to the layperson aristarchuss report

triang Moon ed by For the Earth-Moon-Sun viewed from Earth, and extending one arm toward the Sun and your your r is the Moon's distance, and d is the ll a makes the cosine equation look like this than done! If Figure 8.2 were drawn to actual scale, with the Sun very, very far away-that ishr pha he used analogous geometrical methods to accomplish the procedure described here.) 1. Aristarchus estimated (guessed?) that during the quarter-Moon phase the angle E was 87d ng one arm toward the Sun and your other arm toward the arms is E A bit of algebra makes the cosine equation look like this d/cos(E). Having already deter was measure the angleE during the Moons qted the h not quite, a right angle. ( equation in the previous paragraph to determine how many times farther away the Sunest adopted value for an n the -0,866, and d 1 answer should take the form d-n r, where n is a number. For now, leave r that represents the Moon's distance. For example, if angle E were 30 degrees, then cos (E)merely d. To illustrate, solar distance d derived from Aristarchus's method is exquisitely sensitive to the ad Aelir distrase i Aristare angle merely one degreelarerthat is,8 degres instead e Again express your answer in the formd-nxr Modern measurement reveals that Aristarchus was way off in his estimation of angle E (so far off in pect he just plucked a value 'out of the hat"). The true value of E is 89.85 degrees. Recompute the sola, thaH again in the form d-nr. 4. Now it should be easy to find the true diameter of the Sun in units of Moon-diameters, in other words, how Moons would fit across the face of the Sun? The Moon almost precisely covers the Sun during a solar eclinse o and the Sun appear the same angular diameter in the sky. But the Sun is much farthera er away than the Moon; to be the same diameter as the Moon, it must, in fact, be a much larger body than the Moon. For example, ih 3 times farther than the Moon, yet they appear to be the same diameter, the Sun must really be 3 Moon-diameten Or in general, if the Sun is n times farther than the Moon, yet they appear to be the same diameter, the Sun Moon-diameters wide. a. Using your value for n from Part I, write down an expression for the Sun's diameter in units of Moon b. Aristarchus wen t on to find the Sun's diameter in units of Earth-diameters. To do this, he used his estimate for lunar eclipse observations that the Moon is one third as wide as Earth. Considering this information and you answer to Part 4(a), write down an expression for the Sun's diameter in units of Earth-diameters, according S. To the layperson, Aristarchus's report on his findings is about as exciting as a pamphlet on mixing cement. Not sylable is wasted on commentary or personal reflection. Yet we can hardly imagine that Aristarchus was unmoved the extraordinary result of his calculations. Based on your answers to Part 4, can you explain why Aristarchus mig have concluded that the Sun, and not Earth, lay at the center of the cosmos? What other unique and important featune of the Sun might have supported this conclusion? sly, Aristarchus did not carry out the next logical step: finding the Sun's and Moon's distances in Earth-diameters. With such information, he could have formed a true scale model of the Sun-Earth-Moon system. the same way that a globe shows a scaled-down version of continents and oceans. The essential question is this g the Sun's and Moon's distances in units d diameter (expressed in Earth-diameters), and 6 is the Sun's or Moon's angular diameter in the sky. For examp obet is cully 10 Earth-diameters wide and it spans an angle of 4 degrees in the sky, its distance is equivalent that the Sun is actually Earth-diameters across and the Moon is Earth-diameters across object be from Earth to appear as they do, a half-degree across, in the sky? To answer this question, we apply equation from the previous activity: r- 57.3s/0, where r is the Sun's or Moon's distance, s is the Suns or 143 Earth-diameters. 6. Using the sector equation plus Aristarchus's diameters for the Moon and the Sun, compute the distance of C if a to about and (b) the Sun in units of Earth-diameters, according to Aristarchus. 2 Activity 8 . Now we're ready to construct the scale model of the Sun-Earth-Moon system, as it might have been envisioned by 1 inch Aristarchus more than 2000 years ago. Suppose Earth is represented by a ping pong ball, which is approximately across. Using your answers to Parts 4 and 6, determine to scale the following measurements a. the diameter (in inches) of a ball representing the Moon b. the distance (in feet) of that Moon ball from t c. the diameter (in inches) of a ball representing the Sun d. the distance (in feet) of that Sun ball from the center of the ping-pong ball "Earth Modern measurements reveal that Aristarchus was not too far off in his determination of the Moon's diameter distance from Earth. However, he erred significantly in his estimates of the Sun's diameter and distance: the Sun is more than 100 Earth-diameters across and 11,000 Earth-diameters distant. There's nothing wrong with ods, they were just impossible to carry out reliably in his day. Nevertheless, Aristarchuss erroneous solar distance became the de facto standard into the Middle Ages. For the first time, someone had calculated the size Aristarchus's meth- and distance of a celestial gathered from Earth. In bold strokes, Aristarchus had combined geometry, logic, and obser that it was possible to "lay down a ruler in i vation to effectively free himself of his earthly bonds and to show astronomers t Mternlar the vrneis drthers to the mt basic kind