Chapter 1

Light and Shadows

1. Images of the sun

Dh. sun. when you move through the foliage of the high lime trees.

You drop splashes of light on the ground. So beautiful that I dare not tread on them.

E. Rostand

In the shade of a group of trees the ground is dappled randomly with spots of light. some small. some large. but all regularly elliptical. If you hold a pencil in front of one of these spots. the line connecting pencil and shadow will indicate where the rays of light that form the spot come from; they are. of course. sunlight that falls through some opening in the foliage: all we see here and there between the leaves is a blinding ray of light.

Random dappling of spots of sunlight beneath trees. (photo by Veikko Makela)

M. Minnaert, Light and Color in the Outdoors© Springer-Verlag New York, Inc. 1993

2 Light and shadows

Fig. 1. The sun's rays penetrating dense foliage.

The surprising thing is that all these images have the same shape, although it is obviously impossible that all the holes and slits in the foliage happen to have the same shape. If you intercept one of these rays on a piece of paper held at right angles to the ray, the image is no longer elliptical but round. Hold the piece of paper higher and higher and you will note that the spot of light gets smaller and smaller. This shows that the beam of light causing the spot is conical: the image is only elliptical because the ground cuts the cone at an angle.

The origin of the phenomenon lies in the fact that the sun is not a point source. Each tiny opening P in Fig. 1 gives a sharp image of the sun AB, a somewhat larger opening P' gives a slightly displaced sharp image A'B' (dashed lines); a wider opening that contains both P and P' gives a slightly hazy but clear image A'B. And indeed we see spots of light of all kinds of brightness: of two equally large spots the brighter one is also the less sharp.

:t~ __ -=========~1~O~8============~ll Fig. 2, We see the sun's disk at an angle of ltios radian.

Images of the sun 3

When clouds move in to hide the sun, you will also see them mov­ing across the images of the sun, but in the opposite direction; dur­ing a partial solar eclipse, all spots of light are crescent-shaped. When there is a large sunspot, it is visible on the clearest of the spots of light. Get a clear image of the sun by making a tiny round hole in a sheet of thin cardboard. Then examine the images of the sun formed by a square opening at a number of distances between the card­board and the ground.

The angIe subtended at our eye when we view the sun must be the same as angle APB of the cone forming the image of the sun. Such small angles are measured in radians. We say, for example: 'This is an angle of YlOB radian', which means that the sun looks to us as large as I cm (% in) at a distance of 108 cm (3 ft 6 in), or 10 cm (4 in) at a distance of 1080 cm (35 ft)-see Fig. 2. In the same way, the di­ameter of a sharp spot of light must be YlOBth of its distance to the opening in the foliage; if the image is hazy, the size of the opening in the foliage must be added. Catch faint but sharp images of the sun on a sheet of paper, hold it at right angles to the rays, measure the diameter k of the spot of light. and determine the distance L from the paper to the opening in the foliage with the aid of a length of strtng. Is it true that k = L/ 108?

Measure the minor axis, k, and the major axis, b, of the elliptical spots of light formed on the liqUid of a spirit level; their ratio will be same as that of the height of the tree, H, and the distance L. From this it follows that H = (k/b)L = 108k(k/b). A conspicuously large spot of light under a beech had axes of 53 cm (20 % in) and 33 cm (13 in); the height of the opening in the foliage above the ground was thus 108x33x(33/53) = 2200 cm or 22 m (24 yd). Note that the im­ages of the sun are more elongated in the morning and toward the evening, and rounder at midday.

Fig. 3. Shadows of metal wires in slant­ing rays of the sun: (a) distinct shadow; (b) indistinct shadow. a

b

4 Light and shadows

Fine images of the sun are found in the shade of beeches, lime trees, and sycamores, but seldom in that of poplars, elms, or plane trees.

Pay partlcular attention to the spots of light formed by trees at the banks of shallow water: they are outlined wonderfully at the bottom of the waterl

2. Shadows

When you look at your shadow on the ground, you will notice that the shadow of your feet is clearly defined, whereas that of your head is not. The shadow of the bottom part of a tree or post is sharp, while that of the higher part becomes increasingly unclear toward the top.

Hold your hand spread out in front of a piece of paper: its shadow is sharp. Move it away from the paper and note that the umbra (the dark part of the shadow) of the fingers becomes narrower, while the penumbras (the lighter parts of the shadow) get broader and finally merge.

Again, these findings are a consequence of the sun not being a. point source and correspond to what we have observed earlier with the spots of light. Look at the shadow of a butterfly or a bird (how often do we do that?) and note that it looks more or less like a round spot: a shadow image of the sun.

