SHADOWS

I recommend to those who are new to these games the entertainment of watching the gyrations and transformations of their own shadows while walking at night along a lamplit road. As you pass close to the lamp your shadow will appear short and squat by your side, and slowly turn in the direction of your walk while growing longer and narrower, till the bright lamp of the next lampost will replace it by the shadow that is now behind you.

E.H. Gombrich: Shadows: The Depiction of Cast Shadows in Western Art, National Gallery Publications, 1995, p. 12

2.1 No light without shadow

A shadow is a volume of space not directly illuminated when light is intercepted by an object. Usually we are aware of these shafts of darkness only when they fall upon an illuminated surface, where they are seen as dim, distorted outlines. But if the medium through which the shaft of the shadow passes is filled with particles able to scatter light, such as dust-laden or hazy air, fog, or turbid water, the shaft itself becomes visible.

We are apt to overlook the effect of shadows on the way the world looks to us. If we notice them at all, it is probably as a diversion. We can amuse ourselves by noticing how our shadow changes shape depending on where we stand relative to the source of light, or on the contours of the surface on which the shadow falls. And people have always been beguiled by the dancing shadows produced by the flickering flames of a fire or a candle. But since shadows are cast by objects, they are indirect evidence of space and solidity. Adding shadows to the objects in a painting makes them seem more real, and lends depth to the scene in which they appear.

Although our eyes instinctively seek out light, without the subtle contrasts between brightness and darkness brought about when light meets solidity, the world would look flat and drab. Recall for a moment the monotony of an overcast day, made all the more oppressive by an absence of shadows. In fact, the pattern of light and shade created when a scene is illuminated is one of the best visual clues that we can have as to the shape of things within it. It emphasises the relief of uneven surfaces, and provides

Antisolar point

Figure 2.1 The antisolar point. This is a point on the opposite side of the sky to the Sun. It lies in the direction of the shadow of your head.

Antisolar point clues to depth, enabling us to discern the shape and form of otherwise featureless snow-covered landscapes and clouds. Despite the fact that Galileo's telescope was not powerful enough to reveal precise details of lunar features, Galileo deduced that the Moon's surface was like the Earth's surface, mountainous and uneven, from the elongated lunar shadows that he saw through the telescope.

It hardly needs saying that all shadows point away from the source of light. In the case of solar shadows this direction is known as the antisolar point. This is an imaginary point that lies on the opposite side of the sky from the Sun, on a line that passes through the eye of the observer. This means that your antisolar point is in a different place from mine. We can't share the same antisolar point. The antisolar point is of interest because so many optical phenomena, such as rainbows, glories and heiligenschein, are centred on it.

2.2 Solar shadows

Leaving aside the Moon, which in any case shines by borrowed light, the Sun is our only source of natural light. Because it appears to us as a disc, rather than a point like a faraway star, the Sun illuminates everything with rays of light that are not quite parallel to one another. Such a source of light is known as an extended source. Close to an object the effect of this divergence is negligible, and so shadows are sharpest when they fall near an object. Further away the divergence increases, giving the shadow a fuzzy edge

2.2 Solar shadows known as the penumbra; the darker, central portion is known as the umbra. Although the umbra receives no light directly from the Sun, it is illuminated by airlight, and by sunlight reflected from surrounding surfaces. Together these lighten solar shadows. The fuzziness of the penumbra is due to the fact that the amount of direct sunlight illuminating the penumbra increases towards its outer edge, making it difficult to discern the point at which the shadow gives way to full sunlight.

The penumbra is such an obvious feature of solar shadows that it comes as a surprise to learn that the first person to describe it, and explain how it arises, was Leonardo da Vinci in the fifteenth century. Earlier accounts of shadows used the fact that a shadow has a similar shape to the object that causes it to argue that this was possible only because light travels in straight lines. In other words, shadows were used as proof that light travels in straight lines. The penumbra was ignored because sunlight was represented as if it came from a single point on the Sun, rather than many points spread over its disc. Shadows formed by a point source lack a penumbra. However, Leonardo's analysis of shadows was not widely known, and the world had to wait another hundred years for Johannes Kepler to lay the foundations for the science of optics in its modern form, and in passing explain the penumbra. In fact, we owe the term 'penumbra' to him.

