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Can a large space station, such as an O'Neill Cylinder, orbit the Moon with any degree of stability? I'm not sure if such a construction couldpossibly possess corrective thrusters or not, so I would imagine that, to be safe, an orbit would have to be stable at least on the scale of centuries.

...John...

I think it's possible but why ?

The L4 and L5 points are stable and never occulted by the moon

You can look here for the explain of Lagrange points

and I had my own aswer for the question of this page : if there is no trojan asteroids in the lagrange points of earth/moon, I think it's because the lack of gravity of our planet. Jupiter is huge and the astero??d belt near

When you compare it to Jupiter, I think you are making reference to the L4 & L5 points with respect to Jupiter's solar orbit. I do not believe any of Jupiter's moons are known to have asteroids in their Lagrange Points with respect to Jupiter. Saturn has a few. Jupiter's trojans are in solar orbit, in Jupiter's L4 & L5 points.

The Moon's L4 and L5 points with respect to Earth are not stable, at least not for centuries. At best they're stable for less than 2 years.

The reasons: The solar tide is strong and the Moon's orbit is eccentric. It's difficult to stay 60 degrees ahead of or 60 degrees behind something that's speeding up and slowing down.

Either one of those two reasons by themselves is enough to destabalize the Moon's L4 & L5 points.

For a lunar satellite to remain stable for centuries, you want to avoid low lunar orbit, where uneven mass distribution in the Moon will bring you down, and high lunar orbit where a gravitational tidal force from Earth will strip you away. An altitude of 2000-4000 km should suffice.

Like Jupiter, Earth and Mars are both known to have trojans that share their solar orbits.

tony873004 wrote:Like Jupiter, Earth and Mars are both known to have trojans that share their solar orbits.

Earth has no known trojans. Trojan asteroids are co-orbital asteroids that are located in L4 or L5 points, nowhere else. But there are some other Earth co-orbitals and quasi-satellites.

Ynjevi wrote:

tony873004 wrote:Like Jupiter, Earth and Mars are both known to have trojans that share their solar orbits.

Earth has no known trojans. Trojan asteroids are co-orbital asteroids that are located in L4 or L5 points, nowhere else. But there are some other Earth co-orbitals and quasi-satellites.

Cruithne, which is usually in a horseshoe orbit, sometimes fails to cross the L3 point. During these times, its path is around L4 or L5.

Notice in this rotating frame animated GIF that its path slows significantly at the L3 point, like trying to roll a ball over the top of a hill, it barely has enough energy to crest the hill. From time to time, the Earth's "repelling push" doesn't give Cruithne's kidney-beaned shaped orbit enough energy to crest the hill.

The reason for the differences is that Cruithne's orbit is inclined to Earth's orbit. This is why only one side of the kidney-bean shape seems to have repelling power. The other one, while it appears to come close to Earth, misses badly on the z-axis, and Earth's gravity is consequently weak on Cruithne at that point. Over long periods of time, the nodes of the axes precess, and the side with the "repelling power" switches. In between these two is a medium position where it is repelled, but not enough to clear the L3 point. Cruithne will then spend centuries orbiting as a trojan. The animated GIF won't show that since its just an 800 year simulation repeating over and over again, and not milleniums-long like it would need to be to show the described behavior.

But, you're right, since it is currently in horseshoe behavior.