Planetary Motion In Exercise 65, the polar equation for the elliptical orbit of Neptune

Chapter 10, Problem 67

(choose chapter or problem)

In Exercise 65, the polar equation for the elliptical orbit of Neptune was found. Use the equation and a computer algebra system to perform each of the following.

(a) Approximate the area swept out by a ray from the sun to the planet as  \(\theta\)  increases from 0 to  \(\pi / 9\).  Use this result to determine the number of years required for the planet to move through this arc when the period of one revolution around the sun in 165 years.

(b) By trial and error, approximate the angle  \(\alpha\)  such that the area swept out by a ray from the sun to the planet as  \(\theta\)  increases from  \(\pi\)  to  \(\alpha\)  equals the area found in part (a) (see figure). Does the ray sweep through a larger or smaller angle than in part (a) to generate the same area? Why is this the case?

(c) Approximate the distances the planet traveled in parts (a) and (b). Use these distances to approximate the average number of kilometers per year the planet traveled in the two cases.

Text Transcription:

theta

pi/9

alpha

pi