Chapter 6, Problem 56P

(III) The orbital periods Τ and mean orbital distances r for Jupiter’s four largest moons are given in Table 6–3, on the previous page. (a) Starting with Kepler’s third law in the form

$T^2=\left(\frac{4{\pi }^2}{G{m}_{\text{J}}}\right)r^3.$

where m_J is the mass of Jupiter, show that this relation implies that a plot of log(T) vs. log(r) will yield a straight line. Explain what Kepler’s third law predicts about the slope and y-intercept of this straight-line plot. (b) Using the data for Jupiter’s four moons, plot log(T) vs. log(r) and show that you get a straight line. Determine the slope of this plot and compare it to the value you expect if the data are consistent with Kepler’s third law. Determine, the y-intercept of the plot and use it to compute the mass of Jupiter.