| The Moon: Shape, Size, Nature.
The Apparent Size of the Moon.
Have you, I wonder, ever had the experience of being outdoors early on a September evening and marvelling at just how enormous the full moon looks as it rises above the eastern horizon? Certainly it looks a lot smaller when it is high in the sky! By a show of hands in the classroom, my remarks to this effect usually provoke the response that quite a number of you had noticed this phenomenon at one time or another. What, though, is the cause?
There are several plausible-sounding explanations:
Perhaps this is caused by the fact that we are looking horizontally through a lot of the Earth's atmosphere, rather than straight up through a thinner layer. This might distort and somehow magnify the image of the moon.
Perhaps the moon is not always at the same distance from the Earth, and on occasion its closer proximity causes it to look exceptionally large.
Finally, perhaps the effect is simply an illusion: could it be that the image is merely perceived as bigger when the moon is near the horizon?
There is an interesting point to be made here. Not all of these explanations can be correct, although each one sounds plausible. But all the speculation in the world cannot ever hope to answer the question! What you need to do is make some measurements and see what is really going on, rather than sit and pontificate. (In a very real sense, science became `modern' back in the sixteenth century when it was realised that experiments usually reveal truth more correctly than any philosophical discussion or appeal to revered authority.)
There are a variety of ways to make the necessary measurements, of which the most precise would be to take a photograph of the moon just as it rises and then, later, to use exactly the same combination of camera and lenses to take a photo when the moon is high in the sky. You can then intercompare the sizes of the images captured on the film.
Alternatively, you can make a crude estimate of the apparent size of the moon in a very simple way. Before trying the measurement, ask yourself whether you think you could hide the moon behind your thumb if you held it (your thumb, that is) out in front of you at arm's length, in a classical `thumbs up' position. You might be very surprised to learn that you can! (Try it and see, and look at the figures on page 31 of your text.) Even if the moon were twice as big across as it is now, you could still cover it completely with your thumb in this way. Now, of course, you can use this permanently attached part of your anatomy to gauge whether or not the apparent size of the moon changes during the night (or from night to night) as it changes position.
These two experiments will give the same answer. The moon is no bigger on the horizon than it is when high in the sky. The effect we see is entirely illusory! The origin of this very strong psychological impression is not perfectly understood, by the way. It may originate in the fact that when the moon is low on the horizon we can subjectively compare it to things like trees and houses in the foreground, whereas when it is high in the sky, surrounded on all sides by a very large black void, it is perceived as a rather small source of light. Whatever the cause, it is a compelling illusion.
By the way, the second proposed explanation, above, does have a grain of truth to it. The moon's orbit around the Earth is not a perfect circle, and at various times it is a little closer to us or a little farther away. The consequent variations in its apparent size can be measured, with care, but are much too small to account for the very striking psychological effects we notice. Moreover, the time at which the moon actually looks the largest, as demonstrated by careful measurements, does not always correspond to its being full. Sometimes, for instance, the moon is relatively close to us, and apparently large 'from top to bottom,' when it is in a crescent phase rather than full.
The Cause of the Lunar Phases.In order to reinforce your understanding of why the moon varies through its phases -- a topic thoroughly discussed on pages 41-42 of your text -- try the following simple experiment. (I hope you will also remember the computer visualisation which I showed you using the CD which comes with your textbook, a resource you should certainly be using!) Stand in a darkened room with one very strong source of light (perhaps a spotlight) at the far end. Hold up a ball at arm's length to represent the moon. Let your eyes represent an attentive observer on the Earth, which is itself represented by your head. Now move the ball around in various ways, so that its orientation with respect to the source of light changes, just as the moon orbits the Earth while illuminated from one direction by the sun. What happens? Well, at times you see the moon (ball) as a thin crescent [when it is held nearly in the direction of the lamp]; at other times, it resembles a full moon [when it is held on the side opposite to the lamp], and so on. So, too, as the moon orbits the Earth we observe it in various directions relative to the sun's location, and we see the lit-up side of the moon more or less fully depending upon the geometrical arrangement at the time. In short, despite what many people erroneously believe, the phases of the moon are not caused by the Earth's shadow falling upon it. This correct understanding is not a modern development, by the way. Indeed, Lucretius, writing in the third century BC, said: ...it may be that it [the moon] shines only when the sun's rays fall upon it. Then day by day, as it moves away from the sun's orb, it turns more of its illuminated surface towards our view till in its rising it gazes down face to face upon the setting sun and beams with lustre at the full. Thereafter, it is bound to hide its light bit by bit as it glides round heaven towards the solar fire... In other words, the ancients had a clear and correct understanding of the origin of the phases of the moon - and of the cause of eclipses too. (Indeed, there are historical records of the successful prediction of eclipses in the centuries before Christ, and many indications of a well-developed understanding.)