The angular diameter (see Appendix, page 301) of the Sun is approximately 0.5°, or 1—0 radian, and so the maximum distance at which the umbral shadow can be formed is approximately 120 times the diameter of the object. Beyond this point, only the penumbral shadow persists. To cast an umbral shadow at a particular point, an object must have an angular diameter that is greater than that of the Sun measured from the point at which the shadow falls. Hence neither small objects that are close to a surface, nor large ones that are far from one, can cast an umbral shadow. It's worth keeping this in mind where eclipses and crepuscular rays are concerned. Eclipses, of course, are due to the grandest of all shadows: those that the Earth and Moon cast into the void. We are unaware of the Moon's shadow until we pass through it and witness an eclipse of the Sun. In an eclipse of the Moon, the Moon passes through the Earth's shadow.

The Moon is our only other major source of natural light. At its brightest, moonlight is about half a million times less bright than sunlight. How do shadows cast in moonlight compare with those cast in sunlight? The most striking difference between them is how profound lunar shadows appear to be compared with solar ones. In daylight, scattered and reflected

Figure 2.2 (a) Solar shadows. The Sun is not a point source of light and so shadows formed in sunlight have a fuzzy edge called the penumbra. The central region of the shadow receives no light directly from the Sun, and is called the umbra. In this diagram, the width of the penumbra has been greatly exaggerated to show its relationship to the umbra. In reality, it is much less broad. The penumbra is fuzzy because some light reaches it from the Sun's edge. For example, although no light from A reaches the area between A' and B', this area receives light from B.

(b) Mach bands. Where a darker rectangle meets a lighter one, the edge A of the darker one is noticeably darker than its opposite edge B.

Figure 2.2 (a) Solar shadows. The Sun is not a point source of light and so shadows formed in sunlight have a fuzzy edge called the penumbra. The central region of the shadow receives no light directly from the Sun, and is called the umbra. In this diagram, the width of the penumbra has been greatly exaggerated to show its relationship to the umbra. In reality, it is much less broad. The penumbra is fuzzy because some light reaches it from the Sun's edge. For example, although no light from A reaches the area between A' and B', this area receives light from B.

(b) Mach bands. Where a darker rectangle meets a lighter one, the edge A of the darker one is noticeably darker than its opposite edge B.

sunlight are bright enough to illuminate all but the deepest recesses. At night, only those nooks and crannies that are directly illuminated by moonlight are visible, hence the inky shadows and dramatic contrasts that characterise a moonlit scene. At the same time, a lunar shadow appears sharper than a solar shadow because its penumbra is not very bright. Nevertheless, even in moonlight, the branches at the top of a tree cast shadows with quite obvious penumbras. To see this clearly, catch the shadow on a sheet of white paper.

2.3 | Shadows formed by point sources

Shadows play an important role in the way the Moon appears to us, both to the naked eye and through a telescope. Compared with solar shadows on Earth, those on the Moon are very much darker because of the absence of an atmosphere that would otherwise scatter light into the shadow. There is scattered light on the Moon: sunlight reflected by its surface. But much of this doesn't reach the depths of the pits and hollows that cover the Moon's surface. Solar shadows on the Moon are thus rather like lunar shadows on Earth: very dark. This is why the surface of the crescent Moon appears so much fainter than it does at other times during the cycle of lunar phases (see section 9.18 for a fuller explanation). At the same time, shadows cast by craters and mountains when the Moon is waxing or waning emphasise relief, and this makes the lunar landscape far more real when seen through a telescope than it would otherwise appear. It is this that enabled Galileo to conclude that the lunar surface is uneven, not smooth as his contemporaries believed.

