| The Moving Earth:
Welcome to the Dance!
Suppose you were at a ballet performance, and wanted to figure out the choreography of the dance being performed on stage. If you were sitting in a balcony seat overlooking the stage, this would be fairly straightforward. But just imagine how much more difficult you would find it if you were walking across the stage in a long curved path during the performance. And the problems would be enormously exacerbated if you were carrying out continuous pirouettes and spins as you moved -- perhaps because you were one of the dancers!
In a sense, this is the problem we face. We are on a moving, spinning platform, the Earth, trying to make sense of the motions we see around us, some of which are simple reflections of our own motions rather than fundamental to the organization of the Solar System. How can we deal with this problem?
[ Digression: Using the CDs which come with your textbook, you have access to a nice solution: get off the Earth! Thanks to a modern computer simulation, you can see how the Solar System would look, in speeded-up version, to an intelligent creature located in the depths of space. The remarkably simple clockwork nature of the Solar System then becomes straightforwardly apparent. The ancients, of course, had no such device or understanding.]
The first and most important simplification is to compensate for the diurnal rotation of the sky around us - in other words, stop the Earth's spin, so that you are constantly looking the same direction in space! We cannot really do this, of course, but we can simulate the effect in the following way:
imagine opening your eyes in the middle of the night at a time when you are facing towards a particular star -- Sirius, let us say
take a photograph or make a sketch of what you see, then close your eyes
open them again when the Earth has spun exactly once relative to the stars, so that you are once again facing Sirius
do this repeatedly, day after day, for a year or more. Now piece all these images together to make the equivalent of a movie. What will you see in the final film?
Here is the answer. The stars would not change appreciably in their patterns (although if you watched for many centuries you would see that they do individually shift around very slowly, so that the constellations gradually change shape). You would be reasonably correct, therefore, to think of the stars as defining a large unchanging backdrop. Months pass, however, and the sun gradually drifts into view from the west, coming into view from behind your head around your right side, and then slowly moves eastwards, or to the left, across the field of stars. It is as though the sun were a bright firefly slowly circling your head, once a year, in a counterclockwise sense. (Please note that I am speaking from the point of view of a resident of the Northern Hemisphere! I leave it to you to figure out how things would look to someone in a location like the southern tip of New Zealand.)
In other words, the sun moves independently of the background sphere of stars. You would also notice the moon and a few points of light -- the planets! -- moving independently, also drifting from west to east at varying rates. The moon, for instance, goes once around us every 29 days or so (hence the origin of the word and concept of the `month').
The Greeks worked all this out, carefully `subtracting out' the daily motions to visualise the situation in the way I have just described. Their conclusions were:
that we are completely and absolutely at rest (even the once-a-day spin was ascribed to the motion of the whole cosmos);
that the sun actually does orbit the Earth (like the aforementioned firefly);
that the moon also moves in an orbit centered on the Earth (about which they were right); and
that all the planets move in orbits centered on the Earth (which was wrong! Their motions are centered on the Sun itself).
The correct explanation, which you know already, is well worth repeating! When we look off towards Sirius in December, it is prominently in view in the dark sky. Six months later, however, the entire Earth have moved halfway around our orbit, and the sun is in the way if we look towards Sirius. If we believe incorrectly, as the Greeks did, that the Earth is at rest, it appears to us that the Sun has come drifting slowly into view, moving from west to east across the stellar backdrop.
Even if you think you fully understand the sense of these apparent motions, you probably won't -- or at least not fully! -- unless you try the following sort of experiment: Look constantly in some fixed direction (say, towards Lake Ontario) and walk around and around a tree in a counter-clockwise direction, never changing the direction in which you face. (This requires walking backwards in part of the exercise.) Notice how, from your point of view, the tree itself appears to move, coming in from your right, briefly occupying center stage, and then moving across to your left. (Now imagine trying to interpret the tree's apparent motion across the background scenery if you were to repeat this whole exercise while spinning rapidly on your toes!)
