen for both Protaras and Julich (further north the effect is less pronounced but is still visible). As explained later (Figure 12) these curves must have exactly two maxima and two minima, clearly visible here. Azimuth plays a subtle, significant role in heliostat shading, as the Sun's trajectories move further south in winter and closer to a more standard east-west route in summer. For example, the front-row heliostats, that are not subjected to blocking, may present a small shading. For these heliostats, shading is maximum around spring, because in winter the Sun's azimuth is much further south in the morning and in the afternoon, so there are much less east-west shading effects. In summer, azimuth is close to 90\({}^{\circ}\) at 08:00 and to 270\({}^{\circ}\) at 16:00, as expected. Sunrise and sunset curves for Julich and Protaras for 2024 can be seen in Figure 10 (calculated by iteratively converging altitude to 0\({}^{\circ}\), for a time error \(<10^{-5}\) h = 0.04 s). Relevant data for the four plotted curves in Figure 10 are reported in Table 1. The minimum and maximum of these four curves are always different, only the longer day (at the summer solstice) and the shorter day (at the winter solstice) occur at the same day. Julich and Protaras curves for sunrise and sunset cross each other around spring and autumnal equinoxes (about 1-2 weeks before or after).