In this problem we will explore how modifying the gravitational force law changes orbits. In

order to do this problem, you will need to understand how elliptic orbits result from Newton's

gravitational force F = LmH/r2.

Imagine that the gravitational potential was modified from U(r) = k/r to U(r) = k(e^-r/a)/r where k = LmH and a is a constant with units of length. Bounded orbits, i.e. circular and elliptic orbits, occur for large enough athe easiest way to see this is to take a and we recover Newton's law.

(a) In our world (a = U = k/r), the closest and furthest points that the Earth gets to the Sun are r_close = 1.47 10¹¹ m and r_far = 1.6 10¹¹ m. Now consider the modified potential with a large compared to r_far and r_close, say a = 1.5 10¹⁵ m. If the Earth has the same energy and angular momentum, what are the closest and furthest points of its orbit? Give your answer to four significant figures.

(b) Newton's law emits elliptic orbits. If the Sun is taken to be at the origin, the radius r of an

orbit as a function of the angle from the Sun is given by r = b /(1 + ecos()), where e is known as the eccentricity of the orbit and b is a constant with units of length. For elliptic orbits 0 < e < 1. For the modified potential, if a is large enough, the orbits are very nearly elliptical² and a similar formula holds with only the eccentricity modified:

r =b/(1 + e cos()), What are the eccentricities e and e of the orbits for the two cases considered in the previous

part? Give your answer to three significant figures.

(c) For large a (a b), what is the approximate difference e e e in terms of a, b, and e? In

other words, Taylor expand e to first order in b/a. How well does this formula work for the

numerical values you calculated in the previous part?

• ² it turns out that the elliptic orbit precesses a little bit, which means that changes a little for each revolution