Regardless of which interpetation you adopt, the inevitable conclusion is that the sun, moon, and planets move independently of the stars. The pattern of stars is very much the same as it was a century ago; but the planets, sun and moon will be found, from day to day and night to night, in differing locations on this starry background. Unfortunately, our lives today are controlled by artificial lighting, and very few people even notice the stars and planets, but the ancients noted and accumulated all these bits of information.
What they wanted to do was develop a cosmology (a model of the universe, or a way of picturing how it works, just as you would try to analyse the gears of a watch to see how it works). They assumed that we are at the centre, unmoving, while the whole apparatus whirls around us once a day. The apparatus was taken to consist of spheres of various sizes. One great outer sphere carries a fixed pattern of stars; another has the sun, but must move at a slightly different rate to explain why the sun appears, through the year, to drift across the field of stars; yet others bear the moon and planets. We will see in due course why this simple model had serious problems.
Does the Earth Actually Move Through Space?If the ancients had been able to prove, to their own satisfaction, that the Earth itself is on the move through space, they would obviously have been stimulated to more speculation about our place in space and perhaps have developed a correct understanding of the Sun as the centre of the Solar System. Unfortunately, no proof was possible before the invention of the telescope, as we will see, but you may be interested to know that it can be shown in several independent ways. In class, I challenged you to try to think of ways in which you might hope to show that the Earth itself moves. No one in the class suggested that we would feel the motion: after all, we earlier discounted the notion that we might feel the spin of the Earth, and likewise here. The Earth coasts through space without rumbling and bumping! But there are other possible techniques. Let us ask if they will or will not work. Measure Changes in the Apparent Brightnesses of Stars: Completely Impractical! As we orbit the Sun, we sometimes draw a little closer to certain stars, sometimes move a little farther away. (Imagine a child on a merry-go-round, with an anxious parent standing nearby. The motion brings the child closer to the parent, then away again, in a continuing cycle.) The closer we get to a given star, the brighter it should look. You might expect, therefore, that the stars in the plane of the Earth's orbit would appear to wax and wane in brightness depending on our position. This is quite true in principle, but our back-and-forth motion is over so small a distance, relative to the distances of the stars themselves, that the effects are negligibly small. To quantify this, consider a hypothetical nearby star, just one light year (about 10 trillion, or 10**13, kilometers) away -- and please note that there is no star quite so close to us as this, so that I am overestimating the effects in the following discussion. The Earth is about 150,000,000 kilometers from the sun, so when we are on the side of our orbit closest to the star we are closer to it by merely a few thousandths of a percent than when we are on the other side of our orbit. You know from experience that the apparent brightness of a source of light depends on how close you are to it. As noted earlier, this is an inverse-square law, which means, for instance, that if you stand twice as far away from a lamp as you were before, it will look one quarter as bright as it did (since 1/4 is the inverse of the square of 2). Using this, you can calculate that the star described will slowly change in brightness by less than one one-hundredth of a percent, over a timespan of six months - and this for a relatively nearby star! Most stars are much more remote, and the effects are correspondingly smaller since our back-and-forth motion represents an even smaller fractional change in their distance. This is simply not a reasonable test even with modern technology, not to mention for the ancient astronomers. (There is also the practical difficulty of measuring the brightness of stars which are 'behind the sun' from our point of view, but stars rather near it can be seen shortly after sunset or before sunrise.) Measure The Actual Speed of Our Motion: Yes! Every morning, I drive in to work from west of Kingston, and every night I drive home. There are often speed traps on the road, and a police officer who cared to put the observations together would learn that I move back and forth every day in this fashion, as regular as clockwork. We can do something quite similar to the stars, studying a physical effect called the Doppler effect in the light emitted by them. (This technique will be described later, but here I want only to tell you (a) that it can be done and (b) that it does not require us to bounce radar signals off the remote stars, as people sometimes think. We actually study the starlight itself. It is interesting, though, that the police officer with the radar gun uses the same physical principle.) This kind of study provides