Question

I.       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 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 (1 – e cos f)

where a is the average distance between centers, e is a positive constant that measures the “flatness” of the elliptical orbit, and f is the solution of Kepler’s equation

= f – e sin f

in which T is the time it takes for one complete orbit of the planet.

(a)      Estimate the distance from the planet Mars to the Sun when t = 1 year.

(b)     Estimate the distance from the Earth to the Sun when t = 100 days.

For Mars use a = 228 million km, e = 0.0934, and T = 1.88 years.

For Earth use a = 150 million km, e = 0.0167, and T = 365.25 days.

Graph t as a function of f . Use the graphing calculator to estimate an approximate solution for f . Now use Newton’s method to find a more accurate solution correct to 5 decimal places.

(c)      Calculate dr/dt. Show that r reaches a maximum value when t = T/2.

Answer 1