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Transcript
1 Naked-Eye Astronomy Lab Diameter of the Earth Needed Math: geometry and trigonometry Aristotle knew that the Earth was spherical and not flat. There were several observations that Aristotle used to argue this fact. One such observation was that the position of the North Pole star above the horizon changes as one moves around the Earth. In this lab, you will use this effect not only to prove that the Earth is spherical, but how large a sphere it is. The principle here is that because the surface of the Earth is curved, as you view a celestial object at different positions on the Earth the direction of that object relative to the ground will differ. Consider Figure 1 below, where two observers are located at different places on the Earth and view the Sun at the same time. Figure 1 The curve of the Earth's surface causes the orientation of the ground for the two observers to differ and so the direction of the Sun relative to the ground must differ. This is an easily measurable effect and depends solely on the sharpness of the curvature of the Earth's surface, which is related to the radius of the Earth. This effect, therefore, can be used to measure the size of the Earth. This was realized by an Egyptian astronomer named Eratosthenes about 200 BC. However, the Sun moves around the sky during the year and moves with the sky throughout the day and so using the Sun to measure this effect involves some special timing coordination with observers at two different locations. This measurement is much easier if one uses a celestial object that doesn’t move at all. Since the whole sky rotates around the North Celestial Pole, the North Pole star (which, actually, is a little bit away from the pole and so does move a little) is the best object for this measurement. You can determine the size of the Earth, then, by measuring the angular distance 2 between the horizon and the North Pole star (the angular distance above the horizon is known as the “elevation angle” or “altitude”) at two different locations. If you make this measurement at about the same time of night and about the same time of year, the small movement of the North Pole star will be insignificant. The measurement method is quite simple, so you can make the second measurement during a break in the term if you go someplace far enough away. Alternatively, if you know someone else at another location whom you trust and who is willing to do a small favor for you, you can ask them to make the second measurement. The relation between the elevation angle of the North Pole star at two different locations and the size of the Earth is as follows. First, imagine that the two locations that you use are located such that they and the North Pole star all lie in the same plane and therefore fit on a page, as in Figure 2 below. (Notice that in Figure 2 the North Star is drawn to the lower right, rather than at the top of the page. This figure is a correct picture. It is drawn this way partly to remind you that North is not really “up.”) The angles 1 and 2 are the elevation angles of the North Pole Star and the angle is the angle between the two locations as seen from the center of the Earth. Figure 2 Now note that the separation between the two locations is a fraction/360o of the way around the Earth. Likewise, since the circumference of the Earth is 2R, where R is the radius of the Earth, if the two locations are separated by a distance s, then they are also a fraction s/2R of the way around the Earth. Since these fractions must be equal, /360o=s/2R or R = 360o s/2. So, if we know s (the distance between the two locations), we need only to figure out how relates to the elevation angles, 1 and 2, and we can determine the radius of the Earth. 3 To see how relates to 1 and 2, consider the other two angles labelled in Figure 2. Since the ground makes a 90o angle with the vertical at each location, the angle 90o- is the angle between the line pointing to the North Pole star and the extension of the line pointing to the center of the Earth, as indicated. And, since the North star is, essentially, infinitely far away, the lines pointing to the North star from two locations on Earth are parallel. Now, try the following exercise (taken from “Stephen Hawkin’s Universe Teacher’s Guide”): 1. On a piece of lined paper, draw any two intersecting lines, as shown below. 2. With a protractor, measure the angle each drawn line makes with one of the parallel printed lines (angles ‘a’ and ‘b’ in the diagram). The printed lines represent the parallel lines pointing to a celestial object, like the North Star. 3. Subtract one angle from the other. 4. Measure the angle between the two lines where they intersect (angle ‘c’ in figure). 