Eclipses of Two Kinds.You should know, and can read on pages 44-48 of your text, about the differences between solar and lunar eclipses. Because the topic is nicely and completely covered there, I will not repeat the discussion; but please be sure to develop a good understanding of these important concepts. I will restrict myself to a few brief remarks, and consider what we can conclude about the size of the moon itself from our observations of lunar eclipses in particular. First, though, let's think briefly about solar eclipses, events during which the moon passes between the Earth and the sun. As a consequence, the sun disappears as the shadow of the moon passes over us: we are unexpectedly plunged into the shade, just as if an especially thick cloud were to pass overhead directly between us and the sun. (Photographs taken from space can show the actual shadow on the Earth's surface.) The immediate effect, of course, is that at least some of the people on the ground can no longer see all of the sun, and people in privileged central locations lose sight of it completely, possibly for as long as several minutes. But the eclipse phenomenon is rather more remarkable than just that of a cloud obscuring the sun's light. The striking thing is that the moon does so essentially perfectly - it is just the right size. This is a truly remarkable coincidence! The sun is about four hundred times as big across as the moon is, but it is also about four hundred times as far away. This means that the two objects look almost exactly the same size. Thus when the moon passes between us and the sun, it almost precisely blocks off the light, leaving visible only the tenuous outer parts (the corona of the sun), producing one of the most spectacular phenomena in Nature. (See the photograph on the top left of page 46 of your text.) This precise match is not true for any moon orbiting any other planet in the solar system: we are in a uniquely favoured location. In a lunar eclipse, by contrast, it is the moon which moves into the shadow of the Earth, so that light from the sun no longer reaches it. Since the moon is not hot enough to emit any visible light of its own, it can only ever be seen because of the sunlight it reflects. During the eclipse, then, it vanishes in the same way that a spy does when he slips into a shadowed doorway to avoid detection by the enemy. You should remember that lunar eclipses occur at night, while we watch the moon in the dark sky, and at such times we ourselves are already `in the shadow' of the Earth - that is, we are on the dark side of the Earth which faces away from the sun. During the eclipse, then, the Moon joins us for a time in that shadow. Since the moon can be seen by everyone on the half of the Earth facing the moon, a lunar eclipse can be enjoyed by billions of people. We will see one this very fall, in late October 2004!
