Chapter 13, Problem 13.76P

DATA 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 (106 km)	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 Kepler’s 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 log(T) (with T in seconds) versus log(a) (with a in meters). Explain why the data should also fall close to a straight line in such a plot. (b) According to Kepler’s laws, what should be the slope of your log(T) versus log(a) 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 Vesta’s orbit? Where does this place Vesta’s 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.