Sun Angle and Beam Concentration

If every beam of sunlight reaching Earth has the same amount of energy, why do some areas warm up more than others? To answer this thoroughly, we need to analyze an area’s energy budget: CRACL (Conduction, Radiation, Advection, Convection, and Latent Heat).  But since it all starts with the Sun’s radiation reaching Earth, let’s focus on the concentration of sunlight reaching the ground, which depends on the angle of the sun above the horizon, or sun angle, regardless of one’s latitude, day of the year, and time of the day.

The Sun emits near-constant electromagnetic energy (radiation), and how it intersects with the Earth’s surface determines how it heats. The warmed surface, in turn, warms the air. Imagine a beam of sunlight that has a circular cross-section shining perpendicularly onto the ground. The area of the illuminated ground is the same as the area of the sunbeam. If this same beam hits the ground at a 30° angle, the circular beam of light illuminates an elliptical patch of ground. The ellipse has a major axis twice as large as the sunbeam’s diameter, yet the minor axis remains the same as the sunbeam’s diameter.  The ellipse’s area is twice as large as the sunbeam, so the beam’s energy is spread out over twice as large an area, or the beam concentration is 0.5 (Figure 1). We will call the ratio of the sunbeam’s cross-sectional area to the area of the illuminated ground area beam concentration (Table 1).

If the Sun is on the horizon, the rays don’t hit the ground (they are parallel to it), so the beam is spread out over an infinite piece of real estate, and the beam concentration equals 0%. When the Sun is directly overhead or at the zenith, the beam’s size is the same size illuminating the ground, so the beam concentration is 100%.

Calculating Beam Concentration

We will use the ellipse created when a circular beam of sunlight intersects the flat, level ground at 30º above the horizon (as shown in the following illustration).

     Beam Concentration = Cross-Sectional Area of Sunbeam / Area of Illuminated Ground

where the

     Area of Sunbeam = Area of a Circle

                                      = π (Diameter/2)²

     Area of the Illuminated Ground = Area of Ellipse

                                       = π (Major Axis/2) (Minor Axis/2)

Since the beam of sunlight has a diameter of 1 unit and its angle to the ground is 30º​, then the axes of the illuminated ellipse are:

     Minor Axis = Circle Diameter = 1     and

     Major Axis = Circle Diameter / sin(30º) = 1 / (1/2) = 2

Then,

     Beam Concentration = π (1/2)² / (π (2/2) (1/2)) = 2

We can write a general equation for beam concentration:

     Beam Concentration = π (Radius)² / (π (Radius / sin(Sun Angle)) (Radius))    or

     Beam Concentration = sin(Sun Angle)

This implies that the sunbeam’s cross-sectional shape is unimportant; all we need to know is the Sun’s angle above the horizon!  What happens if you use a square beam of sunlight?  Test the conclusion that the sunbeam’s shape does not affect the beam concentration (view this scenario using the Beam Concentration web app available below).

If you are unfamiliar with the sine function, see Right Triangles and Vectors.

Table 1. When the Sun is higher in the sky, a beam of sunlight illuminates a smaller area of ground, so the intensity of sunlight within the smaller area is more concentrated than when the Sun is lower in the sky.

When the Sun is high in the sky, the energy of a beam of sunlight is more concentrated on level ground.  More energy reaches a unit area of land compared to when the Sun is lower in the sky.  As we saw with the black fabric experiment and the diurnal heating patterns, the more radiation absorbed by a surface, the greater the temperature change.  Beam concentration is an important variable that determines a location’s daily and seasonal temperature patterns.

Use the Web App to Explore Beam Concentration

If you would like to explore the relationship of beam concentration and the Sun’s angle above the horizon in more detail, run the web app, Beam Concentration, below.

Click the button to run the web app for desktops, laptops, tablets, and smartphones.

Acknowledgments

Thanks to Matti Horne for her suggestions during beta testing and for designing our new certificate of completion.  Alan Gould’s insightful suggestions led to a revision of the webpage and several changes in the web app after its initial release, including adding a square beam of sunlight.

The Shape of Sunbeams

Why are two shapes of sunbeams used in the web app?  If you’ve played with a flashlight or a laser, the shape of the beam of light is almost always circular, so conceptually, this may be easier to consider first. However, if you want to calculate the total amount of the Sun’s energy reaching Earth’s surface, you need to use square sunbeams since there won’t be any gaps between adjacent sunbeams reaching the ground.

Animation of the times of the day when the sun reaches 30º in the sky for every 10º of latitude. The blue line represents the times this occurs before noon, and the red line shows the times after local noon.

At 30º sun angle, the beam concentration is 50%.