The Size of the Moon.Now that we understand the situation, can we deduce anything about the relative sizes of the Earth and moon by virtue of the fact that the Earth's shadow can completely hide the moon? You might think that this guarantees that the Earth is bigger than the moon, but it is not that simple! For instance: A relatively small Earth could cast a very large shadow, one in which the entire moon could be swallowed up, if the sun were itself a small bright object rather close to the Earth. Think, for instance, of a moth flying near a candle flame: it can cast an enormous shadow on the wall, much larger than itself. Conversely, a large Earth could cast a quite small shadow if the sun itself is very large but not very far away from us, since the shadow of the Earth would then converge quite sharply. (Look at the figures on page 45 of your text to see what I mean about the convergence of the shadow, but please remember that those figures are not drawn to scale, so the effect is much exaggerated.) This arises from the fact that light from the `top' of the sun and the `bottom' of the sun can get past the edges of the Earth following different, converging paths. To fit into the Earth's small shadow, the moon might have to be very tiny indeed. As it happens, the situation is intermediate: the sun is big, of course, but also at quite a large distance. The relatively large Earth therefore casts a big enough shadow that the moon can still fit entirely into it during a lunar eclipse. [By the way, the great distance of the sun means that objects high above us in the Earth's atmosphere cast shadows which are about the same size as themselves (unless the sun is very low in the sky and the shadows are cast at a steep angle onto the ground, like your own lengthening shadow at sunset). The shadow of a cloud is about the same size as the cloud; the shadow of a plane is about the same size as the plane. In the animated movie A Bug's Life, this bit of basic astronomy was misunderstood when the insects were shown as cutting out a small outline of a bird in a leaf, then flying the leaf high into the air to cast a much larger outline on the ground as a working 'blueprint' for an artificial bird intended to scare off a bunch of predatory grasshoppers.] has already told us the size of the Earth. Now let us collect our thoughts and see if we can work out the size of the moon itself from the considerations we have just discussed. If we assume, to simplify matters, that the sun acts like a point of light very far away, then the shadow cast by the Earth will be roughly the same size as the Earth itself. (That is, we ignore the slow convergence of the Earth's shadow. This is not a bad working assumption.) Since we can see the edge of the Earth's shadow pass over the face of the moon during lunar eclipses, we can compare the apparent size of the circular arc of shadow to the circular shape of the moon - and work out the actual size of the moon itself! (The answer, by the way, is that the moon is about 2000 miles in diameter, about a quarter of the diameter of the Earth. It is, by any standards, a large chunk of rock!) Once again, let me emphasise the qualitative correctness of the results of our ruminations. On rather general grounds, we have concluded that the moon is a large lump comparable in size, but probably a few times smaller, than the Earth itself. This is a profound conclusion!
The Distance of the Moon.The logic flows on inexorably. If you take seriously your estimate of the size of the moon, you can now determine the actual distance of the moon by comparing its true size in miles or kilometers to its apparent size in angular measure (i.e. 'how big it looks'). The real question, easily answered geometrically, is: how far away must the moon be to look as small as it does? (Remember that we have just determined the actual size of this object!) The correct answer is that it is nearly half a million kilometers away. By the way, we now know this distance very precisely because of our ability to bounce radar signals and laser beams off the lunar surface. So far as I know, calculations of this sort were not carried out by the ancients. Perhaps people did not trust the eclipse argument, or feared that it was uncertain. Even so, however, knowledgeable ancient people could have determined the distance to the moon in yet another way, at least in principle, by using triangulation, the basic technique of surveying. This is most easily understood with respect to a simple analogy, as follows: "Suppose you are on the telephone to a friend who lives one kilometer away, and complain to her that you are being pestered by the roar of a helicopter which is hovering high over your house. You may not be able to judge the height of the helicopter very precisely by looking up at it, but let us suppose that your friend now looks in the direction of your house. For her, the helicopter will not be overhead. But suppose she sees it at angle of exactly 45 degrees above the horizon. Some very simple geometry will allow her to work out that the helicopter is one kilometer directly over your head, in the particular situation I have described." There are two things to note. One is that this can only be done if you know how far apart you and your friend are. You need to know the baseline, as it is called. Secondly, you should recognize that the two observations have to be simultaneous, since the helicopter may be moving. If your friend looks out half an hour from now, it will probably be nowhere in sight! The principle is exactly the same when applied to the moon. Two people separated by a wide enough known distance (baseline) will see the moon in slightly different directions. This is most easily measured by noting the position of the moon relative to the stars near it in the sky - for instance, from one person's point of view, the star Sirius (say) might appear to be just on the edge of the moon, while from the other's it might seem to be standing well away from it. (You may remember that in the lecture I demonstrated the way this works with the use of a computer program which simulated what we would see from Kingston and Vancouver at the same moment.) The Greeks did not carry out such parallax determinations, perhaps because of the following complications: at the time of the Greeks, map-making was very rudimentary, and travel was difficult, so the precise separation of the observers would be only crudely known it would be even more difficult to know that the observations were at the same time, since clocks were very primitive in those days and hard to transport. Nor could you telephone to each other to coordinate the observations. The simultaneity must be guaranteed, just as with the helicopter example, because the moon is drifting in front of the field of stars so that its position changes constantly. (Actually, however, if you were to travel apart on a direct North-South line you could both make the observation when the moon passed through the local meridian, or "middle of the sky," to be sure that the time was the same.) the biggest problem is that the moon is quite far away -- about a quarter of a million miles. Thus if you were even as far apart as 500 miles, a long way in those days, you would both be looking almost exactly the same direction in space. From both your points of view, the moon would appear to be in nearly the same position relative to the background of remote stars. Still, in principle, even the ancient Greeks could have worked out quite correct values for the interesting and important size and distance of the moon itself.