A great deal of light reaches the ground, even on days when the sky is completely overcast. Despite this, distinct shadows are not formed because sunlight is scattered as it passes through clouds, so that it is fairly evenly distributed across the whole dome of the sky when it emerges from them. Under these conditions there is as much light coming from one direction of the sky as from any other direction and so a shadow can't be cast. Nevertheless, if the sky is partially blocked off, say by a wall, then a feeble shadow is cast in the direction of the obstacle.

2.3 Shadows formed by point sources

The shadow cast by an object when it is illuminated by a point source differs in several important respects from one that is cast in direct sunlight. To begin with it has no penumbra.

A point source is one that is very much smaller than the object it illuminates. A street lamp or a candle flame both act as point sources for objects that are at some distance from them. The difference between such sources and the Sun is that rays of light from the lamp or candle effectively diverge from a single point, whereas those from the Sun come from several points spread across the solar disc. Consequently, the edge of a shadow cast by a point source is always sharp no matter how far from an object it is formed. The size of the resulting shadow depends on how close the object is to the

Figure 2.3 Shadow due to a point source. When the source of light is a point, or almost so, shadows cast are sharp (they don't have a penumbra) and increase in size with distance from the object, so that even a small body can cast a huge shadow.

Figure 2.3 Shadow due to a point source. When the source of light is a point, or almost so, shadows cast are sharp (they don't have a penumbra) and increase in size with distance from the object, so that even a small body can cast a huge shadow.

Whay Are Shadows Formed

source of light, and on the distance from the object to the surface on which the shadow is formed.

Stars and planets are point sources, though their light is usually too feeble to cast shadows. Nevertheless, at its brightest, Venus casts umbral shadows that are just noticeable to a dark-adapted eye outdoors on clear moonless nights far from city lights. Even Jupiter and Sirius are bright enough to cast noticeable shadows if light from all other sources is rigorously excluded, say by viewing the shadow within a darkened room. Light from the planet or star enters the room through an open window, and the shadow of a suitably positioned object is cast onto a white surface. Your eyes must, of course, be fully dark adapted to notice the very faint change in brightness that marks the boundary of the shadow. A shadow cast by Sirius is more difficult to see than one cast by Jupiter.

On occasion the Sun can act as a point source. Look carefully at a shadow when the Sun's disc is about to be obscured by a sharp-edged cloud: just before the Sun completely disappears behind the cloud, shadows become much sharper because the Sun has been reduced to a point source, if only for an instant. The same thing occurs shortly before and after totality during a solar eclipse: shadows become very sharp.

Figure 2.4 Shadows due to a point source. This photograph was taken shortly before totality. The Sun was a narrow sliver rather than a disc, and resulting shadows lacked a penumbra. (Photo John Naylor)

2.4 Mach bands

Try the following experiment. Fasten a small coin to a windowpane that faces the Sun. Hold a sheet of white card close to the window so that the shadow of the coin falls on it. Slowly move the card away from the window, keeping an eye on the shadow as you do so. You will find that, when the card is approximately 100 coin-diameters from the coin, the umbral shadow appears as a dark ring surrounding a noticeably less dark centre. At the same time the area just beyond the penumbral shadow appears to be brighter than the rest of the card. The same pattern of apparent increased darkness and brightness is seen if the card is held so as to catch the shadow of the window frame or indeed of any object at sufficient distance from the card.

With a little practice, these variations in brightness can be noticed out of doors around almost any shadow, though success depends on the colour and texture of the surface on which the shadow falls. For example, examine the edge of the shadow of your upper torso and head cast on a cement pavement. If the Sun is low in the sky and therefore you cast a long shadow, you should be able to notice that the more distant parts appear to be fringed by a narrow, bright band. This bright band falls just outside the penumbra. Now shift your attention to the umbra: it appears darkest just inside the outer edge.

