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Daily and seasonal changes

The solar illuminations is subject to daily and seasonal changes. We will try to look closer at the positional changes in this section, changes in the spectral distribution will we presented in the next section.

From every day experience we know that the sun rises in the east, assumes a high position in the sky at noon, and sets in the west. In addition to the daily changes there are also seasonal changes which depend to a large extent on the latitude of the observer. For example, in Central Europe the time between sunrise and sunset can vary by almost eight hours. Just as noticable are the different heights above the horizon at noon in summer and in winter.

The calculation of the solar position is a standard task of astronomy and requires a little geometry. The observed position of the sun for an observer on the earth is usually given in two polar coordinates which denote the azimuth angle Φ counted clockwise from true north and the vertical height α above the horizon. Approximate relations for the two angles are given below.

In the equations L denotes the geographical latitude of the observer. The quantity D denotes the solar declination angle which changes sinusodally throughout the year. It is -23.5° at winter solstice, 0° at equinox, and +23.5° at summer solstice.

The quantity t is the hour angle. It runs roughly from -180° to 0° at noon and on to +180° at midnight. To get it a little more precise, we have to apply two important corrections. The first is the difference between local time and standard time which makes 4 minutes for every degree we are ahead or behind the meridian of our time zone. The second correction is called equation of time and takes care of corrections due to the elliptical shape of the earth orbit and due to the inclination of the earth axis with respect to the plane of the orbit around the sun. The equation of time results in a forward correction of more than 16 minutes at the end of October and a backward correction of more than 14 minutes in mid February.

The diagram below presents the solar elevation and azimuth in a polar diagram which is often called solar chart. Note that the corrections due to the equation of time are not included in the chart.

Figure 1: A solar chart shows the position of the sun for an observer with a 180° fisheye. Thus, intersections of the sun trajectory with the outermost circle at 0° denote sunrise or sunset. The symbols correspond to full hours throughout the day, given in local time. Thus, noon is at exactly 180° (looking south). The plot shows the conditions of summer solstice (triangles), winter solstice (squares), and spring/autumn equinox (circles) for a latitude of 47° north.

Another very important aspect of the angular position is shadowing by buildings or trees during the day and throughout the year. For this case it is more appropriate to draw the azimuth and vertical angles in a cylindrical plot. In the plot given below the angles on the vertical axis are not given directly, but they have been scaled by their tangens with the scale running from 0 to π (3.14). The horizontal axis is a full circle wich corresponds to 2π (6.28) in radians. If the diagram is plotted with an aspect ratio of the axes of 2:1 (as shown below) and rolled into a cylinder it gives the view of an observer in the centre on the horizontal plane. Diagrams of this kind allow to locate nearby objects and to judge when they obstruct the sun. For better illustration the box shows the viewing anlge of a standard 50 mm camera lens looking south.

Figure 2: Cylindrical plot of the sun trajectory, the symbols represent full hours. Squares, circles and triangles denote winter solstice, spring/autumn equinox and summer solstice, respectively. The boxed area in the lower part corresponds to the view through a standard 50 mm camera lens with a viewing angle of 39.6° times 26°. Note that this kind of projection of the viewing area is only valid in the vicinity of the horizon.

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Additional infomation:

Some more on the Angular dependence of insolation.

The solar coordinates in the charts above are given only for four dates throughout the year. For the days in between the angles may by straight lines, but due to the equation of time this will off by as much as 15 minutes. The real thing looks more like a number eight and is called analemma.