For each of the eight planets Mercury to Neptune, the

Chapter 13, Problem 13.76

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For each of the eight planets Mercury to Neptune, the semi-major axis a of their orbit and their orbital period T are as follows: Planet Semi-major Axis 1106 km2 Orbital Period (days) Mercury 57.9 88.0 Venus 108.2 224.7 Earth 149.6 365.2 Mars 227.9 687.0 Jupiter 778.3 4331 Saturn 1426.7 10,747 Uranus 2870.7 30,589 Neptune 4498.4 59,800 (a) Explain why these values, when plotted as T2 versus a3 , fall close to a straight line. Which of Keplers laws is being tested? However, the values of T2 and a3 cover such a wide range that this plot is not a very practical way to graph the data. (Try it.) Instead, plot log1T2 (with T in seconds) versus log1a2 (with a in meters). Explain why the data should also fall close to a straight line in such a plot. (b) According to Keplers laws, what should be the slope of your log1T2 versus log1a2 graph in part (a)? Does your graph have this slope? (c) Using G = 6.674 * 10-11 N # m2>kg2 , calculate the mass of the sun from the y-intercept of your graph. How does your calculated value compare with the value given in Appendix F? (d) The only asteroid visible to the naked eye (and then only under ideal viewing conditions) is Vesta, which has an orbital period of 1325.4 days. What is the length of the semi-major axis of Vestas orbit? Where does this place Vestas orbit relative to the orbits of the eight major planets? Some scientists argue that Vesta should be called a minor planet rather than an asteroid.