In all these situations there is no physical reason for what you see. It is a purely physiological phenomenon due to the fact that, when a lightsensitive cell within the eye is stimulated, the response of neighbouring cells to the same stimulus is lessened. This is an evolutionary adaptation that makes it easier for our visual system to detect shapes by emphasising changes in brightness that occur at their edges. The physiological process is known as lateral inhibition. An unwanted consequence of lateral inhibition is that, when you look at the boundary between areas of unequal brightness, the eye's response makes the brighter side of the boundary appear lighter than it really is, while simultaneously making the darker side look less bright. The phenomenon was first systematically investigated by Ernst Mach, an Austrian physicist and philosopher of distinction, and is known as a Mach band. A Mach band is present wherever there is a sudden change in the brightness of a surface. It can sometimes be confused with the afterimage caused when your eyes stray from a bright area to a darker one. You will find that with a little practice you can easily distinguish the one from the other.

Although Mach bands are illusory, they can be photographed in the sense that the eye responds to variations in brightness in the photograph in exactly the same way as it does to the original. In other words, you will see a Mach band in a photograph of a Mach band, although neither has an objective existence.

2.5 Coloured shadows

An umbral shadow out of doors would be totally dark were it not that it is indirectly illuminated by airlight. On a clear day this is, of course, predom-

| Coloured shadows

This shadow is due to light the candle and is illuminated by

Figure 2.5 Coloured shadows. The shadow of an object may be coloured if it is illuminated by light from a different source than the one that produces the shadow. In this diagram a pencil casts two shadows. One is due to airlight coming through a window. This shadow is illuminated by candle light and looks yellowish. The shadow cast by candle light is illuminated by airlight and looks bluish.

This shadow is due to airlight coming through the window and is illuminated by yellowish light from the candle.

bluish airlight from the window

Figure 2.5 Coloured shadows. The shadow of an object may be coloured if it is illuminated by light from a different source than the one that produces the shadow. In this diagram a pencil casts two shadows. One is due to airlight coming through a window. This shadow is illuminated by candle light and looks yellowish. The shadow cast by candle light is illuminated by airlight and looks bluish.

inantly blue, and shadows sometimes have a noticeably blue hue. Shadows on snow are unmistakably blue. With practice, you should be able to notice a blue hue to shadows cast on surfaces such as cement pavements, which are not as white as snow.

Distant shadows take on a bluish hue because airlight scattered by the intervening atmosphere becomes more noticeable when seen against a dark background. The shadow acts as a backdrop against which we can more easily notice the faint blue hue of airlight.

Strongly coloured shadows are also produced if an object is simultaneously illuminated by more than one source. A shadow formed by the light from a north-facing window on a clear day will look distinctly yellow if the object that casts it is simultaneously illuminated by the light of a filament lamp: the shadow cast by the lamp will be blue. Coloured shadows can also be seen outdoors soon after sunset when objects are illuminated from one direction by blue airlight, and from another by a low-pressure sodium streetlight. A shadow cast by the yellowish sodium light is distinctly blue. The colour can be so vivid that at first glance it may appear to be painted on the surface on which it is seen. Low-pressure sodium lights also cast coloured shadows in combination with moonlight or car headlamps.

The colours we ascribe to these shadows depend only partly on the

Figure 2.6 Blue shadows. A shadow cast on snow looks distinctly blue due to the fact that it is illuminated by blue airlight. (Photo John Naylor)

colour of the illuminating light. The eye itself plays a role. When we look at two adjacent surfaces, the colour of one is affected by that of the other. Painters have long known this. To see how this comes about, imagine that you are looking at a circular blue patch on a red surface. Better yet, cut out a circle of bright blue paper and glue it to a sheet of vivid red paper. When you shift your gaze from the red surface to the blue circle, your eye's blue-sensitive cells respond strongly to the blue surface because they were previously unstimulated when illuminated by light from the red surface. At the same time, the red-sensitive cells have been fatigued and so do not respond strongly to any red that might be reflected by the blue patch. The overall effect is to make the blue look bluer than it would if it were not surrounded by a red surface. If the whole surface were the same colour (say all blue or all red), the response of the cells sensitive to that colour would gradually weaken and the surface would look less vivid than it did to previously unstimulated cells, i.e. as it did when you first looked at it. We can generalise this as follows: a patch of a particular colour will appear more vivid if it is surrounded by another colour that does not stimulate the same type of colour-sensitive cells. The phenomenon is known as simultaneous colour contrast, and is similar to lateral inhibition.

