The Relative Distances and Sizes of the Sun and Moon: First Indications. Consider the following very simple question: How do you know the moon is closer to us than the sun is? The obvious answer is that someone told (or taught) you that, but there are actually various plausible ways in which you could reason that this must be so: Solar eclipses occur. The moon occasionally blocks off the light of the sun in an eclipse, so must be closer to us than the sun is. This is a strong argument, but of course it is a bit circular because it assumes that you know what causes eclipses - which really requires already knowing that the moon is closer to us! The moon moves quickly. Relative to the stars, the moon moves fairly quickly, while the sun moves more slowly. On that basis alone (for reasons to be presented in a moment), you might be tempted to speculate that the moon is indeed closer than the sun. Before justifying this line of reasoning, let's first be quite sure what we mean by this idea of `motion relative to the stars.' Astronomers say, in effect, that "The moon goes once across the whole starry background every 28 days, the sun once every 365 days." What precisely do we mean by these statements? Here is a simple experiment to try. Go out some night and look at the moon, making a simple sketch of where it is relative to the brighter stars in its vicinity. A night later, or even just a few hours later, do this again, and you will discover that the moon has moved relative to the backdrop of stars. This is because it is in orbit around the Earth, and moves gradually from right to left -- west to east -- across the pattern of stars as we look from our Northerly location on the Earth. (But don't forget that the rapid spin of the Earth causes the whole pattern -- the moon and the stellar backdrop -- to move from east to west as the night passes! The moon and stars will gradually set together in the west and be lost to view until tomorrow.) The sun does exactly the same thing from our point of view, but much more slowly. Today, for example, if the sun were `lined up between us and the star Sirius', tomorrow it will seem to have shifted a little to the left - and so on, day by day, until it completes a full circuit after one year. In this case, the reason is that we are moving around the sun, rather than the sun moving around us, but the observable consequences are the same. Of course, this steady motion is not so obvious because we cannot see the backdrop of stars when the sun is up. We can, however, infer its motions by considering the slowly-changing pattern of stars in the night-time sky. (See the sketch on page 35 of your text.) This is why we see a constellation like Taurus in the night sky in the winter, but not at all in May and June when it is 'behind the sun' and invisible to us. "All right," you may say, "the moon moves more quickly than the sun across the starry background. But so what?" Well, consider an analogy to some simple motions seen in day-to-day life. If you lie on the grass and gaze up into the air, a fly may zip into, across, and out of your field of view in just a second or two. Meanwhile, a plane high in the sky may take a couple of minutes to cross your field of view, even though it is actually travelling at hundreds of times the speed of the fly. But what do we learn from this analogy? The answer is that nearby objects can seem to change position quickly, even if they are moving at a modest pace, while more remote objects seem to move only slowly, even if they are actually travelling quite fast. Perhaps this applies to the sun, moon, and planets as well? The ancients thought so, ascribing the various speeds of motion to the degree to which they were `caught up' in some influence of the Earth according to their proximity to us. Remarkably, this sort of consideration led the ancients to deduce, correctly, that Jupiter is more remote from us than Mars is, with Saturn farther still, and so on. But the argument is only suggestive, not compelling. (In our everyday analogy, for instance, you could imagine watching a lazy butterfly slowly flutter through your field of view while a jet fighter races across the sky in such a way that its high speed more than makes up for the extra distance it has to cover.) In short, there was a considerable degree of luck in their coming to the right conclusions. But the argument, although not without its dangers, actually led to some important conceptual progress. The sun is imposing. The sun, a brilliant globe of light, seems more imposing and dominant than the moon, so might be an enormous object at some colossal distance, while the moon could be a smaller, more mundane nearby body. Needless to say, this is a weak and only qualitative argument. After all, a bright flashlight shining into your eyes can easily outshine the moon, but is not a cosmic heavyweight. For all we know, the sun could be little more than a very bright bonfire suspended a few miles above our heads. The moon shows phases. Arguments 2 and 3 are only qualitative, but might carry some weight. There is, however, one absolutely compelling argument! If the moon were much farther away than the sun, it would always appear full (or nearly full) to us. The fact that we see a full range of phases, all the way from new moon and narrow crescents to the full face, tells us that the moon must be closer than the sun. The situation is demonstrated on page 42 of your text, in a sketch which features a young woman holding up a ball which represents the moon orbiting her head [the Earth]. The counter-example is shown in the next figure [below]. When you remember that the sun illuminates only the side of the moon which is facing it, you will recognize that we would always see the fully lit-up face of the moon (or most of it) if it were in a very large orbit which carried it beyond the sun. It would never present a thin crescent. The moon occasionally blocks off the light of the sun in an eclipse, so must be closer to us than the sun is. This is a strong argument, but of course it is a bit circular because it assumes that you know what causes eclipses - which really requires already knowing that the moon is closer to us! The moon moves quickly. Relative to the stars, the moon moves fairly quickly, while the sun moves more slowly. On that basis alone (for reasons to be presented in a moment), you might be tempted to speculate that the moon is indeed closer than the sun.

