Problem 4: In class, we've discussed the elliptical orbits of particles in an inverse-square
(Kepler) central force field, in which the center of force is located at a focus of the ellipse.
However, it turns out that a different central force field can also have elliptical orbits, but
in this case with the center of force not at a focus, but rather at the center of the ellipse.
In this problem, consider a particle of mass u moving in an elliptical orbit about a center
of force located at the center of the ellipse. The semi-major and -minor axes are a and b,
respectively. Assume the orbital angular momentum l is given.
(a) Show that the orbit, written in polar coordinates, has the form
$$r(\phi) = \frac{a}{\sqrt{1 + \gamma^2 \sin^2 \phi}}$$
Find an expression for $\gamma$ in terms of $a$ and $b$.
(b) Show that the force law responsible for this orbit has the form $F(r) = -kr^n$. Determine
the value of $n$, and find an expression for $k$ in terms of $\mu$, $a$, $b$, and $l$.
(c) Find an expression for the orbital period $\tau$ in terms of $\mu$, $a$, $b$, and $l$.
(d) Derive an expression for the total energy $E$. Express your answer in terms of $\mu$, $a$,
$b$, and $k$. [Remember that energy is conserved in a central force field, so your answer
should not depend on $r$ or $\phi$!]