Now consider a blue shadow seen out of doors at twilight just after the streetlights have been turned on. The area around the shadow is illuminated by the orange-yellow light of a sodium lamp. The area within the shadow is illuminated by scattered airlight which is predominantly, though not wholly, blue. Hence, when you look at the shadow, the reflected airlight stimulates the blue-sensitive cells much more strongly than those sensitive to yellow. The result is a shadow that looks distinctly blue. The opposite effect occurs when an object is simultaneously illuminated by light from a north-facing window and a light from a lamp. Light from the lamp is deficient in shorter wavelengths, those at the blue end of the spectrum, and so the shadow cast by airlight is illuminated by light that is deficient in blue. The eye's response to blue is lessened by airlight, and so when looking at this shadow it responds more strongly to the longer wavelengths than to the shorter ones, and the shadow takes on a smoky yellowish hue.

J.W. Goethe, the renowned German poet and dramatist, investigated the phenomenon of coloured shadows in some detail. He carried out many experiments during a lengthy series of investigations that he conducted into the nature of colour in the hope of discrediting the Newtonian view of colour. It was a lost cause, in part because Goethe didn't draw a clear distinction between the physical and physiological aspects of light. But he was a meticulous observer, and he recorded a huge number of fascinating colour-related phenomena that are contained in his Farbenlehre, the book he wrote on his theory of colours. Among these is the following description of coloured shadows.

Let a short, lighted candle be placed at twilight on a sheet of white paper. Between it and the declining daylight let a pencil be placed upright, so that its shadow thrown by the candle may be lighted, but not overcome, by the weak daylight: the shadow will appear of the most beautiful blue.

W. Goethe

2.6 The heiligenschein

You may sometimes have noticed a faint sheen, or increased brightness, around the shadow of your head when this falls on a grass lawn, particularly when the Sun is low, and you cast a long shadow. This sheen is known as a heiligenschein, a German word meaning 'holy glow'. It is much more obvious when you are in motion, and its brightness varies from one patch of lawn to another.

To see a heiligenschein, stand on a lawn in which the blades of grass are erect and a few centimetres long. Rock slowly from one foot to the other and

Figure 2.7 Self-shadowing. Each of the white circles casts a shadow represented here by a black circle. These shadows become visible away from your antisolar point. Around the antisolar shadows are hidden by the objects and so this region appears brighter. The effect of self-shadowing in this diagram is seen more clearly if you hold the diagram at arm's length.

keep an eye on the brightness of the region just above your head's shadow. You should notice that, as the shadow moves, the grass above it becomes slightly brighter, while the grass above the point where the shadow fell previously is now noticeably less bright. Moving makes the heiligenschein more obvious because a change in brightness is more easily noticed than constant brightness.

A heiligenschein can also be seen from a considerable distance on fields of wheat and rice, both from ground level and from the air. In Japan the heiligenschein is known as inada no goko or 'halo in the rice fields'. It is frequently seen from balloons and aircraft when flying over woodland. Look for a heiligenschein around the antisolar point of a plane that you are travelling on, just after take off, or as it comes in to land. These are not the only circumstances in which you can see one, as the following report makes clear.

. . . in this mountain region I have frequently seen the halo projected on a grassy slope a mile or more distant, and under these conditions it appears as a circular or elliptical patch of light without the central shadow, the diminution of intensity due to the penumbral shadow of one's head being, of course, quite inappreciable at such a distance.