The Argument of Aristarchus.

For the reasons summarised in the preceding section, we can safely and correctly conclude that the sun is farther away than the moon is -- but by what factor? Is it twice as far away, or a thousand times as far? It would be nice to have a real number here, a quantitative determination. One Greek astronomer, Aristarchus of Samos, developed an amazingly clever means of evaluating the relative distances, as I will now describe. Before we continue, let's first remind ourselves the question of relative distances immediately translates directly into conclusions about the relative true sizes of the sun and moon. Since these two objects have the same angular size as observed from the Earth - that is, they 'look the same size' - then if the sun is one hundred times farther away than the moon, it must be one hundred times as large. Since the arguments presented in the previous section have already convinced us that the sun is farther away than the moon, we certainly know that it is bigger than the moon, but the relative sizes are as yet unknown. But there is an extra consideration: Once we work out how much larger than the moon the sun, we will also know whether the sun is in turn larger than the Earth itself (because we already know, from a consideration of the size of the Earth's shadow during lunar eclipses, roughly how the Earth and moon compare in size). If Aristarchus' ingenious technique works, therefore, we will be able to quantify the difference and determine whether the sun is only marginally bigger than the moon, or if it a really imposing body of colossal size -- perhaps enormously larger than the Earth itself! By the way, the argument I will be presenting here does not appear in your textbook, and indeed I will not necessarily expect you to reproduce it in detail, although you should certainly know that such an argument was used -- and appreciate the cleverness of it!! My presentation of Aristarchus's argument again invokes the physicist's trick of thinking of 'limiting cases.' (You may recall that I introduced this kind of reasoning in the discussion of the here on the Earth.) Here is the logic: 1 Aristarchus realized that when the moon appears to be exactly half-lit, as at first quarter moon or third quarter moon (see page 42 of the text for these definitions), it is because the angle from the earth to the moon to the sun is precisely 90 degrees. This is shown in the attached figure: In understanding this and the subsequent figures, you must remember that exactly one half of the spherical moon is always brightly lit up by the sun (except during the rare and short-lived lunar eclipses, when the moon moves briefly into the Earth's shadow). In general, of course, we can't see all of the lit-up hemisphere of the moon. When we do, we call it a full moon , but at other times we see other phases. At the time of the quarter moon, for instance, we are looking at it exactly sideways-on, as shown in the figure. At such times, the moon presents the familiar half-pie shape. [ An important digression: the terminology can be a bit confusing! When we speak of a quarter moon, we actually mean that the moon looks half lit, like a pie cut neatly down the middle. The third-quarter moon has the same shape as the first-quarter moon (but it is the other half of the moon's face which is then lit up: see page 42 of your text). The word "quarter" is misleading, and has nothing to do with how much of it appears to be illuminated at that time! It comes from the fact that we see the moon in these phases a quarter of a way and three-quarters of the way through its month-long cycle. Similarly, halfway through the cycle we see the moon as completely illuminated, or full. Imagine the confusion which might result if we referred to that as the half moon!] 2 Aristarchus began by imagining what you would see if the moon were very close to the Earth, as shown in the following figure, one in which the sun is about twice as far from the Earth as the moon is (and thus about twice as big). In the figure, you should imagine the moon travelling counterclockwise from point P (first quarter) to point Q (full moon) and then to R (third quarter), after which it completes its orbit and winds up back at point P a month after it started. While it is travelling between R and P, we will lose sight of it for a time in the daytime sky - at the time of "new moon". The moon's orbit is very nearly a perfect circle, along which the moon itself moves at a nearly uniform pace. (We can tell this without reference to the phases, simply by watching how uniformly it drifts across the field of remote background stars.) In other words, you can imagine the moon as behaving like the tip of the hand of a clock moving steadily around the clock face (although in the way I have drawn it, the clock is running backwards!). Please look at the figure until you are sure about what it is showing you. Another important digression: of course I have had to draw the figure out of scale to make the situation clear. In particular, the Earth and moon, as shown, are much too big. (The distance between the moon and the Earth is actually about thirty times the diameter of the Earth itself.) When you look at the figure as I have drawn it, you might reasonably expect that there would be eclipses every time there was a full or a new moon. But the actual small size of the Earth and moon, combined with the fact that the moon's orbit is slightly tipped with respect to the plane of the Earth's orbit around the sun, means that perfect alignments are very rare. 3 The figure shows that the first-quarter moon will not happen exactly midway between the new moon and the subsequent full moon. As you can appreciate, if the moon is moving smoothly, like the hand of a clock, it will get to point P (starting from the "new moon" position) in less than half the time it takes to get to point Q. Likewise, after