Shrinking Heads & Sharper Sundials
Strangely behaving shadows captured by Andrew Kirk.
"It was shortly after sunrise. Shadows were very long with distinct penumbras. As I jogged along, I noticed the shadow of my head shrink each time my shadow passed over the shadow of a utility pole. So I stopped to document it, and returned next day, too."
"The pole was ~35 mm diameter at chest height. While photographing I
stood about 25 meters away from it."
The width across the shoulders of Andrew's shadow stays the same. (1) Why does the width of the head change? (2) Why is the narrow head shadow much sharper than the normal one on the right?
'Because we are close enough to the sun to see it as a disk' is the easy explanation. The 'how' takes longer!
Look at the head shadow away from the pole. The sun's disk is 0.5° in diameter and its rays diverge from parallel by that amount. There is a volume of darkness extending from the head in which no sunlight can reach - this is the umbra. The umbra narrows farther from the head. Outside of the umbra only part of the sun is obscured and the shadow is less intense - this is the penumbra.
The resulting shadow on a wall or the ground has a dark core (umbra) narrower than the object casting it and a broad fuzzy edge (penumbra).
|The fun starts when the head forms a second shadow inside the broad pole shadow zone. Look at the upper edge of the head in the figure at left. The sun is almost blocked by the pole. Only rays traveling in the direction 'A' from the unobscured edge of the sun are available to define the edge of the second shadow. There are no rays in direction B because they are blocked by the pole. The result - There is a sharp and dark shadow edge formed by rays like A and there is almost no penumbra.
The penumbra-less shadow is therefore narrower than the stand alone shadow and its sharp edges perhaps enhance the narrowness.
Experiment with a pencil in and out of the shadow of a strip of opaque tape pasted on a window. The geometry is the same. Then try a large pinhole in a card inside a shadow - that is the basis of a shadow sharpener to tell the time on a sundial better! Subtle optical effects like this are all around us.
|The pole shadow is similar. There is a narrowing dark central umbra unlit by any direct sunlight surrounded by a penumbra of graded intensity. The disks at left show the sun's appearance at different points inside the penumbra. Closer to the shadow centerline the pole cuts off - eclipses - more and more of the sun.|