J. Evershed

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Figure 2.8 The heiligenschein. A faint brightness is often seen around the shadow of your head when this falls on grass. It is most pronounced when you cast a long shadow, say in the early morning or late afternoon. The brightness is due to self-shadowing and is known as a heiligenschein.

Figure 2.8 The heiligenschein. A faint brightness is often seen around the shadow of your head when this falls on grass. It is most pronounced when you cast a long shadow, say in the early morning or late afternoon. The brightness is due to self-shadowing and is known as a heiligenschein.

Grass near the edge of the shadow of your head looks bright because a blade of grass at or near the antisolar point gets in the way of its own shadow. This is known as shadow hiding. Within this region, you see mainly the illuminated surfaces of blades, and few shadows. When you look away from the antisolar point, you see grass blades from the side, and so can see their shadows. The overall brightness of the grass thus diminishes when you shift your gaze away from the edge of your shadow. Given that shadow hiding is most effective at the antisolar point, it follows that you can't see a heiligenschein around someone else's shadow, and that they can't see yours.

The overall shape of the bright patch due to shadow hiding is determined by the general shape of the objects on which you cast your shadow (figure 2.8). Elongated objects such as blades of grass produce elongated bright patches. Hence the heiligenschein seen in grass is most pronounced just above the shadow of your head. In the case of a forest canopy seen from a plane or balloon, the heiligenschein is circular because the canopy of individual trees seen from above is also approximately circular.

Shadow hiding also plays a role in the variation of the Moon's apparent brightness as it orbits the Earth. The Moon's phases are due to an increase or decrease in the portion of its illuminated surface visible from the Earth. As the area of this visible surface increases from one day to the next, the Moon reflects more and more light in the Earth's direction.

Figure 2.9 Heiligenschein. The shadow of the photographer is surrounded by a bright glow due to light reflected back in his direction by dew drops on the grass. This glow is called a heiligenschein. The photograph can't show the heiligenschein around the shadow of his companion because light reflected in that direction doesn't enter the camera lens. (Photo John Naylor)

Figure 2.9 Heiligenschein. The shadow of the photographer is surrounded by a bright glow due to light reflected back in his direction by dew drops on the grass. This glow is called a heiligenschein. The photograph can't show the heiligenschein around the shadow of his companion because light reflected in that direction doesn't enter the camera lens. (Photo John Naylor)

However, when the Moon is a crescent, this visible surface is illuminated by a low Sun, which leads to a great deal of self-shadowing. Self-shadowing makes the Moon's brightness less than it would be if it were a smooth sphere. As the Moon approaches opposition, the point along its orbit when it is on the opposite side of the Earth from the Sun, it gets closer to the antisolar point, and shadows on its surface become shorter until, at opposition, they vanish from sight altogether, hidden behind the objects which cast them. In their place we see only the illuminated surfaces of these objects, i.e. shadow hiding takes place. This dramatic increase in the Moon's brightness is known as opposition brightening. Note that the Moon is not any brighter at opposition than it would be if it were perfectly smooth. The fact that it has a rough surface means that it is much less bright at phases other than full than it would be if it were smooth. Because Mars also has a rough surface, it is the only other astronomical body that undergoes noticeable opposition brightening. The other superior planets don't do so because they don't have solid surfaces and therefore have no shadow-casting features.

Self-shadowing and shadow hiding also play a part in the appearance of landscapes here on Earth. In the morning or in the afternoon, when the Sun is low in the sky, the sunward side of a rough surface is always noticeably darker than it is in the direction away from the Sun.

A heiligenschein seen on dry grass is sometimes referred to as a dry heiligenschein, to distinguish it from a heiligenschein seen on grass covered in dew. A dew heiligenschein is brighter than a dry heiligenschein.

If you examine dew on a blade of grass with a low-power microscope you will see that most dewdrops are small and spherical. This enables them to act as miniature lenses that focus sunlight on the surface of the leaf. The leaf reflects this light which then is channelled back more or less in the direction from which it came. Dew is deposited as spherical drops on some grasses more easily than on others. Hence dew heiligenschein are more pronounced on some lawns, or particular parts of a lawn, than on others.