5. Compare angle c with the difference between a and b. Figure 3 Compare the situation of the lines and angles drawn on the lined paper and that in Figure 2 above. From this analogy, can you see that in Figure 2, we should have = (90o-1) - (90o-2)? With just a little simple algebra, you should be able to then discover that = 2 - 1. Here’s a geometrical proof of this equation, for those of you not satisfied (hold on and follow closely if you feel weak in geometry). First, we make use of a little trick. Add to Figure 2 a line from the center of the Earth to the North Pole star, as shown in Figure 3, below. The lines pointing to the star from each location and from the center of the Earth are all essentially parallel, so by the geometrical rule of alternate interior angles, the angles 90o-1 and 90o-2 are equal to the angles between this new line and the lines from the center of the Earth to each city, as shown. Then, by examining the angles at 4 Figure 4 the center of the Earth, you should be able to see that = (90o-1) - (90o-2), and hence = 2 - 1. In this lab, you'll measure 1 and 2 directly and then by determining s you can use the equation for R, above, and get the Earth's radius. The trick, now, is to reduce your observing situation to the one described above, i.e. that the two locations and the North Pole star all lie in the same plane. Since the North Celestial Pole is due North of any location on the Earth, you get the desired situation by picking two locations that are due North and South of each other. It is unlikely, however, that you’ll be able to pick a second location that is, conveniently, due North or South of your first location. That’s OK, though, provided you account for the East-West separation properly, as follows. First, imagine if another measurement was made in a town due East or West of your second location and due North or South of your first location. Let's call this other (fictitious) town Dorksville just for discussion-sake. What First Location N orth West East South Second Location D orksville Figure 5 is the elevation angle of the North Pole star in Dorksville compared to what you measure in your second location? Answer: the same. All that matters for the North Star’s elevation angle is how far above the northern horizon it is. To an observer 5 standing at the North Pole, the North Star will be straight overhead, and the so the elevation angle of the North Star will be 90o. As one moves towards the equator, the North Star appears closer to the northern horizon. Movement East and West, though, does not change the position of the North Star. Therefore, two observers that are due East and West of each other measure the same elevation angle of the North Star. Therefore, you can use your measurement of the elevation angle of the North Star in your second location as if you had measured it from Dorksville. But, you need to follow through with the rest of the analysis as if this measurement was made in Dorksville. This means that you must also use the distance, s, between the two locations due North and South of each other, i.e. from your first location to Dorksville. This distance, though, is also just the North-South component of the total distance between your two locations. So, you get what you want by measuring just the N-S distance between your two locations, not the total distance. To measure this distance you can either use a flat map or, if you have the opportunity, try cloking it on the odometer when you go (that would be the truly honest approach). Do not use a globe or make use of the latitudes; either of these methods would be cheating since they implicitly assume prior knowledge of the Earth's radius. Lastly, you need a way of measuring the elevation angle of the North Pole star. A good and easy method involves a easily made device known as a “quadrant (see figure on next page):” Thumbtack a protractor to the side of a stick so that its flat edge is parallel with the edge of the stick. Attach some string to the same thumbtack so that when a weight is attached, it points straight down from the center of the protractor. Now place one end of the stick to your cheek, just under your eye and sight the North Star along the length of the stick, so that the stick points at the star. While holding the stick in this position, have a friend read what angle the string crosses on the protractor. However, this number alone is not the correct answer. You also need to "calibrate" your instrument. When the stick is horizontal (pointing to the horizon, for example), 6 the elevation angle is 0o. You should find, though, that it does not read 0o when it is horizontal. What does it read? This is your zero-point. Now, when you point to the North Star, you need to compare your reading with the zero-point reading to infer the angle of elevation. Procedure: 1. Make a quadrant from a piece of wood with a straight edge, a protractor, thumbtack, string, and weight. 2. Measure the elevation angle of the North Pole star. 3. Travel to another location (at least 100 miles North or South) and measure the elevation angle of the North Pole star there around the same time of night. 4. Determine the N-S distance between your two locations. 5. Use your data and the equations above to determine the radius of the Earth. Additional questions to consider discussing: 1. Can you think of how to use the above data to determine the latitude of each location? (Latitude is the distance from the equator of a particular location defined in terms of the angular distance as seen from the center of the Earth (see figure below).) What are your measured latitudes? 2. How do your results prove that the Earth is not flat?