The lore of shadows, and its application to eclipses
Shadows happen when the illumination comes from a limited source, and convinces us that light travels in straight lines, or rays. Of course, we know that light is actually a wave, but that is another story that must be told elsewhere. We must look at shadows very closely indeed to discover any of the effects of the wave nature of light, and for now we can neglect it as an inconsiderable effect. Illumination that produces shadows has the first traces of coherence, in that all the rays proceed from approximately one point, or are stigmatic. This is a small regularity indeed, but it is enough to give us shadows. If, in addition, the light was of a narrow band of wavelengths, the further coherence would show us interesting frills and bands at the boundaries of the shadows. With laser illumination, these details are easily seen.
There are positive shadows, when an opaque obstacle is located in the stigmatic illumination. We see them every sunny day, in the nearly parallel rays of the sun, which casts the most familiar sharp shadows. There are also negative shadows, when the obstacle is replaced by an aperture in an opaque screen. In either case, the form of the shadow can be found by tracing rays from the source through points on the boundary of the obstacle or aperture. When these rays fall on a surface, they mark the separation of light and shade. In the wide sense, a shadow is essentially a three-dimensional object, not a two-dimensional figure.
Shadows allow us to determine the altitude or declination of the sun, and the heights of objects too high to scale easily. We simply apply the principles of similar triangles. If you have never actually done this, try it!
In most cases, the source of illumination is not a point. Still, one can consider that each point of the source forms its own shadow, and the shadows overlap to form the complete shadow. There may be a region which never extends beyond any of the shadows, or, from which no point of the source can be seen. This region is the umbra, Latin for "shade." There is a region in which none of the shadows extends. From this region, the complete source can be seen, and we are outside the shadow. In between, some of the shadows fall, and some do not. In this region, only part of the source can be seen, and the intensity of illumination depends on how much of the source is visible. grading from the umbra to the illuminated region. This area is called the penumbra, from the Latin for "almost-shadow."
The shadow of an object in the sun clearly shows umbra and penumbra. As we go farther from the object, the umbra decreases in size while the penumbra increases. The shadow becomes "fuzzy." At a certain distance, the shadow is all penumbra. At large distances, the penumbra becomes less dark, as a smaller portion of the sun is obscured, until finally no difference in illumination can be perceived, and the shadow disappears. The negative shadow of a small aperture first has a sharp umbra surrounding an illuminated patch the same size as the aperture. Farther away, the penumbra softens the edges. Finally, we have a cone of light that is actually a shadow image of the sun, as in a pinhole camera. If it is projected on a plane surface, the image is elliptical. There are many interesting observations to be made with solar images of this type. The sun has an apparent diameter of about half a degree, or about 1/115 radian. From this figure one can quantitatively analyze solar shadows.
On a recent afternoon, I noticed that the decorative ironwork on my security door was sharply shadowed on the wall of a room. The edges of the shadows were quite sharp, much sharper than any shadow cast directly by the sun over such a distance. The source of light casting the shadow had to be close to a point source, and had to originate with the sun. How could this happen? I looked for the source, and found it was the chrome trim on a car parked across the street. The lowering sun was reflected from it into my door, and the image was dazzling. In this case, the sun was being reflected from a convex mirror, and a quick sketch convinced me that the size of the solar image in this mirror was much reduced, to a width equal to the subtense of a half a degree in half the radius of curvature of the chrome trim, so indeed the solar image was practically a point source, and cast a sharp shadow.
The earth and moon cast shadows in space, the umbras forming long cones directed away from the sun. When the moon enters the earth's shadow, which it can do when full and directly opposite the sun, there is a lunar eclipse. When the earth enters the moon's shadow, which can happen when the moon is new, there is a solar eclipse. The data that are necessary for calculating the shadows are the radii of sun, earth and moon, and their distances apart. The radius of the sun is 6.96 x 108 m, of the earth, 6.37 x 106 m, and of the moon, 1.738 x 106 m. The earth is 1.497 x 1011 m from the sun, and the moon is 3.84 x 108 m from the earth. The distances vary slightly, since the orbits of the earth and moon are not exactly circular. The earth and sun are slightly oblate, but this can be neglected as well in approximate calculations.
The Figure shows the geometry for finding the size of the shadows. The properties of similar triangles is all that is needed. The penumbra and umbra are noted. Try to imagine what the sun looks like from points in the penumbra. The umbra ends at point P. Beyond that, there is only penumbra since the earth never covers all the sun. The angle θ of the cone of the umbra is easily calculated from the radii of the two bodies, and their distance apart. The angle of the earth's umbra is about 0.26°, as expected. At the distance of the moon, the radius of the umbra is 4.61 x 106 m, sufficient to cover the moon 2.65 times. In a lunar eclipse, we observe the shadow of the earth on the moon, so it appears about the same to everyone that can see the moon.