absolute proof that we are moving back and forth as we orbit the sun, sometimes toward certain stars and at other times away. Of course, if you saw this effect for only one star, it could be argued that it might be just a remarkable coincidence. Perhaps we really are at rest, and that star is rushing back and forth through space exactly once a year. (This could happen, for instance, in a binary system in which the other star is orbiting, once a year, around some other massive object.) We would then be detecting not our own changing speed, but the speed of that star in its orbit. This would be extraordinary bad luck, but fortunately we can rule out such a circumstance because all the stars in one part of the sky show the same effect. In June, for instance, the stars in one part of the sky will display an average velocity of about 30 km/sec towards us, as though they have all decided to come rushing our way. In December, they seem to have changed their minds and are now rushing away. Moreover, such back-and-forth effects are seen for all the stars in the plane of the ecliptic, the plane within which the Earth itself is hypothesised to be moving. (Much above or below the ecliptic, the changing velocities are just not seen.) These cooperative back-and-forth motions must, therefore, be a simple reflection of our orbital motion around the sun. We really do move! This method has an important by-product: the Doppler shift doesn't merely demonstrate, in qualitative terms, that we are moving, but it also yields the actual speed (which is why the police use it). The Doppler effect tells us, as noted above, that we orbit the sun at a speed of about 30 km/sec. Of course, we also know that it takes us one full year to go once around. Knowing how fast we travel and for how long, we can work out the length of the path that we trace out in our nearly circular orbit. From this, we can work out how far we are from the sun. (As it happens, there are more direct ways to do this, but the principle is correct. More to the point, the different answers agree!) There is one more important consideration. The tell-tale evidence of this back-and-forth motion will be seen in the light of any star which is bright enough for us to acquire a spectrum. (To get a spectrum, we have to collect enough light that we can afford to spread it out into its various colours and still see the light clearly at all wavelengths. We will say more about this later.) In other words, a very remote star, if it is extremely luminous (or if we aim the world's largest telescope towards it!), will show the effect just as well as a nearby, more typical star. This test is much more promising than the first one I described above, which would produce at best a marginal effect, and even then only for the very nearest stars. Measure Stellar Aberration: Yes! Have you, I wonder, ever noticed how stupidly moths and other flying insects seem to behave at night when you are driving down the highway? As you approach, they dive right into your headlights or windshield, rather than trying to fly out of the way. Or do they? In fact, the insects are flying around in random directions, quite unaware of the speedy approach of your car. You are moving so quickly (no fly can outrun a car moving at 100 kph) that even the ones flying in the same direction as you are travelling will be hit. From your point of view, the insects seem to be flying at the car; but they are not. A similar effect explains why vertically-falling snow seems to come directly into your windshield as you travel, as if a strong blizzard were blowing into your face. This effect, called aberration, is even better understood in the following experiment. Imagine standing under a big umbrella, held perfectly horizontally over your head, during a steady vertical rain. You stay dry, and will continue to do so even if you walk about slowly. But what happens if you run? Consider a raindrop which just misses the leading edge of your umbrella. By the time it reaches the ground in its continuing fall, your foot or leg may be there because of your rapid motion forward, and your pants get wet. As you know from experience, you can keep dry by tipping the umbrella in the forward direction; and the faster you run, the more you have to tip it. Now consider running around a track in the pouring rain. As you run down the backstretch (the far side of the track), you have the umbrella tipped in the forward direction. After the bend, you reach the homestretch, and the umbrella - still tipped in the forward direction, to keep you dry - is now tipped in the opposite sense to what it was while you were running the backstretch. In other words, as you circle the track you have to adjust the direction of the tip of the umbrella (relative to the surroundings) to keep dry. And so too with the stars! For simplicity, think of a star `above' the solar system, in a direction perpendicular to the plane of the Earth's orbit. The light of that star, of course, represents the falling rain. Now think about the telescope, which is like a very tall cylindrical cup. If the Earth were not moving, the light would fall straight down to the bottom of the tube, and be focussed into your eye. But if the telescope is being carried