4. A planet follows a Kepler orbit (i.e., it satisfies Kepler's three laws) on an ellipse of semi-major axis a and eccentricity \(e\) about a star located at a focus of the ellipse.
a) Calculate the mean distance from the star to the planet when averaged over the arc length of the ellipse.
b) Let the planet be at position P, the star is at the focus F, and let the position of closest approach (called perihelion if the star is the sun) be C. Let \(\theta\) be the angle PFC. Calculate the mean distance from the star to the planet when averaged over \(\theta\) for a full revolution.
[Hint: Look up the Weierstrass substitution for evaluating integrals of rational functions of sine and cosine.]
Bonus (4 points):
c) Calculate the mean distance from the star to the planet when averaged over time for a full revolution.