Position of the Sun at Any Time of Day Throughout the Year (at 40°N Latitude)
Solar Elevation Angle (0°=Horizontal)
Solar Azimuth Angle (0°=Due South)
The position of the sun in the sky at any instant is described by two angles. Imagine pointing at the sun and then swinging your arm down to horizontal. That angle describes the height of the sun, or the "solar elevation angle". Now swing your arm horizontally to point Due South. That angle is the "solar azimuth angle", which is 0° at solar noon and negative in the morning. These two angles, and therefore the position of the sun, can be calculated at any time of the day or day of the year based on the geometry of the Earth-Sun system. The apparent path of the sun through the sky depends only on one's latitude, and these figures show the solar elevation and azimuth angles versus time of day at 40°N latitude for the 21st day of each month. Solar noon is when the sun is Due South, and it differs from clock time because solar time depends on how far you are into your time zone, plus the sun doesn't recognize daylight savings time. The sun's angular speed is 15°/hour (360°/24 hours), so if your longitude is 110°W, then you are 110/15 = 7.33 hours West of Greenwich, England (0° longitude), yet everyone in your time zone has their clock set 7 hours ahead of GMT (Greenwich Mean Time). You are 1/3 hour (20 mins) farther into your time zone, so at your location the sun still has to move westward (actually, it is the Earth spinning eastward) until solar noon at 12:20pm, or 1:20pm if it is daylight savings time.
Knowing the position of the sun at any time has many useful applications. The shadow calculations in Fig. 3 are based on combining the sun's position with information about the location and height of a tree and the solar panels. These plots can be used directly (at 40°N latitude) to calculate the length of a shadow on flat ground at any time of day given the height of the shadow-maker. For example, consider a 50' high tree and the length of its shadow on flat ground at 9am on the Summer and Winter Solstices (6/21 and 12/21).
The inset shows a right triangle formed by the tree and its shadow, where the length of the shadow ("L") depends on the solar elevation angle ("a"). From the figure at left above, the solar elevation angle at 9am on 6/21 is 48°, and on 12/21 it is 13°. If "H" is the height of the tree, then from trigonometry the length of the shadow is given by: L = H/tan(a), where "tan" is the tangent function applied to the solar elevation angle. For a 50' tree, the length of the shadow will be 45' on the Summer Solstice (shortest of the year), and 216' on the Winter Solstice (longest of the year). Note that the shadow length will be shorter above ground level where the panels are. Similar calculations can be used to design such things as the length of overhang on the roof of a Passive Solar house that is needed to block the high summer sun from entering a window while allowing the lower winter sun to enter and warm the house. By the way, I'll mention that I'm a self-employed scientist/engineer, in case someone sees promise with these methods and wants to discuss my consulting on a project and/or doing the math and writing the code.
The inset shows a right triangle formed by the tree and its shadow, where the length of the shadow ("L") depends on the solar elevation angle ("a"). From the figure at left above, the solar elevation angle at 9am on 6/21 is 48°, and on 12/21 it is 13°. If "H" is the height of the tree, then from trigonometry the length of the shadow is given by: L = H/tan(a), where "tan" is the tangent function applied to the solar elevation angle. For a 50' tree, the length of the shadow will be 45' on the Summer Solstice (shortest of the year), and 216' on the Winter Solstice (longest of the year). Note that the shadow length will be shorter above ground level where the panels are. Similar calculations can be used to design such things as the length of overhang on the roof of a Passive Solar house that is needed to block the high summer sun from entering a window while allowing the lower winter sun to enter and warm the house. By the way, I'll mention that I'm a self-employed scientist/engineer, in case someone sees promise with these methods and wants to discuss my consulting on a project and/or doing the math and writing the code.