A very distinct heiligenschein is formed on road signs that have been painted with reflective paint. This type of paint contains large quantities of tiny glass spheres to make it more reflective and these act as very efficient retro-reflectors. The signs are usually above head height. Your shadow can fall on such a surface only when you are illuminated by a light that is below head-height, say by the headlamps of a car. Occasionally you may see a very bright heiligenschein on street signs at sunset or sunrise. The Sun is then low enough for your shadow to fall on street signs that are close to the ground.

You can photograph your heiligenschein, but keep in mind that when you do so it will be formed around the shadow of the camera and so will only appear around your head if you hold the camera to your eye.

2.7 Shadows on water

Distinct shadows are cast on turbid water but not on clear water. On the other hand, reflections can be seen more clearly on water that is not turbid. The effects are related. If you examine shadows cast on turbid water you can see that they are formed within the water and not on the surface. The reason for this is that the suspended particles that make the water turbid also reflect a great deal of light back towards the surface. Those that fall

Figure 2.10 Shadows in water that is slightly turbid can be fringed with colour. Light from A appears reddish because some blue light has been scattered by the particles that make the water turbid. Light from B appears a smoky-blue because scattered light is seen against the dark background of the shadow.

Figure 2.10 Shadows in water that is slightly turbid can be fringed with colour. Light from A appears reddish because some blue light has been scattered by the particles that make the water turbid. Light from B appears a smoky-blue because scattered light is seen against the dark background of the shadow.

Trough

Crest Dark patch

Bright patch

Sunlight

Water

Figure 2.11 Patterns of light and shade in water. A crests acts like a converging lens, and concentrates light forming a small bright patch on the surface below. A trough has the opposite effect: it spreads light so that the surface below is faintly illuminated.

within the shadow are less brightly illuminated than those in direct sunlight and this provides the contrast which is necessary if you are to see a shadow.

On the other hand, the brightness of turbid water makes it difficult to see reflections in it. The best surface reflections are seen in pools of clear

Figure 2.12 Shadows in water. The streaks of light that stream out from the shadow of the hat in this photograph are due to shafts of light within the pool of water in which they were seen. (Photo John Naylor)

water with dark bottoms. The amount of light reflected in this case represents only some 5% of the incident light. The remainder passes into the water where some is absorbed by the material at the bottom of the pool and the rest is reflected back towards the surface. It is possible, however, to see reflections in the surface of turbid water where the surface is in shadow. In this situation, the particles within the water are not illuminated and so reflect no light to overwhelm the faint reflections from the surface.

You will sometimes see a faint orange fringe around the edge of a shadow cast in slightly turbid water. The colour is due to selective scattering by particles suspended within the water, which are small enough to scatter shorter wavelengths preferentially. You see the orange hue within the shadow formed on the surface of the water because no surface reflections are possible from the area covered by this shadow and this allows you to see light reflected from the bottom of the pool. This light is deficient in shorter wavelengths because of selective scattering by the suspended particles. At the same time the scattered short-wave light is seen through the shaft of the shadow within the water, which, consequently, will have a faint smoky-blue hue. You will see this blue hue more distinctly if you use a polarising filter to reduce surface reflections. The effect may be more noticeable in a particular pond or stream on one day but not on another because of changes in the size and concentration of particles suspended in the water.

A final effect to look out for is a faint pattern of darting rays that appears to radiate from the shadow of your head if this falls on slightly turbid water when its surface is ruffled by a light breeze. Although the rays appear to be on the surface, they are in fact shafts of light within the water caused by the focusing action of waves criss-crossing the surface. These shafts of light are parallel to one another, but because of perspective they converge on the antisolar point and so they appear to radiate from the shadow of your head. Since the phenomenon is seen in slightly turbid water, your shadow will sport a faint orange fringe. If you can't see these rays when your shadow has a telltale orange fringe, you can produce the rays by briefly stirring the surface with a stick. The pattern is seen best in water that is at least 40 cm deep, against a dark background and with waves criss-crossing the surface in random directions.

