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Astronomical Distances 13 April 2012 Astronomical Distances 13 April 2012 1/27

Last Time We ve been discussing methods to measure lengths and objects such as mountains, trees, and rivers. Astronomical Distances 13 April 2012 2/27

Last Time We ve been discussing methods to measure lengths and objects such as mountains, trees, and rivers. Today we ll look at some more difficult problems. Namely, we ll see how Eratosthenes estimated the size of the earth, and how Aristarchus came up with methods to estimate the distances to the sun and moon and their radii. People of the ancient world were thus able to find useful methods for making indirect measurements and did not need modern technology to do this. Their methods do often hinge on estimating something accurately, such as an angle, which is not always easy to do. Astronomical Distances 13 April 2012 2/27

Eratosthenes Astronomical Distances 13 April 2012 3/27

Eratosthenes Eratosthenes was born in Cyrene (now in Libya) around 275 B.C. He was the third chief librarian of the great library in Alexandria, Egypt, the center of learning at that time. Eratosthenes thought the earth was round. Moreover, he discovered a method for calculating the size of the earth, using the belief that it was round. Later in history people thought the earth was flat, and didn t come back to the belief that the earth was round until Columbus travels. Astronomical Distances 13 April 2012 3/27

Eratosthenes realized, as Thales did, that the sun is so large and far away from the earth that its rays are parallel as they hit the earth. Astronomical Distances 13 April 2012 4/27

Eratosthenes realized, as Thales did, that the sun is so large and far away from the earth that its rays are parallel as they hit the earth. Eratosthenes knew that on the summer solstice, the sun was directly overhead Syene (present day Aswan). He also knew that the sun wasn t directly overhead Alexandria. He was able to determine that, on the summer solstice, a stick in the ground cast a shadow of an angle about 1/50 of a full circle, which is about 7 in our measure of degrees. Astronomical Distances 13 April 2012 4/27

Eratosthenes was able to determine how far it is between Alexandria and Syene. He did this by having somebody pace off the distance. Astronomical Distances 13 April 2012 5/27

Eratosthenes was able to determine how far it is between Alexandria and Syene. He did this by having somebody pace off the distance. He found that the distance is about 5000 stades, which is about 500 miles. Astronomical Distances 13 April 2012 5/27

Clicker Question How long do you think it would take you to walk 500 miles? Enter a number of hours on your clicker. If you go along I25, you ll reach the Colorado border in about 450 miles. You d nearly reach Yuma AZ on the California border in that distance. You would also get to Abilene TX in that distance. Astronomical Distances 13 April 2012 6/27

Answer Of course, the answer depends on how fast you can walk. A typical walking pace is 3 mph. If you could keep up that pace it would only take you 500 miles 3 mph = 167 hours which is about 7 days. Astronomical Distances 13 April 2012 7/27

Eratosthenes Method How did Eratosthenes use this information to determine the size of the earth? Here is a picture to represent part of the information we have. Astronomical Distances 13 April 2012 8/27

Eratosthenes Method How did Eratosthenes use this information to determine the size of the earth? Here is a picture to represent part of the information we have. The angle marked with a blue arc is the shadow angle he measured. This is about 1/50 of a full revolution. The red line represents a ray from the sun. The line segments represent (really long) sticks placed in Alexandria and Syene. Astronomical Distances 13 April 2012 8/27

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In this picture the line segments are extended to reach the center of the earth. Eratosthenes knew some geometry which he was able to apply to this situation. One thing to notice about this picture is that the red line is parallel to the stick in Syene. This represents the fact that sun s rays are parallel and that the sun was directly overhead Syene at the time of Eratosthenes measurement. Astronomical Distances 13 April 2012 9/27

Drawing just these parallel lines and the line connecting them gives the following picture. Astronomical Distances 13 April 2012 10/27

Drawing just these parallel lines and the line connecting them gives the following picture. What Eratosthenes knew was that when you draw a line crossing parallel lines, you create equal-sized angles. That is, the two blue-marked angles have the same measure. Astronomical Distances 13 April 2012 10/27

Eratosthenes knew another geometric fact, one about angle measurement and circumference. Astronomical Distances 13 April 2012 11/27

Eratosthenes knew another geometric fact, one about angle measurement and circumference. If you have an angle, then the fraction of the circumference taken up by the arc length is the same as the fraction the angle is of a full revolution. For example, this picture shows an angle which is 1/10 of a full revolution. The corresponding arc length is then 1/10 of the circumference of the circle. Astronomical Distances 13 April 2012 11/27

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What Eratosthenes then figured out was that, since the blue-marked angle is 1/50 of a full revolution, the corresponding arc length, drawn in blue above, is 1/50 of the full circumference of the earth. Since he found out that this distance is about 500 miles, he could estimate that the full radius of the earth is about 50 500 = 25, 000 miles. Astronomical Distances 13 April 2012 12/27

Using the fact that the circumference C of a circle is found by the formula C = 2πr, where r is the radius of the circle, Eratosthenes could then calculate the radius of the earth as which is approximately 4000 miles. 25000 2π This is quite close to today s value of about 3960 miles. Note that the earth isn t perfectly round, so sometimes the radius is listed as a small range of values. Astronomical Distances 13 April 2012 13/27

