Bodes Law Problem: In 1766, Johann Titus, a German astronomer, discovered that the

Chapter 14, Problem C2

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Bodes Law Problem: In 1766, Johann Titus, a German astronomer, discovered that the distances of the planets from the Sun are proportional to the terms of a rather simple sequence: 1 Mercury 4 2 Venus 7 3 Earth 10 4 Mars 16 5 Ceres 28 6 Jupiter 52 7 Saturn 100 (Mercury, at 4, does not fit the sequence. Ceres is one of the asteroids in the asteroid belt.) a. Describe the pattern followed by the terms. Use this pattern to find the term for Uranus, planet 8. b. Planets 9 and 10, Neptune and Pluto, have distances corresponding to the numbers 305 and 388, respectively. Are these numbers terms in the sequence? Justify your answer. c. Find a formula for tn, the term value, in terms of n, the planet number. For what values of n is the formula valid? d. The asteroid Ceres was found by looking at a distance from the Sun calculated by Bodes law. If you were to look beyond Pluto for planet 11, how far from the Sun would Bodes law suggest that you look? (Earth is 93 million miles from the Sun.) e. Consult the Internet or some other reference(such as Scientific American magazine, July 1977, page 128) to see why Tituss discovery is called Bodes law.