The Geometry of Sunlight

Marissa Romano and John Noé

Laser Teaching Center
Department of Physics and Astronomy
Stony Brook University



It's straightforward to demonstrate that sunlight that passes through a small hole, or that is reflected from a small mirror, forms a disk of light with a diameter that steadily increases with increasing distance. Physics students with some knowledge of optics often incorrectly attribute this "spreading" of sunlight to the wave-optical phenomenon of diffraction. However, simple geometry is sufficient to explain these effects as well as the blurring of shadows and the three parts of a solar eclipse (umbra, penumbra, antumbra). Consider two light rays from two different parts of the sun that pass through a hole. The direction of each ray, and hence the angle between them, is unaffected by the hole. The maximum angle possible, α, is referred to as the angular size of the sun. According to the small angle approximation, this angle, expressed in radians, is given by the ratio of the solar diameter and the solar distance. Similarly, the angular size of the solar image is equal to the ratio of its diameter to the distance from the hole or mirror to the viewing screen.

In this project [1] we determined the angular size of the sun using a simple yet highly effective method that utilized a measuring template with six concentric circles ranging from nominally 5.0 to 30 mm in diameter. (We created the template with the xfig program and later took into account that the actual diameters are all slightly smaller than intended.) To make each measurement one of us (MR) moved the template until the disk of sunlight matched one of the circles while the other (JN) measured the image distance with a tape measure. The six measurements are shown in red in the graph below. These results are in very good agreement with the relationship (black line) predicted by the known angular size of the sun (α = 9.36 mR) on the date of the measurement (12 March) and the known (3.0 mm) diameter of the hole through which the sunlight passed. A least-squares analysis shows that the method has a sensitivity of a few tenths of one percent, more than sufficient to observe seasonal variations in α due to the ellipticity of the earth's orbit.