Aristarchus and the Distances to the Sun and Moon Aristarchus of Samos, born around 300 B.C., was a Greek astronomer and mathematician. He was the first person known to propose that the sun was at the center of the solar system. Using some measurements, information from lunar eclipses, and some geometry, Aristarchus was able to obtain estimates of the distance to the sun and to the moon. Astronomical Distances 13 April 2012 14/27

Only one work of Aristarchus, On the Sizes and Distances (of the Sun and Moon) survives today. We will outline some of the ideas he described in this work. Astronomical Distances 13 April 2012 15/27

Only one work of Aristarchus, On the Sizes and Distances (of the Sun and Moon) survives today. We will outline some of the ideas he described in this work. Aristarchus did not actually compute the distances to the sun and moon, but described them in terms of the radius of the earth. With Eratosthenes determine of the radius of the earth, we can then get actual estimations of these distances. Astronomical Distances 13 April 2012 15/27

Aristarchus Ideas The first thing Aristarchus did was to consider when the moon was exactly half full. The following picture represents this situation. Astronomical Distances 13 April 2012 16/27

Aristarchus Ideas The first thing Aristarchus did was to consider when the moon was exactly half full. The following picture represents this situation. Aristarchus reasoned that at this point, the line from the moon to the sun and the line from the earth to the moon makes a right angle. Aristarchus attempted to measure the angle ϕ, and estimated it to be 87, in our current mode of measurement. Astronomical Distances 13 April 2012 16/27

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By using methods written by Euclid, Aristarchus was able to conclude from his estimation that the ratio S/L satisfied 18 S L 20 With more modern methods (S/L is equal to the cosine of 87 ), we can conclude from his estimation that the sun is around 19 times as far away as the moon. Astronomical Distances 13 April 2012 17/27

This gives an idea of the relative distances to the sun and the moon, but it doesn t tell the actual distances of either. Astronomical Distances 13 April 2012 18/27

This gives an idea of the relative distances to the sun and the moon, but it doesn t tell the actual distances of either. It is difficult to determine when the moon is exactly half full. It is also very hard to estimate the angle ϕ that Arisarchus determined was 87. In large part because of these difficulties, this turns out to be a fairly poor estimate. The actual angle is is about 89.8 and the ratio is close to 400 rather than 19. Astronomical Distances 13 April 2012 18/27

Aristarchus then considered what happens during a lunar eclipse. The following picture represents such an eclipse. Astronomical Distances 13 April 2012 19/27

Aristarchus then considered what happens during a lunar eclipse. The following picture represents such an eclipse. In this diagram, s is the radius of the sun, t the radius of the earth, and l is the radius of the moon. Also, S is the distance to the sun and L is the distance to the moon. Astronomical Distances 13 April 2012 19/27

One thing Aristarchus knew was that the apparent size of the sun and moon is the same. That is, when we look at them they appear the same size. During a solar eclipse the moon can just about exactly cover the sun. Aristarchus reasoned, from knowing about similar triangles, that this means S/s = L/l. That is, the ratio of distance to radius is the same for both the sun and the moon. Astronomical Distances 13 April 2012 20/27

Aristarchus also realized that there are several similar triangles in his picture. Below are two of them. Astronomical Distances 13 April 2012 21/27

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By seeing how long a lunar eclipse lasts, the distance d in this picture can be estimated. Recall that Aristarchus estimated the ratio S/L as about 19. Astronomical Distances 13 April 2012 22/27

Because of the detailed algebra used, we won t go through the derivation Aristarchus did. You can see the details at wikipedia.org/wiki/aristarchus On the Sizes and Distances We ll summarize what he found. If we abbreviate the ratios d/l as n and S/L as x, then Aristarchus formulas are radius of the moon radius of the earth = l t = 1 + x x(1 + n) and radius of the sun radius of the earth = s t = 1 + x 1 + n Astronomical Distances 13 April 2012 23/27

The values Aristarchus estimated were x = 19 and n = 2. With these values he got l t = 1 + x x(1 + n) = 20 19(3) =.35 and s t = 1 + x 1 + n = 20 3 = 6.67 Astronomical Distances 13 April 2012 24/27

The values Aristarchus estimated were x = 19 and n = 2. With these values he got l t = 1 + x x(1 + n) = 20 19(3) =.35 and s t = 1 + x 1 + n = 20 3 = 6.67 In other words, Aristarchus estimated that the radius of the moon was about a third as big as that of the earth, and the sun s radius was nearly 7 times that of the earth. Astronomical Distances 13 April 2012 24/27

Aristarchus estimations of the distance to the moon and to the sun were 20 times the earth radius and 380 times the earth radius, respectively. The known values today are about 60 and 23, 500. With Eratosthenes estimation of the radius of the earth, we can put that together with Aristarchus work to get actual estimates of the radii of the moon and sun and the distances to each. While his values are not really close to the actual values, just the fact that he was able to come up with estimates is amazing. And, his values would be much more accurate if he could have found a better estimate of the angle ϕ. Astronomical Distances 13 April 2012 25/27

Homework 8 and Next Week We ll consider the topic of statistics next week, and focus on some ideas that will help us understand the barrage of statistical information we see in the media. Astronomical Distances 13 April 2012 26/27

Quiz Question True or False: We need modern technology to come up with estimates of distances and heights, such as heights of mountains and the distance to the moon. A True B False Astronomical Distances 13 April 2012 27/27