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PHY 121 Astronomy, chapter 4: The Origin of Modern Astronomy Homework: RQ. 2, 4, 7; PROBL. 2, 3 RQ 4-2 Why did classical astronomers conclude that Earth had to be motionless? Solution: Classical astronomers concluded that Earth had to be motionless because they could not see any parallax on the stars. They started with the wrong premise that the stars are on a sphere which is not too large in its diameter and so the stars were assumed to be much closer than they actually are. Starting with this wrong assumption, they concluded that the appearance of the constellations close to the ecliptic should change dramatically (parallax) during one year if Earth would actually move relative to the sphere with the stars. In addition, the fact that we do not feel any motion of Earth was taken to be as proof that Earth is motionless. RQ 4-4 In what ways were the models of Ptolemy and Copernicus similar? Solution: -Both Ptolemaic and Copernican system assumed uniform circular motion and needed equants and epicycles. -Ptolemy’s system required them to account for the variation in the orbital speed of a given planet and retrograde motion. Copernicus’ system required them to account only for the variation in the orbital speed of the planets. -Both Ptolemy and Copernicus assumed a celestial sphere at a great distance. RQ 4-7 Why did Tycho Brahe expect the new star of 1572 to show parallax? Why was the lack of parallax evidence against the Ptolemaic model? Solution: Greek philosophy taught that “new stars” had to be closer to Earth than the Moon because the heavens were perfect and unchangeable. This new star of 1572 should therefore be close to Earth and show daily parallax as do the Moon and the planets. However, Tycho Brahe observed no parallax and reasoned that it was further away than the Moon and probably as far away as the distant stars. Also the Ptolemaic system taught that the heavens were perfect and unchangeable. Since a new star appeared to be located on the celestial sphere, the basic assumptions of the Ptolemaic system were not valid. PROBL. 4-2 Galileo’s telescope showed him that Venus has a large angular diameter (61 seconds of arc) when it is a crescent and a small angular diameter (10 seconds of arc) when it is nearly full. Use the small-angle formula to find the ratio of its maximum distance to its minimum distance. Is this ratio compatible with the Ptolemaic universe shown on page 45? Solution: Small-angle formula: angular diameter linear diameter = 206,265 dis tan ce 〈1〉 The farthest away an object is, the smaller its angular diameter. So when the angular diameter is 10 seconds of arc, the distance between Venus and Earth is the farthest. linear diameter 10 206,265 = dis tan ce MAX 〈1〉 ⇒ 61 linear diameter = dis tan ceMIN 206,265 ⇒ dis tan ceMAX 6.1 = dis tan ceMIN 1 In the Ptolemaic model, Venus is always between Earth and Sun. So according to the Ptolemaic universe, shown on page 45, the same ratio is about 1.5 1. PROBL. 4-3 Galileo’s telescope were not of high quality by modern standards. He was able to see the Moons of Jupiter, but he never reported seeing features on Mars. Use the small-angle formula to find the angular diameter of Mars when it is closest to Earth. How does that compare with the maximum diameter of Jupiter? Solution: angular diameter linear diameter = 206,265 dis tan ce Distance between Mars and Sun: dist. MARS − SUN = 2.279 ⋅ 10 8 km Small-angle formula: 1 Distance between Jupiter and Sun: dist. JUPITER − SUN = 7.783 ⋅ 10 8 km Distance between Earth and Sun: dist. EARTH − SUN = 1.496 ⋅ 10 8 km Hence, Also, dist.MARS − EARTH = 0.783 ⋅ 10 8 km MARS : linear diameter = 6.796km 2 dist. JUPITER − EARTH = 6.287 ⋅ 10 8 km JUPITER : linear diameter = 142.900km 1 ⇒ ang. diam.MARS = 206,265 ⋅ 3 linear dist.MARS 2 → ang . diam.MARS = 18 sec of arc dist.MARS − EARTH 1 ⇒ ang . diam. JUPITER = 206,265 ⋅ linear dist. JUPITER 3 → ang . diam. JUPITER = 48 sec of arc dist. JUPITER − EARTH Although Jupiter is in longer distance from Earth comparing to Mars, it looks bigger to us for its angular diameter is 2.7 times bigger than Mars’s angular diameter. That is why Galileo could see details on Jupiter’s surface.