Hexagonal system

If n bodies of equal mass are placed at the corners of an n-polygon they may rotate together in a circular orbit provided they have exactly the right velocity.
In this simulation we have six bodies at the corners of a hexagon.

In the right column we show that the velocity must be

where k = 1.827350.

With m = 1·10²⁵ kg, R = 1·10¹⁰ m and G = 6.67·10^-11 we get v = 349.1193 m/s.

Exercise

1. Start the simulation. The orbits seem stable, but what happens after about two revolutions?
   To make the orbits stable for many revolutions, the initial positions and velocities must have even higher precision, and the time interval between calculations must be small enough. In this simulation the inaccuracies in the initial conditions dominate. You can verify this by restarting the simulation and decreasing the time interval.
2. Change some of the initial conditions slightly. You may for instance change the velocity of body 1 and 4 to 349 m/s. When do the orbits begin to deviate visibly from a circle now?
3. Imagine that we add a star with mass 1·10³⁰ kg at the center of the hexagon. Of course body 1 to 6 must now have much greater velocities to orbit in a circle. Do you think this would increase the stability of the orbits?
   Perhaps you would like to make a new simulation and investigate? If not, you may look up the simulation "Hexagonal system with central star".