An Exploration of the Lagrange Points

I was hoping to have a different post ready for this week, but recent solar activity kept me directed on other projects this week.

However, while browsing around some of my favorite sites, I discovered an entry in Wired's dot Physics column that was very relevant to topics I've discussed recently on this blog:

James Webb Space Telescope and L2 Orbits

Dr. Allain goes into a fair amount of detail on analyzing the gravitational and non-gravitational forces that will act on JWST at the L2 Lagrange point, even including the question of stability of bodies at this location.  A similar analysis could be done for L1 and L3.  We already have a small fleet of spacecraft at the Sun-Earth L1 point (YouTube), and WMAP operated at L2, but don't yet have a good use for the Sun-Earth L3 point (maybe a full-time solar observatory for maintaining 360 degree coverage of solar activity?).

Perhaps the Geocentrists might find this analysis useful in meeting the Lagrange Point challenge?

I'll use this opportunity to show off a few tests of my explorations of the Lagrange points using my n-body code (see Doin' Real Science: Simulating Particles).  I've run some simulations of the Sun (red) & Jupiter (green) and placed test particles (cyan x) of negligible mass (spacecraft?) near the unstable Lagrange points (L1, L2, L3) and similar test particles (magenta +) near the stable Lagrange points at L4 & L5.

Click for larger image or download movie here (11MB)

In the movie above, I've set the origin of the coordinate system at the center of the Sun and turned off the particle trails.  As the movie plays, we see the particles near the stable Lagrange Points (magenta) pretty much stay on the orbit of Jupiter, while the particles at the unstable points (cyan) move substantially.  But we can get a much better idea of what's happening if we transform to a coordinate system that rotates around the Sun with Jupiter.

Click for larger image or download movie here (11MB)

Above we see the motions in a frame that rotates around the Sun with Jupiter.  We get a better idea of how the unstable points either migrate away from the planet moving into a more solar orbit, or go into orbit around Jupiter.  We see that the particles at the stable points (magenta) actually move very little.  Of the two sets of five particles ahead and behind the orbit of Jupiter, the central particle is at maximum point of stability and we see they hardly move at all.

Click for larger image or download movie here (11MB)

If I zoom in close to Jupiter, in the simulation above, we see the two satellites nearest on either side of Jupiter move into complex orbits around the planet.  The two satellites furthest from Jupiter on either side of the planet move outward into solar orbits.  But we also have two remaining points on either side of the planet, which correspond to the satellites I placed precisely at L1 & L2.  For a simulation of one orbit of Jupiter (12 years), they've done a great job of staying in place since these two points are unstable, but I suspect they would start migrating away if I ran the simulation for longer.

Enjoy!