Chapter 4, Problem 72RE

According to Kepler’s law, the planets in our solar system move in elliptical orbits around the Sun. If a planet’s closest approach to the Sun occurs at time $t=0,$ then the distance $r$ from the center of the planet to the center of the Sun at some later time $t$ can be determined from the equation $r=a\left(1-e\cos \varphi \right)$ where $a$ is the average distance between centers, $e$ is a positive constant that measures the “flatness� of the elliptical orbit, and $\varphi$ is the solution of Kepler’s equation $\frac{2\pi t}{T}=\varphi -e\sin \varphi$ in which $T$ is the time it takes for one complete orbit of the planet. Estimate the distance from the Earth to the Sun when $t=90$ days. [First find $\varphi$ from Kepler’s equation, and then use this value of $\varphi$ to find the distance. Use $a=150\times {10}^6\text{ km},e=0.0167,$ and $T=365$ days.]