the moment of complete fullness, as the moon sweeps through point Q, it will take a fairly long time to get to point R but then a relatively short time to get back to the next "new" moon. Can we quantify these qualitative expectations in some useful way? 4 Yes, we can! Now that you understand the behaviour, I can add some more information to the diagram without fear of confusing you completely. In the next figure, for instance, you will find some hypothetical dates, indicating that we might see a first-quarter moon on Oct 1 (when the moon is at point P), a full moon on Oct 11 (point Q), a third-quarter moon on Oct 21 (point R), and a return to the next first-quarter moon on Oct 31 (back to point P). In this scenario, it takes 20 days for the moon to go from first quarter to third quarter (P to Q to R), but then only 10 days to get back around to the next first-quarter phase, with the whole orbit taking exactly 30 days. 5 Given these dates, let us do some simple geometry. Since it takes 20 days to go from point P through point Q to point R, but then only 10 more days to go back from point R to point P, then clearly the angle marked as a large green arrow must be one-third of a full circle, or 120 degrees; and the angle marked by the red arrow, half that size, must be 60 degrees. 6 To complete the exercise, we focus on the small triangle with the three corners which are marked by the moon at point R, by the Earth itself, and by the Sun (triangle R-S-E). (See the next figure.) We now know two angles: angle Y is sixty degrees, and angle Z is ninety degrees because the moon is in the third quarter phase [remember point 1 of the discussion, above]. This means that we know all three angles, since they must add up to 180 degrees in a triangle. The precise shape of the triangle is thus completely known. 7 What does this tell us? Even if we do not know any of the distances in absolute terms, like miles, knowing the shape of the triangle allows us to determine the relative lengths of its sides. For a triangle with the angles I have used, for instance, the longest side (that joining the Earth and the Sun) is exactly twice that of the shortest side (that joining the Earth and the moon). In other words, we now know how many times farther away the sun is than the moon. ` Please note just how simple this method is, at least in principle. You merely take note of the precise moment at which the moon is in the first quarter (Oct 1, in my example), in the third quarter (Oct 21), and then back in the first quarter again (Oct 31). The calculations follow straightforwardly. ONE IMPORTANT POINT: please note that the diagram above does not depict the actual situation. (Every year I get some students who refer to this argument to say that "...it allows us to prove that the sun is twice as far away as the moon." That would be correct if the dates of the various phases were as I have given them in this hypothetical case -- but that's not what happens in reality! I have presented this case (the figure just above) as an example of how dramatically different things would be if the sun was a lot closer to us! The reality is that the four phases of the moon -- first quarter, full, third quarter, new -- are separated by almost exactly the same time interval (about a week each), which tells us that the sun must be much farther away than the moon is. Indeed, using this method, Aristarchus estimated that the Sun is about 90 times the distance of the moon. The actual number is 400, so this is not correct in an absolute sense. What I want to emphasise, however, is not that Aristarchus was wrong, but rather that he was right in spirit! The number was incorrect, but the implication was profound: his analysis tells us that the sun is huge and remote, a truly imposing body of considerable grandeur. Moreover, since the moon and sun have the same angular size, the sun must be (by his reckoning) ninety times as big across as the moon in actual size. Since we already know from eclipse studies that the Earth is only a few times larger than the moon, the implication is that the sun is also enormously larger than the Earth itself! Indeed, Aristarchus could justifiably have concluded that the size and splendour of the sun made it the dominant object in the solar system, and probably the center. You may wonder why Aristarchus got the wrong numerical answer. The basic problem is that the geometry is not nearly as favourable as I have drawn it! Try drawing the sketch to true scale and you will see the difficulty. (This is the other "limiting case", with a very remote sun rather than one very nearby.) The moon is about 240,000 miles away; the sun is about 93,000,000 miles away; the diameter of the Earth and Moon are about 8000 and 2000 miles, respectively; and the sun is 860,000 miles in diameter. You will see that the times of first and third quarter come when the moon is just about at right angles to us. This is shown schematically in the next figure, which is still not to scale - for one thing, the image of the sun should be 400 times as far from the Earth as the moon is, and big enough to swallow up the entire Earth-moon system! Consequently, the rays of light from the very remote sun arrive almost exactly parallel, as shown, and the quarter phases really are almost exactly a quarter of the way (in time) through the lunar cycle. A very small error in estimating the time when the moon is exactly half-lit translates into a very large uncertainty in the deduced distance, so Aristarchus had a tough job! There is one more complication. The moon is not a featureless ball: it has mountains and craters, bright and dark areas. Trying to decide when it is "exactly half lit up" is very subjective. Despite its imprecision, however, there is no denying the cleverness of the approach! 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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