Exercise 2.1: Simulation of the orbit

1. Write a program to simulate motion in a central force field. Verify the case of circular orbit using (in astronomical units) ($x_0=1$, $y_0=0$ and $v_x(t=0)=0$. Use the condition (32 to calculate $v_y(t=0)$ for a circular orbit. Choose a value of $\Delta t$ such that to a good approximation the total energy $E$ is conserved. Is your value of $\Delta t$ small enough to reproduce the orbit over several periods?

2. Run the program for different sets of initial conditions $x_0$ and $v_y(t=0)$ consistent with the condition for a circular orbit. Set $y_0=0$ and $v_x(t=0)=0$. For each orbit, measure the radius and the period to verify Kepler's third law ( $T^2/a^3=\mathrm{const.}$). Think of a simple condition which allows you to find the numerical value of the period.

3. Show that Euler's method does not shield stable orbits for the same choice of $\Delta t$ used in the previous items. Is it sufficient to simply choose a smaller $\Delta t$ or Euler's method is no stable for this dynamical system? Use the average velocity $1/2(v_n+v_{n+1})$ to obtain $x_{n+1}$. Are the results any better?

4. Set $y_0=0$ and $v_x(t=0)=0$. By trial and error find several choices of $x_0$ and $v_y(t=0)$ which yield coonvinient elliptical orbits. Determine total energy, angular momentum, semi-major and semi-nimor axes, eccentricity, and period for each orbit.

5. You probably noticed that Euler's algorithm with a fixed $\Delta t$ breaks down if you get to close to the sun. How are you able to visually confirm this? What is the cause of the failure of the method? Think of a simple modification of your program that can improve your results.