of siermuremsent. triang Moon ed by For the Earth-Moon-Sun viewed from Earth, and extending one arm toward the Sun and your your r is the Moon's distance, and d is the ll a makes the cosine equation look like this than done! If Figure 8.2 were drawn to actual scale, with the Sun very, very far away-that ishr pha he used analogous geometrical methods to accomplish the procedure described here.) 1. Aristarchus estimated (guessed?) that during the quarter-Moon phase the angle E was 87d ng one arm toward the Sun and your other arm toward the arms is E A bit of algebra makes the cosine equation look like this d/cos(E). Having already deter was measure the angleE during the Moons qted the h not quite, a right angle. ( equation in the previous paragraph to determine how many times farther away the Sunest adopted value for an n the -0,866, and d 1 answer should take the form d-n r, where n is a number. For now, leave r that represents the Moon's distance. For example, if angle E were 30 degrees, then cos (E)merely d. To illustrate, solar distance d derived from Aristarchus's method is exquisitely sensitive to the ad Aelir distrase i Aristare angle merely one degreelarerthat is,8 degres instead e Again express your answer in the formd-nxr Modern measurement reveals that Aristarchus was way off in his estimation of angle E (so far off in pect he just plucked a value 'out of the hat"). The true value of E is 89.85 degrees. Recompute the sola, thaH again in the form d-nr. 4. Now it should be easy to find the true diameter of the Sun in units of Moon-diameters, in other words, how Moons would fit across the face of the Sun? The Moon almost precisely covers the Sun during a solar eclinse o and the Sun appear the same angular diameter in the sky. But the Sun is much farthera er away than the Moon; to be the same diameter as the Moon, it must, in fact, be a much larger body than the Moon. For example, ih 3 times farther than the Moon, yet they appear to be the same diameter, the Sun must really be 3 Moon-diameten Or in general, if the Sun is n times farther than the Moon, yet they appear to be the same diameter, the Sun Moon-diameters wide. a. Using your value for n from Part I, write down an expression for the Sun's diameter in units of Moon b. Aristarchus wen t on to find the Sun's diameter in units of Earth-diameters. To do this, he used his estimate for lunar eclipse observations that the Moon is one third as wide as Earth. Considering this information and you answer to Part 4(a), write down an expression for the Sun's diameter in units of Earth-diameters, according S. To the layperson, Aristarchus's report on his findings is about as exciting as a pamphlet on mixing cement. Not sylable is wasted on commentary or personal reflection. Yet we can hardly imagine that Aristarchus was unmoved the extraordinary result of his calculations. Based on your answers to Part 4, can you explain why Aristarchus mig have concluded that the Sun, and not Earth, lay at the center of the cosmos? What other unique and important featune of the Sun might have supported this conclusion? sly, Aristarchus did not carry out the next logical step: finding the Sun's and Moon's distances in Earth-diameters. With such information, he could have formed a true scale model of the Sun-Earth-Moon system. the same way that a globe shows a scaled-down version of continents and oceans. The essential question is this g the Sun's and Moon's distances in units d diameter (expressed in Earth-diameters), and 6 is the Sun's or Moon's angular diameter in the sky. For examp obet is cully 10 Earth-diameters wide and it spans an angle of 4 degrees in the sky, its distance is equivalent that the Sun is actually Earth-diameters across and the Moon is Earth-diameters across object be from Earth to appear as they do, a half-degree across, in the sky? To answer this question, we apply equation from the previous activity: r- 57.3s/0, where r is the Sun's or Moon's distance, s is the Suns or 143 Earth-diameters. 6. Using the sector equation plus Aristarchus's diameters for the Moon and the Sun, compute the distance of C if a to about and (b) the Sun in units of Earth-diameters, according to Aristarchus. 2 Activity 8 . Now we're ready to construct the scale model of the Sun-Earth-Moon system, as it might have been envisioned by 1 inch Aristarchus more than 2000 years ago. Suppose Earth is represented by a ping pong ball, which is approximately across. Using your answers to Parts 4 and 6, determine to scale the following measurements a. the diameter (in inches) of a ball representing the Moon b. the distance (in feet) of that Moon ball from t c. the diameter (in inches) of a ball representing the Sun d. the distance (in feet) of that Sun ball from the center of the ping-pong ball "Earth Modern measurements reveal that Aristarchus was not too far off in his determination of the Moon's diameter distance from Earth. However, he erred significantly in his estimates of the Sun's diameter and distance: the Sun is more than 100 Earth-diameters across and 11,000 Earth-diameters distant. There's nothing wrong with ods, they were just impossible to carry out reliably in his day. Nevertheless, Aristarchuss erroneous solar distance became the de facto standard into the Middle Ages. For the first time, someone had calculated the size Aristarchus's meth- and distance of a celestial gathered from Earth. In bold strokes, Aristarchus had combined geometry, logic, and obser that it was possible to "lay down a ruler in i vation to effectively free himself of his earthly bonds and to show astronomers t Mternlar the vrneis drthers to the mt basic kind of siermuremsent