Epimetheus (horseshoe orbits)
Grab a mass 100000 star, and put it down stationary. Grab a mass 10 planet and put it in a circular orbit around the star - Not too close, I'd say around 3 star-diameters away.
Use CoM to remove any drift.
Let the planet go one orbit, and grab an asteroid.
Put it in a circular orbit going in the same direction as the planet, but slightly above the planet's orbit. Not much further than the planet's radius. Start it opposite the planet.
Lagrange track the star and planet. Wait until something interesting happens.
Think it worked?
SPOILER
SPOILER_SHOW
If you did this successfully, the asteroid should move toward the planet in a circle... but slow down and stop before it hits.
(If the asteroid gets too close, it may turn around and keep going, but in a bumpy pattern. Try again.)
The asteroid turns around and starts going away from the planet, slightly lower than its previous orbit. It does the same thing on the opposite side, going back up. Then it repeats.
The end result is a horseshoe like shape, with the ends floating near the planet.
(If the asteroid gets too close, it may turn around and keep going, but in a bumpy pattern. Try again.)
The asteroid turns around and starts going away from the planet, slightly lower than its previous orbit. It does the same thing on the opposite side, going back up. Then it repeats.
The end result is a horseshoe like shape, with the ends floating near the planet.
SPOILER
SPOILER_SHOW
In a higher orbit, the asteroid lags behind the planet. Once the planet gets close enough, it slows the asteroid down (reducing kinetic and total energy). However, at a certain point, the asteroid is in a low enough orbit that it can "run away" from the planet. It eventually catches up to the planet on the opposite side, and gets pulled "forward", increasing its energy and putting it into a higher orbit. Then it starts to slow down, and the process repeats.
Can you put multiple asteroids down on similar paths? Can you replace the asteroid with a (small; mass ~4) planet?*
Are horseshoes stable? Start over, but put down two planets, one with about twice the orbital radius than the other. Try to make a horseshoe around both planets.
*This would actually cause both planets to drift in a horseshoe, with the less massive one moving more. This can be interpreted as an exchange of energies. See Epimetheus link below.
Links:
http://en.wikipedia.org/wiki/Horseshoe_orbit
http://en.wikipedia.org/wiki/Epimetheus_(moon)
Got any more ideas for experiments? Post them!
