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.
|