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Determining Optimum Solar Panel Tilt Angle

Copyright 2015 by Northern Arizona Wind and Sun

Solar Panel Tilt Angle:
How to Determine the Optimum Position for your Solar Panel

One of the most common questions when installing a solar system is where to point the panel to get the most energy from the system. This is true of solar thermal, reflective collectors and photovoltaic alike. It's really a question of optics and planetary science, but you won’t need an advanced scientific degree to understand.In this article, I am going to break this all down to the basics so you can all understand the source of the light, how this affects the nature of the light at our planet, and the characteristics of our planet’s orientation during its trip around the sun that determine the optimum angle for your solar panel. For those of you looking to skip the science lesson, the answer for a fixed collector is your location's degrees latitude and pointing true South. Your latitude can be found by searching your address here: http://mynasadata.larc.nasa.gov/latitudelongitude-finder/. Your declination, or the difference between Magnetic North and True North, the polar opposite of True South, can be found here: http://www.ngdc.noaa.gov/geomag-web/.

For the rest, who are so inclined as to set up their own energy source, and so, obviously interested to know how it all works, we will begin at the source, our Sun. The Sun is a giant sphere of boiling gas, about 1,392,000km in diameter and burning at a temperature of about 5778K. In standard units, this translates to 864,327 miles and about 9941ºF. As a mostly spherical ball, the Sun emits light radially, equally in all directions. We can imagine this light as coming in rays from the center of the sun. Now, imagine we have another Sun sized object directly next to the Sun.

With a little geometry, we can see that the tangent line with the second object's surface, where the furthest north radial line from the sun intersects the object, is perpendicular to the radial line from the second object’s center. The angle then, from the line joining the centers, to this furthest north line, is the inverse sine of one solar radius over two radii.

The same would be true for the bottom of the object, so all light rays leaving the Sun within an angle of 60º, would hit this object. If we start moving that object away, that angle, in which all rays are captured, becomes smaller. For instance, when the object is one solar diameter away from the Sun, that angle is reduced to about 28.95º.

If the object is moved away from the sun, at a distance of 107.5 times the diameter of the Sun (the distance from the Sun to the Earth), that angle becomes about 0.5º. If we shrink that object down to the size of the Earth, that angle becomes 0.00488º, or basically 0º.

So we see, the Sun's rays reaching the Earth all come in at essentially the same angle. In other words, the sunlight reaching the Earth comes in parallel rays. In order to maximize the rays intercepted by the collector, we want the panel to be perpendicular, or normal, to the incident Sun rays. This makes our geometry much simpler. We only have to concern ourselves with right angles and complimentary angles, no matter where we are on the Earth’s surface. For the purpose of a fixed angle collector, let’s begin the discussion with the line joining the center of the Sun and the center of the Earth passing through the Equator. In reality, this situation only happens two times a year, the spring and autumnal equinoxes, but this position is an average for all other times of the year, and for a fixed angle collector, the average is the angle of interest. In this situation, the line between the center of the Earth and the Equator is parallel to the Sun’s rays. The line from the center of the Earth to any point on the surface, the radial line, is perpendicular with the surface itself. Finally, we want the solar collector to be perpendicular to the incoming Sun rays, which in turn, makes it perpendicular to the line connecting the centers of the Earth and Sun and passing through the Equator. We begin with the right triangle formed with the center of the Earth, the location on the Earth’s surface of our panel, and the intersection of the line joining the centers and the plane of the panel.

Since we know that the angle at the intersection is 90º, and the angle at the center of the Earth is the latitude of the panel, the other angle in the triangle, we’ll call it , is the compliment of the latitude. By continuing the radial line out, we get a normal line to the Earth’s surface at the panel location. The angle between our panel, in its proper position, and this normal line is the same as . We will call it , to indicate that it is on the outside of the Earth. Since the normal line and the surface of the Earth are perpendicular, and , the angle between the panel and the surface, , being the compliment of , is equal to the latitude of the panel’s position. Thus, the optimum angle to mount the panel, from horizontal, is the latitude at the location of the panel.

To determine where to point the panel with respect to cardinal directions North, South, East, and West, we have to change our point of view or frame of reference. The Earth rotates around a central axis of rotation. This axis defines True North and True South. Unfortunately, True North is not the same as Magnetic North, the direction a compass will indicate. This difference is known as the declination and it varies with your location on the Earth’s surface. The declination can be found on navigation maps, as well as through other resources, such as the tools available on the National Oceanic and Administration website found here: http://www.ngdc.noaa.gov/.

There is an experimental method to determine True North, and thereby True South, as well. If you look up the times of sunrise and sunset for a given day, you can determine the time of Solar Noon, the time when the Sun is halfway through its transit across the sky. Calculate the length of time between sunrise and sunset and divide by two. Add this amount of time to the time of sunrise to find the time of Solar Noon. If a pipe is planted and plumbed, exactly vertical, or direct normal to the Earth’s surface, the shadow it casts at Solar Noon will point True North (in the Northern Hemisphere).

If we look at the Earth as a spinning sphere with a light shining at it, we see that, regardless of the orientation of the axis of rotation, the light will trace a circle, around the Earth, at the same latitude, over the course of one rotation. With the actual tilt of the Earth’s axis of 23º, this is true to within less than an eighth of a degree. So, the Sun rises at the same angle with respect to True South as it sets on any given day. This means that, for a fixed angle collector, the best direction to point your panel is True South in the Northern Hemisphere, and True North in the Southern Hemisphere.

As a result, we see that for a fixed angle collector, the optimum angle from horizontal is the latitude of the installation, and the ideal direction to point the collector is towards the axis of rotation, True South in the north and True North in the south. The tilt of the Earth’s axis does not affect the orientation of these types of collectors. It will, however, have an effect on systems which adjust for seasonal changes, and tracking systems, which will be discussed in the articles that follow.