2.8 Shadows formed by clouds

Clouds cast shadows into the air. These are visible as faint reductions in sky brightness that fan out from the edge of any cloud that lies directly between the Sun and an observer. The shadows are most noticeable in a clean atmosphere because airlight due to a haze may be bright enough to mask the contrast that renders them visible. They appear to fan out from the parent cloud because of perspective. In reality, their sides are almost parallel because they are formed by sunlight, the rays of which do not diverge greatly. Perspective causes the part of the shadow that is nearest to you to look larger than the one furthest from you.

Dramatic examples of this effect may sometimes be seen above the horizon when the Sun is about to rise or at which it has just set. These twilight shadows are known as crepuscular rays (section 5.5). At other times of the day, the alternating pattern of light and shade emanating from a cloud are variously known as 'sunbeams', 'the Sun drawing water' or 'the backstays of the Sun'.

Looking at these diverging shafts of light, it is tempting to conclude that the Sun acts as a point source, and that its light spreads out in all directions. In other words, it appears as if sunlight illuminates the Earth from a single

2.8 | Shadows formed by clouds

Jet Casting Shadow Clouds Images
Figure 2.13 Cloud shadows. Clouds cast shadows that are visible as diverging shafts of light and dark in the sky. (Photo John Naylor)

point. Yet, when we look at the way in which light and shadow are distributed over a cloud, it is obvious that this cannot be the case. Ignoring the slight divergence of the Sun's rays due to the fact that it is an extended source, we may assume that the Sun's rays are more or less parallel to one another. Hence the Sun illuminates everything around us from the same direction. This can be confirmed by looking at cumulus clouds that lie at some distance to the Sun's disc. Note which parts are in shadow and which are directly illuminated. If the Sun were a point source, it would illuminate the sunward side of these clouds, whereas these may well be in shadow because, from the point of view of the clouds, the Sun is really behind them, rather than off to one side.

Contrails are those long plumes of cloud that sometimes form in the wake of aircraft. They are due to condensation of the water vapour emitted by aircraft engines when the aircraft is flying through cool or moist air. If there is a thin layer of cloud below the aircraft, the shadow of a contrail may be visible from the ground as a faint dark streak in the cloud layer. The contrail shadow is subject to the same perspective considerations as any other cloud shadow and so to someone on the ground it will appear to lie further from the Sun than the contrail. Very occasionally, when an aircraft is flying directly towards or away from the observer, the contrail shadow lies directly between the observer and the contrail. If the Sun is low in the sky, the contrail shadow will fall ahead of the contrail, and it will seem as if the plane is following the line of the shadow. Contrail shadows can also be seen from an aircraft. You can find out more about this in section 6.4, which deals with glories.

2.9 Shadows under trees in leaf

Look at the shadow of any tree in leaf, and you will notice that many of the patches of sunlight within its shadow are oval. This is not accidental. The gaps between overlapping leaves act as pinhole cameras and each produces a separate image of the Sun. These images are not perfectly sharp because the gaps between the leaves are not small enough to be true pinholes. Furthermore, although the Sun is circular, its image in this situation is usually oval because its light strikes the ground obliquely. Where the Sun is overhead, the patches of light are circular. During a partial solar eclipse the oval patterns become crescent-shaped, and during an annular eclipse (see section 10.2) they form rings. The rarity of these patterns makes it well worth while recording them on film.

2.9 Shadows under trees in leaf

Figure 2.15 Shadow of a tree during an eclipse. The usual ovals of light that are seen within the shadow of a tree become crescents during the partial phase of an eclipse of the Sun. (Photo John Naylor)

You will find that the size of the leaf affects the coarseness of the patches of light. Compare those formed under a tree that has small leaves with those formed under a tree with large leaves.

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