The photograph, from the BBC, shows the shadow of the earth on the moon in the recent eclipse of January 2001. It is noticeably circular, which shows that the earth is a sphere. This was commonly adduced as evidence of the spherical earth in ancient astronomy. The shape of the earth's shadow is different from the shape of the dark area in lunar phases. In the classical Roman-Greek world, the earth was regarded as spherical, and the fact was well-established. The point P at the end of the umbra is about 1.00 x 109 m from the earth, a million kilometers. When the moon is in the umbra, it is only illuminated, if at all, by light coming from a ring around the earth, which has been scattered in the earth's atmosphere. This light is generally reddish, as was seen in the January 2001 eclipse. The view is telescopic, with the isolated circular Mare Crisium visible at the left, and the connected Maria Fecunditatis, Tranquillitatis and Serenitatis from top to bottom beside it.
The moon also casts a shadow. Its umbra extends a distance of about 3.7 x 108 m behind it. Since the distance to the moon is about 3.8 x 108 m, the earth just about escapes the umbra altogether. In a total solar eclipse, the tip of the umbra, the point P, just grazes the surface of the earth, and only in a narrow strip is the sun seen completely covered by the moon. In many cases, the umbra does not reach the earth at all, and an annular eclipse results. These conditions make solar eclipses rather rare, and difficult to predict. In a solar eclipse, we are actually in the shadow, so it appears differently to observers at different locations. Although a total eclipse is very infrequent, and requires travelling to places where it can be seen, a penumbral or partial eclipse is much easier to observe. The photograph shows a total eclipse. The bright area is a Baily's Bead, the sun showing through a gap in the rough surface of the moon. The white area surrounding the moon is the solar corona, not visible when the sun's disc is visible.
The paths of the sun and moon are separate, and cross only at the nodes of the moon's orbit, where it crosses the sun's path, the ecliptic. Eclipses can occurr only when the moon is at a node, and the sun is either near the node, for a solar eclipse, or near the opposite node, for a lunar eclipse. This is what makes eclipses rare, not occurring at every new and full moon. The sun is near a node only twice a year, at dates separated by about six months, so there are about two lunar and two solar eclipses each year. The orbits cross at only about 5°, so eclipses can occur in an interval before and after the sun is actually at a node. There may be more than one eclipse as the sun passes each node. It is easy to predict an eclipse if you know where the sun, moon and nodes are. A lunar eclipse can be seen everywhere that the moon is visible, which means at night, while a solar eclipse can be seen only over restricted areas of the earth. For a total eclipse, these areas are very restricted, as has been noted. The sun returns to the same node in an eclipse year, which is only about 346 days, because the nodes are moving westward to meet the sun.
We have noted that the umbra of the earth's shadow is considerably wider than the moon, so there is no difficulty in understanding why a total lunar eclipse may occur every time the sun is at a node. We have also noted that the sun and moon appear about the same size, so for one to fit over the other they must be aligned fairly exactly. This means that for an eclipse, both sun and moon must simultaneously occupy the node. Since this is a very unlikely event, we might think that solar eclipses would not occur. However, they do occur, and about twice a year, every time the sun is at a node. The reason for this is that the moon does not appear in the same place relative to the sun at different locations on the earth, because of lunar parallax, as shown in the diagram on the right. The moon is close enough to the earth that this parallax is about 1°, twice the size of the moon itself. The tip of the umbra of the moon's shadow goes swinging about in space, and may touch the earth somewhere whenever the sun is at a node.
The diagram at the left shows that parallax can put the center of the moon on the ecliptic for about 23 days when the sun is near a node. Since the sidereal month is only 27 days, a solar eclipse of some kind is almost guaranteed somewhere on the earth, and occasionally even two eclipses in successive months. A solar eclipse at a given location is indeed a rare event; a solar eclipse somewhere on earth is rather common. Note that parallax does not affect lunar eclipses in the same way, since we are observing the illumination of the moon, not its location. It will appear eclipsed no matter where one is, since the umbra shares that same parallactic displacement as the moon. In a solar eclipse, the sun has a negligible parallactic displacement of only about 8". Oddly, elementary astronomy texts do not mention the effect of parallax on solar eclipses, which may leave the student dissatisfied.