sideways by the motion of the Earth, the light from the star you are aiming at will not make it into the eyepiece. Instead, it will run into the `side of the cup' unless you tip the telescope to compensate. Just as in the case of the runner on the track, it will be necessary to constantly change the tip of the telescope as the months pass, since the Earth moves 'around the track' in very nearly a circular path. To clarify this understanding, let us once again consider this situation on the basis of 'limiting cases' in physics. (I warned you that we would run into such reasoning again and again!) Consider two extremes: 1 Suppose, first, that the rain is falling very very slowly, just misting down with almost no speed whatever - or, if you like, suppose you had the power to stop the motion of the raindrops entirely, so that they are literally suspended in the air, like a very fine mist. To stay dry under these circumstances, you would have to tip your umbrella horizontally and walk along behind it, in the `tunnel' you have excavated through the suspended raindrops. 2 If, on the other hand, the rain falls very quickly - absolutely pelting down - you need to tip the umbrella only a little as you walk, as a bit of thought will show. In the astronomical context, we note that light itself moves very quickly, and we are closer to the second extreme than the first. But the Earth itself is moving quickly enough around the Sun that the effect is noticeable. We do indeed need to tip our telescopes through a measureable, although small, angle! Indeed, stellar aberration was detected in 1725, by an astronomer named Bradley, and provided the first real proof that the Earth is on the move as it orbits the sun (although by that time no one doubted it). Measure Shifting Star Patterns: Impractical! In the lecture, I demonstrated that there are a number of ways in which I can tell that I am walking about at the front of the theatre. One of these is to look overhead at the pattern of lights in the ceiling. I may find, for instance, that I am `directly beneath' a particular light bulb - perhaps one which is a dud, for instance. Minutes later, I look up and discover that the dud is no longer directly overhead: I have wandered over to some new location, under some other bulb. If I am moving in a repetitious `orbit' of some sort, pacing back and forth like a caged zoo animal, I will find myself back under the dud from time to time, in a predictable, periodic way. Can we do something similar with the stars? Does the Earth move around `under the stars' in a way which is revealed by a wholesale back-and-forth shift in the entire pattern? Are we sometimes under one star, and at other times of the year under another? The answer is no, we do not detect such a shift of the whole pattern. The reason is that, on average, the stars are so fantastically far away that our little back-and-forth around the sun is inconsequentially small. (In fact, some of the very largest stars would be able to swallow up the entire Earth-Sun system! Our entire orbital motion is less than the size of such stars.) To understand the difficulty of the suggested observation, imagine standing inside the Skydome and looking up at the curved ceiling very far above you. Position yourself in such a way that you can see a pinprick hole in the roof directly above your right eye. (That is, if a drop of rainwater fell through it, it would land in the very centre of your upward looking eye.) Now shift your whole body an inch to your left. A falling raindrop would now miss your eye, but how well could you judge that you are no longer directly under the hole? Since the ceiling as a whole looks the same as before, it is very hard to make that judgement. Although this may sound like an unprofitable line of exploration, we are in fact on the verge of an important breakthrough. The critical thing to notice is that the shift we have been talking about here is one in the whole pattern of stars, as though all the stars were at the same distance (like the lights in the ceiling). Since the average distance of the stars is very large, the effect is immeasureably small. But the stars are not distributed in that way, even though that's how it looks. In reality, there are a small number of relatively nearby stars and many more remote ones. To see that this makes a difference, suppose that you were once again in the Skydome, looking up at a series of sunlit pinholes in the roof. Imagine a single bright firefly hovering halfway between the floor and the ceiling. If you were to move about a little, would you notice a change in the pattern of dots of light? If so, does this suggest an astronomical analogy? Measure Stellar Parallax: Yes, Yes, Yes! The reasoning of the previous section indeed suggests something! We will try to observe and measure the stellar parallax of a nearby star -- the apparent shift of one star's position relative to a bunch of others. Let's start, though, by remembering our earlier discussion of parallax. We noted that, in principle, even the ancient Greeks could have worked out the distance of the moon by noticing that, at the same moment, it appeared in different locations (relative to the background stars) when observed by two