Last time, I showed you what a number of circular planetary orbits looked like. Today, I'm looking at a bunch of orbits passing through a single point with different speeds.
Reading up on this stuff on wikipedia, I was reminded of how the period of the orbit depended only on the length of the major axis. My first thought was that since all these orbits had the same aphelion point, they would have the same semi-major axis and the same period.
I was puzzled when all my dots didn't meet up at the starting point all at the same time. My first reaction was that my equations were off, so I spent a bit of time checking them again and again, but it looked like everyting added up. Even changing the step size had no impact.
Eventually I left the tracks of the particles on screen, and the source of my confusion became clear. I forgot to picture the other side of the ellipse correctly. As the speed of the particles increased, the perihelion point moved further and further out from the central body, until the orbit was circular. Then the two points switched roles and the starting point became the closest approach to the star.
Once I saw how the geometry was really changing, the change in orbital period became clear. While I've calculated the solutions to these differential equations many times over the years, without really playing with it myself, I really didn't understand what was going on.
That's what this is all about. I'm trying to take these complex systems and build up my physical intuition and understanding at a gut level, much like Montessori builds up understanding through multiple senses.