Chapter 10.6, Problem 24ES

Use the following values, where needed:

$\begin{array}{|l|}\hline \text{radius of the Earth}=\text{4000 mi}=\text{6440 km}\hfill \\ \text{1 year }\left(\text{Earth year}\right)=\text{365 days }\left(\text{Earth days}\right)\hfill \\ \text{1 AU}=\text{92}\text{.9}\times {\text{10}}^{\text{6}}\text{mi}=\text{150}\times {\text{10}}^{\text{6}}\text{km}\hfill \\ \hline\end{array}$

(a) Let a be the semimajor axis of a planet's orbit around the Sun, and let T be its period. Show that if T is measured in days and a is measured in kilometers, then $T=\left(365\times {10}^{-9}\right){\left(a/150\right)}^{3/2}.$

(b) Use the result in part (a) to find the period of the planet Mercury in days, given that its semimajor axis is $a=57.95\times {10}^6\text{ km}.$

(c) Choose a polar coordinate system with the Sun at the pole, and find an equation for the orbit of Mercury in that coordinate system given that the eccentricity of the orbit is $e=\mathrm{0.206.}$

(d) Use a graphing utility to generate the orbit of Mercury from the equation obtained in part (c).