people in very different locations. The necessary separation would be large - some hundreds of kilometers at least - but the effect is certainly detectable. This is an example of what we call geocentric parallax, since it relies on the fact that we have two observers at two different sites on the Earth ("geo"). This will never do for the stars! Even the nearest stars are so fantastically far away that two simultaneous observations, even from very widely-separated locations on the Earth, will never reveal any stellar parallax effect. To see the shift, we need to look at the stars from even more widely-spaced points of view. This sounds impossible, unless you have spaceships - but it's not! Since the Earth is moving around the sun, we are on opposite sides of the Sun every six months. With a three-hundred-million kilometer baseline, one can successfully measure heliocentric parallaxes for the nearer stars! (The word `heliocentric' means that the effect arises because we are moving back-and-forth, with constantly changing perspectives, around the sun. The word `helios' is the Greek for `sun.') Look at the figure on page 525 of your text, therefore. As we move from one side of the sun to the other, we should see a periodic shifting, back and forth, in the position of a nearby star against the backdrop of more remote stars as the months progress. It is important to note that the effects shown in the figure are outrageously exaggerated! The shifts are really very tiny indeed. On photographs taken six months apart, even the nearest stars are seen to shift by only about the size of their own images - just the barest indication of a shift. (We can see the effects better from outside the Earth's atmosphere, because some of the blurring of the image itself is due to the passage of starlight through the air above us.) The detection of stellar parallax is, therefore, anything but easy. Even the closest stars are so remote that the shifts are almost immeasureably small, and indeed were first detected only in 1837, after centuries of endeavour. The effects cannot be seen without a telescope, and even then require tremendously careful and precise measurements. They were, consequently, far beyond the capabilities of the ancients. In summary, please note that the determination of parallaxes is really only made possible by the fact that the stars are not all at the same distance. The more remote stars provide an ideal essentially unmoving background reference frame against which we can notice and measure subtle postional changes of those in the foreground. It is much easier to detect the periodic shifting back and forth of some stars, relative to the background, than to measure a tiny shift of the whole pattern! As we orbit the Sun, we sometimes draw a little closer to certain stars, sometimes move a little farther away. (Imagine a child on a merry-go-round, with an anxious parent standing nearby. The motion brings the child closer to the parent, then away again, in a continuing cycle.) The closer we get to a given star, the brighter it should look. You might expect, therefore, that the stars in the plane of the Earth's orbit would appear to wax and wane in brightness depending on our position. 4 This is quite true in principle, but our back-and-forth motion is over so small a distance, relative to the distances of the stars themselves, that the effects are negligibly small. To quantify this, consider a hypothetical nearby star, just one light year (about 10 trillion, or 10**13, kilometers) away -- and please note that there is no star quite so close to us as this, so that I am overestimating the effects in the following discussion. The Earth is about 150,000,000 kilometers from the sun, so when we are on the side of our orbit closest to the star we are closer to it by merely a few thousandths of a percent than when we are on the other side of our orbit. 5 You know from experience that the apparent brightness of a source of light depends on how close you are to it. As noted earlier, this is an inverse-square law, which means, for instance, that if you stand twice as far away from a lamp as you were before, it will look one quarter as bright as it did (since 1/4 is the inverse of the square of 2). Using this, you can calculate that the star described will slowly change in brightness by less than one one-hundredth of a percent, over a timespan of six months - and this for a relatively nearby star! Most stars are much more remote, and the effects are correspondingly smaller since our back-and-forth motion represents an even smaller fractional change in their distance. This is simply not a reasonable test even with modern technology, not to mention for the ancient astronomers. (There is also the practical difficulty of measuring the brightness of stars which are 'behind the sun' from our point of view, but stars rather near it can be seen shortly after sunset or before sunrise.) Success! Taken all together, the heliocentric parallax, Doppler-shift and stellar aberration observations prove unquestionably that the Earth itself moves back and forth around the sun. 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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(Sunday, 25 February, 2018.)