Given the following equations and parameters: does the orbit repeat? If not, why? and what would make it repeat?
Any clarification on what makes an orbit periodic would be greatly appreciated!
I solved the two-body problem with ode45 in Matlab and obtained the following graphs for the position and velocity of the orbit through multiple periods. From the graphs, I see no reason why the orbit would not repeat, but I want to make sure I am not missing something due to the relative/absolute error limits in ode45.
Orbit Position/Velocity:
Equations of Motion: $$\ddot x + \mu\frac{x}{r^3} = 0 $$ $$\ddot y + \mu\frac{y}{r^3} = 0 $$ $$\ddot z + \mu\frac{z}{r^3} = 0 $$
where: $r^2 = x^2 + y^2 + z^2$ and $\mu = 3.986004 \space\space * \space\space 10^{14} \space m^3/s^2$
Rewrite the equation in its vector form,
$$\ddot {\vec{r}} = -\mu\frac{\vec{r}}{r^3} $$
which is the equation for orbiting movements due to gravitational force. It follows an elliptical trajectory in a plane formed by the radial vector and the velocity. It repeats itself.