Please attempt to complete one of the following exercises before the interview. Don't worry if you cannot complete these--they are meant to be challenging. I am looking for effort and curiosity as much as the right solution. We will discuss the solutions at the interview.

1. Download and install the rebound N-body code from https://rebound.readthedocs.io/en/latest/index.html and try running one of the example problems here -- any of the C or Python problems is fine. Make plots of the initial and final positions of the particles in the simulation.

2. Consider a particle on a circular orbit with radius $r_o$ in a Keplerian ($1/r$) potential. Now imagine you push the particle to a slightly larger radius $r_o+\delta$ (where $\delta\ll r_o$), but leave its angular momentum unchanged. The particle's radius will now oscillate about $r_o$. What is the frequency of this oscillation? What shape is the new orbit? Hint: Write down the equation of motion in the radial direction as a function of $r_o+\delta$ and then Taylor expand this equation.