This theory of eclipses is one of the triumphs of Greek astronomy, and permitted a more or less accurate prediction of eclipses, but probably not an accurate prediction of the path of totality of a solar eclipse. We must not think that other astronomies had the same understanding and used similar methods based on theory. Generally, they regarded the moon as self-luminous and at the same distance as the sun. The Chaldeans thought the moon had a bright side and a dark side, and rotated through the month to show the different phases. The only way to predict eclipses in such astronomies was to use empirical intervals between eclipses. One such interval, the saros, seems to have been discovered by the Chaldeans, and used for the prediction of eclipses. The saros depends on an approximate commensurability of the eclipse year, the month, and the 18.6 year period of the precession of the nodes of the moon's orbit. The saros is 18 years, 11-1/3 days, after which an eclipse repeats, a third of the way farther around the earth. After three such periods, about 54 years, the eclipse recurs at the same longitude.
Legend says that Thales predicted the total solar eclipse of 28 May 585 BC, possibly by using the saros, but this is probably only one of the likely stories created in ancient times to fill gaps in history. A recent astronomy text said that the eclipse affected a battle between the Lydians and Medes, which it calls two Greek factions. Actually, neither Lydians or Medes were Greek, and the Medes were really Persians. It also says saros comes from a word for "repetition." It actually comes from sarow, meaning "sweep clean." In fact, eclipses, especially solar ones, were as rare then as now, probably because of the efficiency of the religious observances used to prevent them. Astronomers were far more likely to claim to have evaded an eclipse than to have predicted one. At any rate, lunar eclipses could be predicted with some success, but solar eclipses were a difficult problem.
There is still more to be said about shadows in general. If you look carefully at the edges of shadows, you may perceive brighter or darker fringes where the illumination changes. As long as the rate of change of intensity with distance is uniform, the eye follows it, but when the rate changes, as at the edges of the penumbra, the eye tends to overcorrect. If the intensity begins to increase less rapidly, a bright fringe is seen. If it decreases less rapidly, a dark fringe is seen. That is, there seems to be an inertia in the perception. These are called contrast fringes, and are an effect of visual perception.
Where a shadow is the result of the overlapping of two penumbras, as with two tree branches, say, about in the same direction, a brightening will be seen in the middle of the shadow, produced when the two branches exactly overlap and block the light less than the two branches separately. This phenomenon, and contrast fringes, should not be confused with the effects of interference.
Often there are two sources of illumination. By day, there may be the sun and the blue sky, for example. A shadow cast by the sun is still illuminated by the sky, so it tends to appear blue. This is seen to the best advantage in a snow-covered scene, where the shadows can appear definitely blue against the brilliant white of the snow. The skylight is diffuse, and does not cast distinct shadows. The true colors of a shadow can be masked by the contrast colors that are a result of the visual sense. A shadow can appear blue when there is no blue light involved at all.
The two sources may be two street lamps, or the moon and a street lamp. With two point sources, dual shadows are cast, which may be of different colors and characters. What happens to your shadow while you walk from beneath one street light to beneath the next one? Some people have thought that their shadow grew fainter as they walked away from one light, then disappeared when the lights were equally distant, and finally deepened as the next light was approached. Check what actually happens by experiment. The difference in colors of two street lamps, or between the moon and a street lamp, can give rise to interesting contrast appearances. Since the illumination decreases as the square of the distance from a point source, you can compare the brightness of two sources by finding the point when the shadows are equally dark, and measuring the distances to the sources. The full moon, incidentally, produces a normal illumination of about 0.20 lux (lumens per square metre).
Strange things happen to the shadows when the source is a line rather than a point, as when a total solar eclipse is approaching and the illumination comes from a narrow bright crescent. All the shadows take on an unusual aspect in this case. Similar observations can be made just as the sun is setting at a clear horizon, when again the source is a narrow line, or during a partial eclipse. Among other things, the density variations in the air can be seen as striae, bright and dark shadows racing over the ground.
The three-dimensional shape of a shadow can sometimes be seen, when the atmosphere is misty and scatters light. The shadows of the posts of streetlights are three-dimensional in the fog. In the evening, the hazy sky shows where the rays of the sun penetrate the clouds, as crepuscular rays. These appear to form a fan, but really they are parallel and extend from horizon to horizon. Your shadow cast by the sun behind you into mountaintop mist may show your head encircled by rings of light, the Brocken-spectre or glory.
Shadows can be cast on the surface of water. Here, we must consider not only the light returned from the surface, but also that that penetrates into the water and is returned from the depths. These two sources of illumination may differ in color. Agitation of the surface, and turbidity of the water, play important parts in the appearances. Sometimes the surface is smooth, and true shadows cannot be seen.
Finally, clouds may cast shadows on themselves, and on other clouds. The direction of the light is often mistaken. It does not come from the sun as seen in the sky, but is parallel to the rays falling on you. This can lead to many curious impressions in which the clouds are viewed as illuminated by a nearby source like a lamp on a table, and not by a distant searchlight.
Many of the observations just mentioned are discussed in more detail in
M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954).
Composed by J. B. Calvert
Created 12 January 2001
Last revised 30 July 2003