Partlcularly noteworthy is the shadow of a wiremesh fence when the sun is low in the sky: only the vertical wires throw a shadow, the horizontal ones do not. Hold a piece of paper with a small hole in it at right angles to the sun and note the resulting elliptical spot of light on the ground. Think of the shadow of the wire netting as being caused by a number of such ellipses in juxtaposition, but dark ones, not light ones; this shadow becomes sharp when the wire lies along the major axis and hazy when it lies along the minor axis (Fig. 3).

Hold a piece of paper hOrizontal behind the fence on the side opposite tht~ sun and then move it away, so that the gradual development of the remark-· able shadows can be observed. Try this with the sun falling on to the wire netting at different angles, tilt the piece of paper, etc.

Shadows have played an important role in popular beliefs. It used to be conSidered a terrible punishment for anyone to lose his shadow. and anyone possessing a headless shadow would die within a yearl Tales like these, which are common among all peoples and through·· out history, are also of interest to us, because they show how care·· ful one has to be with statements of untrained observers, however numerous and unanimous they may be.

Images of the sun and shadows during solar eclipses 5

3. Images of the sun and shadows during solar eclipses and at sunset

During a solar eclipse. the dark moon is seen to move in front of the solar disk until only a thin crescent is left. At that instant it is worth while noting that all spots of light under a tree resemble crescents. large or small. bright or dim.

Again. the shape of shadows corresponds to the crescent shape. For instance. the shadows of our fingers take on an extraordinary claw­like shape. Each small dark object throws a crescentlike shadow; the shadow of a small rod consists of a number of such crescents. while a curvature appears at its ends.

A good example of a single dark object is an air balloon; it has been observed that during a solar eclipse both its shadow and that of its basket are crescent-shaped. An aircraft at a given height also throws a curved shadow.

Solar eclipses. even partial ones. are rare. It is therefore interest­ing to know that similar distortions of shadows can be seen to be caused during sunset in clear weather at the seaside if you stick coins or disks of cardboard of varying sizes to a window or suspend them from a thin thread. The shape of the shadows and the distribution of light vary according to the size of the coins or disks and how far the sun has sunk below the horizon.

4. Double shadows

When the trees have lost their leaves. you may happen to see the shadows of two equally thick branches covering each other. The shadow of the branch closer to you is sharp and black; that of the other is broader and grayish. There is something peculiar here: where the shadows cover each other exactly. a bright thin line is visible within the narrower shadow. which gives it the appearance of being doubled. Why should this be so?

Assume that your eye is successively at A. B. C. D, and E in Fig. 4. Every time it will see the solar disk with the two branches in front as shown in the lower part of the figure. In order to distinguish between them. assume that the more distant of the branches appears some­what thicker than the other. It is clear that in positions B and D your eye will see the solar disk covered in two places. but in position C in only one place. because the branches then cover each other exactly. This explains the existence of the thin line of light.

It appears that this phenomenon will occur whenever the angle be­tween the two branches subtended at the eye is smaller than that of the solar disk. This occurrence may also be seen when the shadows of two telephone wires cover each other.

6 Light and shadows

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Fig. 4. How double shadows arise.

The shadows in spots of light on the ground underneath tall trees vary in a typical manner. Hold a book in such a way that its shadow falls in the dim light of a large spot of sunlight: the shadow will be much clearer than usual. Now look for a situation where high. thin twigs throw a faint shadow and hold a pencil above it: you will note that you get double or multiple shadows. of which one component is sometimes fainter and sometimes clearer than the other. In this sit­uation a book will throw a shadow with exceptionally clear penum,­bra. All these typical events are easily explained.

I once strolled along the beach on an evening. late in March. The sun set in the west across the sea and the moon was bright in the east. The sunset, which lasted for quite a while. made my shadow fall eastward. But then there was a period when I had no shadow until the brightness of the moon became stronger than the light from the red evening sky and my shadow fell westward.

From the Icelandic AlJUr of Windhael by S. Nordal

Is this an accurate observation?

5. The shadow of an aircraft trail

During certain weather conditions. an aircraft leaves an ever-widen­ing vapor trail against a clear sky. It is sometimes possible to see the shadow of this trail on a lower layer of cloud. Note how the distance between trail and shadow slowly changes.

The shadow of an aircraft trail 7

Vapor trail of an aircraft and its shadow on a cloud layer. (Photo by Pekka Parviainen)

6. Lengthening shadows

At night when you pass a streetlight you can see how your shadow lengthens faster and faster as you walk away from the light. Faster and faster? It cannot be true: the speed of lengthening is constant­see Fig. 5.

For more about shadows. see §§ 59. Ill. 115. 208. 247. 251. and 261.

8 Light and shadows

~ ~ ~ ~ ~ Fig. 5. The lengthening of our shadow when we pass a streetlight.