The Flatness of the Sky: Depth Perception and the 'Travelling Moon'?We've been considering surveying principles by which the ancients could have measured the distance of the moon. This leads me to an apparently unrelated remark. Sad to say, the night sky does not look `three-dimensional.' Why is this so, and what, if anything, does it tell us? When we look out into space, we cannot readily judge the relative distances of the various stars and planets. Our impression is that they are all equally remote, and consequently when we look at the dark sky it is rather like looking at a painted backdrop with specks of white paint on it -- it is very 'flat' in appearance. Imagine how wonderful it would be to gaze out into a field of stars which looked like a swarm of fireflies, some close to us, some farther away, so that you had a real sense of the depths of space surrounding us. Unfortunately, that is not the case - but why not? The answer to this question is closely related to my discussion of the hypothetical measurements of the distance of the moon, a subject which we will now generalize to one of the most important concepts and tools in all astronomy: stellar parallax. We will consider this subject in detail a little later, but let us anticipate matters by considering just how we gauge distances in our day-to-day lives. When we reach out to pick up our cup of coffee, how do we know just where to put our hand? The most effective tool we have is our binocular vision, a direct consequence of the fact that we have two eyes. To understand the effect, hold a finger up in front of your face, about a foot away, and look at it with just your right eye, covering or closing your left eye. Move your hand around until your finger seems, from the point of view of your right eye, to be exactly lined up with some much more distant object - a tree in the background, for instance. Now close your right eye and open the left, and you will see that the finger seems to have shifted relative to the tree, simply because you are looking at it from a different direction. This effect is called parallax . If you blink your eyes alternately, the finger will seem to jump back and forth against the background. If, however, you now move your finger out to arm's length and repeat the experiment, you will discover that the effect is much reduced. The parallactic shift is greater for nearby objects, and your brain would correctly infer that any object which shows a great deal of parallactic shift when viewed by alternating glances from your two eyes must be quite nearby. But is this how we watch the world? Of course not! We do not spend all our time blinking our eyes in strict alternation; instead, we keep both eyes wide open. But the effects of parallax are still present, and as a consequence, the brain sees two slightly different images rather than a flickering back and forth from one viewpoint to the other. These independent pictures of the world suffer from a degree of misregistration, therefore, depending on the proximity of the objects around you. The brain merges them into a single image and uses the information about the misregistration to give you depth perception. The problem with the stars, of course, is that they are all so fantastically remote that we can see no parallax effects at all. One star may be much closer to us than another, but we will not see the tell-tale parallactic shift if we blink our eyes alternately. Nor does the brain detect any slight misregistration of the two images of the sky provided by our two eyes, and we have no celestial depth perception at all. In day-to-day life, we can resort to many other ways of judging the distances of remote objects, things that are too far away for our binocular vision to be of direct use. For instance, we can rely on the fact that distant objects look smaller than when they are nearby. If we know the size of the object we are looking at - say, a car farther down the highway -- we use the apparent angular size to infer distance. But the stars are so remote that they appear only as points of light, and this sort of distance estimation technique is of no use. The consequence of all this is that the stars, and even the much closer planets, appear to be located on a flat, dark background. In fact, it takes quite a leap of imagination to realize that we are looking out into a realm in which the various stars may be at enormously different distances! These considerations also explain why the moon seems to `travel along with you' when you drive in a car at night. It is a simple consequence of the moon's great distance. Suppose, for instance, you have to look due South to see the moon. If you now drive ten miles West, you will still have to look due South to see it. Your position on the Earth has changed so little that there is no apparent change in the moon's direction relative to your car and you: it seems to hover `just off your left shoulder.' By contrast, everything else -- trees, houses, and so on -- will seem to drift back past you as you drive. Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Friday, 15 January, 2021.)